Section 7: the corrected R = 1 elementary-stage branch

This file lowers the residual sec7_fourierBranch_of_frattiniTrivial from a full arbitrary Fourier-branch packet to the specific elementary-stage packet that the manuscript uses when R = Φ(K) = 1.

Manuscript anchors: in thm:closedrecursion / the inductive step, if R = 1, then the middle quotient B = Y/R is just Y, so no nontrivial R-obstruction or scalar-pushout Fourier layer remains. The branch is the elementary stage (lem:elementarystage, eq:Mstage) with the lower quotient child C = Y/K and the proper exact-image children J < Y mapping onto C.

Lean shape: the obstruction space is fixed to the one-point 𝔽₂-module PUnit, so W = 0, dimExp = 0, and zR = 1. The residual is now split into a smaller core packet, where the partial-count identities are recorded only at the unique character of PUnit; the ordinary all-character elementary-stage data is assembled theorem-level from the subsingleton dual.

def Q2Presentation.Induction.sec7ZeroObstruction (X : Type) : X → PUnit.{1}

The zero obstruction map into the one-point 𝔽₂-module.

Equations

• Q2Presentation.Induction.sec7ZeroObstruction X x✝ = 0

@[simp]

theorem Q2Presentation.Induction.sec7ZeroObstruction_apply (X : Type) (x : X) : sec7ZeroObstruction X x = 0

theorem Q2Presentation.Induction.sec7ZeroObstruction_zeroFiber_card (X : Type) [Fintype X] : Nat.card { x : X // sec7ZeroObstruction X x = 0 } = Nat.card X

The zero-obstruction fibre is the whole elementary lift index.

theorem Q2Presentation.Induction.sec7ZeroObstruction_zeroDualFiber_card (X : Type) [Fintype X] : Nat.card { x : X // 0 (sec7ZeroObstruction X x) = 0 } = Nat.card X

At the unique zero character, the elementary partial-count fibre is also the whole lift index.

structure Q2Presentation.Induction.Sec7FrattiniTrivialElementaryStageData (p : FramedPair) : Type 1

The lowered R = 1 residual: the elementary-stage data for the case B = Y/R = Y.

It records only the data actually used by lem:elementarystage / eq:Mstage:

• the lower child C = Y/K;
• the proper exact-image subtraction children J < Y mapping onto C;
• the common elementary partial-count recipe for the zero obstruction space;
• the candidate and local source elementary lift-index sets and the corresponding zero-obstruction realization identities.

This is strictly smaller than an arbitrary FourierBranchData p: the obstruction space, its dimension exponent, and the lift scalar are fixed as PUnit, 0, and 1.

• lowerChild : FramedPair

  The lower quotient child C = Y/K.

• lower_small : framedMeasure self.lowerChild < framedMeasure p

  Strict decrease of the lower quotient child.

• properChildren : List FramedPair

  The proper exact-image children in the elementary-stage subtraction.

• proper_small (c : FramedPair) : c ∈ self.properChildren → framedMeasure c < framedMeasure p

  Strict decrease of every proper exact-image child.

• partialN : Module.Dual (ZMod 2) PUnit.{1} → List ℕ → ℕ

  Source-independent elementary partial-count recipe.

• XA : Type

  Candidate-source elementary lift index.

• finXA : Fintype self.XA
• XQ : Type

  Local-source elementary lift index.

• finXQ : Fintype self.XQ
• realizeA : framedCountA p = ↑(Nat.card { x : self.XA // sec7ZeroObstruction self.XA x = 0 })

  Candidate elementary lift identity; the obstruction is identically zero and z_R = 1.

• realizeQ : framedCountQ p = ↑(Nat.card { x : self.XQ // sec7ZeroObstruction self.XQ x = 0 })

  Local elementary lift identity; the obstruction is identically zero and z_R = 1.

• partialA (χ : Module.Dual (ZMod 2) PUnit.{1}) : Nat.card { x : self.XA // χ (sec7ZeroObstruction self.XA x) = 0 } = self.partialN χ (List.map (⇑Cardinal.toNat) (List.map framedCountA (self.lowerChild :: self.properChildren)))

  Candidate elementary partial-count realization.

• partialQ (χ : Module.Dual (ZMod 2) PUnit.{1}) : Nat.card { x : self.XQ // χ (sec7ZeroObstruction self.XQ x) = 0 } = self.partialN χ (List.map (⇑Cardinal.toNat) (List.map framedCountQ (self.lowerChild :: self.properChildren)))

  Local elementary partial-count realization.

structure Q2Presentation.Induction.Sec7FrattiniTrivialElementaryStageCoreData (p : FramedPair) : Type 1

The smaller elementary-stage core packet for the R = 1 branch.

Compared with Sec7FrattiniTrivialElementaryStageData, it stores the elementary partial-count identities only for the zero character. Since Module.Dual (ZMod 2) PUnit is subsingleton, the all-character fields of the ordinary stage data are recovered below by theorem-level assembly. This closes a concrete part of lem:elementarystage: there is no genuine Fourier family of characters in the W = 0 elementary stage.

• lowerChild : FramedPair

  The lower quotient child C = Y/K.

• lower_small : framedMeasure self.lowerChild < framedMeasure p

  Strict decrease of the lower quotient child.

• properChildren : List FramedPair

  The proper exact-image children in the elementary-stage subtraction.

• proper_small (c : FramedPair) : c ∈ self.properChildren → framedMeasure c < framedMeasure p

  Strict decrease of every proper exact-image child.

• partialN : Module.Dual (ZMod 2) PUnit.{1} → List ℕ → ℕ

  Source-independent elementary partial-count recipe.

• XA : Type

  Candidate-source elementary lift index.

• finXA : Fintype self.XA
• XQ : Type

  Local-source elementary lift index.

• finXQ : Fintype self.XQ
• realizeA : framedCountA p = ↑(Nat.card { x : self.XA // sec7ZeroObstruction self.XA x = 0 })

  Candidate elementary lift identity; the obstruction is identically zero and z_R = 1.

• realizeQ : framedCountQ p = ↑(Nat.card { x : self.XQ // sec7ZeroObstruction self.XQ x = 0 })

  Local elementary lift identity; the obstruction is identically zero and z_R = 1.

• partialA_zero : Nat.card { x : self.XA // 0 (sec7ZeroObstruction self.XA x) = 0 } = self.partialN 0 (List.map (⇑Cardinal.toNat) (List.map framedCountA (self.lowerChild :: self.properChildren)))

  Candidate elementary partial-count realization at the unique dual character.

• partialQ_zero : Nat.card { x : self.XQ // 0 (sec7ZeroObstruction self.XQ x) = 0 } = self.partialN 0 (List.map (⇑Cardinal.toNat) (List.map framedCountQ (self.lowerChild :: self.properChildren)))

  Local elementary partial-count realization at the unique dual character.

structure Q2Presentation.Induction.Sec7FrattiniTrivialElementaryChildData (p : FramedPair) : Type 1

The exact-image child packet for the R = 1 elementary stage. This isolates the manuscript's child geometry (C = Y/K and the proper exact-image subtraction children) from the source-specific elementary lift counts.

• lowerChild : FramedPair

  The lower quotient child C = Y/K.

• lowerPostMap : p.fst.Y →* self.lowerChild.fst.Y

  The finite target-level projection Y → C used for candidate postcomposition.

• lowerPostMap_surj : Function.Surjective ⇑self.lowerPostMap

  The finite target-level projection Y → C is onto.

• lowerPostMap_boundary (y : p.fst.Y) (b : ↑Boundary.toProfinite.toTop) : p.fst.qYMap y = (ProfiniteGrp.Hom.hom p.snd.beta) b → self.lowerChild.fst.qYMap (self.lowerPostMap y) = (ProfiniteGrp.Hom.hom self.lowerChild.snd.beta) b

  The finite projection carries the top boundary frame to the inherited lower frame.

• lower_small : framedMeasure self.lowerChild < framedMeasure p

  Strict decrease of the lower quotient child.

• properChildren : List FramedPair

  The proper exact-image children in the elementary-stage subtraction.

• proper_small (c : FramedPair) : c ∈ self.properChildren → framedMeasure c < framedMeasure p

  Strict decrease of every proper exact-image child.

Child geometry for the R = 1 elementary stage

theorem Q2Presentation.Induction.sec7_framedMeasure_pos (p : FramedPair) : 0 < framedMeasure p

Every framed target has positive marked-kernel measure.

theorem Q2Presentation.Induction.sec7_Ksub_le_piY_ker {p : FramedPair} {chief : NonScalarChiefFactor p} (K : KernelMinimal chief) : K.Ksub ≤ p.fst.piY.ker

The selected kernel lies in the tame-quotient kernel ker pi_Y.

theorem Q2Presentation.Induction.sec7_Ksub_ne_bot {p : FramedPair} {chief : NonScalarChiefFactor p} (K : KernelMinimal chief) : K.Ksub ≠ ⊥

The manuscript kernel K selected over a genuine chief factor is nontrivial.

If K = 1, then the condition KS = P in KernelMinimal would collapse to S = P, contradicting the strictness of the chief factor S < P. This closes the nontriviality part of the former lower-quotient residual rather than recording it as an assumption.

theorem Q2Presentation.Induction.sec7_Ksub_card_gt_one {p : FramedPair} {chief : NonScalarChiefFactor p} (K : KernelMinimal chief) : 1 < Nat.card ↥K.Ksub

The elementary layer K has cardinality greater than one.

This is theorem-level Section 7 bookkeeping: it follows only from the corrected KernelMinimal join condition over the nontrivial chief factor, not from any remaining elementary-stage residual.

def Q2Presentation.Induction.sec7ElementaryLowerTarget {p : FramedPair} {chief : NonScalarChiefFactor p} (K : KernelMinimal chief) (hθ : K.Ksub ≤ p.fst.thetaY.ker) : BoundaryFramedTarget

The canonical lower framed target C = Y/K, assuming the decoration kills K.

The hypothesis is exactly the framed quotient constructor missing from the current KernelMinimal API: the manuscript supplies it by lem:decorationblock. The group, marked kernel, tame quotient map, and decoration are all the literal quotient data.

Equations

• One or more equations did not get rendered due to their size.

def Q2Presentation.Induction.sec7ElementaryLowerFrame {p : FramedPair} {chief : NonScalarChiefFactor p} (K : KernelMinimal chief) (hθ : K.Ksub ≤ p.fst.thetaY.ker) : BoundaryFrame (sec7ElementaryLowerTarget K hθ)

The lower quotient inherits the original boundary frame because H and E are preserved definitionally.

Equations

• Q2Presentation.Induction.sec7ElementaryLowerFrame K hθ = { alpha := p.snd.alpha, alpha_surj := ⋯, psi := p.snd.psi, beta := p.snd.beta, beta_def := ⋯ }

def Q2Presentation.Induction.sec7ElementaryLowerChild {p : FramedPair} {chief : NonScalarChiefFactor p} (K : KernelMinimal chief) (hθ : K.Ksub ≤ p.fst.thetaY.ker) : FramedPair

The lower child C = Y/K as a framed pair.

Equations

• Q2Presentation.Induction.sec7ElementaryLowerChild K hθ = ⟨Q2Presentation.Induction.sec7ElementaryLowerTarget K hθ, Q2Presentation.Induction.sec7ElementaryLowerFrame K hθ⟩

def Q2Presentation.Induction.sec7ElementaryLowerMap {p : FramedPair} {chief : NonScalarChiefFactor p} (K : KernelMinimal chief) (hθ : K.Ksub ≤ p.fst.thetaY.ker) : p.fst.Y →* (sec7ElementaryLowerChild K hθ).fst.Y

The quotient map Y -> C = Y/K.

Equations

• Q2Presentation.Induction.sec7ElementaryLowerMap K hθ = QuotientGroup.mk' K.Ksub

theorem Q2Presentation.Induction.sec7ElementaryLowerMap_surjective {p : FramedPair} {chief : NonScalarChiefFactor p} (K : KernelMinimal chief) (hθ : K.Ksub ≤ p.fst.thetaY.ker) : Function.Surjective ⇑(sec7ElementaryLowerMap K hθ)

The lower quotient map Y → C = Y/K is onto.

theorem Q2Presentation.Induction.sec7ElementaryLowerMap_qYMap {p : FramedPair} {chief : NonScalarChiefFactor p} (K : KernelMinimal chief) (hθ : K.Ksub ≤ p.fst.thetaY.ker) (y : p.fst.Y) : (sec7ElementaryLowerChild K hθ).fst.qYMap ((sec7ElementaryLowerMap K hθ) y) = p.fst.qYMap y

The lower quotient map preserves the boundary coordinates q_Y = (π_Y, θ_Y). Both coordinates are the defining quotient lifts.

def Q2Presentation.Induction.sec7ElementaryExactImageChild (p : FramedPair) (J : Subgroup p.fst.Y) (hJ : Function.Surjective ⇑(p.fst.piY.comp J.subtype)) : FramedPair

A proper exact-image subgroup J ≤ Y inherits the boundary-framed structure from Y; in the R = 1 branch the middle target B = Y, so these are precisely the subtraction children of eq:Mstage.

Equations

• Q2Presentation.Induction.sec7ElementaryExactImageChild p J hJ = ⟨p.fst.exactImageTarget J hJ, p.fst.exactImageFrame J hJ p.snd⟩

structure Q2Presentation.Induction.Sec7FrattiniTrivialElementaryChildGeometryData {p : FramedPair} (h : ¬ScalarTerminal p) (chief : FirstNonScalarChiefFactor p) (K : KernelMinimal chief.toNonScalarChiefFactor) (hR : Nat.card ↥(kernelFrattini K) = 1) : Type 1

Manuscript-true child-geometry packet for the elementary stage when R = 1.

This is deliberately only the target/child half of lem:elementarystage / eq:Mstage:

• K is killed by the decoration, so the lower child is the literal quotient C = Y/K;
• the proper subtraction terms are actual subgroups J < Y mapping onto C;
• the strict-decrease estimates are proved below from the recorded cardinal equalities |L_Y| = |K| |L_C| and |J ∩ L_Y| = |J ∩ K| |L_C|, together with 2 |J ∩ K| ≤ |K|.

It contains no source-specific lift-count identities.

• Ksub_le_thetaY_ker : K.Ksub ≤ p.fst.thetaY.ker

  lem:decorationblock in the elementary branch: the boundary decoration kills K.

• Ksub_card_gt_one : 1 < Nat.card ↥K.Ksub

  The elementary layer M = K is nontrivial.

• lower_measure_mul_Kcard : framedMeasure (sec7ElementaryLowerChild K ⋯) * Nat.card ↥K.Ksub = framedMeasure p

  Lower-target cardinal identity |L_Y| = |K| |L_C|, written in measure form.

• properImages : Type

  Finite index of proper exact-image subgroups in the M-stage subtraction.

• finProperImages : Fintype self.properImages
• subgroup : self.properImages → Subgroup p.fst.Y

  The subgroup J < B = Y.

• subgroup_surj (i : self.properImages) : Function.Surjective ⇑(p.fst.piY.comp (self.subgroup i).subtype)

  Each proper image still projects onto the fixed tame quotient H.

• proper_subgroup (i : self.properImages) : self.subgroup i < ⊤

  Manuscript condition J < B = Y.

• maps_onto_lower (i : self.properImages) : Function.Surjective ⇑((sec7ElementaryLowerMap K ⋯).comp (self.subgroup i).subtype)

  Manuscript condition J -> C = Y/K is onto.

• kernelPartCard : self.properImages → ℕ

  The cardinal |J ∩ K| in the exact sequence for J ∩ L_Y.

• kernelPart_bound (i : self.properImages) : 2 * self.kernelPartCard i ≤ Nat.card ↥K.Ksub

  Properness gives 2 |J ∩ K| ≤ |K|.

• child_measure_eq (i : self.properImages) : framedMeasure (sec7ElementaryExactImageChild p (self.subgroup i) ⋯) = self.kernelPartCard i * framedMeasure (sec7ElementaryLowerChild K ⋯)

  Manuscript |J ∩ L_Y| = |J ∩ K| |L_C|, written as a framed-measure identity.

def Q2Presentation.Induction.Sec7FrattiniTrivialElementaryChildGeometryData.properChildren {p : FramedPair} {h : ¬ScalarTerminal p} {chief : FirstNonScalarChiefFactor p} {K : KernelMinimal chief.toNonScalarChiefFactor} {hR : Nat.card ↥(kernelFrattini K) = 1} (D : Sec7FrattiniTrivialElementaryChildGeometryData h chief K hR) : List FramedPair

The proper exact-image children as a concrete finite list.

Equations

• D.properChildren = List.map (fun (i : D.properImages) => Q2Presentation.Induction.sec7ElementaryExactImageChild p (D.subgroup i) ⋯) Finset.univ.toList

theorem Q2Presentation.Induction.Sec7FrattiniTrivialElementaryChildGeometryData.lower_small {p : FramedPair} {h : ¬ScalarTerminal p} {chief : FirstNonScalarChiefFactor p} {K : KernelMinimal chief.toNonScalarChiefFactor} {hR : Nat.card ↥(kernelFrattini K) = 1} (D : Sec7FrattiniTrivialElementaryChildGeometryData h chief K hR) : framedMeasure (sec7ElementaryLowerChild K ⋯) < framedMeasure p

Strict decrease of the lower quotient child C = Y/K.

theorem Q2Presentation.Induction.Sec7FrattiniTrivialElementaryChildGeometryData.proper_small {p : FramedPair} {h : ¬ScalarTerminal p} {chief : FirstNonScalarChiefFactor p} {K : KernelMinimal chief.toNonScalarChiefFactor} {hR : Nat.card ↥(kernelFrattini K) = 1} (D : Sec7FrattiniTrivialElementaryChildGeometryData h chief K hR) (c : FramedPair) : c ∈ D.properChildren → framedMeasure c < framedMeasure p

The eq:Mdescent strict-decrease estimate for every proper exact-image child.

def Q2Presentation.Induction.Sec7FrattiniTrivialElementaryChildGeometryData.toChildData {p : FramedPair} {h : ¬ScalarTerminal p} {chief : FirstNonScalarChiefFactor p} {K : KernelMinimal chief.toNonScalarChiefFactor} {hR : Nat.card ↥(kernelFrattini K) = 1} (D : Sec7FrattiniTrivialElementaryChildGeometryData h chief K hR) : Sec7FrattiniTrivialElementaryChildData p

Forget the semantic child-geometry packet to the split child-data interface used by Sec7FrattiniTrivialElementaryStageSplitData.

Equations

• One or more equations did not get rendered due to their size.

structure Q2Presentation.Induction.Sec7FrattiniTrivialElementaryLowerQuotientData {p : FramedPair} (h : ¬ScalarTerminal p) (chief : FirstNonScalarChiefFactor p) (K : KernelMinimal chief.toNonScalarChiefFactor) (hR : Nat.card ↥(kernelFrattini K) = 1) : Prop

The lower-quotient half of the child geometry in the R = 1 elementary stage.

This isolates the manuscript's C = Y/K construction from the proper exact-image subtraction. Its fields are exactly the target-side facts used to build and bound the lower child: the decoration kills K, the elementary layer K is nontrivial, and the marked-kernel measure satisfies |L_Y| = |K| |L_C|.

• Ksub_le_thetaY_ker : K.Ksub ≤ p.fst.thetaY.ker

  lem:decorationblock in the elementary branch: the boundary decoration kills K.

• Ksub_card_gt_one : 1 < Nat.card ↥K.Ksub

  The elementary layer M = K is nontrivial.

• lower_measure_mul_Kcard : framedMeasure (sec7ElementaryLowerChild K ⋯) * Nat.card ↥K.Ksub = framedMeasure p

  Lower-target cardinal identity |L_Y| = |K| |L_C|, written in measure form.

structure Q2Presentation.Induction.Sec7FrattiniTrivialElementaryLowerQuotientCoreData {p : FramedPair} (h : ¬ScalarTerminal p) (chief : FirstNonScalarChiefFactor p) (K : KernelMinimal chief.toNonScalarChiefFactor) (hR : Nat.card ↥(kernelFrattini K) = 1) : Prop

The narrowed lower-quotient residual for the R = 1 elementary stage.

The nontriviality field 1 < |K| has been removed from the residual and is proved by sec7_Ksub_card_gt_one. What remains is exactly the quotient-construction content of lem:decorationblock / the first identity in lem:strictdecrease: the decoration kills K, and the marked-kernel measure of the literal quotient C = Y/K satisfies |L_Y| = |K| |L_C|.

• Ksub_le_thetaY_ker : K.Ksub ≤ p.fst.thetaY.ker

  lem:decorationblock in the elementary branch: the boundary decoration kills K.

• lower_measure_mul_Kcard : framedMeasure (sec7ElementaryLowerChild K ⋯) * Nat.card ↥K.Ksub = framedMeasure p

  Lower-target cardinal identity |L_Y| = |K| |L_C|, written in measure form.

def Q2Presentation.Induction.Sec7FrattiniTrivialElementaryLowerQuotientCoreData.toLowerQuotientData {p : FramedPair} {h : ¬ScalarTerminal p} {chief : FirstNonScalarChiefFactor p} {K : KernelMinimal chief.toNonScalarChiefFactor} {hR : Nat.card ↥(kernelFrattini K) = 1} (D : Sec7FrattiniTrivialElementaryLowerQuotientCoreData h chief K hR) : Sec7FrattiniTrivialElementaryLowerQuotientData h chief K hR

Assemble the former lower-quotient packet from the narrowed quotient-construction packet and the theorem-level nontriviality of K.

Equations

• ⋯ = ⋯

structure Q2Presentation.Induction.Sec7FrattiniTrivialElementaryProperImageData {p : FramedPair} (h : ¬ScalarTerminal p) (chief : FirstNonScalarChiefFactor p) (K : KernelMinimal chief.toNonScalarChiefFactor) (hR : Nat.card ↥(kernelFrattini K) = 1) (lower : Sec7FrattiniTrivialElementaryLowerQuotientData h chief K hR) : Type 1

The proper exact-image half of the child geometry in the R = 1 elementary stage, relative to an already constructed lower quotient C = Y/K.

This is the finite subgroup enumeration in eq:Mstage: the subgroups J < Y mapping onto C, together with the exact-sequence cardinal estimate |J ∩ L_Y| = |J ∩ K| |L_C| and the properness bound 2 |J ∩ K| ≤ |K|.

• properImages : Type

  Finite index of proper exact-image subgroups in the M-stage subtraction.

• finProperImages : Fintype self.properImages
• subgroup : self.properImages → Subgroup p.fst.Y

  The subgroup J < B = Y.

• subgroup_surj (i : self.properImages) : Function.Surjective ⇑(p.fst.piY.comp (self.subgroup i).subtype)

  Each proper image still projects onto the fixed tame quotient H.

• proper_subgroup (i : self.properImages) : self.subgroup i < ⊤

  Manuscript condition J < B = Y.

• maps_onto_lower (i : self.properImages) : Function.Surjective ⇑((sec7ElementaryLowerMap K ⋯).comp (self.subgroup i).subtype)

  Manuscript condition J -> C = Y/K is onto.

• kernelPartCard : self.properImages → ℕ

  The cardinal |J ∩ K| in the exact sequence for J ∩ L_Y.

• kernelPart_bound (i : self.properImages) : 2 * self.kernelPartCard i ≤ Nat.card ↥K.Ksub

  Properness gives 2 |J ∩ K| ≤ |K|.

• child_measure_eq (i : self.properImages) : framedMeasure (sec7ElementaryExactImageChild p (self.subgroup i) ⋯) = self.kernelPartCard i * framedMeasure (sec7ElementaryLowerChild K ⋯)

  Manuscript |J ∩ L_Y| = |J ∩ K| |L_C|, written as a framed-measure identity.

structure Q2Presentation.Induction.Sec7FrattiniTrivialElementaryProperImageFamilyData {p : FramedPair} (h : ¬ScalarTerminal p) (chief : FirstNonScalarChiefFactor p) (K : KernelMinimal chief.toNonScalarChiefFactor) (hR : Nat.card ↥(kernelFrattini K) = 1) (lower : Sec7FrattiniTrivialElementaryLowerQuotientData h chief K hR) : Type 1

The narrowed proper-image family residual for the R = 1 elementary stage.

This is only the finite exact-image enumeration in eq:Mstage: the proper subgroups J < B = Y that still project to the lower quotient C = Y/K. It contains no cardinal estimates.

• properImages : Type

  Finite index of proper exact-image subgroups in the M-stage subtraction.

• finProperImages : Fintype self.properImages
• subgroup : self.properImages → Subgroup p.fst.Y

  The subgroup J < B = Y.

• subgroup_surj (i : self.properImages) : Function.Surjective ⇑(p.fst.piY.comp (self.subgroup i).subtype)

  Each proper image still projects onto the fixed tame quotient H.

• proper_subgroup (i : self.properImages) : self.subgroup i < ⊤

  Manuscript condition J < B = Y.

• maps_onto_lower (i : self.properImages) : Function.Surjective ⇑((sec7ElementaryLowerMap K ⋯).comp (self.subgroup i).subtype)

  Manuscript condition J -> C = Y/K is onto.

@[reducible, inline]

abbrev Q2Presentation.Induction.Sec7FrattiniTrivialElementaryProperExactImageSubgroups {p : FramedPair} {h : ¬ScalarTerminal p} {chief : FirstNonScalarChiefFactor p} {K : KernelMinimal chief.toNonScalarChiefFactor} {hR : Nat.card ↥(kernelFrattini K) = 1} (lower : Sec7FrattiniTrivialElementaryLowerQuotientData h chief K hR) : Type

The canonical finite index of proper exact-image subgroups in the R = 1 elementary stage. An element is an actual subgroup J ≤ B = Y satisfying exactly the conditions used in eq:Mstage: J < Y, it still projects onto the fixed tame quotient H, and its composite with the lower quotient map Y → C = Y/K is onto.

Equations

• One or more equations did not get rendered due to their size.

instance Q2Presentation.Induction.instFintypeSec7FrattiniTrivialElementaryProperExactImageSubgroups {p : FramedPair} {h : ¬ScalarTerminal p} {chief : FirstNonScalarChiefFactor p} {K : KernelMinimal chief.toNonScalarChiefFactor} {hR : Nat.card ↥(kernelFrattini K) = 1} (lower : Sec7FrattiniTrivialElementaryLowerQuotientData h chief K hR) : Fintype (Sec7FrattiniTrivialElementaryProperExactImageSubgroups lower)

Equations

• Q2Presentation.Induction.instFintypeSec7FrattiniTrivialElementaryProperExactImageSubgroups lower = id inferInstance

def Q2Presentation.Induction.sec7ElementaryProperImageFamilyData_canonical {p : FramedPair} {h : ¬ScalarTerminal p} {chief : FirstNonScalarChiefFactor p} {K : KernelMinimal chief.toNonScalarChiefFactor} {hR : Nat.card ↥(kernelFrattini K) = 1} (lower : Sec7FrattiniTrivialElementaryLowerQuotientData h chief K hR) : Sec7FrattiniTrivialElementaryProperImageFamilyData h chief K hR lower

The canonical exact-image family, with no residual enumeration choice.

Equations

• One or more equations did not get rendered due to their size.

structure Q2Presentation.Induction.Sec7FrattiniTrivialElementaryProperImageDescentData {p : FramedPair} (h : ¬ScalarTerminal p) (chief : FirstNonScalarChiefFactor p) (K : KernelMinimal chief.toNonScalarChiefFactor) (hR : Nat.card ↥(kernelFrattini K) = 1) (lower : Sec7FrattiniTrivialElementaryLowerQuotientData h chief K hR) (family : Sec7FrattiniTrivialElementaryProperImageFamilyData h chief K hR lower) : Type

The narrowed proper-image descent residual for the R = 1 elementary stage.

This is only the cardinality half of eq:Mdescent, relative to an already enumerated family of proper exact images: |J ∩ L_Y| = |J ∩ K| |L_C| and the properness bound 2 |J ∩ K| ≤ |K|.

• kernelPartCard : family.properImages → ℕ

  The cardinal |J ∩ K| in the exact sequence for J ∩ L_Y.

• kernelPart_bound (i : family.properImages) : 2 * self.kernelPartCard i ≤ Nat.card ↥K.Ksub

  Properness gives 2 |J ∩ K| ≤ |K|.

• child_measure_eq (i : family.properImages) : framedMeasure (sec7ElementaryExactImageChild p (family.subgroup i) ⋯) = self.kernelPartCard i * framedMeasure (sec7ElementaryLowerChild K ⋯)

  Manuscript |J ∩ L_Y| = |J ∩ K| |L_C|, written as a framed-measure identity.

structure Q2Presentation.Induction.Sec7FrattiniTrivialElementaryProperImageCanonicalDescentData {p : FramedPair} (h : ¬ScalarTerminal p) (chief : FirstNonScalarChiefFactor p) (K : KernelMinimal chief.toNonScalarChiefFactor) (hR : Nat.card ↥(kernelFrattini K) = 1) (lower : Sec7FrattiniTrivialElementaryLowerQuotientData h chief K hR) : Type

Canonical eq:Mdescent cardinal data for the actual subtype-indexed exact-image family. This is strictly narrower than asking for descent data for an arbitrary chosen enumeration of the same subgroups: all enumeration bookkeeping is theorem-level.

• kernelPartCard : Sec7FrattiniTrivialElementaryProperExactImageSubgroups lower → ℕ

  The cardinal |J ∩ K| for the canonical exact-image subgroup J.

• kernelPart_bound (i : Sec7FrattiniTrivialElementaryProperExactImageSubgroups lower) : 2 * self.kernelPartCard i ≤ Nat.card ↥K.Ksub

  Properness gives 2 |J ∩ K| ≤ |K|.

• child_measure_eq (i : Sec7FrattiniTrivialElementaryProperExactImageSubgroups lower) : framedMeasure (sec7ElementaryExactImageChild p ↑i ⋯) = self.kernelPartCard i * framedMeasure (sec7ElementaryLowerChild K ⋯)

  Manuscript |J ∩ L_Y| = |J ∩ K| |L_C|, for the canonical exact-image index.

def Q2Presentation.Induction.Sec7FrattiniTrivialElementaryProperImageCanonicalDescentData.canonicalIndexOfFamily {p : FramedPair} {h : ¬ScalarTerminal p} {chief : FirstNonScalarChiefFactor p} {K : KernelMinimal chief.toNonScalarChiefFactor} {hR : Nat.card ↥(kernelFrattini K) = 1} {lower : Sec7FrattiniTrivialElementaryLowerQuotientData h chief K hR} (family : Sec7FrattiniTrivialElementaryProperImageFamilyData h chief K hR lower) (i : family.properImages) : Sec7FrattiniTrivialElementaryProperExactImageSubgroups lower

Embed a chosen family member into the canonical exact-image subtype.

Equations

• Q2Presentation.Induction.Sec7FrattiniTrivialElementaryProperImageCanonicalDescentData.canonicalIndexOfFamily family i = ⟨family.subgroup i, ⋯⟩

def Q2Presentation.Induction.Sec7FrattiniTrivialElementaryProperImageCanonicalDescentData.toDescentData {p : FramedPair} {h : ¬ScalarTerminal p} {chief : FirstNonScalarChiefFactor p} {K : KernelMinimal chief.toNonScalarChiefFactor} {hR : Nat.card ↥(kernelFrattini K) = 1} {lower : Sec7FrattiniTrivialElementaryLowerQuotientData h chief K hR} (D : Sec7FrattiniTrivialElementaryProperImageCanonicalDescentData h chief K hR lower) (family : Sec7FrattiniTrivialElementaryProperImageFamilyData h chief K hR lower) : Sec7FrattiniTrivialElementaryProperImageDescentData h chief K hR lower family

Pull canonical eq:Mdescent data back along any chosen enumeration of proper exact-image children.

Equations

• One or more equations did not get rendered due to their size.

structure Q2Presentation.Induction.Sec7FrattiniTrivialElementaryLowerDecorationData {p : FramedPair} (h : ¬ScalarTerminal p) (chief : FirstNonScalarChiefFactor p) (K : KernelMinimal chief.toNonScalarChiefFactor) (hR : Nat.card ↥(kernelFrattini K) = 1) : Prop

The decoration-kernel half of the lower-quotient construction in the R = 1 elementary stage. This is the precise target-side assertion from lem:decorationblock: the boundary decoration kills the selected elementary layer K, so the quotient C = Y/K inherits the same decoration.

• Ksub_le_thetaY_ker : K.Ksub ≤ p.fst.thetaY.ker

  lem:decorationblock in the elementary branch: the boundary decoration kills K.

structure Q2Presentation.Induction.Sec7FrattiniTrivialElementaryLowerMeasureData {p : FramedPair} (h : ¬ScalarTerminal p) (chief : FirstNonScalarChiefFactor p) (K : KernelMinimal chief.toNonScalarChiefFactor) (hR : Nat.card ↥(kernelFrattini K) = 1) (decor : Sec7FrattiniTrivialElementaryLowerDecorationData h chief K hR) : Prop

The finite exact-sequence/cardinality half of the lower quotient C = Y/K. Relative to the already supplied decoration-kernel datum, this records exactly the marked-kernel measure identity |L_Y| = |K| |L_C|.

• lower_measure_mul_Kcard : framedMeasure (sec7ElementaryLowerChild K ⋯) * Nat.card ↥K.Ksub = framedMeasure p

  Lower-target cardinal identity |L_Y| = |K| |L_C|, written in measure form.

Decoration and lower marked-kernel exact sequence for the R = 1 child

def Q2Presentation.Induction.Sec7FrattiniTrivialElementaryDecorationProofs.thetaOnK {p : FramedPair} {chief : FirstNonScalarChiefFactor p} (K : KernelMinimal chief.toNonScalarChiefFactor) : ↥K.Ksub →* Multiplicative p.fst.E

The restriction of the elementary decoration to the selected kernel K.

Equations

• Q2Presentation.Induction.Sec7FrattiniTrivialElementaryDecorationProofs.thetaOnK K = p.fst.thetaY.comp K.Ksub.subtype

theorem Q2Presentation.Induction.Sec7FrattiniTrivialElementaryDecorationProofs.frattini_le_thetaOnK_ker {p : FramedPair} {chief : FirstNonScalarChiefFactor p} (K : KernelMinimal chief.toNonScalarChiefFactor) : frattini ↥K.Ksub ≤ (thetaOnK K).ker

Any decoration into an elementary multiplicative vector group kills Φ(K).

def Q2Presentation.Induction.Sec7FrattiniTrivialElementaryDecorationProofs.quotientDecorationHom {p : FramedPair} {chief : FirstNonScalarChiefFactor p} (K : KernelMinimal chief.toNonScalarChiefFactor) : towerMgrp K →* Multiplicative p.fst.E

The decoration descended to the Frattini quotient K/Φ(K).

Equations

• One or more equations did not get rendered due to their size.

@[simp]

theorem Q2Presentation.Induction.Sec7FrattiniTrivialElementaryDecorationProofs.quotientDecorationHom_mk {p : FramedPair} {chief : FirstNonScalarChiefFactor p} (K : KernelMinimal chief.toNonScalarChiefFactor) (k : ↥K.Ksub) : (quotientDecorationHom K) ((QuotientGroup.mk' (frattini ↥K.Ksub)) k) = p.fst.thetaY ↑k

def Q2Presentation.Induction.Sec7FrattiniTrivialElementaryDecorationProofs.decorationAddHom {p : FramedPair} {chief : FirstNonScalarChiefFactor p} (K : KernelMinimal chief.toNonScalarChiefFactor) : towerM K →+ p.fst.E

The descended decoration, read as an additive homomorphism on M = K/Φ(K).

Equations

• One or more equations did not get rendered due to their size.

def Q2Presentation.Induction.Sec7FrattiniTrivialElementaryDecorationProofs.decorationLinearMap {p : FramedPair} {chief : FirstNonScalarChiefFactor p} (K : KernelMinimal chief.toNonScalarChiefFactor) : towerM K →ₗ[ZMod 2] p.fst.E

The descended decoration as a ZMod 2-linear map M → E.

Equations

• One or more equations did not get rendered due to their size.

@[simp]

theorem Q2Presentation.Induction.Sec7FrattiniTrivialElementaryDecorationProofs.decorationLinearMap_mk {p : FramedPair} {chief : FirstNonScalarChiefFactor p} (K : KernelMinimal chief.toNonScalarChiefFactor) (k : ↥K.Ksub) : Multiplicative.ofAdd ((decorationLinearMap K) (Additive.ofMul ((QuotientGroup.mk' (frattini ↥K.Ksub)) k))) = p.fst.thetaY ↑k

On a represented kernel element, the descended linear map recovers thetaY.

theorem Q2Presentation.Induction.Sec7FrattiniTrivialElementaryDecorationProofs.decorationLinearMap_invariant {p : FramedPair} {chief : FirstNonScalarChiefFactor p} (K : KernelMinimal chief.toNonScalarChiefFactor) (c : towerC K) : decorationLinearMap K ∘ₗ ↑((towerActM K) c) = decorationLinearMap K

The descended decoration is invariant for the quotient conjugation action.

theorem Q2Presentation.Induction.Sec7FrattiniTrivialElementaryDecorationProofs.invariant_linearMap_eq_zero {p : FramedPair} {chief : FirstNonScalarChiefFactor p} (K : KernelMinimal chief.toNonScalarChiefFactor) (f : towerM K →ₗ[ZMod 2] p.fst.E) (hf : ∀ (c : towerC K), f ∘ₗ ↑((towerActM K) c) = f) : f = 0

A C-invariant map M → E is zero, by the proved dual-invariant vanishing (M^∨)^C = 0.

theorem Q2Presentation.Induction.Sec7FrattiniTrivialElementaryDecorationProofs.Ksub_le_thetaY_ker {p : FramedPair} {chief : FirstNonScalarChiefFactor p} (K : KernelMinimal chief.toNonScalarChiefFactor) : K.Ksub ≤ p.fst.thetaY.ker

Manuscript lem:decorationblock for the selected elementary layer: the decoration kills K.

theorem Q2Presentation.Induction.sec7_frattiniTrivial_elementaryLowerDecorationData {p : FramedPair} (h : ¬ScalarTerminal p) (chief : FirstNonScalarChiefFactor p) (K : KernelMinimal chief.toNonScalarChiefFactor) (hR : Nat.card ↥(kernelFrattini K) = 1) : Nonempty (Sec7FrattiniTrivialElementaryLowerDecorationData h chief K hR)

Former lower-decoration residual, now theorem-level. The decoration-kernel statement follows from the already proved Section 7 simple-head mechanism: thetaY descends to a C-invariant linear map on M = K/Φ(K), and (M^∨)^C = 0 forces that map to vanish.

structure Q2Presentation.Induction.Sec7FrattiniTrivialElementaryLowerMarkedExactData {p : FramedPair} (h : ¬ScalarTerminal p) (chief : FirstNonScalarChiefFactor p) (K : KernelMinimal chief.toNonScalarChiefFactor) (hR : Nat.card ↥(kernelFrattini K) = 1) (decor : Sec7FrattiniTrivialElementaryLowerDecorationData h chief K hR) : Type

Exact-sequence data on marked kernels for the lower quotient C = Y/K.

This is the mechanism behind the former lower-measure atom: a map from L_Y onto L_C whose kernel is the selected elementary layer K.

• lowerMarkedMap : ↥p.fst.LY →* ↥(sec7ElementaryLowerChild K ⋯).fst.LY

  The marked-kernel map induced by Y → Y/K.

• lowerMarkedMap_surj : Function.Surjective ⇑self.lowerMarkedMap

  The induced marked-kernel map is onto L_C.

• lowerMarkedKernelEquiv : ↥self.lowerMarkedMap.ker ≃* ↥K.Ksub

  Its kernel is the selected layer K.

The lower marked-kernel exact sequence 1 → K → L_Y → L_C → 1

def Q2Presentation.Induction.sec7ElementaryLowerMarkedMap {p : FramedPair} {chief : NonScalarChiefFactor p} (K : KernelMinimal chief) (hθ : K.Ksub ≤ p.fst.thetaY.ker) : ↥p.fst.LY →* ↥(sec7ElementaryLowerChild K hθ).fst.LY

The quotient map Y → Y/K, restricted and corestricted to the marked kernels L_Y → L_C. The codomain is exactly the marked kernel of the literal lower child C = Y/K, namely the image of L_Y in the quotient.

Equations

• Q2Presentation.Induction.sec7ElementaryLowerMarkedMap K hθ = { toFun := fun (y : ↥p.fst.LY) => ⟨(QuotientGroup.mk' K.Ksub) ↑y, ⋯⟩, map_one' := ⋯, map_mul' := ⋯ }

theorem Q2Presentation.Induction.sec7ElementaryLowerMarkedMap_surjective {p : FramedPair} {chief : NonScalarChiefFactor p} (K : KernelMinimal chief) (hθ : K.Ksub ≤ p.fst.thetaY.ker) : Function.Surjective ⇑(sec7ElementaryLowerMarkedMap K hθ)

The lower marked-kernel map is onto, because L_C is defined as the image of L_Y in Y/K.

def Q2Presentation.Induction.sec7ElementaryLowerMarkedKernelEquiv {p : FramedPair} {chief : NonScalarChiefFactor p} (K : KernelMinimal chief) (hθ : K.Ksub ≤ p.fst.thetaY.ker) : ↥(sec7ElementaryLowerMarkedMap K hθ).ker ≃* ↥K.Ksub

The kernel of L_Y → L_C is the selected layer K, viewed inside L_Y.

Equations

• One or more equations did not get rendered due to their size.

theorem Q2Presentation.Induction.sec7_frattiniTrivial_elementaryLowerMarkedExactData {p : FramedPair} (h : ¬ScalarTerminal p) (chief : FirstNonScalarChiefFactor p) (K : KernelMinimal chief.toNonScalarChiefFactor) (hR : Nat.card ↥(kernelFrattini K) = 1) (decor : Sec7FrattiniTrivialElementaryLowerDecorationData h chief K hR) : Nonempty (Sec7FrattiniTrivialElementaryLowerMarkedExactData h chief K hR decor)

Closed lower marked-kernel exactness theorem (R = 1, child geometry). The exact sequence is the literal quotient sequence 1 → K → L_Y → L_C → 1 induced by Y → Y/K.

theorem Q2Presentation.Induction.sec7_frattiniTrivial_elementaryLowerMeasureData {p : FramedPair} (h : ¬ScalarTerminal p) (chief : FirstNonScalarChiefFactor p) (K : KernelMinimal chief.toNonScalarChiefFactor) (hR : Nat.card ↥(kernelFrattini K) = 1) (decor : Sec7FrattiniTrivialElementaryLowerDecorationData h chief K hR) : Nonempty (Sec7FrattiniTrivialElementaryLowerMeasureData h chief K hR decor)

Former lower-measure residual, now theorem-level from exactness. The marked-kernel measure identity |L_Y| = |K| |L_C| is obtained by cardinality of a finite surjective homomorphism and the kernel equivalence with K.

theorem Q2Presentation.Induction.sec7_frattiniTrivial_elementaryLowerQuotientCoreData {p : FramedPair} (h : ¬ScalarTerminal p) (chief : FirstNonScalarChiefFactor p) (K : KernelMinimal chief.toNonScalarChiefFactor) (hR : Nat.card ↥(kernelFrattini K) = 1) : Nonempty (Sec7FrattiniTrivialElementaryLowerQuotientCoreData h chief K hR)

Former lower-quotient core residual, now theorem-level from two smaller packets. The old core packet is assembled from the decoration-kernel fact and the lower quotient marked-kernel cardinal identity. The nontriviality field of the still older packet is proved separately by sec7_Ksub_card_gt_one.

theorem Q2Presentation.Induction.sec7_frattiniTrivial_elementaryLowerQuotientData {p : FramedPair} (h : ¬ScalarTerminal p) (chief : FirstNonScalarChiefFactor p) (K : KernelMinimal chief.toNonScalarChiefFactor) (hR : Nat.card ↥(kernelFrattini K) = 1) : Nonempty (Sec7FrattiniTrivialElementaryLowerQuotientData h chief K hR)

Former lower-quotient residual, now theorem-level. The public packet is assembled from the narrowed quotient-construction residual and the proved nontriviality of the minimal kernel K.

theorem Q2Presentation.Induction.sec7_frattiniTrivial_elementaryProperImageFamilyData {p : FramedPair} (h : ¬ScalarTerminal p) (chief : FirstNonScalarChiefFactor p) (K : KernelMinimal chief.toNonScalarChiefFactor) (hR : Nat.card ↥(kernelFrattini K) = 1) (lower : Sec7FrattiniTrivialElementaryLowerQuotientData h chief K hR) : Nonempty (Sec7FrattiniTrivialElementaryProperImageFamilyData h chief K hR lower)

Former proper-image family residual, now theorem-level. The finite exact-image family is the canonical subtype of subgroups J < B = Y satisfying the two surjectivity conditions from eq:Mstage; no residual enumeration choice remains.

structure Q2Presentation.Induction.Sec7FrattiniTrivialElementaryProperImageCanonicalRestrictedExactData {p : FramedPair} (h : ¬ScalarTerminal p) (chief : FirstNonScalarChiefFactor p) (K : KernelMinimal chief.toNonScalarChiefFactor) (hR : Nat.card ↥(kernelFrattini K) = 1) (lower : Sec7FrattiniTrivialElementaryLowerQuotientData h chief K hR) : Type

Exact-sequence data for every canonical proper exact-image subgroup in the R = 1 elementary stage.

For each J < Y mapping onto C = Y/K, this records the restricted exact sequence 1 → J ∩ K → J ∩ L_Y → L_C → 1 behind eq:Mdescent.

• kernelPart : Sec7FrattiniTrivialElementaryProperExactImageSubgroups lower → Subgroup ↥K.Ksub

  The subgroup playing the role of J ∩ K for a canonical exact image.

• restrictedMap (i : Sec7FrattiniTrivialElementaryProperExactImageSubgroups lower) : ↥(sec7ElementaryExactImageChild p ↑i ⋯).fst.LY →* ↥(sec7ElementaryLowerChild K ⋯).fst.LY

  The map J ∩ L_Y → L_C induced by Y → C.

• restrictedMap_surj (i : Sec7FrattiniTrivialElementaryProperExactImageSubgroups lower) : Function.Surjective ⇑(self.restrictedMap i)

  The restricted map is onto the lower marked kernel.

• restrictedKernelEquiv (i : Sec7FrattiniTrivialElementaryProperExactImageSubgroups lower) : ↥(self.restrictedMap i).ker ≃* ↥(self.kernelPart i)

  Its kernel is the selected J ∩ K subgroup.

• kernelPart_bound (i : Sec7FrattiniTrivialElementaryProperExactImageSubgroups lower) : 2 * Nat.card ↥(self.kernelPart i) ≤ Nat.card ↥K.Ksub

  Properness of J < Y gives the manuscript bound 2 |J ∩ K| ≤ |K|.

Restricted exact sequences for canonical proper exact images

theorem Q2Presentation.Induction.sec7_subgroup_eq_top_of_ker_le_of_surjective_restrict {G : Type u_1} {H : Type u_2} [Group G] [Group H] (f : G →* H) (J : Subgroup G) (hJ : Function.Surjective ⇑(f.comp J.subtype)) (hker : f.ker ≤ J) : J = ⊤

If J maps onto a quotient and contains the kernel, then J is the whole group. This is the elementary finite-group argument behind the proper-image kernel bound.

def Q2Presentation.Induction.sec7ProperImageKernelPart {p : FramedPair} {h : ¬ScalarTerminal p} {chief : FirstNonScalarChiefFactor p} {K : KernelMinimal chief.toNonScalarChiefFactor} {hR : Nat.card ↥(kernelFrattini K) = 1} (lower : Sec7FrattiniTrivialElementaryLowerQuotientData h chief K hR) (i : Sec7FrattiniTrivialElementaryProperExactImageSubgroups lower) : Subgroup ↥K.Ksub

The literal kernel part J ∩ K, viewed as a subgroup of K.

Equations

• Q2Presentation.Induction.sec7ProperImageKernelPart lower i = (↑i).subgroupOf K.Ksub

theorem Q2Presentation.Induction.sec7ProperImageKernelPart_proper {p : FramedPair} {h : ¬ScalarTerminal p} {chief : FirstNonScalarChiefFactor p} {K : KernelMinimal chief.toNonScalarChiefFactor} {hR : Nat.card ↥(kernelFrattini K) = 1} (lower : Sec7FrattiniTrivialElementaryLowerQuotientData h chief K hR) (i : Sec7FrattiniTrivialElementaryProperExactImageSubgroups lower) : sec7ProperImageKernelPart lower i < ⊤

If J < Y maps onto Y/K, then J ∩ K is a proper subgroup of K.

theorem Q2Presentation.Induction.sec7ProperImageKernelPart_bound {p : FramedPair} {h : ¬ScalarTerminal p} {chief : FirstNonScalarChiefFactor p} {K : KernelMinimal chief.toNonScalarChiefFactor} {hR : Nat.card ↥(kernelFrattini K) = 1} (lower : Sec7FrattiniTrivialElementaryLowerQuotientData h chief K hR) (i : Sec7FrattiniTrivialElementaryProperExactImageSubgroups lower) : 2 * Nat.card ↥(sec7ProperImageKernelPart lower i) ≤ Nat.card ↥K.Ksub

The proper-subgroup cardinal estimate 2 |J ∩ K| ≤ |K|.

def Q2Presentation.Induction.sec7ProperImageRestrictedDomainVal {p : FramedPair} {h : ¬ScalarTerminal p} {chief : FirstNonScalarChiefFactor p} {K : KernelMinimal chief.toNonScalarChiefFactor} {hR : Nat.card ↥(kernelFrattini K) = 1} {lower : Sec7FrattiniTrivialElementaryLowerQuotientData h chief K hR} {i : Sec7FrattiniTrivialElementaryProperExactImageSubgroups lower} (x : ↥(sec7ElementaryExactImageChild p ↑i ⋯).fst.LY) : p.fst.Y

The underlying element of the proper-image marked kernel J ∩ L_Y, read in Y.

Equations

• Q2Presentation.Induction.sec7ProperImageRestrictedDomainVal x = id (fun (x : ↥(p.fst.LY.subgroupOf ↑i)) => ↑((p.fst.subgroupOfInclToLY ↑i) x)) x

theorem Q2Presentation.Induction.sec7ProperImageRestrictedDomainVal_mem_LY {p : FramedPair} {h : ¬ScalarTerminal p} {chief : FirstNonScalarChiefFactor p} {K : KernelMinimal chief.toNonScalarChiefFactor} {hR : Nat.card ↥(kernelFrattini K) = 1} {lower : Sec7FrattiniTrivialElementaryLowerQuotientData h chief K hR} {i : Sec7FrattiniTrivialElementaryProperExactImageSubgroups lower} (x : ↥(sec7ElementaryExactImageChild p ↑i ⋯).fst.LY) : sec7ProperImageRestrictedDomainVal x ∈ p.fst.LY

The underlying element of J ∩ L_Y really lies in L_Y.

theorem Q2Presentation.Induction.sec7ElementaryLower_piY_lowerMap {p : FramedPair} {chief : NonScalarChiefFactor p} (K : KernelMinimal chief) (hθ : K.Ksub ≤ p.fst.thetaY.ker) (y : p.fst.Y) : (sec7ElementaryLowerChild K hθ).fst.piY ((sec7ElementaryLowerMap K hθ) y) = p.fst.piY y

Compatibility of the lower quotient map with the tame quotient map.

def Q2Presentation.Induction.sec7ProperImageRestrictedMap {p : FramedPair} {h : ¬ScalarTerminal p} {chief : FirstNonScalarChiefFactor p} {K : KernelMinimal chief.toNonScalarChiefFactor} {hR : Nat.card ↥(kernelFrattini K) = 1} (lower : Sec7FrattiniTrivialElementaryLowerQuotientData h chief K hR) (i : Sec7FrattiniTrivialElementaryProperExactImageSubgroups lower) : ↥(sec7ElementaryExactImageChild p ↑i ⋯).fst.LY →* ↥(sec7ElementaryLowerChild K ⋯).fst.LY

The canonical restricted map J ∩ L_Y → L_C, induced by the quotient map Y → Y/K.

Equations

• One or more equations did not get rendered due to their size.

theorem Q2Presentation.Induction.sec7ProperImageRestrictedMap_surjective {p : FramedPair} {h : ¬ScalarTerminal p} {chief : FirstNonScalarChiefFactor p} {K : KernelMinimal chief.toNonScalarChiefFactor} {hR : Nat.card ↥(kernelFrattini K) = 1} (lower : Sec7FrattiniTrivialElementaryLowerQuotientData h chief K hR) (i : Sec7FrattiniTrivialElementaryProperExactImageSubgroups lower) : Function.Surjective ⇑(sec7ProperImageRestrictedMap lower i)

The canonical restricted map is onto L_C. If j ∈ J maps to a marked-kernel class in C, then compatibility with π_Y shows j ∈ L_Y.

def Q2Presentation.Induction.sec7ProperImageRestrictedKernelEquiv {p : FramedPair} {h : ¬ScalarTerminal p} {chief : FirstNonScalarChiefFactor p} {K : KernelMinimal chief.toNonScalarChiefFactor} {hR : Nat.card ↥(kernelFrattini K) = 1} (lower : Sec7FrattiniTrivialElementaryLowerQuotientData h chief K hR) (i : Sec7FrattiniTrivialElementaryProperExactImageSubgroups lower) : ↥(sec7ProperImageRestrictedMap lower i).ker ≃* ↥(sec7ProperImageKernelPart lower i)

The kernel of the restricted map is the literal intersection J ∩ K.

Equations

• One or more equations did not get rendered due to their size.

theorem Q2Presentation.Induction.sec7_frattiniTrivial_elementaryProperImageCanonicalRestrictedExactData {p : FramedPair} (h : ¬ScalarTerminal p) (chief : FirstNonScalarChiefFactor p) (K : KernelMinimal chief.toNonScalarChiefFactor) (hR : Nat.card ↥(kernelFrattini K) = 1) (lower : Sec7FrattiniTrivialElementaryLowerQuotientData h chief K hR) : Nonempty (Sec7FrattiniTrivialElementaryProperImageCanonicalRestrictedExactData h chief K hR lower)

Closed canonical restricted-exactness theorem (R = 1, child geometry). For every canonical proper exact-image subgroup J < Y mapping onto C = Y/K, the restricted quotient map gives the exact sequence 1 → J ∩ K → J ∩ L_Y → L_C → 1; the half-cardinality bound comes from J ∩ K < K.

theorem Q2Presentation.Induction.sec7_frattiniTrivial_elementaryProperImageCanonicalDescentData {p : FramedPair} (h : ¬ScalarTerminal p) (chief : FirstNonScalarChiefFactor p) (K : KernelMinimal chief.toNonScalarChiefFactor) (hR : Nat.card ↥(kernelFrattini K) = 1) (lower : Sec7FrattiniTrivialElementaryLowerQuotientData h chief K hR) : Nonempty (Sec7FrattiniTrivialElementaryProperImageCanonicalDescentData h chief K hR lower)

Former canonical descent residual, now theorem-level from restricted exactness. The cardinal formula |J ∩ L_Y| = |J ∩ K| |L_C| follows from finite exact-sequence cardinality applied to each canonical restricted map.

theorem Q2Presentation.Induction.sec7_frattiniTrivial_elementaryProperImageDescentData {p : FramedPair} (h : ¬ScalarTerminal p) (chief : FirstNonScalarChiefFactor p) (K : KernelMinimal chief.toNonScalarChiefFactor) (hR : Nat.card ↥(kernelFrattini K) = 1) (lower : Sec7FrattiniTrivialElementaryLowerQuotientData h chief K hR) (family : Sec7FrattiniTrivialElementaryProperImageFamilyData h chief K hR lower) : Nonempty (Sec7FrattiniTrivialElementaryProperImageDescentData h chief K hR lower family)

Former proper-image descent residual, now theorem-level. The old statement allowed an arbitrary chosen finite family. It is assembled by pulling the canonical eq:Mdescent data back along that family's subgroup map into the canonical subtype.

theorem Q2Presentation.Induction.sec7_frattiniTrivial_elementaryProperImageData {p : FramedPair} (h : ¬ScalarTerminal p) (chief : FirstNonScalarChiefFactor p) (K : KernelMinimal chief.toNonScalarChiefFactor) (hR : Nat.card ↥(kernelFrattini K) = 1) (lower : Sec7FrattiniTrivialElementaryLowerQuotientData h chief K hR) : Nonempty (Sec7FrattiniTrivialElementaryProperImageData h chief K hR lower)

Former proper-image residual, now theorem-level. The public exact-image packet is assembled from the independent finite enumeration and cardinal-descent pieces of eq:Mstage / eq:Mdescent.

theorem Q2Presentation.Induction.sec7_frattiniTrivial_elementaryChildGeometryData {p : FramedPair} (h : ¬ScalarTerminal p) (chief : FirstNonScalarChiefFactor p) (K : KernelMinimal chief.toNonScalarChiefFactor) (hR : Nat.card ↥(kernelFrattini K) = 1) : Nonempty (Sec7FrattiniTrivialElementaryChildGeometryData h chief K hR)

Former child-geometry residual, now theorem-level. The broad child-geometry packet is assembled from the manuscript's two independent target-side pieces: the lower quotient C = Y/K and the proper exact-image enumeration in the M-stage subtraction.

theorem Q2Presentation.Induction.sec7_frattiniTrivial_elementaryChildData {p : FramedPair} (h : ¬ScalarTerminal p) (chief : FirstNonScalarChiefFactor p) (K : KernelMinimal chief.toNonScalarChiefFactor) (hR : Nat.card ↥(kernelFrattini K) = 1) : Nonempty (Sec7FrattiniTrivialElementaryChildData p)

The exact-image child half of the R = 1 elementary-stage split packet is now assembled theorem-level from the child-geometry residual.

structure Q2Presentation.Induction.Sec7FrattiniTrivialElementaryNormalizedCountData {p : FramedPair} (D : Sec7FrattiniTrivialElementaryChildData p) : Type 1

The normalized source-count packet for the R = 1 elementary stage, relative to a fixed child packet. The obstruction space has already collapsed to PUnit, so the residual stores literal lift-index cardinalities rather than cardinalities of vacuous zero-obstruction subtypes. The subtype-to-index conversion is proved below.

• partialN : Module.Dual (ZMod 2) PUnit.{1} → List ℕ → ℕ

  Source-independent elementary partial-count recipe.

• XA : Type

  Candidate-source elementary lift index.

• finXA : Fintype self.XA
• XQ : Type

  Local-source elementary lift index.

• finXQ : Fintype self.XQ
• realizeA_card : framedCountA p = ↑(Nat.card self.XA)

  Candidate elementary lift identity, with the vacuous zero-obstruction fibre simplified to the full lift index.

• realizeQ_card : framedCountQ p = ↑(Nat.card self.XQ)

  Local elementary lift identity, with the vacuous zero-obstruction fibre simplified to the full lift index.

• partialA_zero_card : Nat.card self.XA = self.partialN 0 (List.map (⇑Cardinal.toNat) (List.map framedCountA (D.lowerChild :: D.properChildren)))

  Candidate zero-character partial-count identity, normalized to the full lift index.

• partialQ_zero_card : Nat.card self.XQ = self.partialN 0 (List.map (⇑Cardinal.toNat) (List.map framedCountQ (D.lowerChild :: D.properChildren)))

  Local zero-character partial-count identity, normalized to the full lift index.

structure Q2Presentation.Induction.Sec7FrattiniTrivialElementaryStageSplitData (p : FramedPair) : Type 1

A split residual packet: child geometry plus normalized source-count data.

• child : Sec7FrattiniTrivialElementaryChildData p

  The lower/proper exact-image child data.

• counts : Sec7FrattiniTrivialElementaryNormalizedCountData self.child

  The normalized elementary count data for those children.

def Q2Presentation.Induction.Sec7FrattiniTrivialElementaryChildData.children {p : FramedPair} (D : Sec7FrattiniTrivialElementaryChildData p) : List FramedPair

The elementary-stage child list attached to the split child packet.

Equations

• D.children = D.lowerChild :: D.properChildren

theorem Q2Presentation.Induction.Sec7FrattiniTrivialElementaryChildData.small {p : FramedPair} (D : Sec7FrattiniTrivialElementaryChildData p) (c : FramedPair) : c ∈ D.children → framedMeasure c < framedMeasure p

Strict decrease for the split child packet.

def Q2Presentation.Induction.Sec7FrattiniTrivialElementaryNormalizedCountData.toCoreData {p : FramedPair} {D : Sec7FrattiniTrivialElementaryChildData p} (C : Sec7FrattiniTrivialElementaryNormalizedCountData D) : Sec7FrattiniTrivialElementaryStageCoreData p

Assemble the old core packet from child data plus normalized count data. The only work here is the theorem-level conversion from the vacuous zero-obstruction subtype to the full lift index.

Equations

• One or more equations did not get rendered due to their size.

def Q2Presentation.Induction.Sec7FrattiniTrivialElementaryStageSplitData.toCoreData {p : FramedPair} (D : Sec7FrattiniTrivialElementaryStageSplitData p) : Sec7FrattiniTrivialElementaryStageCoreData p

Assemble the old core residual from the split elementary-stage packet.

Equations

• D.toCoreData = D.counts.toCoreData

def Q2Presentation.Induction.Sec7FrattiniTrivialElementaryStageData.children {p : FramedPair} (D : Sec7FrattiniTrivialElementaryStageData p) : List FramedPair

The elementary-stage child list: the lower quotient child followed by the proper-image subtraction children.

Equations

• D.children = D.lowerChild :: D.properChildren

theorem Q2Presentation.Induction.Sec7FrattiniTrivialElementaryStageData.small {p : FramedPair} (D : Sec7FrattiniTrivialElementaryStageData p) (c : FramedPair) : c ∈ D.children → framedMeasure c < framedMeasure p

Strict decrease for the assembled elementary-stage child list.

theorem Q2Presentation.Induction.Sec7FrattiniTrivialElementaryStageData.card_dual_punit : Fintype.card (Module.Dual (ZMod 2) PUnit.{1}) = 1

The dual of the zero obstruction space has one element.

def Q2Presentation.Induction.Sec7FrattiniTrivialElementaryStageData.toFourierBranchData {p : FramedPair} (D : Sec7FrattiniTrivialElementaryStageData p) : FourierBranchData p

Assemble the ordinary FourierBranchData packet from the lowered R = 1 elementary-stage data.

Equations

• One or more equations did not get rendered due to their size.

def Q2Presentation.Induction.Sec7FrattiniTrivialElementaryStageCoreData.children {p : FramedPair} (D : Sec7FrattiniTrivialElementaryStageCoreData p) : List FramedPair

The elementary-stage child list attached to the core packet.

Equations

• D.children = D.lowerChild :: D.properChildren

theorem Q2Presentation.Induction.Sec7FrattiniTrivialElementaryStageCoreData.small {p : FramedPair} (D : Sec7FrattiniTrivialElementaryStageCoreData p) (c : FramedPair) : c ∈ D.children → framedMeasure c < framedMeasure p

Strict decrease for the core elementary-stage child list.

theorem Q2Presentation.Induction.Sec7FrattiniTrivialElementaryStageCoreData.partialA {p : FramedPair} (D : Sec7FrattiniTrivialElementaryStageCoreData p) (χ : Module.Dual (ZMod 2) PUnit.{1}) : Nat.card { x : D.XA // χ (sec7ZeroObstruction D.XA x) = 0 } = D.partialN χ (List.map (⇑Cardinal.toNat) (List.map framedCountA (D.lowerChild :: D.properChildren)))

The all-character candidate partial identity follows from the single zero-character identity because the dual of PUnit is subsingleton.

theorem Q2Presentation.Induction.Sec7FrattiniTrivialElementaryStageCoreData.partialQ {p : FramedPair} (D : Sec7FrattiniTrivialElementaryStageCoreData p) (χ : Module.Dual (ZMod 2) PUnit.{1}) : Nat.card { x : D.XQ // χ (sec7ZeroObstruction D.XQ x) = 0 } = D.partialN χ (List.map (⇑Cardinal.toNat) (List.map framedCountQ (D.lowerChild :: D.properChildren)))

The all-character local partial identity follows from the single zero-character identity because the dual of PUnit is subsingleton.

def Q2Presentation.Induction.Sec7FrattiniTrivialElementaryStageCoreData.toElementaryStageData {p : FramedPair} (D : Sec7FrattiniTrivialElementaryStageCoreData p) : Sec7FrattiniTrivialElementaryStageData p

Assemble the previous elementary-stage data packet from the smaller core packet.

Equations

• One or more equations did not get rendered due to their size.

def Q2Presentation.Induction.Sec7FrattiniTrivialElementaryStageCoreData.toFourierBranchData {p : FramedPair} (D : Sec7FrattiniTrivialElementaryStageCoreData p) : FourierBranchData p

Direct Fourier-branch assembly from the smaller core packet.

Equations

• D.toFourierBranchData = D.toElementaryStageData.toFourierBranchData

def Q2Presentation.Induction.sec7MStageFunctional (zM : ℕ) : List ℕ → ℕ

The elementary-stage numerical functional from eq:Mstage:

N([e(C), e(J₁), ..., e(Jₙ)]) = z_M * e(C) - Σ e(Jᵢ).

The empty-list value is unused by the assembled child packet, but making the function total keeps the normalized-count interface independent of a separate nonempty-list subtype.

Equations

• Q2Presentation.Induction.sec7MStageFunctional zM [] = 0
• Q2Presentation.Induction.sec7MStageFunctional zM (lower :: proper) = zM * lower - proper.sum

structure Q2Presentation.Induction.Sec7FrattiniTrivialElementaryMStageNormalizedCountData {p : FramedPair} (D : Sec7FrattiniTrivialElementaryChildData p) : Type 1

A manuscript-shaped normalized source-count packet for the R = 1 elementary stage.

Compared with Sec7FrattiniTrivialElementaryNormalizedCountData, this stores the actual eq:Mstage functional: the common lift multiplicity is 2^(2 dim M), and the candidate and local lift-index cardinalities are obtained from the lower child and the proper-image subtraction children by the same formula.

• dimM : ℕ

  The dimension of the elementary layer M over 𝔽₂.

• liftMultiplicity : ℕ

  The common lift multiplicity |Z¹_{Γ,ρ}(M)| = 2^(2 dim M).

• liftMultiplicity_eq : self.liftMultiplicity = 2 ^ (2 * self.dimM)

  The manuscript multiplicity formula.

• XA : Type

  Candidate-source elementary lift index.

• finXA : Fintype self.XA
• XQ : Type

  Local-source elementary lift index.

• finXQ : Fintype self.XQ
• realizeA_card : framedCountA p = ↑(Nat.card self.XA)

  Candidate elementary lift identity, normalized to the full lift index.

• realizeQ_card : framedCountQ p = ↑(Nat.card self.XQ)

  Local elementary lift identity, normalized to the full lift index.

• partialA_mstage : Nat.card self.XA = sec7MStageFunctional self.liftMultiplicity (List.map (⇑Cardinal.toNat) (List.map framedCountA (D.lowerChild :: D.properChildren)))

  Candidate eq:Mstage: lower quotient contribution minus proper exact-image terms.

• partialQ_mstage : Nat.card self.XQ = sec7MStageFunctional self.liftMultiplicity (List.map (⇑Cardinal.toNat) (List.map framedCountQ (D.lowerChild :: D.properChildren)))

  Local eq:Mstage: lower quotient contribution minus proper exact-image terms.

structure Q2Presentation.Induction.Sec7FrattiniTrivialElementaryMStageMultiplicityData {p : FramedPair} (D : Sec7FrattiniTrivialElementaryChildData p) : Type

The lift-multiplicity half of the R = 1 elementary source-count calculation.

This is the numerical part of lem:elementarystage: the common size of the cocycle torsor for an M-lift is 2^(2 dim M). It is separated from the source-specific lift-index realizations and from exact-image subtraction.

• dimM : ℕ

  The dimension of the elementary layer M over 𝔽₂.

• liftMultiplicity : ℕ

  The common lift multiplicity |Z¹_{Γ,ρ}(M)| = 2^(2 dim M).

• liftMultiplicity_eq : self.liftMultiplicity = 2 ^ (2 * self.dimM)

  The manuscript multiplicity formula.

structure Q2Presentation.Induction.Sec7FrattiniTrivialElementaryMStageSourceIndexData {p : FramedPair} (D : Sec7FrattiniTrivialElementaryChildData p) (mult : Sec7FrattiniTrivialElementaryMStageMultiplicityData D) : Type 1

The source-specific elementary lift indices, after the multiplicity has been fixed.

This records only the realization of the candidate and local exact-image counts as the cardinalities of their elementary lift-index sets. It contains no subtraction formula.

• XA : Type

  Candidate-source elementary lift index.

• finXA : Fintype self.XA
• XQ : Type

  Local-source elementary lift index.

• finXQ : Fintype self.XQ
• realizeA_card : framedCountA p = ↑(Nat.card self.XA)

  Candidate elementary lift identity, normalized to the full lift index.

• realizeQ_card : framedCountQ p = ↑(Nat.card self.XQ)

  Local elementary lift identity, normalized to the full lift index.

structure Q2Presentation.Induction.Sec7FrattiniTrivialElementaryMStageExactCountData {p : FramedPair} (D : Sec7FrattiniTrivialElementaryChildData p) (mult : Sec7FrattiniTrivialElementaryMStageMultiplicityData D) (src : Sec7FrattiniTrivialElementaryMStageSourceIndexData D mult) : Prop

The exact-image subtraction half of eq:Mstage, relative to fixed lift indices.

This is the common source-independent functional z_M · e(C) - Σ_{J<C} e(J), instantiated for the candidate and local sources.

• partialA_mstage : Nat.card src.XA = sec7MStageFunctional mult.liftMultiplicity (List.map (⇑Cardinal.toNat) (List.map framedCountA (D.lowerChild :: D.properChildren)))

  Candidate eq:Mstage: lower quotient contribution minus proper exact-image terms.

• partialQ_mstage : Nat.card src.XQ = sec7MStageFunctional mult.liftMultiplicity (List.map (⇑Cardinal.toNat) (List.map framedCountQ (D.lowerChild :: D.properChildren)))

  Local eq:Mstage: lower quotient contribution minus proper exact-image terms.

def Q2Presentation.Induction.Sec7FrattiniTrivialElementaryMStageNormalizedCountData.toNormalizedCountData {p : FramedPair} {D : Sec7FrattiniTrivialElementaryChildData p} (C : Sec7FrattiniTrivialElementaryMStageNormalizedCountData D) : Sec7FrattiniTrivialElementaryNormalizedCountData D

Forget the eq:Mstage-shaped count packet to the normalized zero-obstruction count interface used by the existing elementary-stage assembly.

Equations

• One or more equations did not get rendered due to their size.

def Q2Presentation.Induction.sec7MStageDim {p : FramedPair} {chief : FirstNonScalarChiefFactor p} (K : KernelMinimal chief.toNonScalarChiefFactor) : ℕ

The actual elementary-layer dimension used by the R = 1 M-stage.

In the manuscript notation this is dim_{𝔽₂} M, with M = K/Φ(K). The current concrete Section 7 tower realizes this as towerM K; when R = Φ(K) = 1, this is the elementary layer lifted in eq:Mstage.

Equations

• Q2Presentation.Induction.sec7MStageDim K = Module.finrank (ZMod 2) (Q2Presentation.Induction.towerM K)

structure Q2Presentation.Induction.Sec7FrattiniTrivialElementaryMStageCocycleMultiplicityData {p : FramedPair} (h : ¬ScalarTerminal p) (chief : FirstNonScalarChiefFactor p) (K : KernelMinimal chief.toNonScalarChiefFactor) (hR : Nat.card ↥(kernelFrattini K) = 1) (child : Sec7FrattiniTrivialElementaryChildData p) : Type 1

The source-level cocycle groups controlling the elementary M-lift torsors.

This is the source-theoretic content behind the old multiplicity residual: for both sources, the relevant Z¹_{Γ,ρ}(M) groups have cardinality 2^(2 dim M), with dim M fixed to the literal concrete module towerM K. It is smaller than Sec7FrattiniTrivialElementaryMStageMultiplicityData: it no longer allows an arbitrary dimension or an arbitrary common multiplicity detached from the manuscript module layer.

• ZA : Type

  Candidate Z¹_{Γ_A,ρ}(M) additive group.

• addZA : AddCommGroup self.ZA
• finZA : Fintype self.ZA
• ZQ : Type

  Local Z¹_{G_{ℚ₂},ρ}(M) additive group.

• addZQ : AddCommGroup self.ZQ
• finZQ : Fintype self.ZQ
• candidate_card : Nat.card self.ZA = 2 ^ (2 * sec7MStageDim K)

  Candidate Euler/duality cardinality formula.

• local_card : Nat.card self.ZQ = 2 ^ (2 * sec7MStageDim K)

  Local Euler/Tate cardinality formula.

structure Q2Presentation.Induction.Sec7FrattiniTrivialElementaryMStageCandidateCocycleGroupData {p : FramedPair} (h : ¬ScalarTerminal p) (chief : FirstNonScalarChiefFactor p) (K : KernelMinimal chief.toNonScalarChiefFactor) (hR : Nat.card ↥(kernelFrattini K) = 1) (child : Sec7FrattiniTrivialElementaryChildData p) : Type

Candidate-side constructor for the actual word-complex cocycle group in the R = 1 M-stage.

The three endomorphisms are the evaluated candidate word-complex operators on the literal module M = towerM K: Sinv = ρ(σ)⁻¹, T = ρ(τ), and P is the norm operator appearing in Lifting.Z1. The current framed-pair interface does not yet carry the lower source map/marking from which these endomorphisms are computed, so this is the smallest constructor atom left by the existing API: it pins the candidate cocycle object to Q2Presentation.Lifting.Z1 rather than to an arbitrary additive group.

• Sinv : Module.End (ZMod 2) (towerM K)

  The candidate operator ρ(σ)⁻¹ on M = towerM K.

• T : Module.End (ZMod 2) (towerM K)

  The candidate operator ρ(τ) on M = towerM K.

• P : Module.End (ZMod 2) (towerM K)

  The candidate norm operator P = 1 + T + ⋯ on M = towerM K.

• finZA : Fintype ↥(Lifting.Z1 self.Sinv self.T self.P)

  Finiteness of the literal candidate cocycle group Lifting.Z1 Sinv T P.

structure Q2Presentation.Induction.Sec7FrattiniTrivialElementaryMStageCandidateCocycleCardinalityData {p : FramedPair} (h : ¬ScalarTerminal p) (chief : FirstNonScalarChiefFactor p) (K : KernelMinimal chief.toNonScalarChiefFactor) (hR : Nat.card ↥(kernelFrattini K) = 1) (child : Sec7FrattiniTrivialElementaryChildData p) (cand : Sec7FrattiniTrivialElementaryMStageCandidateCocycleGroupData h chief K hR child) : Prop

Candidate-side cardinality formula for the actual word-complex cocycle group.

• candidate_card : Nat.card ↥(Lifting.Z1 cand.Sinv cand.T cand.P) = 2 ^ (2 * sec7MStageDim K)

  Candidate Euler/Fox--Heisenberg cardinality formula.

structure Q2Presentation.Induction.Sec7FrattiniTrivialElementaryMStageLocalCocycleGroupData {p : FramedPair} (h : ¬ScalarTerminal p) (chief : FirstNonScalarChiefFactor p) (K : KernelMinimal chief.toNonScalarChiefFactor) (hR : Nat.card ↥(kernelFrattini K) = 1) (child : Sec7FrattiniTrivialElementaryChildData p) : Type 1

Local-source constructor for the actual continuous local cocycle group in the R = 1 M-stage.

The bundled finite discrete representation is the local G_{ℚ₂}-module whose underlying 𝔽₂-space is linearly identified with the concrete target module towerM K. The continuous local cocycle group is then A.Z1 from Local.Cohomology.

• A : Local.FiniteDiscreteRep GQ2Profinite

  The local finite discrete G_{ℚ₂}-module corresponding to M = towerM K.

• module_equiv : self.A.V ≃ₗ[ZMod 2] towerM K

  Identification of the local coefficient module with the concrete Section 7 module.

• source : ↑GQ2Profinite.toProfinite.toTop →* towerC K

  The local source quotient through which this coefficient representation factors.

  This is the Lean form of the source-interface assertion that the local G_{ℚ₂} action on the selected coefficient module is the same finite C = Y/K action as the target-side tower action.

• source_surjective : Function.Surjective ⇑self.source

  The source-interface map realizes the whole finite quotient C = towerC K.

• module_equiv_action (g : ↑GQ2Profinite.toProfinite.toTop) (v : self.A.V) : self.module_equiv ((self.A.ρ g) v) = ((towerActM K) (self.source g)) (self.module_equiv v)

  Equivariance of the chosen linear identification with the concrete tower action.

• finZQ : Fintype ↥self.A.Z1

  Finiteness of the literal local continuous cocycle group A.Z1.

structure Q2Presentation.Induction.Sec7FrattiniTrivialElementaryMStageLocalCocycleCardinalityData {p : FramedPair} (h : ¬ScalarTerminal p) (chief : FirstNonScalarChiefFactor p) (K : KernelMinimal chief.toNonScalarChiefFactor) (hR : Nat.card ↥(kernelFrattini K) = 1) (child : Sec7FrattiniTrivialElementaryChildData p) (loc : Sec7FrattiniTrivialElementaryMStageLocalCocycleGroupData h chief K hR child) : Prop

Local-source cardinality formula for the actual continuous local cocycle group.

• local_card : Nat.card ↥loc.A.Z1 = 2 ^ (2 * sec7MStageDim K)

  Local Tate/Euler cardinality formula.

structure Q2Presentation.Induction.Sec7FrattiniTrivialElementaryMStageCandidateSourceIndexData {p : FramedPair} (D : Sec7FrattiniTrivialElementaryChildData p) (mult : Sec7FrattiniTrivialElementaryMStageMultiplicityData D) : Type 1

Candidate-source elementary lift-index realization, separated from the local side.

• XA : Type

  Candidate-source elementary M-lift index.

• finXA : Fintype self.XA
• realizeA_card : framedCountA p = ↑(Nat.card self.XA)

  Candidate boundary-framed count realized as the cardinality of the lift index.

structure Q2Presentation.Induction.Sec7FrattiniTrivialElementaryMStageLocalSourceIndexData {p : FramedPair} (D : Sec7FrattiniTrivialElementaryChildData p) (mult : Sec7FrattiniTrivialElementaryMStageMultiplicityData D) : Type 1

Local-source elementary lift-index realization, separated from the candidate side.

• XQ : Type

  Local-source elementary M-lift index.

• finXQ : Fintype self.XQ
• realizeQ_card : framedCountQ p = ↑(Nat.card self.XQ)

  Local boundary-framed count realized as the cardinality of the lift index.

structure Q2Presentation.Induction.Sec7FrattiniTrivialElementaryMStageCandidateExactCountData {p : FramedPair} (D : Sec7FrattiniTrivialElementaryChildData p) (mult : Sec7FrattiniTrivialElementaryMStageMultiplicityData D) (src : Sec7FrattiniTrivialElementaryMStageSourceIndexData D mult) : Prop

Candidate eq:Mstage exact-image subtraction, separated from the local side.

• partialA_mstage : Nat.card src.XA = sec7MStageFunctional mult.liftMultiplicity (List.map (⇑Cardinal.toNat) (List.map framedCountA (D.lowerChild :: D.properChildren)))

  Candidate eq:Mstage: lower quotient contribution minus proper exact-image terms.

structure Q2Presentation.Induction.Sec7FrattiniTrivialElementaryMStageLocalExactCountData {p : FramedPair} (D : Sec7FrattiniTrivialElementaryChildData p) (mult : Sec7FrattiniTrivialElementaryMStageMultiplicityData D) (src : Sec7FrattiniTrivialElementaryMStageSourceIndexData D mult) : Prop

Local eq:Mstage exact-image subtraction, separated from the candidate side.

• partialQ_mstage : Nat.card src.XQ = sec7MStageFunctional mult.liftMultiplicity (List.map (⇑Cardinal.toNat) (List.map framedCountQ (D.lowerChild :: D.properChildren)))

  Local eq:Mstage: lower quotient contribution minus proper exact-image terms.

structure Q2Presentation.Induction.Sec7FrattiniTrivialElementaryMStageCandidateWordOperatorData {p : FramedPair} (h : ¬ScalarTerminal p) (chief : FirstNonScalarChiefFactor p) (K : KernelMinimal chief.toNonScalarChiefFactor) (hR : Nat.card ↥(kernelFrattini K) = 1) (child : Sec7FrattiniTrivialElementaryChildData p) : Type

Candidate-side word-operator datum for the literal word complex on M = towerM K.

This is the smaller candidate-side semantic input behind the former cocycle-group constructor. It records the evaluated candidate action operators and their finite-word linear shadows (lem:jacobian, lem:h0shadow) but does not include either the cocycle group itself or its cardinality. Finiteness of the corresponding Lifting.Z1 is theorem-level because towerM K is a finite 𝔽₂-module.

• S : Module.End (ZMod 2) (towerM K)

  The candidate operator ρ(σ) on M = towerM K.

• Sinv : Module.End (ZMod 2) (towerM K)

  The candidate operator ρ(σ)⁻¹ on M = towerM K.

• T : Module.End (ZMod 2) (towerM K)

  The candidate operator ρ(τ) on M = towerM K.

• X₀ : Module.End (ZMod 2) (towerM K)

  The candidate operator ρ(x₀) on M = towerM K.

• X₁ : Module.End (ZMod 2) (towerM K)

  The candidate operator ρ(x₁) on M = towerM K.

• P : Module.End (ZMod 2) (towerM K)

  The candidate norm operator P = 1 + T + ⋯ appearing in Lifting.Z1.

• sinv_left : self.Sinv * self.S = 1

  Sinv is the left inverse of the evaluated σ-operator.

• tame_shadow : self.Sinv * self.T * self.S = self.T ^ 2

  The tame row shadow of the candidate word complex.

• norm_shadow : self.P * (self.T - 1) = 0

  The norm row shadow P·(T-1)=0.

• wild_shadow : (self.P + self.Sinv) * (self.X₁ - 1) = 0

  The wild row shadow (P+Sinv)·(X₁-1)=0.

structure Q2Presentation.Induction.Sec7FrattiniTrivialElementaryMStageCandidateCocycleFinrankData {p : FramedPair} (h : ¬ScalarTerminal p) (chief : FirstNonScalarChiefFactor p) (K : KernelMinimal chief.toNonScalarChiefFactor) (hR : Nat.card ↥(kernelFrattini K) = 1) (child : Sec7FrattiniTrivialElementaryChildData p) (cand : Sec7FrattiniTrivialElementaryMStageCandidateCocycleGroupData h chief K hR child) : Prop

Candidate-side finrank formula for the literal word-complex cocycle group.

This is the linear-algebraic content of lem:elementarystage on the candidate side: candidate deformation duality plus (M^∨)^C = 0 gives dim Z¹_{Γ_A,ρ}(M) = 2·dim M. The old cardinality packet is recovered below from this rank formula by the finite-vector-space identity |V| = 2^(finrank V) over 𝔽₂.

• candidate_finrank : Module.finrank (ZMod 2) ↥(Lifting.Z1 cand.Sinv cand.T cand.P) = 2 * sec7MStageDim K

  Candidate Fox--Heisenberg/Euler dimension formula.

noncomputable def Q2Presentation.Induction.sec7MStageCandidateActionEnd {p : FramedPair} {chief : FirstNonScalarChiefFactor p} (K : KernelMinimal chief.toNonScalarChiefFactor) : towerC K →* Module.End (ZMod 2) (towerM K)

The lower candidate action on the literal Section 7 module M = towerM K, viewed as a monoid representation into linear endomorphisms.

The already-proved concrete tower gives towerActM K : towerC K →* (towerM K ≃ₗ towerM K). This definition forgets the inverse map of each linear equivalence, producing the representation used by the candidate word complex.

Equations

• Q2Presentation.Induction.sec7MStageCandidateActionEnd K = { toFun := fun (c : Q2Presentation.Induction.towerC K) => ↑((Q2Presentation.Induction.towerActM K) c), map_one' := ⋯, map_mul' := ⋯ }

structure Q2Presentation.Induction.Sec7FrattiniTrivialElementaryMStageCandidateLowerSourceData {p : FramedPair} (h : ¬ScalarTerminal p) (chief : FirstNonScalarChiefFactor p) (K : KernelMinimal chief.toNonScalarChiefFactor) (hR : Nat.card ↥(kernelFrattini K) = 1) (child : Sec7FrattiniTrivialElementaryChildData p) : Type

The missing candidate lower source/marking for the R = 1 M-stage.

This is deliberately smaller and more source-specific than the former operator packet. It records a genuine candidate epimorphism to the lower quotient C = towerC K, its four marked values σ, τ, x₀, x₁, and the candidate relators seen in that lower quotient. No linear operators or cocycle cardinalities are chosen here.

• source : Profinite.SurjContHom GammaA (ProfiniteGrp.ofFiniteGrp (FiniteGrp.of (towerC K)))

  The candidate lower-source epimorphism Γ_A ↠ C = Y/K.

• sigma : towerC K

  The value of the lower candidate marking on σ.

• tau : towerC K

  The value of the lower candidate marking on τ.

• x₀ : towerC K

  The value of the lower candidate marking on x₀.

• x₁ : towerC K

  The value of the lower candidate marking on x₁.

• eval_sigma : (ProfiniteGrp.Hom.hom self.source.hom) (gammaGen Gen.sigma) = self.sigma

  Evaluation of the source map on the universal candidate generator σ.

• eval_tau : (ProfiniteGrp.Hom.hom self.source.hom) (gammaGen Gen.tau) = self.tau

  Evaluation of the source map on the universal candidate generator τ.

• eval_x₀ : (ProfiniteGrp.Hom.hom self.source.hom) (gammaGen Gen.x0) = self.x₀

  Evaluation of the source map on the universal candidate generator x₀.

• eval_x₁ : (ProfiniteGrp.Hom.hom self.source.hom) (gammaGen Gen.x1) = self.x₁

  Evaluation of the source map on the universal candidate generator x₁.

• tame_rel : self.sigma⁻¹ * self.tau * self.sigma = self.tau ^ 2

  The tame candidate relator in the lower quotient, σ⁻¹ τ σ = τ².

• wild_rel : (Q2Word.wildRelator.eval fun (a : Gen) => match a with | Gen.sigma => self.sigma | Gen.tau => self.tau | Gen.x0 => self.x₀ | Gen.x1 => self.x₁) = 1

  The full candidate wild relator in the lower quotient.

  The next structure records the finite-word/Fox calculation that turns this relator, via the evaluated action on M, into the wild linear shadow.

def Q2Presentation.Induction.Sec7FrattiniTrivialElementaryMStageCandidateLowerSourceData.marking {p : FramedPair} {h : ¬ScalarTerminal p} {chief : FirstNonScalarChiefFactor p} {K : KernelMinimal chief.toNonScalarChiefFactor} {hR : Nat.card ↥(kernelFrattini K) = 1} {child : Sec7FrattiniTrivialElementaryChildData p} (src : Sec7FrattiniTrivialElementaryMStageCandidateLowerSourceData h chief K hR child) : Marking (towerC K)

The lower marking Gen → towerC K attached to the source datum.

Equations

• src.marking Q2Presentation.Gen.sigma = src.sigma
• src.marking Q2Presentation.Gen.tau = src.tau
• src.marking Q2Presentation.Gen.x0 = src.x₀
• src.marking Q2Presentation.Gen.x1 = src.x₁

@[simp]

theorem Q2Presentation.Induction.Sec7FrattiniTrivialElementaryMStageCandidateLowerSourceData.marking_sigma {p : FramedPair} {h : ¬ScalarTerminal p} {chief : FirstNonScalarChiefFactor p} {K : KernelMinimal chief.toNonScalarChiefFactor} {hR : Nat.card ↥(kernelFrattini K) = 1} {child : Sec7FrattiniTrivialElementaryChildData p} (src : Sec7FrattiniTrivialElementaryMStageCandidateLowerSourceData h chief K hR child) : src.marking Gen.sigma = src.sigma

@[simp]

theorem Q2Presentation.Induction.Sec7FrattiniTrivialElementaryMStageCandidateLowerSourceData.marking_tau {p : FramedPair} {h : ¬ScalarTerminal p} {chief : FirstNonScalarChiefFactor p} {K : KernelMinimal chief.toNonScalarChiefFactor} {hR : Nat.card ↥(kernelFrattini K) = 1} {child : Sec7FrattiniTrivialElementaryChildData p} (src : Sec7FrattiniTrivialElementaryMStageCandidateLowerSourceData h chief K hR child) : src.marking Gen.tau = src.tau

@[simp]

theorem Q2Presentation.Induction.Sec7FrattiniTrivialElementaryMStageCandidateLowerSourceData.marking_x0 {p : FramedPair} {h : ¬ScalarTerminal p} {chief : FirstNonScalarChiefFactor p} {K : KernelMinimal chief.toNonScalarChiefFactor} {hR : Nat.card ↥(kernelFrattini K) = 1} {child : Sec7FrattiniTrivialElementaryChildData p} (src : Sec7FrattiniTrivialElementaryMStageCandidateLowerSourceData h chief K hR child) : src.marking Gen.x0 = src.x₀

@[simp]

theorem Q2Presentation.Induction.Sec7FrattiniTrivialElementaryMStageCandidateLowerSourceData.marking_x1 {p : FramedPair} {h : ¬ScalarTerminal p} {chief : FirstNonScalarChiefFactor p} {K : KernelMinimal chief.toNonScalarChiefFactor} {hR : Nat.card ↥(kernelFrattini K) = 1} {child : Sec7FrattiniTrivialElementaryChildData p} (src : Sec7FrattiniTrivialElementaryMStageCandidateLowerSourceData h chief K hR child) : src.marking Gen.x1 = src.x₁

def Q2Presentation.Induction.Sec7FrattiniTrivialElementaryMStageCandidateLowerSourceData.S {p : FramedPair} {h : ¬ScalarTerminal p} {chief : FirstNonScalarChiefFactor p} {K : KernelMinimal chief.toNonScalarChiefFactor} {hR : Nat.card ↥(kernelFrattini K) = 1} {child : Sec7FrattiniTrivialElementaryChildData p} (src : Sec7FrattiniTrivialElementaryMStageCandidateLowerSourceData h chief K hR child) : Module.End (ZMod 2) (towerM K)

The evaluated σ-operator on M = towerM K.

Equations

• src.S = (Q2Presentation.Induction.sec7MStageCandidateActionEnd K) src.sigma

def Q2Presentation.Induction.Sec7FrattiniTrivialElementaryMStageCandidateLowerSourceData.Sinv {p : FramedPair} {h : ¬ScalarTerminal p} {chief : FirstNonScalarChiefFactor p} {K : KernelMinimal chief.toNonScalarChiefFactor} {hR : Nat.card ↥(kernelFrattini K) = 1} {child : Sec7FrattiniTrivialElementaryChildData p} (src : Sec7FrattiniTrivialElementaryMStageCandidateLowerSourceData h chief K hR child) : Module.End (ZMod 2) (towerM K)

The evaluated σ⁻¹-operator on M = towerM K.

Equations

• src.Sinv = (Q2Presentation.Induction.sec7MStageCandidateActionEnd K) src.sigma⁻¹

def Q2Presentation.Induction.Sec7FrattiniTrivialElementaryMStageCandidateLowerSourceData.T {p : FramedPair} {h : ¬ScalarTerminal p} {chief : FirstNonScalarChiefFactor p} {K : KernelMinimal chief.toNonScalarChiefFactor} {hR : Nat.card ↥(kernelFrattini K) = 1} {child : Sec7FrattiniTrivialElementaryChildData p} (src : Sec7FrattiniTrivialElementaryMStageCandidateLowerSourceData h chief K hR child) : Module.End (ZMod 2) (towerM K)

The evaluated τ-operator on M = towerM K.

Equations

• src.T = (Q2Presentation.Induction.sec7MStageCandidateActionEnd K) src.tau

def Q2Presentation.Induction.Sec7FrattiniTrivialElementaryMStageCandidateLowerSourceData.X₀ {p : FramedPair} {h : ¬ScalarTerminal p} {chief : FirstNonScalarChiefFactor p} {K : KernelMinimal chief.toNonScalarChiefFactor} {hR : Nat.card ↥(kernelFrattini K) = 1} {child : Sec7FrattiniTrivialElementaryChildData p} (src : Sec7FrattiniTrivialElementaryMStageCandidateLowerSourceData h chief K hR child) : Module.End (ZMod 2) (towerM K)

The evaluated x₀-operator on M = towerM K.

Equations

• src.X₀ = (Q2Presentation.Induction.sec7MStageCandidateActionEnd K) src.x₀

def Q2Presentation.Induction.Sec7FrattiniTrivialElementaryMStageCandidateLowerSourceData.X₁ {p : FramedPair} {h : ¬ScalarTerminal p} {chief : FirstNonScalarChiefFactor p} {K : KernelMinimal chief.toNonScalarChiefFactor} {hR : Nat.card ↥(kernelFrattini K) = 1} {child : Sec7FrattiniTrivialElementaryChildData p} (src : Sec7FrattiniTrivialElementaryMStageCandidateLowerSourceData h chief K hR child) : Module.End (ZMod 2) (towerM K)

The evaluated x₁-operator on M = towerM K.

Equations

• src.X₁ = (Q2Presentation.Induction.sec7MStageCandidateActionEnd K) src.x₁

noncomputable def Q2Presentation.Induction.Sec7FrattiniTrivialElementaryMStageCandidateLowerSourceData.P {p : FramedPair} {h : ¬ScalarTerminal p} {chief : FirstNonScalarChiefFactor p} {K : KernelMinimal chief.toNonScalarChiefFactor} {hR : Nat.card ↥(kernelFrattini K) = 1} {child : Sec7FrattiniTrivialElementaryChildData p} (src : Sec7FrattiniTrivialElementaryMStageCandidateLowerSourceData h chief K hR child) : Module.End (ZMod 2) (towerM K)

The norm operator P = 1 + T + ⋯ of the evaluated τ-orbit.

Equations

• src.P = Q2Presentation.Lifting.repNorm (Q2Presentation.Induction.sec7MStageCandidateActionEnd K) src.tau

theorem Q2Presentation.Induction.Sec7FrattiniTrivialElementaryMStageCandidateLowerSourceData.sinv_left {p : FramedPair} {h : ¬ScalarTerminal p} {chief : FirstNonScalarChiefFactor p} {K : KernelMinimal chief.toNonScalarChiefFactor} {hR : Nat.card ↥(kernelFrattini K) = 1} {child : Sec7FrattiniTrivialElementaryChildData p} (src : Sec7FrattiniTrivialElementaryMStageCandidateLowerSourceData h chief K hR child) : src.Sinv * src.S = 1

The inverse shadow follows from the source action, not from an arbitrary choice.

theorem Q2Presentation.Induction.Sec7FrattiniTrivialElementaryMStageCandidateLowerSourceData.tame_shadow {p : FramedPair} {h : ¬ScalarTerminal p} {chief : FirstNonScalarChiefFactor p} {K : KernelMinimal chief.toNonScalarChiefFactor} {hR : Nat.card ↥(kernelFrattini K) = 1} {child : Sec7FrattiniTrivialElementaryChildData p} (src : Sec7FrattiniTrivialElementaryMStageCandidateLowerSourceData h chief K hR child) : src.Sinv * src.T * src.S = src.T ^ 2

The tame shadow follows by applying the lower action to the candidate tame relator.

theorem Q2Presentation.Induction.Sec7FrattiniTrivialElementaryMStageCandidateLowerSourceData.norm_shadow {p : FramedPair} {h : ¬ScalarTerminal p} {chief : FirstNonScalarChiefFactor p} {K : KernelMinimal chief.toNonScalarChiefFactor} {hR : Nat.card ↥(kernelFrattini K) = 1} {child : Sec7FrattiniTrivialElementaryChildData p} (src : Sec7FrattiniTrivialElementaryMStageCandidateLowerSourceData h chief K hR child) : src.P * (src.T - 1) = 0

The norm shadow is the standard telescoping identity for the norm of τ.

theorem Q2Presentation.Induction.sec7_tameRel_sigma_tau {G : Type u_1} [Group G] [Finite G] {q : Marking G} (h : tameRel q) : (q Gen.sigma)⁻¹ * q Gen.tau * q Gen.sigma = q Gen.tau ^ 2

The tame relator in the word-language implies the explicit right-conjugation relation on the marked generators. This is the tiny local form needed to turn an admissible lower marking into the lower-source packet below.

structure Q2Presentation.Induction.Sec7FrattiniTrivialElementaryMStageCandidateLowerAdmissibleMarkingData {p : FramedPair} (h : ¬ScalarTerminal p) (chief : FirstNonScalarChiefFactor p) (K : KernelMinimal chief.toNonScalarChiefFactor) (hR : Nat.card ↥(kernelFrattini K) = 1) (child : Sec7FrattiniTrivialElementaryChildData p) : Type

A smaller source-side residual for the candidate lower quotient in the R = 1 M-stage.

Instead of postulating a continuous epimorphism Γ_A ↠ C = towerC K together with its four generator values and the two relators, this atom records only an admissible finite marking of the lower quotient. The actual epimorphism and its generator values are then recovered theorem-level from the already-proved finite-quotient semantics GammaA_epi_equiv_admissible (manuscript prop:epi-semantics).

• q : Marking (towerC K)

  The admissible candidate marking of the lower quotient C = towerC K.

• admissible : Admissible self.q

  The marking satisfies the two candidate relators, generates C, and has wild values in O₂(C).

structure Q2Presentation.Induction.Sec7FrattiniTrivialElementaryMStageCandidateLowerFinitePresentedMarkingData {p : FramedPair} (h : ¬ScalarTerminal p) (chief : FirstNonScalarChiefFactor p) (K : KernelMinimal chief.toNonScalarChiefFactor) (hR : Nat.card ↥(kernelFrattini K) = 1) (child : Sec7FrattiniTrivialElementaryChildData p) : Type

The finite presented lower marking of C = towerC K.

This is the presentation half of admissibility: a concrete finite marking of the lower quotient, generation, and the two candidate relators. It intentionally does not record wild-core membership; that is the separate finite O₂(C) datum below.

• q : Marking (towerC K)

  The finite lower candidate marking of C = towerC K.

• generated : generates self.q

  The marked images generate C.

• tame : tameRel self.q

  The tame candidate relator holds in C.

• wild : wildRel self.q

  The wild candidate relator holds in C.

• x0_mem_twoCore : self.q Gen.x0 ∈ twoCore (towerC K)

  The lower image of x₀ lies in O₂(C); stored here so the later core projections do not quantify over an arbitrary unrelated presented marking.

• x1_mem_twoCore : self.q Gen.x1 ∈ twoCore (towerC K)

  The lower image of x₁ lies in O₂(C); stored here so the later core projections do not quantify over an arbitrary unrelated presented marking.

structure Q2Presentation.Induction.Sec7FrattiniTrivialElementaryMStageCandidateLowerFiniteWildCoreData {p : FramedPair} {h : ¬ScalarTerminal p} {chief : FirstNonScalarChiefFactor p} {K : KernelMinimal chief.toNonScalarChiefFactor} {hR : Nat.card ↥(kernelFrattini K) = 1} {child : Sec7FrattiniTrivialElementaryChildData p} (presented : Sec7FrattiniTrivialElementaryMStageCandidateLowerFinitePresentedMarkingData h chief K hR child) : Prop

The finite wild-core half of the lower admissible marking.

For the already-chosen finite presented lower marking, this records exactly the remaining condition from prop:epi-semantics: the two wild marked values land in O₂(C).

• x0_mem_twoCore : presented.q Gen.x0 ∈ twoCore (towerC K)

  The lower image of x₀ lies in O₂(C).

• x1_mem_twoCore : presented.q Gen.x1 ∈ twoCore (towerC K)

  The lower image of x₁ lies in O₂(C).

Route245 split of the Route244 lower finite candidate-marking atoms

structure Q2Presentation.Induction.Sec7FrattiniTrivialElementaryMStageCandidateLowerFiniteMarkingData {p : FramedPair} (h : ¬ScalarTerminal p) (chief : FirstNonScalarChiefFactor p) (K : KernelMinimal chief.toNonScalarChiefFactor) (hR : Nat.card ↥(kernelFrattini K) = 1) (child : Sec7FrattiniTrivialElementaryChildData p) : Type

The bare finite lower candidate marking of C = towerC K.

This is strictly smaller than the former presented-marking atom: it contains only the four elements of the lower quotient, with no generation or relator assertions.

• q : Marking (towerC K)

  The finite lower candidate marking of C = towerC K.

• hom : GammaA ⟶ ProfiniteGrp.ofFiniteGrp (FiniteGrp.of (towerC K))

  The candidate lower map whose generator values define q.

• surj : Function.Surjective ⇑(ProfiniteGrp.Hom.hom self.hom)

  The lower candidate map has exact image all of C = towerC K.

• q_eq : self.q = fun (a : Gen) => (ProfiniteGrp.Hom.hom self.hom) (gammaGen a)

  The marking is the image of the universal candidate generators under hom.

structure Q2Presentation.Induction.Sec7FrattiniTrivialElementaryMStageCandidateLowerFiniteGenerationData {p : FramedPair} {h : ¬ScalarTerminal p} {chief : FirstNonScalarChiefFactor p} {K : KernelMinimal chief.toNonScalarChiefFactor} {hR : Nat.card ↥(kernelFrattini K) = 1} {child : Sec7FrattiniTrivialElementaryChildData p} (mark : Sec7FrattiniTrivialElementaryMStageCandidateLowerFiniteMarkingData h chief K hR child) : Prop

The generation half of the finite lower presented marking.

• generated : generates mark.q

  The marked images generate C.

structure Q2Presentation.Induction.Sec7FrattiniTrivialElementaryMStageCandidateLowerFiniteTameRelatorData {p : FramedPair} {h : ¬ScalarTerminal p} {chief : FirstNonScalarChiefFactor p} {K : KernelMinimal chief.toNonScalarChiefFactor} {hR : Nat.card ↥(kernelFrattini K) = 1} {child : Sec7FrattiniTrivialElementaryChildData p} (mark : Sec7FrattiniTrivialElementaryMStageCandidateLowerFiniteMarkingData h chief K hR child) : Prop

The tame-relator half of the finite lower presented marking.

• tame : tameRel mark.q

  The tame candidate relator holds in C.

structure Q2Presentation.Induction.Sec7FrattiniTrivialElementaryMStageCandidateLowerFiniteWildRelatorData {p : FramedPair} {h : ¬ScalarTerminal p} {chief : FirstNonScalarChiefFactor p} {K : KernelMinimal chief.toNonScalarChiefFactor} {hR : Nat.card ↥(kernelFrattini K) = 1} {child : Sec7FrattiniTrivialElementaryChildData p} (mark : Sec7FrattiniTrivialElementaryMStageCandidateLowerFiniteMarkingData h chief K hR child) : Prop

The wild-relator half of the finite lower presented marking.

• wild : wildRel mark.q

  The wild candidate relator holds in C.

structure Q2Presentation.Induction.Sec7FrattiniTrivialElementaryMStageCandidateLowerFiniteSourceMapData {p : FramedPair} (h : ¬ScalarTerminal p) (chief : FirstNonScalarChiefFactor p) (K : KernelMinimal chief.toNonScalarChiefFactor) (hR : Nat.card ↥(kernelFrattini K) = 1) (child : Sec7FrattiniTrivialElementaryChildData p) : Type

The source-map residual behind the finite lower candidate marking.

This is the corrected provenance datum for the Route245 finite-marking split: instead of asserting generation/relators/core membership for an arbitrary bare function Gen → towerC K, it records the actual lower candidate epimorphism Γ_A ↠ C = towerC K. The six formerly visible finite-marking atoms are then recovered from GammaA_epi_equiv_admissible.

• hom : GammaA ⟶ ProfiniteGrp.ofFiniteGrp (FiniteGrp.of (towerC K))

  The candidate lower homomorphism to C = towerC K.

• surj : Function.Surjective ⇑(ProfiniteGrp.Hom.hom self.hom)

  The lower candidate map has exact image all of C = towerC K.

structure Q2Presentation.Induction.Sec7FrattiniTrivialElementaryMStageCandidateLowerBoundarySurjData {p : FramedPair} (h : ¬ScalarTerminal p) (chief : FirstNonScalarChiefFactor p) (K : KernelMinimal chief.toNonScalarChiefFactor) (hR : Nat.card ↥(kernelFrattini K) = 1) : Type

A boundary-framed source map to the literal lower quotient C = Y/K.

This is a stricter manuscript-shaped provenance datum than a bare map Γ_A ↠ towerC K: it records an actual candidate boundary-framed surjection to the lower child constructed from the same quotient and the same inherited frame. The bare finite source-map packet below is recovered by forgetting the boundary-compatibility proof and the framing predicate.

• hθ : K.Ksub ≤ p.fst.thetaY.ker

  The decoration-kernel hypothesis used to form the lower child C = Y/K.

• f : boundaryFramedSurj boundaryPackage_GammaA (sec7ElementaryLowerTarget K ⋯) (sec7ElementaryLowerFrame K ⋯)

  A candidate boundary-framed surjection to the inherited lower child.

structure Q2Presentation.Induction.Sec7FrattiniTrivialElementaryMStageCandidateLowerBoundaryMarkingData {p : FramedPair} (h : ¬ScalarTerminal p) (chief : FirstNonScalarChiefFactor p) (K : KernelMinimal chief.toNonScalarChiefFactor) (hR : Nat.card ↥(kernelFrattini K) = 1) (decor : Sec7FrattiniTrivialElementaryLowerDecorationData h chief K hR) : Type

Finite marking form of the lower boundary-source datum.

The decoration-kernel hypothesis is supplied by the already-proved lem:decorationblock theorem (sec7_frattiniTrivial_elementaryLowerDecorationData). This residual is therefore strictly finite and candidate-side: it gives an admissible marking of the literal lower quotient C = Y/K whose generator values already satisfy the inherited boundary equation. The continuous boundary-framed epimorphism is then recovered theorem-level from GammaA_epi_equiv_admissible and generator-density.

• q : Marking (sec7ElementaryLowerTarget K ⋯).Y

  The finite candidate marking of the literal lower quotient C = Y/K.

• admissible : Admissible self.q

  The marking satisfies the candidate finite-quotient presentation.

• boundary_gen (a : Gen) : (sec7ElementaryLowerTarget K ⋯).qYMap (self.q a) = (ProfiniteGrp.Hom.hom (sec7ElementaryLowerFrame K ⋯).beta) ((ProfiniteGrp.Hom.hom boundaryPackage_GammaA.toBoundary) (gammaGen a))

  The marking is compatible with the inherited boundary frame on the four universal generators.

structure Q2Presentation.Induction.Sec7FrattiniTrivialElementaryMStageCandidateLowerBoundaryAdmissibleMarkingData {p : FramedPair} (h : ¬ScalarTerminal p) (chief : FirstNonScalarChiefFactor p) (K : KernelMinimal chief.toNonScalarChiefFactor) (hR : Nat.card ↥(kernelFrattini K) = 1) (decor : Sec7FrattiniTrivialElementaryLowerDecorationData h chief K hR) : Type

The finite admissible-marking half of the lower boundary source.

This separates the manuscript's finite Γ_A-quotient datum from the inherited boundary frame: the marking already satisfies the candidate presentation, but no boundary equation is stored here.

• q : Marking (sec7ElementaryLowerTarget K ⋯).Y

  The finite candidate marking of the literal lower quotient C = Y/K.

• hom : GammaA ⟶ ProfiniteGrp.ofFiniteGrp (FiniteGrp.of (sec7ElementaryLowerTarget K ⋯).Y)

  The actual candidate lower-boundary homomorphism whose generator values define q.

• surj : Function.Surjective ⇑(ProfiniteGrp.Hom.hom self.hom)

  The source map has exact image all of the lower quotient.

• q_eq : self.q = fun (a : Gen) => (ProfiniteGrp.Hom.hom self.hom) (gammaGen a)

  The marking is exactly the image of the universal candidate generators.

• admissible : Admissible self.q

  The marking satisfies the finite candidate presentation.

• pi_boundary_gen (a : Gen) : (sec7ElementaryLowerTarget K ⋯).piY ((ProfiniteGrp.Hom.hom self.hom) (gammaGen a)) = ((ProfiniteGrp.Hom.hom (sec7ElementaryLowerFrame K ⋯).beta) ((ProfiniteGrp.Hom.hom boundaryPackage_GammaA.toBoundary) (gammaGen a))).1

  The tame quotient coordinate of the inherited-boundary generator equation.

• theta_boundary_gen (a : Gen) : (sec7ElementaryLowerTarget K ⋯).thetaY ((ProfiniteGrp.Hom.hom self.hom) (gammaGen a)) = ((ProfiniteGrp.Hom.hom (sec7ElementaryLowerFrame K ⋯).beta) ((ProfiniteGrp.Hom.hom boundaryPackage_GammaA.toBoundary) (gammaGen a))).2

  The elementary-decoration coordinate of the inherited-boundary generator equation.

structure Q2Presentation.Induction.Sec7FrattiniTrivialElementaryMStageCandidateLowerBoundaryGeneratorCompatibilityData {p : FramedPair} {h : ¬ScalarTerminal p} {chief : FirstNonScalarChiefFactor p} {K : KernelMinimal chief.toNonScalarChiefFactor} {hR : Nat.card ↥(kernelFrattini K) = 1} {decor : Sec7FrattiniTrivialElementaryLowerDecorationData h chief K hR} (mark : Sec7FrattiniTrivialElementaryMStageCandidateLowerBoundaryAdmissibleMarkingData h chief K hR decor) : Prop

The inherited-boundary compatibility half of the lower source.

For the already chosen finite admissible marking, this is exactly the generator-level boundary-frame equation. The all-elements boundary-framed surjection is reconstructed below by GammaA_epi_equiv_admissible and gammaGen_hom_ext.

• boundary_gen (a : Gen) : (sec7ElementaryLowerTarget K ⋯).qYMap (mark.q a) = (ProfiniteGrp.Hom.hom (sec7ElementaryLowerFrame K ⋯).beta) ((ProfiniteGrp.Hom.hom boundaryPackage_GammaA.toBoundary) (gammaGen a))

  Compatibility with the inherited boundary frame on the four universal generators.

Route251 split of the lower-boundary finite source atoms

structure Q2Presentation.Induction.Sec7FrattiniTrivialElementaryMStageCandidateLowerBoundaryMarkingCarrierData {p : FramedPair} (h : ¬ScalarTerminal p) (chief : FirstNonScalarChiefFactor p) (K : KernelMinimal chief.toNonScalarChiefFactor) (hR : Nat.card ↥(kernelFrattini K) = 1) (decor : Sec7FrattiniTrivialElementaryLowerDecorationData h chief K hR) : Type

The carrier half of the lower-boundary finite marking: only the four marked candidate images in the literal lower quotient C = Y/K.

• q : Marking (sec7ElementaryLowerTarget K ⋯).Y

  The finite candidate marking of the lower quotient C = Y/K.

• hom : GammaA ⟶ ProfiniteGrp.ofFiniteGrp (FiniteGrp.of (sec7ElementaryLowerTarget K ⋯).Y)

  The actual candidate lower-boundary homomorphism whose generator values define q.

• surj : Function.Surjective ⇑(ProfiniteGrp.Hom.hom self.hom)

  The source map has exact image all of the lower quotient.

• q_eq : self.q = fun (a : Gen) => (ProfiniteGrp.Hom.hom self.hom) (gammaGen a)

  The carrier marking is exactly the image of the universal candidate generators.

• pi_boundary_gen (a : Gen) : (sec7ElementaryLowerTarget K ⋯).piY ((ProfiniteGrp.Hom.hom self.hom) (gammaGen a)) = ((ProfiniteGrp.Hom.hom (sec7ElementaryLowerFrame K ⋯).beta) ((ProfiniteGrp.Hom.hom boundaryPackage_GammaA.toBoundary) (gammaGen a))).1

  The tame quotient coordinate of the inherited-boundary generator equation.

• theta_boundary_gen (a : Gen) : (sec7ElementaryLowerTarget K ⋯).thetaY ((ProfiniteGrp.Hom.hom self.hom) (gammaGen a)) = ((ProfiniteGrp.Hom.hom (sec7ElementaryLowerFrame K ⋯).beta) ((ProfiniteGrp.Hom.hom boundaryPackage_GammaA.toBoundary) (gammaGen a))).2

  The elementary-decoration coordinate of the inherited-boundary generator equation.

structure Q2Presentation.Induction.Sec7FrattiniTrivialElementaryMStageCandidateLowerBoundaryGenerationData {p : FramedPair} {h : ¬ScalarTerminal p} {chief : FirstNonScalarChiefFactor p} {K : KernelMinimal chief.toNonScalarChiefFactor} {hR : Nat.card ↥(kernelFrattini K) = 1} {decor : Sec7FrattiniTrivialElementaryLowerDecorationData h chief K hR} (carrier : Sec7FrattiniTrivialElementaryMStageCandidateLowerBoundaryMarkingCarrierData h chief K hR decor) : Prop

The generation component of admissibility for the lower-boundary marking.

• generated : generates carrier.q

  The marked images generate C.

structure Q2Presentation.Induction.Sec7FrattiniTrivialElementaryMStageCandidateLowerBoundaryTameRelatorData {p : FramedPair} {h : ¬ScalarTerminal p} {chief : FirstNonScalarChiefFactor p} {K : KernelMinimal chief.toNonScalarChiefFactor} {hR : Nat.card ↥(kernelFrattini K) = 1} {decor : Sec7FrattiniTrivialElementaryLowerDecorationData h chief K hR} (carrier : Sec7FrattiniTrivialElementaryMStageCandidateLowerBoundaryMarkingCarrierData h chief K hR decor) : Prop

The tame-relator component of admissibility for the lower-boundary marking.

• tame : tameRel carrier.q

  The tame candidate relator holds in the lower quotient.

structure Q2Presentation.Induction.Sec7FrattiniTrivialElementaryMStageCandidateLowerBoundaryWildRelatorData {p : FramedPair} {h : ¬ScalarTerminal p} {chief : FirstNonScalarChiefFactor p} {K : KernelMinimal chief.toNonScalarChiefFactor} {hR : Nat.card ↥(kernelFrattini K) = 1} {decor : Sec7FrattiniTrivialElementaryLowerDecorationData h chief K hR} (carrier : Sec7FrattiniTrivialElementaryMStageCandidateLowerBoundaryMarkingCarrierData h chief K hR decor) : Prop

The wild-relator component of admissibility for the lower-boundary marking.

• wild : wildRel carrier.q

  The wild candidate relator holds in the lower quotient.

structure Q2Presentation.Induction.Sec7FrattiniTrivialElementaryMStageCandidateLowerBoundaryX0WildCoreData {p : FramedPair} {h : ¬ScalarTerminal p} {chief : FirstNonScalarChiefFactor p} {K : KernelMinimal chief.toNonScalarChiefFactor} {hR : Nat.card ↥(kernelFrattini K) = 1} {decor : Sec7FrattiniTrivialElementaryLowerDecorationData h chief K hR} (carrier : Sec7FrattiniTrivialElementaryMStageCandidateLowerBoundaryMarkingCarrierData h chief K hR decor) : Prop

The x₀ two-core component of admissibility for the lower-boundary marking.

• x0_mem_twoCore : carrier.q Gen.x0 ∈ twoCore (sec7ElementaryLowerTarget K ⋯).Y

  The lower image of x₀ lies in O₂(C).

structure Q2Presentation.Induction.Sec7FrattiniTrivialElementaryMStageCandidateLowerBoundaryX1WildCoreData {p : FramedPair} {h : ¬ScalarTerminal p} {chief : FirstNonScalarChiefFactor p} {K : KernelMinimal chief.toNonScalarChiefFactor} {hR : Nat.card ↥(kernelFrattini K) = 1} {decor : Sec7FrattiniTrivialElementaryLowerDecorationData h chief K hR} (carrier : Sec7FrattiniTrivialElementaryMStageCandidateLowerBoundaryMarkingCarrierData h chief K hR decor) : Prop

The x₁ two-core component of admissibility for the lower-boundary marking.

• x1_mem_twoCore : carrier.q Gen.x1 ∈ twoCore (sec7ElementaryLowerTarget K ⋯).Y

  The lower image of x₁ lies in O₂(C).

structure Q2Presentation.Induction.Sec7FrattiniTrivialElementaryMStageCandidateLowerBoundarySourceMapData {p : FramedPair} (h : ¬ScalarTerminal p) (chief : FirstNonScalarChiefFactor p) (K : KernelMinimal chief.toNonScalarChiefFactor) (hR : Nat.card ↥(kernelFrattini K) = 1) (decor : Sec7FrattiniTrivialElementaryLowerDecorationData h chief K hR) : Type

The actual lower-boundary candidate source map behind the finite carrier. This is the manuscript source datum: an epimorphism Γ_A ↠ C = Y/K together with the two coordinate forms of the inherited boundary equation on the four distinguished generators.

• hom : GammaA ⟶ ProfiniteGrp.ofFiniteGrp (FiniteGrp.of (sec7ElementaryLowerTarget K ⋯).Y)

  The candidate lower-boundary homomorphism to the literal lower quotient.

• surj : Function.Surjective ⇑(ProfiniteGrp.Hom.hom self.hom)

  The lower-boundary map has exact image all of C = Y/K.

• pi_boundary_gen (a : Gen) : (sec7ElementaryLowerTarget K ⋯).piY ((ProfiniteGrp.Hom.hom self.hom) (gammaGen a)) = ((ProfiniteGrp.Hom.hom (sec7ElementaryLowerFrame K ⋯).beta) ((ProfiniteGrp.Hom.hom boundaryPackage_GammaA.toBoundary) (gammaGen a))).1

  The tame quotient coordinate of the inherited-boundary generator equation.

• theta_boundary_gen (a : Gen) : (sec7ElementaryLowerTarget K ⋯).thetaY ((ProfiniteGrp.Hom.hom self.hom) (gammaGen a)) = ((ProfiniteGrp.Hom.hom (sec7ElementaryLowerFrame K ⋯).beta) ((ProfiniteGrp.Hom.hom boundaryPackage_GammaA.toBoundary) (gammaGen a))).2

  The elementary-decoration coordinate of the inherited-boundary generator equation.

Checkpoint 254 split of the lower-boundary finite-frame input

structure Q2Presentation.Induction.Sec7FrattiniTrivialElementaryMStageCandidateLowerBoundaryFiniteFrameSourceData {p : FramedPair} (h : ¬ScalarTerminal p) (chief : FirstNonScalarChiefFactor p) (K : KernelMinimal chief.toNonScalarChiefFactor) (hR : Nat.card ↥(kernelFrattini K) = 1) (decor : Sec7FrattiniTrivialElementaryLowerDecorationData h chief K hR) : Type

The source-level finite lower-boundary frame: an actual candidate epimorphism to the inherited lower quotient, together with the generator-level inherited boundary equation. This is the remaining candidate-side source datum; the finite presentation and O₂ components below are recovered theorem-level from GammaA_epi_equiv_admissible, not assumed separately.

• hom : GammaA ⟶ ProfiniteGrp.ofFiniteGrp (FiniteGrp.of (sec7ElementaryLowerTarget K ⋯).Y)

  The candidate lower-boundary homomorphism to the literal lower quotient.

• surj : Function.Surjective ⇑(ProfiniteGrp.Hom.hom self.hom)

  The lower-boundary map has exact image all of C = Y/K.

• boundary_gen (a : Gen) : (sec7ElementaryLowerTarget K ⋯).qYMap ((ProfiniteGrp.Hom.hom self.hom) (gammaGen a)) = (ProfiniteGrp.Hom.hom (sec7ElementaryLowerFrame K ⋯).beta) ((ProfiniteGrp.Hom.hom boundaryPackage_GammaA.toBoundary) (gammaGen a))

  Compatibility with the inherited boundary frame on the four universal generators.

structure Q2Presentation.Induction.Sec7FrattiniTrivialElementaryMStageCandidateLowerBoundaryFiniteFrameCarrierData {p : FramedPair} (h : ¬ScalarTerminal p) (chief : FirstNonScalarChiefFactor p) (K : KernelMinimal chief.toNonScalarChiefFactor) (hR : Nat.card ↥(kernelFrattini K) = 1) (decor : Sec7FrattiniTrivialElementaryLowerDecorationData h chief K hR) : Type

The finite lower-boundary carrier: a marking of the inherited lower quotient that is known to come from an actual candidate epimorphism. The source map fields are not extra finite presentation assumptions; they are exactly what lets the generation, relator, and O₂ components be proved from prop:epi-semantics.

• q : Marking (sec7ElementaryLowerTarget K ⋯).Y

  The finite candidate marking of the literal lower quotient C = Y/K.

• hom : GammaA ⟶ ProfiniteGrp.ofFiniteGrp (FiniteGrp.of (sec7ElementaryLowerTarget K ⋯).Y)

  The actual candidate lower-boundary homomorphism whose generator values define q.

• surj : Function.Surjective ⇑(ProfiniteGrp.Hom.hom self.hom)

  The source map has exact image all of the lower quotient.

• q_eq : self.q = fun (a : Gen) => (ProfiniteGrp.Hom.hom self.hom) (gammaGen a)

  The carrier marking is exactly the image of the universal candidate generators.

• boundary_gen (a : Gen) : (sec7ElementaryLowerTarget K ⋯).qYMap (self.q a) = (ProfiniteGrp.Hom.hom (sec7ElementaryLowerFrame K ⋯).beta) ((ProfiniteGrp.Hom.hom boundaryPackage_GammaA.toBoundary) (gammaGen a))

  Compatibility with the inherited boundary frame on the four universal generators.

structure Q2Presentation.Induction.Sec7FrattiniTrivialElementaryMStageCandidateLowerBoundaryFiniteFrameGenerationData {p : FramedPair} {h : ¬ScalarTerminal p} {chief : FirstNonScalarChiefFactor p} {K : KernelMinimal chief.toNonScalarChiefFactor} {hR : Nat.card ↥(kernelFrattini K) = 1} {decor : Sec7FrattiniTrivialElementaryLowerDecorationData h chief K hR} (carrier : Sec7FrattiniTrivialElementaryMStageCandidateLowerBoundaryFiniteFrameCarrierData h chief K hR decor) : Prop

The generation component of the finite lower-boundary marking.

• generated : generates carrier.q

structure Q2Presentation.Induction.Sec7FrattiniTrivialElementaryMStageCandidateLowerBoundaryFiniteFrameTameData {p : FramedPair} {h : ¬ScalarTerminal p} {chief : FirstNonScalarChiefFactor p} {K : KernelMinimal chief.toNonScalarChiefFactor} {hR : Nat.card ↥(kernelFrattini K) = 1} {decor : Sec7FrattiniTrivialElementaryLowerDecorationData h chief K hR} (carrier : Sec7FrattiniTrivialElementaryMStageCandidateLowerBoundaryFiniteFrameCarrierData h chief K hR decor) : Prop

The tame-relator component of the finite lower-boundary marking.

• tame : tameRel carrier.q

structure Q2Presentation.Induction.Sec7FrattiniTrivialElementaryMStageCandidateLowerBoundaryFiniteFrameWildData {p : FramedPair} {h : ¬ScalarTerminal p} {chief : FirstNonScalarChiefFactor p} {K : KernelMinimal chief.toNonScalarChiefFactor} {hR : Nat.card ↥(kernelFrattini K) = 1} {decor : Sec7FrattiniTrivialElementaryLowerDecorationData h chief K hR} (carrier : Sec7FrattiniTrivialElementaryMStageCandidateLowerBoundaryFiniteFrameCarrierData h chief K hR decor) : Prop

The wild-relator component of the finite lower-boundary marking.

• wild : wildRel carrier.q

structure Q2Presentation.Induction.Sec7FrattiniTrivialElementaryMStageCandidateLowerBoundaryFiniteFrameX0CoreData {p : FramedPair} {h : ¬ScalarTerminal p} {chief : FirstNonScalarChiefFactor p} {K : KernelMinimal chief.toNonScalarChiefFactor} {hR : Nat.card ↥(kernelFrattini K) = 1} {decor : Sec7FrattiniTrivialElementaryLowerDecorationData h chief K hR} (carrier : Sec7FrattiniTrivialElementaryMStageCandidateLowerBoundaryFiniteFrameCarrierData h chief K hR decor) : Prop

The x₀ ∈ O₂(C) component of the finite lower-boundary marking.

• x0_mem_twoCore : carrier.q Gen.x0 ∈ twoCore (sec7ElementaryLowerTarget K ⋯).Y

structure Q2Presentation.Induction.Sec7FrattiniTrivialElementaryMStageCandidateLowerBoundaryFiniteFrameX1CoreData {p : FramedPair} {h : ¬ScalarTerminal p} {chief : FirstNonScalarChiefFactor p} {K : KernelMinimal chief.toNonScalarChiefFactor} {hR : Nat.card ↥(kernelFrattini K) = 1} {decor : Sec7FrattiniTrivialElementaryLowerDecorationData h chief K hR} (carrier : Sec7FrattiniTrivialElementaryMStageCandidateLowerBoundaryFiniteFrameCarrierData h chief K hR decor) : Prop

The x₁ ∈ O₂(C) component of the finite lower-boundary marking.

• x1_mem_twoCore : carrier.q Gen.x1 ∈ twoCore (sec7ElementaryLowerTarget K ⋯).Y

structure Q2Presentation.Induction.Sec7FrattiniTrivialElementaryMStageCandidateLowerBoundaryFiniteFrameBoundaryMapData {p : FramedPair} (h : ¬ScalarTerminal p) (chief : FirstNonScalarChiefFactor p) (K : KernelMinimal chief.toNonScalarChiefFactor) (hR : Nat.card ↥(kernelFrattini K) = 1) (decor : Sec7FrattiniTrivialElementaryLowerDecorationData h chief K hR) : Type

Full boundary-framed source map for the inherited lower finite frame.

This is one step closer to Definition 4.1 than the checkpoint-255 generator-only source atom: it records an actual candidate epimorphism Γ_A ↠ C = Y/K satisfying the inherited boundary-frame equation for every element of Γ_A. The generator-level finite-frame source datum used below is recovered by evaluating this equality on gammaGen.

• hom : GammaA ⟶ ProfiniteGrp.ofFiniteGrp (FiniteGrp.of (sec7ElementaryLowerTarget K ⋯).Y)

  The candidate lower-boundary homomorphism to the literal lower quotient.

• surj : Function.Surjective ⇑(ProfiniteGrp.Hom.hom self.hom)

  The lower-boundary map has exact image all of C = Y/K.

• boundary_all (γ : ↑GammaA.toProfinite.toTop) : (sec7ElementaryLowerTarget K ⋯).qYMap ((ProfiniteGrp.Hom.hom self.hom) γ) = (ProfiniteGrp.Hom.hom (sec7ElementaryLowerFrame K ⋯).beta) ((ProfiniteGrp.Hom.hom boundaryPackage_GammaA.toBoundary) γ)

  Full inherited-boundary equation, not merely its four generator specializations.

structure Q2Presentation.Induction.Sec7FrattiniTrivialElementaryMStageCandidateLowerBoundaryFiniteFrameBoundarySurjData {p : FramedPair} (h : ¬ScalarTerminal p) (chief : FirstNonScalarChiefFactor p) (K : KernelMinimal chief.toNonScalarChiefFactor) (hR : Nat.card ↥(kernelFrattini K) = 1) (decor : Sec7FrattiniTrivialElementaryLowerDecorationData h chief K hR) : Type

Canonical source object for the inherited lower finite frame.

Instead of storing the homomorphism, surjectivity, and boundary equation as separate custom fields, this packet stores an actual member of the canonical boundary-framed surjection type e_{Γ_A}^β(C). The full map datum above is recovered by projecting this subtype.

• source : boundaryFramedSurj boundaryPackage_GammaA (sec7ElementaryLowerTarget K ⋯) (sec7ElementaryLowerFrame K ⋯)

  The actual lower-child candidate boundary-framed surjection.

structure Q2Presentation.Induction.Sec7FrattiniTrivialElementaryMStageCandidateLowerBoundaryFiniteFrameAdmissibleQuotientData {p : FramedPair} (h : ¬ScalarTerminal p) (chief : FirstNonScalarChiefFactor p) (K : KernelMinimal chief.toNonScalarChiefFactor) (hR : Nat.card ↥(kernelFrattini K) = 1) (decor : Sec7FrattiniTrivialElementaryLowerDecorationData h chief K hR) : Type

Canonical finite admissible quotient for the inherited lower frame.

This is below the checkpoint-257 source atom: it contains only the finite marking of the literal lower quotient C = Y/K together with the candidate finite-quotient presentation conditions. The continuous epimorphism from Γ_A is reconstructed theorem-level by GammaA_epi_equiv_admissible.

• source : boundaryFramedSurj boundaryPackage_GammaA (sec7ElementaryLowerTarget K ⋯) (sec7ElementaryLowerFrame K ⋯)

  The actual inherited lower-child boundary-framed candidate surjection.

• q : Marking (sec7ElementaryLowerTarget K ⋯).Y

  The finite candidate marking of the inherited lower quotient.

• q_eq : self.q = fun (a : Gen) => (ProfiniteGrp.Hom.hom (↑self.source).hom) (gammaGen a)

  The marking is the image of the universal generators under source.

• admissible : Admissible self.q

  The marking satisfies the candidate finite-quotient presentation.

structure Q2Presentation.Induction.Sec7FrattiniTrivialElementaryMStageCandidateLowerBoundaryFiniteFrameBoundaryGeneratorData {p : FramedPair} {h : ¬ScalarTerminal p} {chief : FirstNonScalarChiefFactor p} {K : KernelMinimal chief.toNonScalarChiefFactor} {hR : Nat.card ↥(kernelFrattini K) = 1} {decor : Sec7FrattiniTrivialElementaryLowerDecorationData h chief K hR} (mark : Sec7FrattiniTrivialElementaryMStageCandidateLowerBoundaryFiniteFrameAdmissibleQuotientData h chief K hR decor) : Prop

Generator-level inherited-boundary compatibility for the chosen finite quotient.

This is the second half of the lower-frame source construction: once the admissible marking is fixed, the boundary equation is only required on the four dense universal generators. The all-elements boundary-framed equation is recovered below by gammaGen_hom_ext.

• boundary_gen (a : Gen) : (sec7ElementaryLowerTarget K ⋯).qYMap (mark.q a) = (ProfiniteGrp.Hom.hom (sec7ElementaryLowerFrame K ⋯).beta) ((ProfiniteGrp.Hom.hom boundaryPackage_GammaA.toBoundary) (gammaGen a))

  Compatibility with the inherited boundary frame on the four universal generators.

structure Q2Presentation.Induction.Sec7FrattiniTrivialElementaryMStageCandidateLowerBoundaryFiniteFrameAdmissibleCarrierData {p : FramedPair} (h : ¬ScalarTerminal p) (chief : FirstNonScalarChiefFactor p) (K : KernelMinimal chief.toNonScalarChiefFactor) (hR : Nat.card ↥(kernelFrattini K) = 1) (decor : Sec7FrattiniTrivialElementaryLowerDecorationData h chief K hR) : Type

Carrier for the finite admissible quotient of the inherited lower frame.

This is the next split below the checkpoint-258 admissible-quotient atom: first choose only the literal finite marking of C = Y/K; the five finite-presentation clauses are kept as separate manuscript-shaped inputs below.

• source : boundaryFramedSurj boundaryPackage_GammaA (sec7ElementaryLowerTarget K ⋯) (sec7ElementaryLowerFrame K ⋯)

  The actual inherited lower-child boundary-framed candidate surjection.

• q : Marking (sec7ElementaryLowerTarget K ⋯).Y

  The finite candidate marking of the inherited lower quotient.

• q_eq : self.q = fun (a : Gen) => (ProfiniteGrp.Hom.hom (↑self.source).hom) (gammaGen a)

  The marking is the image of the universal generators under source.

• admissible : Admissible self.q

  The source epimorphism makes this marking admissible.

structure Q2Presentation.Induction.Sec7FrattiniTrivialElementaryMStageCandidateLowerBoundaryFiniteFrameAdmissibleGenerationData {p : FramedPair} {h : ¬ScalarTerminal p} {chief : FirstNonScalarChiefFactor p} {K : KernelMinimal chief.toNonScalarChiefFactor} {hR : Nat.card ↥(kernelFrattini K) = 1} {decor : Sec7FrattiniTrivialElementaryLowerDecorationData h chief K hR} (carrier : Sec7FrattiniTrivialElementaryMStageCandidateLowerBoundaryFiniteFrameAdmissibleCarrierData h chief K hR decor) : Prop

Generation clause for the split finite admissible quotient.

• generated : generates carrier.q

structure Q2Presentation.Induction.Sec7FrattiniTrivialElementaryMStageCandidateLowerBoundaryFiniteFrameAdmissibleTameData {p : FramedPair} {h : ¬ScalarTerminal p} {chief : FirstNonScalarChiefFactor p} {K : KernelMinimal chief.toNonScalarChiefFactor} {hR : Nat.card ↥(kernelFrattini K) = 1} {decor : Sec7FrattiniTrivialElementaryLowerDecorationData h chief K hR} (carrier : Sec7FrattiniTrivialElementaryMStageCandidateLowerBoundaryFiniteFrameAdmissibleCarrierData h chief K hR decor) : Prop

Tame-relator clause for the split finite admissible quotient.

• tame : tameRel carrier.q

structure Q2Presentation.Induction.Sec7FrattiniTrivialElementaryMStageCandidateLowerBoundaryFiniteFrameAdmissibleWildData {p : FramedPair} {h : ¬ScalarTerminal p} {chief : FirstNonScalarChiefFactor p} {K : KernelMinimal chief.toNonScalarChiefFactor} {hR : Nat.card ↥(kernelFrattini K) = 1} {decor : Sec7FrattiniTrivialElementaryLowerDecorationData h chief K hR} (carrier : Sec7FrattiniTrivialElementaryMStageCandidateLowerBoundaryFiniteFrameAdmissibleCarrierData h chief K hR decor) : Prop

Wild-relator clause for the split finite admissible quotient.

• wild : wildRel carrier.q

structure Q2Presentation.Induction.Sec7FrattiniTrivialElementaryMStageCandidateLowerBoundaryFiniteFrameAdmissibleX0CoreData {p : FramedPair} {h : ¬ScalarTerminal p} {chief : FirstNonScalarChiefFactor p} {K : KernelMinimal chief.toNonScalarChiefFactor} {hR : Nat.card ↥(kernelFrattini K) = 1} {decor : Sec7FrattiniTrivialElementaryLowerDecorationData h chief K hR} (carrier : Sec7FrattiniTrivialElementaryMStageCandidateLowerBoundaryFiniteFrameAdmissibleCarrierData h chief K hR decor) : Prop

x₀ ∈ O₂(C) clause for the split finite admissible quotient.

• x0_mem_twoCore : carrier.q Gen.x0 ∈ twoCore (sec7ElementaryLowerTarget K ⋯).Y

structure Q2Presentation.Induction.Sec7FrattiniTrivialElementaryMStageCandidateLowerBoundaryFiniteFrameAdmissibleX1CoreData {p : FramedPair} {h : ¬ScalarTerminal p} {chief : FirstNonScalarChiefFactor p} {K : KernelMinimal chief.toNonScalarChiefFactor} {hR : Nat.card ↥(kernelFrattini K) = 1} {decor : Sec7FrattiniTrivialElementaryLowerDecorationData h chief K hR} (carrier : Sec7FrattiniTrivialElementaryMStageCandidateLowerBoundaryFiniteFrameAdmissibleCarrierData h chief K hR decor) : Prop

x₁ ∈ O₂(C) clause for the split finite admissible quotient.

• x1_mem_twoCore : carrier.q Gen.x1 ∈ twoCore (sec7ElementaryLowerTarget K ⋯).Y

structure Q2Presentation.Induction.Sec7FrattiniTrivialElementaryMStageCandidateLowerBoundaryTowerMarkingCompatibilityData {p : FramedPair} (h : ¬ScalarTerminal p) (chief : FirstNonScalarChiefFactor p) (K : KernelMinimal chief.toNonScalarChiefFactor) (hR : Nat.card ↥(kernelFrattini K) = 1) (decor : Sec7FrattiniTrivialElementaryLowerDecorationData h chief K hR) : Type

Finite tower-level source marking with inherited boundary coordinates.

This is the strictly smaller manuscript datum below the generator-level source-map leaf: it contains no continuous homomorphism Γ_A → C and no surjectivity assertion. It only records an admissible finite quotient marking of the concrete tower quotient towerC K = Y/K, plus the two inherited boundary coordinates of that marking on the four distinguished generators. The actual epimorphism and its exact image are recovered below from GammaA_epi_equiv_admissible.

• q : Marking (towerC K)

  The finite candidate marking of the concrete tower quotient C = towerC K.

• admissible : Admissible self.q

  The marking is admissible, so Γ_A ↠ towerC K is supplied by GammaA_epi_equiv_admissible rather than by this datum.

• pi_boundary_gen (a : Gen) : (sec7ElementaryLowerTarget K ⋯).piY (self.q a) = ((ProfiniteGrp.Hom.hom (sec7ElementaryLowerFrame K ⋯).beta) ((ProfiniteGrp.Hom.hom boundaryPackage_GammaA.toBoundary) (gammaGen a))).1

  Tame-quotient coordinate of the inherited boundary equation on generators.

• theta_boundary_gen (a : Gen) : (sec7ElementaryLowerTarget K ⋯).thetaY (self.q a) = ((ProfiniteGrp.Hom.hom (sec7ElementaryLowerFrame K ⋯).beta) ((ProfiniteGrp.Hom.hom boundaryPackage_GammaA.toBoundary) (gammaGen a))).2

  Elementary-decoration coordinate of the inherited boundary equation on generators.

structure Q2Presentation.Induction.Sec7FrattiniTrivialElementaryMStageCandidateLowerBoundaryTowerTamePro2CompatibilityData {p : FramedPair} (h : ¬ScalarTerminal p) (chief : FirstNonScalarChiefFactor p) (K : KernelMinimal chief.toNonScalarChiefFactor) (hR : Nat.card ↥(kernelFrattini K) = 1) (decor : Sec7FrattiniTrivialElementaryLowerDecorationData h chief K hR) : Type

Lower tame/pro-2 tower compatibility datum for the inherited lower finite frame.

This is one step below the checkpoint-336 tower-marking packet. It still records only a finite admissible marking of the concrete tower quotient towerC K, but the boundary coordinates are not phrased through the already-paired boundary map β : ∂_bd → H × E. Instead, they are stated separately against the two primitive boundary factors: the tame map α : T_tame → H and the pro-2 decoration map ψ : Π → E. The paired β-coordinate equalities of the old packet are recovered theorem-level using BoundaryFrame.beta_def, boundaryBeta_apply, and the two projections of boundaryPackage_GammaA.toBoundary.

• q : Marking (towerC K)

  The finite candidate marking of the concrete tower quotient C = towerC K.

• admissible : Admissible self.q

  The marking is admissible; the epimorphism from Γ_A is still reconstructed by GammaA_epi_equiv_admissible, not assumed here.

• pi_tame_gen (a : Gen) : (sec7ElementaryLowerTarget K ⋯).piY (self.q a) = (ProfiniteGrp.Hom.hom (sec7ElementaryLowerFrame K ⋯).alpha) ((ProfiniteGrp.Hom.hom boundaryPackage_GammaA.tameMap) (gammaGen a))

  Tame coordinate, stated directly through the tame boundary factor α.

• theta_pro2_gen (a : Gen) : (sec7ElementaryLowerTarget K ⋯).thetaY (self.q a) = (ProfiniteGrp.Hom.hom (sec7ElementaryLowerFrame K ⋯).psi) ((ProfiniteGrp.Hom.hom boundaryPackage_GammaA.pro2Map) (gammaGen a))

  Elementary-decoration coordinate, stated directly through the pro-2 boundary factor ψ.

structure Q2Presentation.Induction.Sec7FrattiniTrivialElementaryMStageCandidateLowerBoundaryTowerTamePro2PointData {p : FramedPair} (h : ¬ScalarTerminal p) (chief : FirstNonScalarChiefFactor p) (K : KernelMinimal chief.toNonScalarChiefFactor) (hR : Nat.card ↥(kernelFrattini K) = 1) (decor : Sec7FrattiniTrivialElementaryLowerDecorationData h chief K hR) : Type

Pointwise tame/pro-2 factor compatibility datum for the inherited lower finite frame.

This is below the previous factor-compatibility packet: it keeps the same finite admissible tower marking, but it no longer stores the two boundary-factor equations as functions ∀ a : Gen. Instead it records the eight literal generator rows separately. The functional statement is reconstructed theorem-level by case analysis on Gen.

• q : Marking (towerC K)

  The finite candidate marking of the concrete tower quotient C = towerC K.

• admissible : Admissible self.q

  The marking is admissible; the epimorphism from Γ_A is reconstructed by GammaA_epi_equiv_admissible.

• pi_tame_sigma : (sec7ElementaryLowerTarget K ⋯).piY (self.q Gen.sigma) = (ProfiniteGrp.Hom.hom (sec7ElementaryLowerFrame K ⋯).alpha) ((ProfiniteGrp.Hom.hom boundaryPackage_GammaA.tameMap) (gammaGen Gen.sigma))

  Tame coordinate on σ.

• pi_tame_tau : (sec7ElementaryLowerTarget K ⋯).piY (self.q Gen.tau) = (ProfiniteGrp.Hom.hom (sec7ElementaryLowerFrame K ⋯).alpha) ((ProfiniteGrp.Hom.hom boundaryPackage_GammaA.tameMap) (gammaGen Gen.tau))

  Tame coordinate on τ.

• pi_tame_x0 : (sec7ElementaryLowerTarget K ⋯).piY (self.q Gen.x0) = (ProfiniteGrp.Hom.hom (sec7ElementaryLowerFrame K ⋯).alpha) ((ProfiniteGrp.Hom.hom boundaryPackage_GammaA.tameMap) (gammaGen Gen.x0))

  Tame coordinate on x₀.

• pi_tame_x1 : (sec7ElementaryLowerTarget K ⋯).piY (self.q Gen.x1) = (ProfiniteGrp.Hom.hom (sec7ElementaryLowerFrame K ⋯).alpha) ((ProfiniteGrp.Hom.hom boundaryPackage_GammaA.tameMap) (gammaGen Gen.x1))

  Tame coordinate on x₁.

• theta_pro2_sigma : (sec7ElementaryLowerTarget K ⋯).thetaY (self.q Gen.sigma) = (ProfiniteGrp.Hom.hom (sec7ElementaryLowerFrame K ⋯).psi) ((ProfiniteGrp.Hom.hom boundaryPackage_GammaA.pro2Map) (gammaGen Gen.sigma))

  Pro-2 decoration coordinate on σ.

• theta_pro2_tau : (sec7ElementaryLowerTarget K ⋯).thetaY (self.q Gen.tau) = (ProfiniteGrp.Hom.hom (sec7ElementaryLowerFrame K ⋯).psi) ((ProfiniteGrp.Hom.hom boundaryPackage_GammaA.pro2Map) (gammaGen Gen.tau))

  Pro-2 decoration coordinate on τ.

• theta_pro2_x0 : (sec7ElementaryLowerTarget K ⋯).thetaY (self.q Gen.x0) = (ProfiniteGrp.Hom.hom (sec7ElementaryLowerFrame K ⋯).psi) ((ProfiniteGrp.Hom.hom boundaryPackage_GammaA.pro2Map) (gammaGen Gen.x0))

  Pro-2 decoration coordinate on x₀.

• theta_pro2_x1 : (sec7ElementaryLowerTarget K ⋯).thetaY (self.q Gen.x1) = (ProfiniteGrp.Hom.hom (sec7ElementaryLowerFrame K ⋯).psi) ((ProfiniteGrp.Hom.hom boundaryPackage_GammaA.pro2Map) (gammaGen Gen.x1))

  Pro-2 decoration coordinate on x₁.

Checkpoint 339: one pro-2 row is theorem-level from admissibility

structure Q2Presentation.Induction.Sec7FrattiniTrivialElementaryMStageCandidateLowerBoundaryTowerTamePro2PointReducedData {p : FramedPair} (h : ¬ScalarTerminal p) (chief : FirstNonScalarChiefFactor p) (K : KernelMinimal chief.toNonScalarChiefFactor) (hR : Nat.card ↥(kernelFrattini K) = 1) (decor : Sec7FrattiniTrivialElementaryLowerDecorationData h chief K hR) : Type

Seven-row pointwise tame/pro-2 factor datum for the inherited lower finite frame.

Compared with the checkpoint-338 point packet, the pro-2 decoration row on τ is no longer primitive: it is recovered theorem-level from admissibility by sec7_lowerBoundary_theta_tau_row_from_admissible.

• q : Marking (towerC K)

  The finite candidate marking of the concrete tower quotient C = towerC K.

• admissible : Admissible self.q

  The marking is admissible; the epimorphism from Γ_A is reconstructed by GammaA_epi_equiv_admissible.

• pi_tame_sigma : (sec7ElementaryLowerTarget K ⋯).piY (self.q Gen.sigma) = (ProfiniteGrp.Hom.hom (sec7ElementaryLowerFrame K ⋯).alpha) ((ProfiniteGrp.Hom.hom boundaryPackage_GammaA.tameMap) (gammaGen Gen.sigma))

  Tame coordinate on σ.

• pi_tame_tau : (sec7ElementaryLowerTarget K ⋯).piY (self.q Gen.tau) = (ProfiniteGrp.Hom.hom (sec7ElementaryLowerFrame K ⋯).alpha) ((ProfiniteGrp.Hom.hom boundaryPackage_GammaA.tameMap) (gammaGen Gen.tau))

  Tame coordinate on τ.

• pi_tame_x0 : (sec7ElementaryLowerTarget K ⋯).piY (self.q Gen.x0) = (ProfiniteGrp.Hom.hom (sec7ElementaryLowerFrame K ⋯).alpha) ((ProfiniteGrp.Hom.hom boundaryPackage_GammaA.tameMap) (gammaGen Gen.x0))

  Tame coordinate on x₀.

• pi_tame_x1 : (sec7ElementaryLowerTarget K ⋯).piY (self.q Gen.x1) = (ProfiniteGrp.Hom.hom (sec7ElementaryLowerFrame K ⋯).alpha) ((ProfiniteGrp.Hom.hom boundaryPackage_GammaA.tameMap) (gammaGen Gen.x1))

  Tame coordinate on x₁.

• theta_pro2_sigma : (sec7ElementaryLowerTarget K ⋯).thetaY (self.q Gen.sigma) = (ProfiniteGrp.Hom.hom (sec7ElementaryLowerFrame K ⋯).psi) ((ProfiniteGrp.Hom.hom boundaryPackage_GammaA.pro2Map) (gammaGen Gen.sigma))

  Pro-2 decoration coordinate on σ.

• theta_pro2_x0 : (sec7ElementaryLowerTarget K ⋯).thetaY (self.q Gen.x0) = (ProfiniteGrp.Hom.hom (sec7ElementaryLowerFrame K ⋯).psi) ((ProfiniteGrp.Hom.hom boundaryPackage_GammaA.pro2Map) (gammaGen Gen.x0))

  Pro-2 decoration coordinate on x₀.

• theta_pro2_x1 : (sec7ElementaryLowerTarget K ⋯).thetaY (self.q Gen.x1) = (ProfiniteGrp.Hom.hom (sec7ElementaryLowerFrame K ⋯).psi) ((ProfiniteGrp.Hom.hom boundaryPackage_GammaA.pro2Map) (gammaGen Gen.x1))

  Pro-2 decoration coordinate on x₁.

Checkpoint 340: tame x₀/x₁ rows from lower-kernel membership

structure Q2Presentation.Induction.Sec7FrattiniTrivialElementaryMStageCandidateLowerBoundaryTowerTamePro2PointKernelReducedData {p : FramedPair} (h : ¬ScalarTerminal p) (chief : FirstNonScalarChiefFactor p) (K : KernelMinimal chief.toNonScalarChiefFactor) (hR : Nat.card ↥(kernelFrattini K) = 1) (decor : Sec7FrattiniTrivialElementaryLowerDecorationData h chief K hR) : Type

A smaller pointwise packet for the inherited lower finite frame.

Compared with the checkpoint-339 seven-row packet, the tame rows on x₀ and x₁ are not assumed as boundary equalities. Instead this packet records the strictly local statement that the two chosen wild generator values land in the marked lower kernel L_C = ker(π_C). Since the candidate tame quotient kills the universal x₀ and x₁, the two tame rows are recovered theorem-level below.

• q : Marking (towerC K)

  The finite candidate marking of the concrete tower quotient C = towerC K.

• admissible : Admissible self.q

  The marking is admissible; the epimorphism from Γ_A is reconstructed by GammaA_epi_equiv_admissible.

• pi_tame_sigma : (sec7ElementaryLowerTarget K ⋯).piY (self.q Gen.sigma) = (ProfiniteGrp.Hom.hom (sec7ElementaryLowerFrame K ⋯).alpha) ((ProfiniteGrp.Hom.hom boundaryPackage_GammaA.tameMap) (gammaGen Gen.sigma))

  Tame coordinate on σ.

• pi_tame_tau : (sec7ElementaryLowerTarget K ⋯).piY (self.q Gen.tau) = (ProfiniteGrp.Hom.hom (sec7ElementaryLowerFrame K ⋯).alpha) ((ProfiniteGrp.Hom.hom boundaryPackage_GammaA.tameMap) (gammaGen Gen.tau))

  Tame coordinate on τ.

• x0_mem_lower_LY : self.q Gen.x0 ∈ (sec7ElementaryLowerTarget K ⋯).LY

  Lower-kernel membership for the x₀ image.

• x1_mem_lower_LY : self.q Gen.x1 ∈ (sec7ElementaryLowerTarget K ⋯).LY

  Lower-kernel membership for the x₁ image.

• theta_pro2_sigma : (sec7ElementaryLowerTarget K ⋯).thetaY (self.q Gen.sigma) = (ProfiniteGrp.Hom.hom (sec7ElementaryLowerFrame K ⋯).psi) ((ProfiniteGrp.Hom.hom boundaryPackage_GammaA.pro2Map) (gammaGen Gen.sigma))

  Pro-2 decoration coordinate on σ.

• theta_pro2_x0 : (sec7ElementaryLowerTarget K ⋯).thetaY (self.q Gen.x0) = (ProfiniteGrp.Hom.hom (sec7ElementaryLowerFrame K ⋯).psi) ((ProfiniteGrp.Hom.hom boundaryPackage_GammaA.pro2Map) (gammaGen Gen.x0))

  Pro-2 decoration coordinate on x₀.

• theta_pro2_x1 : (sec7ElementaryLowerTarget K ⋯).thetaY (self.q Gen.x1) = (ProfiniteGrp.Hom.hom (sec7ElementaryLowerFrame K ⋯).psi) ((ProfiniteGrp.Hom.hom boundaryPackage_GammaA.pro2Map) (gammaGen Gen.x1))

  Pro-2 decoration coordinate on x₁.

structure Q2Presentation.Induction.Sec7FrattiniTrivialElementaryMStageCandidateLowerBoundaryTowerTamePro2PointConcreteKernelReducedData {p : FramedPair} (h : ¬ScalarTerminal p) (chief : FirstNonScalarChiefFactor p) (K : KernelMinimal chief.toNonScalarChiefFactor) (hR : Nat.card ↥(kernelFrattini K) = 1) (decor : Sec7FrattiniTrivialElementaryLowerDecorationData h chief K hR) : Type

Concrete-generator version of the kernel-reduced pointwise packet.

Compared with Sec7FrattiniTrivialElementaryMStageCandidateLowerBoundaryTowerTamePro2PointKernelReducedData, this datum no longer mentions the candidate boundary maps on the five remaining tame/pro-2 generator rows. It records the same lower finite marking against the concrete boundary generators tameSigma, tameTau, piSigma, piX0, and piX1; the old rows are recovered below using gammaTameMap_apply_sigma, gammaTameMap_apply_tau, and the p_gammaGen_* pro-2 generator theorems.

• q : Marking (towerC K)

  The finite candidate marking of the concrete tower quotient C = towerC K.

• admissible : Admissible self.q

  The marking is admissible; the epimorphism from Γ_A is reconstructed by GammaA_epi_equiv_admissible.

• pi_tame_sigma_concrete : (sec7ElementaryLowerTarget K ⋯).piY (self.q Gen.sigma) = (ProfiniteGrp.Hom.hom (sec7ElementaryLowerFrame K ⋯).alpha) tameSigma

  Tame coordinate on σ, stated against the concrete tame generator.

• pi_tame_tau_concrete : (sec7ElementaryLowerTarget K ⋯).piY (self.q Gen.tau) = (ProfiniteGrp.Hom.hom (sec7ElementaryLowerFrame K ⋯).alpha) tameTau

  Tame coordinate on τ, stated against the concrete tame generator.

• x0_mem_lower_LY : self.q Gen.x0 ∈ (sec7ElementaryLowerTarget K ⋯).LY

  Lower-kernel membership for the x₀ image.

• x1_mem_lower_LY : self.q Gen.x1 ∈ (sec7ElementaryLowerTarget K ⋯).LY

  Lower-kernel membership for the x₁ image.

• theta_pro2_sigma_concrete : (sec7ElementaryLowerTarget K ⋯).thetaY (self.q Gen.sigma) = (ProfiniteGrp.Hom.hom (sec7ElementaryLowerFrame K ⋯).psi) piSigma

  Pro-2 decoration coordinate on σ, stated against the concrete pro-2 generator.

• theta_pro2_x0_concrete : (sec7ElementaryLowerTarget K ⋯).thetaY (self.q Gen.x0) = (ProfiniteGrp.Hom.hom (sec7ElementaryLowerFrame K ⋯).psi) piX0

  Pro-2 decoration coordinate on x₀, stated against the concrete pro-2 generator.

• theta_pro2_x1_concrete : (sec7ElementaryLowerTarget K ⋯).thetaY (self.q Gen.x1) = (ProfiniteGrp.Hom.hom (sec7ElementaryLowerFrame K ⋯).psi) piX1

  Pro-2 decoration coordinate on x₁, stated against the concrete pro-2 generator.

Checkpoint 342: the O₂(C) part of admissibility is theorem-level

structure Q2Presentation.Induction.Sec7FrattiniTrivialElementaryMStageCandidateLowerBoundaryTowerTamePro2PointConcreteKernelPresentedData {p : FramedPair} (h : ¬ScalarTerminal p) (chief : FirstNonScalarChiefFactor p) (K : KernelMinimal chief.toNonScalarChiefFactor) (hR : Nat.card ↥(kernelFrattini K) = 1) (decor : Sec7FrattiniTrivialElementaryLowerDecorationData h chief K hR) : Type

Presented concrete-generator kernel packet for the inherited lower finite frame.

Compared with the checkpoint-341 concrete-kernel packet, this datum no longer assumes full Admissible q. It records only the presentation part of admissibility (generation plus the tame and wild relators), together with the already local lower-kernel memberships for the wild generators. The two O₂(C) clauses required for Admissible q are recovered theorem-level below from the normal 2-subgroup L_C ◁ C: since L_C is a normal 2-subgroup, L_C ≤ O₂(C).

• q : Marking (towerC K)

  The finite candidate marking of the concrete tower quotient C = towerC K.

• presented : generates self.q ∧ tameRel self.q ∧ wildRel self.q

  The presentation part of admissibility: generation and the two candidate relators.

• pi_tame_sigma_concrete : (sec7ElementaryLowerTarget K ⋯).piY (self.q Gen.sigma) = (ProfiniteGrp.Hom.hom (sec7ElementaryLowerFrame K ⋯).alpha) tameSigma

  Tame coordinate on σ, stated against the concrete tame generator.

• pi_tame_tau_concrete : (sec7ElementaryLowerTarget K ⋯).piY (self.q Gen.tau) = (ProfiniteGrp.Hom.hom (sec7ElementaryLowerFrame K ⋯).alpha) tameTau

  Tame coordinate on τ, stated against the concrete tame generator.

• x0_mem_lower_LY : self.q Gen.x0 ∈ (sec7ElementaryLowerTarget K ⋯).LY

  Lower-kernel membership for the x₀ image.

• x1_mem_lower_LY : self.q Gen.x1 ∈ (sec7ElementaryLowerTarget K ⋯).LY

  Lower-kernel membership for the x₁ image.

• theta_pro2_sigma_concrete : (sec7ElementaryLowerTarget K ⋯).thetaY (self.q Gen.sigma) = (ProfiniteGrp.Hom.hom (sec7ElementaryLowerFrame K ⋯).psi) piSigma

  Pro-2 decoration coordinate on σ, stated against the concrete pro-2 generator.

• theta_pro2_x0_concrete : (sec7ElementaryLowerTarget K ⋯).thetaY (self.q Gen.x0) = (ProfiniteGrp.Hom.hom (sec7ElementaryLowerFrame K ⋯).psi) piX0

  Pro-2 decoration coordinate on x₀, stated against the concrete pro-2 generator.

• theta_pro2_x1_concrete : (sec7ElementaryLowerTarget K ⋯).thetaY (self.q Gen.x1) = (ProfiniteGrp.Hom.hom (sec7ElementaryLowerFrame K ⋯).psi) piX1

  Pro-2 decoration coordinate on x₁, stated against the concrete pro-2 generator.

Checkpoint 343: the source map now determines q and the presentation rows

structure Q2Presentation.Induction.Sec7FrattiniTrivialElementaryMStageCandidateLowerBoundaryTowerTamePro2PointConcreteKernelSourceData {p : FramedPair} (h : ¬ScalarTerminal p) (chief : FirstNonScalarChiefFactor p) (K : KernelMinimal chief.toNonScalarChiefFactor) (hR : Nat.card ↥(kernelFrattini K) = 1) (decor : Sec7FrattiniTrivialElementaryLowerDecorationData h chief K hR) : Type

Source-level concrete-generator datum for the inherited lower finite frame.

This is strictly below the checkpoint-342 presented concrete-kernel packet. The primitive no longer chooses an arbitrary q : Gen → towerC K, and it no longer assumes generates q, tameRel q, or wildRel q. Instead it records the actual candidate source epimorphism Γ_A ↠ towerC K; the lower marking is defined theorem-level as its four generator values, and the presentation part is recovered from GammaA_epi_equiv_admissible.

The fields that remain primitive are precisely the lower-boundary source data not provided by finite-quotient semantics: the two concrete tame rows, the two lower-kernel memberships, and the three concrete pro-2 rows.

• source : Profinite.SurjContHom GammaA (ProfiniteGrp.ofFiniteGrp (FiniteGrp.of (towerC K)))

  The actual candidate source epimorphism to the concrete tower quotient C = towerC K.

• pi_tame_sigma_concrete : (sec7ElementaryLowerTarget K ⋯).piY ((ProfiniteGrp.Hom.hom self.source.hom) (gammaGen Gen.sigma)) = (ProfiniteGrp.Hom.hom (sec7ElementaryLowerFrame K ⋯).alpha) tameSigma

  Tame coordinate on σ, stated against the concrete tame generator.

• pi_tame_tau_concrete : (sec7ElementaryLowerTarget K ⋯).piY ((ProfiniteGrp.Hom.hom self.source.hom) (gammaGen Gen.tau)) = (ProfiniteGrp.Hom.hom (sec7ElementaryLowerFrame K ⋯).alpha) tameTau

  Tame coordinate on τ, stated against the concrete tame generator.

• x0_mem_lower_LY : (ProfiniteGrp.Hom.hom self.source.hom) (gammaGen Gen.x0) ∈ (sec7ElementaryLowerTarget K ⋯).LY

  Lower-kernel membership for the x₀ image.

• x1_mem_lower_LY : (ProfiniteGrp.Hom.hom self.source.hom) (gammaGen Gen.x1) ∈ (sec7ElementaryLowerTarget K ⋯).LY

  Lower-kernel membership for the x₁ image.

• theta_pro2_sigma_concrete : (sec7ElementaryLowerTarget K ⋯).thetaY ((ProfiniteGrp.Hom.hom self.source.hom) (gammaGen Gen.sigma)) = (ProfiniteGrp.Hom.hom (sec7ElementaryLowerFrame K ⋯).psi) piSigma

  Pro-2 decoration coordinate on σ, stated against the concrete pro-2 generator.

• theta_pro2_x0_concrete : (sec7ElementaryLowerTarget K ⋯).thetaY ((ProfiniteGrp.Hom.hom self.source.hom) (gammaGen Gen.x0)) = (ProfiniteGrp.Hom.hom (sec7ElementaryLowerFrame K ⋯).psi) piX0

  Pro-2 decoration coordinate on x₀, stated against the concrete pro-2 generator.

• theta_pro2_x1_concrete : (sec7ElementaryLowerTarget K ⋯).thetaY ((ProfiniteGrp.Hom.hom self.source.hom) (gammaGen Gen.x1)) = (ProfiniteGrp.Hom.hom (sec7ElementaryLowerFrame K ⋯).psi) piX1

  Pro-2 decoration coordinate on x₁, stated against the concrete pro-2 generator.

axiom Q2Presentation.Induction.sec7_frattiniTrivial_elementaryMStageCandidateLowerBoundaryTowerTamePro2PointConcreteKernelSourceData {p : FramedPair} (h : ¬ScalarTerminal p) (chief : FirstNonScalarChiefFactor p) (K : KernelMinimal chief.toNonScalarChiefFactor) (hR : Nat.card ↥(kernelFrattini K) = 1) (decor : Sec7FrattiniTrivialElementaryLowerDecorationData h chief K hR) : Nonempty (Sec7FrattiniTrivialElementaryMStageCandidateLowerBoundaryTowerTamePro2PointConcreteKernelSourceData h chief K hR decor)

The new primitive source-level concrete-generator kernel datum.

It is below the checkpoint-342 residual because it does not assume the lower marking or any of the three finite presentation clauses; those are reconstructed below from the source epimorphism by GammaA_epi_equiv_admissible.

theorem Q2Presentation.Induction.sec7_frattiniTrivial_elementaryMStageCandidateLowerBoundaryTowerTamePro2PointConcreteKernelPresentedData {p : FramedPair} (h : ¬ScalarTerminal p) (chief : FirstNonScalarChiefFactor p) (K : KernelMinimal chief.toNonScalarChiefFactor) (hR : Nat.card ↥(kernelFrattini K) = 1) (decor : Sec7FrattiniTrivialElementaryLowerDecorationData h chief K hR) : Nonempty (Sec7FrattiniTrivialElementaryMStageCandidateLowerBoundaryTowerTamePro2PointConcreteKernelPresentedData h chief K hR decor)

The checkpoint-342 presented concrete-generator datum is theorem-level from the source-level datum. The marking is the generator-value marking of the source epimorphism, and generation plus both candidate relators are obtained from finite-quotient semantics.

theorem Q2Presentation.Induction.sec7_frattiniTrivial_elementaryMStageCandidateLowerBoundaryTowerTamePro2PointConcreteKernelReducedData {p : FramedPair} (h : ¬ScalarTerminal p) (chief : FirstNonScalarChiefFactor p) (K : KernelMinimal chief.toNonScalarChiefFactor) (hR : Nat.card ↥(kernelFrattini K) = 1) (decor : Sec7FrattiniTrivialElementaryLowerDecorationData h chief K hR) : Nonempty (Sec7FrattiniTrivialElementaryMStageCandidateLowerBoundaryTowerTamePro2PointConcreteKernelReducedData h chief K hR decor)

The checkpoint-341 concrete-generator kernel packet is theorem-level from the presented concrete-kernel datum. The only lost field is full admissibility, whose O₂(C) clauses are reconstructed from the two lower-kernel memberships.

theorem Q2Presentation.Induction.sec7_frattiniTrivial_elementaryMStageCandidateLowerBoundaryTowerTamePro2PointKernelReducedData {p : FramedPair} (h : ¬ScalarTerminal p) (chief : FirstNonScalarChiefFactor p) (K : KernelMinimal chief.toNonScalarChiefFactor) (hR : Nat.card ↥(kernelFrattini K) = 1) (decor : Sec7FrattiniTrivialElementaryLowerDecorationData h chief K hR) : Nonempty (Sec7FrattiniTrivialElementaryMStageCandidateLowerBoundaryTowerTamePro2PointKernelReducedData h chief K hR decor)

The checkpoint-340 kernel-reduced packet is theorem-level from the concrete-generator packet. The five remaining tame/pro-2 rows are transported along the already-proved generator-image theorems for the candidate tame and pro-2 boundary maps.

theorem Q2Presentation.Induction.sec7_frattiniTrivial_elementaryMStageCandidateLowerBoundaryTowerTamePro2PointReducedData {p : FramedPair} (h : ¬ScalarTerminal p) (chief : FirstNonScalarChiefFactor p) (K : KernelMinimal chief.toNonScalarChiefFactor) (hR : Nat.card ↥(kernelFrattini K) = 1) (decor : Sec7FrattiniTrivialElementaryLowerDecorationData h chief K hR) : Nonempty (Sec7FrattiniTrivialElementaryMStageCandidateLowerBoundaryTowerTamePro2PointReducedData h chief K hR decor)

The checkpoint-339 seven-row point packet is theorem-level from the kernel-reduced packet: the tame rows on x₀ and x₁ are forced by lower-kernel membership and by the candidate tame quotient identities γ_A(x₀)=γ_A(x₁)=1 in Ttame.

theorem Q2Presentation.Induction.sec7_frattiniTrivial_elementaryMStageCandidateLowerBoundaryTowerTamePro2PointData {p : FramedPair} (h : ¬ScalarTerminal p) (chief : FirstNonScalarChiefFactor p) (K : KernelMinimal chief.toNonScalarChiefFactor) (hR : Nat.card ↥(kernelFrattini K) = 1) (decor : Sec7FrattiniTrivialElementaryLowerDecorationData h chief K hR) : Nonempty (Sec7FrattiniTrivialElementaryMStageCandidateLowerBoundaryTowerTamePro2PointData h chief K hR decor)

The checkpoint-338 eight-row point packet is theorem-level from the seven-row packet: the omitted pro-2 decoration row on τ is forced by oddness of tame inertia and the elementary 2-group decoration target.

theorem Q2Presentation.Induction.sec7_frattiniTrivial_elementaryMStageCandidateLowerBoundaryTowerTamePro2CompatibilityData {p : FramedPair} (h : ¬ScalarTerminal p) (chief : FirstNonScalarChiefFactor p) (K : KernelMinimal chief.toNonScalarChiefFactor) (hR : Nat.card ↥(kernelFrattini K) = 1) (decor : Sec7FrattiniTrivialElementaryLowerDecorationData h chief K hR) : Nonempty (Sec7FrattiniTrivialElementaryMStageCandidateLowerBoundaryTowerTamePro2CompatibilityData h chief K hR decor)

The previous tame/pro-2 factor packet is theorem-level from the literal generator rows.

theorem Q2Presentation.Induction.sec7_frattiniTrivial_elementaryMStageCandidateLowerBoundaryTowerMarkingCompatibilityData {p : FramedPair} (h : ¬ScalarTerminal p) (chief : FirstNonScalarChiefFactor p) (K : KernelMinimal chief.toNonScalarChiefFactor) (hR : Nat.card ↥(kernelFrattini K) = 1) (decor : Sec7FrattiniTrivialElementaryLowerDecorationData h chief K hR) : Nonempty (Sec7FrattiniTrivialElementaryMStageCandidateLowerBoundaryTowerMarkingCompatibilityData h chief K hR decor)

The checkpoint-336 tower-marking packet is now theorem-level from the still lower tame/pro-2 factor compatibility datum above.

theorem Q2Presentation.Induction.sec7_frattiniTrivial_elementaryMStageCandidateLowerBoundaryFiniteFrameBoundarySourceMapData {p : FramedPair} (h : ¬ScalarTerminal p) (chief : FirstNonScalarChiefFactor p) (K : KernelMinimal chief.toNonScalarChiefFactor) (hR : Nat.card ↥(kernelFrattini K) = 1) (decor : Sec7FrattiniTrivialElementaryLowerDecorationData h chief K hR) : Nonempty (Sec7FrattiniTrivialElementaryMStageCandidateLowerBoundarySourceMapData h chief K hR decor)

Primitive source-map mechanism for the inherited lower finite frame, now theorem-level from the finite tower-level marking compatibility datum above.

The candidate epimorphism is obtained by applying GammaA_epi_equiv_admissible to the admissible marking of towerC K; the two generator boundary coordinates are transported along the resulting generator-value identity. This avoids assuming the epimorphism or its surjectivity as primitive data and does not reintroduce the split finite-frame admissibility fields.

theorem Q2Presentation.Induction.sec7_frattiniTrivial_elementaryMStageCandidateLowerBoundaryFiniteFrameBoundarySurjData {p : FramedPair} (h : ¬ScalarTerminal p) (chief : FirstNonScalarChiefFactor p) (K : KernelMinimal chief.toNonScalarChiefFactor) (hR : Nat.card ↥(kernelFrattini K) = 1) (decor : Sec7FrattiniTrivialElementaryLowerDecorationData h chief K hR) : Nonempty (Sec7FrattiniTrivialElementaryMStageCandidateLowerBoundaryFiniteFrameBoundarySurjData h chief K hR decor)

The checkpoint-334 semantic leaf is now theorem-level from the more primitive source-map mechanism: the full boundary-framed equation follows because the two finite maps to H × E agree on the dense universal generators.

theorem Q2Presentation.Induction.sec7_frattiniTrivial_elementaryMStageCandidateLowerBoundaryFiniteFrameAdmissibleCarrierData {p : FramedPair} (h : ¬ScalarTerminal p) (chief : FirstNonScalarChiefFactor p) (K : KernelMinimal chief.toNonScalarChiefFactor) (hR : Nat.card ↥(kernelFrattini K) = 1) (decor : Sec7FrattiniTrivialElementaryLowerDecorationData h chief K hR) : Nonempty (Sec7FrattiniTrivialElementaryMStageCandidateLowerBoundaryFiniteFrameAdmissibleCarrierData h chief K hR decor)

Former finite marking-carrier primitive, now theorem-level from the actual lower-child boundary-framed candidate surjection.

theorem Q2Presentation.Induction.sec7_frattiniTrivial_elementaryMStageCandidateLowerBoundaryFiniteFrameAdmissibleGenerationData {p : FramedPair} {h : ¬ScalarTerminal p} {chief : FirstNonScalarChiefFactor p} {K : KernelMinimal chief.toNonScalarChiefFactor} {hR : Nat.card ↥(kernelFrattini K) = 1} {decor : Sec7FrattiniTrivialElementaryLowerDecorationData h chief K hR} (carrier : Sec7FrattiniTrivialElementaryMStageCandidateLowerBoundaryFiniteFrameAdmissibleCarrierData h chief K hR decor) : Nonempty (Sec7FrattiniTrivialElementaryMStageCandidateLowerBoundaryFiniteFrameAdmissibleGenerationData carrier)

Former generation primitive for the split finite admissible quotient, now recovered from GammaA_epi_equiv_admissible applied to the carrier's actual source epimorphism.

theorem Q2Presentation.Induction.sec7_frattiniTrivial_elementaryMStageCandidateLowerBoundaryFiniteFrameAdmissibleTameData {p : FramedPair} {h : ¬ScalarTerminal p} {chief : FirstNonScalarChiefFactor p} {K : KernelMinimal chief.toNonScalarChiefFactor} {hR : Nat.card ↥(kernelFrattini K) = 1} {decor : Sec7FrattiniTrivialElementaryLowerDecorationData h chief K hR} (carrier : Sec7FrattiniTrivialElementaryMStageCandidateLowerBoundaryFiniteFrameAdmissibleCarrierData h chief K hR decor) : Nonempty (Sec7FrattiniTrivialElementaryMStageCandidateLowerBoundaryFiniteFrameAdmissibleTameData carrier)

Former tame-relator primitive for the split finite admissible quotient, now recovered from GammaA_epi_equiv_admissible applied to the carrier's actual source epimorphism.

theorem Q2Presentation.Induction.sec7_frattiniTrivial_elementaryMStageCandidateLowerBoundaryFiniteFrameAdmissibleWildData {p : FramedPair} {h : ¬ScalarTerminal p} {chief : FirstNonScalarChiefFactor p} {K : KernelMinimal chief.toNonScalarChiefFactor} {hR : Nat.card ↥(kernelFrattini K) = 1} {decor : Sec7FrattiniTrivialElementaryLowerDecorationData h chief K hR} (carrier : Sec7FrattiniTrivialElementaryMStageCandidateLowerBoundaryFiniteFrameAdmissibleCarrierData h chief K hR decor) : Nonempty (Sec7FrattiniTrivialElementaryMStageCandidateLowerBoundaryFiniteFrameAdmissibleWildData carrier)

Former wild-relator primitive for the split finite admissible quotient, now recovered from GammaA_epi_equiv_admissible applied to the carrier's actual source epimorphism.

theorem Q2Presentation.Induction.sec7_frattiniTrivial_elementaryMStageCandidateLowerBoundaryFiniteFrameAdmissibleX0CoreData {p : FramedPair} {h : ¬ScalarTerminal p} {chief : FirstNonScalarChiefFactor p} {K : KernelMinimal chief.toNonScalarChiefFactor} {hR : Nat.card ↥(kernelFrattini K) = 1} {decor : Sec7FrattiniTrivialElementaryLowerDecorationData h chief K hR} (carrier : Sec7FrattiniTrivialElementaryMStageCandidateLowerBoundaryFiniteFrameAdmissibleCarrierData h chief K hR decor) : Nonempty (Sec7FrattiniTrivialElementaryMStageCandidateLowerBoundaryFiniteFrameAdmissibleX0CoreData carrier)

Former x₀ ∈ O₂(C) primitive for the split finite admissible quotient, now recovered from GammaA_epi_equiv_admissible applied to the carrier's actual source epimorphism.

theorem Q2Presentation.Induction.sec7_frattiniTrivial_elementaryMStageCandidateLowerBoundaryFiniteFrameAdmissibleX1CoreData {p : FramedPair} {h : ¬ScalarTerminal p} {chief : FirstNonScalarChiefFactor p} {K : KernelMinimal chief.toNonScalarChiefFactor} {hR : Nat.card ↥(kernelFrattini K) = 1} {decor : Sec7FrattiniTrivialElementaryLowerDecorationData h chief K hR} (carrier : Sec7FrattiniTrivialElementaryMStageCandidateLowerBoundaryFiniteFrameAdmissibleCarrierData h chief K hR decor) : Nonempty (Sec7FrattiniTrivialElementaryMStageCandidateLowerBoundaryFiniteFrameAdmissibleX1CoreData carrier)

Former x₁ ∈ O₂(C) primitive for the split finite admissible quotient, now recovered from GammaA_epi_equiv_admissible applied to the carrier's actual source epimorphism.

theorem Q2Presentation.Induction.sec7_frattiniTrivial_elementaryMStageCandidateLowerBoundaryFiniteFrameAdmissibleQuotientData {p : FramedPair} (h : ¬ScalarTerminal p) (chief : FirstNonScalarChiefFactor p) (K : KernelMinimal chief.toNonScalarChiefFactor) (hR : Nat.card ↥(kernelFrattini K) = 1) (decor : Sec7FrattiniTrivialElementaryLowerDecorationData h chief K hR) : Nonempty (Sec7FrattiniTrivialElementaryMStageCandidateLowerBoundaryFiniteFrameAdmissibleQuotientData h chief K hR decor)

Former finite admissible-quotient atom, now theorem-level from the carrier plus the five literal finite-presentation and O₂ clauses.

structure Q2Presentation.Induction.Sec7FrattiniTrivialElementaryMStageCandidateLowerBoundaryFiniteFrameBoundaryPiGeneratorData {p : FramedPair} {h : ¬ScalarTerminal p} {chief : FirstNonScalarChiefFactor p} {K : KernelMinimal chief.toNonScalarChiefFactor} {hR : Nat.card ↥(kernelFrattini K) = 1} {decor : Sec7FrattiniTrivialElementaryLowerDecorationData h chief K hR} (mark : Sec7FrattiniTrivialElementaryMStageCandidateLowerBoundaryFiniteFrameAdmissibleQuotientData h chief K hR decor) : Prop

Tame-quotient coordinate of inherited-boundary compatibility on the four generators.

• pi_boundary_gen (a : Gen) : (sec7ElementaryLowerTarget K ⋯).piY (mark.q a) = ((ProfiniteGrp.Hom.hom (sec7ElementaryLowerFrame K ⋯).beta) ((ProfiniteGrp.Hom.hom boundaryPackage_GammaA.toBoundary) (gammaGen a))).1

structure Q2Presentation.Induction.Sec7FrattiniTrivialElementaryMStageCandidateLowerBoundaryFiniteFrameBoundaryThetaGeneratorData {p : FramedPair} {h : ¬ScalarTerminal p} {chief : FirstNonScalarChiefFactor p} {K : KernelMinimal chief.toNonScalarChiefFactor} {hR : Nat.card ↥(kernelFrattini K) = 1} {decor : Sec7FrattiniTrivialElementaryLowerDecorationData h chief K hR} (mark : Sec7FrattiniTrivialElementaryMStageCandidateLowerBoundaryFiniteFrameAdmissibleQuotientData h chief K hR decor) : Prop

Elementary-decoration coordinate of inherited-boundary compatibility on the four generators.

• theta_boundary_gen (a : Gen) : (sec7ElementaryLowerTarget K ⋯).thetaY (mark.q a) = ((ProfiniteGrp.Hom.hom (sec7ElementaryLowerFrame K ⋯).beta) ((ProfiniteGrp.Hom.hom boundaryPackage_GammaA.toBoundary) (gammaGen a))).2

theorem Q2Presentation.Induction.sec7_frattiniTrivial_elementaryMStageCandidateLowerBoundaryFiniteFrameBoundaryPiGeneratorData {p : FramedPair} {h : ¬ScalarTerminal p} {chief : FirstNonScalarChiefFactor p} {K : KernelMinimal chief.toNonScalarChiefFactor} {hR : Nat.card ↥(kernelFrattini K) = 1} {decor : Sec7FrattiniTrivialElementaryLowerDecorationData h chief K hR} (mark : Sec7FrattiniTrivialElementaryMStageCandidateLowerBoundaryFiniteFrameAdmissibleQuotientData h chief K hR decor) : Nonempty (Sec7FrattiniTrivialElementaryMStageCandidateLowerBoundaryFiniteFrameBoundaryPiGeneratorData mark)

Former tame-quotient coordinate primitive, now obtained by projecting the full boundary-framed equation carried by mark.source.

theorem Q2Presentation.Induction.sec7_frattiniTrivial_elementaryMStageCandidateLowerBoundaryFiniteFrameBoundaryThetaGeneratorData {p : FramedPair} {h : ¬ScalarTerminal p} {chief : FirstNonScalarChiefFactor p} {K : KernelMinimal chief.toNonScalarChiefFactor} {hR : Nat.card ↥(kernelFrattini K) = 1} {decor : Sec7FrattiniTrivialElementaryLowerDecorationData h chief K hR} (mark : Sec7FrattiniTrivialElementaryMStageCandidateLowerBoundaryFiniteFrameAdmissibleQuotientData h chief K hR decor) : Nonempty (Sec7FrattiniTrivialElementaryMStageCandidateLowerBoundaryFiniteFrameBoundaryThetaGeneratorData mark)

Former elementary-decoration coordinate primitive, now obtained by projecting the full boundary-framed equation carried by mark.source.

theorem Q2Presentation.Induction.sec7_frattiniTrivial_elementaryMStageCandidateLowerBoundaryFiniteFrameBoundaryGeneratorData {p : FramedPair} {h : ¬ScalarTerminal p} {chief : FirstNonScalarChiefFactor p} {K : KernelMinimal chief.toNonScalarChiefFactor} {hR : Nat.card ↥(kernelFrattini K) = 1} {decor : Sec7FrattiniTrivialElementaryLowerDecorationData h chief K hR} (mark : Sec7FrattiniTrivialElementaryMStageCandidateLowerBoundaryFiniteFrameAdmissibleQuotientData h chief K hR decor) : Nonempty (Sec7FrattiniTrivialElementaryMStageCandidateLowerBoundaryFiniteFrameBoundaryGeneratorData mark)

Former full generator-level inherited-boundary atom, now theorem-level from the two coordinates of the boundary map qYMap = (π_Y, θ_Y).

theorem Q2Presentation.Induction.sec7_frattiniTrivial_elementaryMStageCandidateLowerBoundaryFiniteFrameBoundaryMapData {p : FramedPair} (h : ¬ScalarTerminal p) (chief : FirstNonScalarChiefFactor p) (K : KernelMinimal chief.toNonScalarChiefFactor) (hR : Nat.card ↥(kernelFrattini K) = 1) (decor : Sec7FrattiniTrivialElementaryLowerDecorationData h chief K hR) : Nonempty (Sec7FrattiniTrivialElementaryMStageCandidateLowerBoundaryFiniteFrameBoundaryMapData h chief K hR decor)

Former full lower-boundary map datum, now theorem-level from the canonical boundary-framed surjection subtype.

theorem Q2Presentation.Induction.sec7_frattiniTrivial_elementaryMStageCandidateLowerBoundaryFiniteFrameSourceData {p : FramedPair} (h : ¬ScalarTerminal p) (chief : FirstNonScalarChiefFactor p) (K : KernelMinimal chief.toNonScalarChiefFactor) (hR : Nat.card ↥(kernelFrattini K) = 1) (decor : Sec7FrattiniTrivialElementaryLowerDecorationData h chief K hR) : Nonempty (Sec7FrattiniTrivialElementaryMStageCandidateLowerBoundaryFiniteFrameSourceData h chief K hR decor)

Former generator-level lower finite-frame source atom, now theorem-level from a full boundary-framed candidate epimorphism to the inherited lower quotient.

theorem Q2Presentation.Induction.sec7_frattiniTrivial_elementaryMStageCandidateLowerBoundaryFiniteFrameCarrierData {p : FramedPair} (h : ¬ScalarTerminal p) (chief : FirstNonScalarChiefFactor p) (K : KernelMinimal chief.toNonScalarChiefFactor) (hR : Nat.card ↥(kernelFrattini K) = 1) (decor : Sec7FrattiniTrivialElementaryLowerDecorationData h chief K hR) : Nonempty (Sec7FrattiniTrivialElementaryMStageCandidateLowerBoundaryFiniteFrameCarrierData h chief K hR decor)

Former carrier atom for the inherited lower finite frame, now theorem-level from the source-level finite frame datum.

theorem Q2Presentation.Induction.sec7_frattiniTrivial_elementaryMStageCandidateLowerBoundaryFiniteFrameGenerationData {p : FramedPair} {h : ¬ScalarTerminal p} {chief : FirstNonScalarChiefFactor p} {K : KernelMinimal chief.toNonScalarChiefFactor} {hR : Nat.card ↥(kernelFrattini K) = 1} {decor : Sec7FrattiniTrivialElementaryLowerDecorationData h chief K hR} (carrier : Sec7FrattiniTrivialElementaryMStageCandidateLowerBoundaryFiniteFrameCarrierData h chief K hR decor) : Nonempty (Sec7FrattiniTrivialElementaryMStageCandidateLowerBoundaryFiniteFrameGenerationData carrier)

Former generation atom for the inherited lower finite frame, now theorem-level from GammaA_epi_equiv_admissible applied to the carrier source map.

theorem Q2Presentation.Induction.sec7_frattiniTrivial_elementaryMStageCandidateLowerBoundaryFiniteFrameTameData {p : FramedPair} {h : ¬ScalarTerminal p} {chief : FirstNonScalarChiefFactor p} {K : KernelMinimal chief.toNonScalarChiefFactor} {hR : Nat.card ↥(kernelFrattini K) = 1} {decor : Sec7FrattiniTrivialElementaryLowerDecorationData h chief K hR} (carrier : Sec7FrattiniTrivialElementaryMStageCandidateLowerBoundaryFiniteFrameCarrierData h chief K hR decor) : Nonempty (Sec7FrattiniTrivialElementaryMStageCandidateLowerBoundaryFiniteFrameTameData carrier)

Former tame-relator atom for the inherited lower finite frame, now theorem-level from GammaA_epi_equiv_admissible.

theorem Q2Presentation.Induction.sec7_frattiniTrivial_elementaryMStageCandidateLowerBoundaryFiniteFrameWildData {p : FramedPair} {h : ¬ScalarTerminal p} {chief : FirstNonScalarChiefFactor p} {K : KernelMinimal chief.toNonScalarChiefFactor} {hR : Nat.card ↥(kernelFrattini K) = 1} {decor : Sec7FrattiniTrivialElementaryLowerDecorationData h chief K hR} (carrier : Sec7FrattiniTrivialElementaryMStageCandidateLowerBoundaryFiniteFrameCarrierData h chief K hR decor) : Nonempty (Sec7FrattiniTrivialElementaryMStageCandidateLowerBoundaryFiniteFrameWildData carrier)

Former wild-relator atom for the inherited lower finite frame, now theorem-level from GammaA_epi_equiv_admissible.

theorem Q2Presentation.Induction.sec7_frattiniTrivial_elementaryMStageCandidateLowerBoundaryFiniteFrameX0CoreData {p : FramedPair} {h : ¬ScalarTerminal p} {chief : FirstNonScalarChiefFactor p} {K : KernelMinimal chief.toNonScalarChiefFactor} {hR : Nat.card ↥(kernelFrattini K) = 1} {decor : Sec7FrattiniTrivialElementaryLowerDecorationData h chief K hR} (carrier : Sec7FrattiniTrivialElementaryMStageCandidateLowerBoundaryFiniteFrameCarrierData h chief K hR decor) : Nonempty (Sec7FrattiniTrivialElementaryMStageCandidateLowerBoundaryFiniteFrameX0CoreData carrier)

Former x₀ ∈ O₂(C) atom for the inherited lower finite frame, now theorem-level from GammaA_epi_equiv_admissible.

theorem Q2Presentation.Induction.sec7_frattiniTrivial_elementaryMStageCandidateLowerBoundaryFiniteFrameX1CoreData {p : FramedPair} {h : ¬ScalarTerminal p} {chief : FirstNonScalarChiefFactor p} {K : KernelMinimal chief.toNonScalarChiefFactor} {hR : Nat.card ↥(kernelFrattini K) = 1} {decor : Sec7FrattiniTrivialElementaryLowerDecorationData h chief K hR} (carrier : Sec7FrattiniTrivialElementaryMStageCandidateLowerBoundaryFiniteFrameCarrierData h chief K hR decor) : Nonempty (Sec7FrattiniTrivialElementaryMStageCandidateLowerBoundaryFiniteFrameX1CoreData carrier)

Former x₁ ∈ O₂(C) atom for the inherited lower finite frame, now theorem-level from GammaA_epi_equiv_admissible.

theorem Q2Presentation.Induction.sec7_frattiniTrivial_elementaryMStageCandidateLowerBoundaryFiniteFrameMarkingData {p : FramedPair} (h : ¬ScalarTerminal p) (chief : FirstNonScalarChiefFactor p) (K : KernelMinimal chief.toNonScalarChiefFactor) (hR : Nat.card ↥(kernelFrattini K) = 1) (decor : Sec7FrattiniTrivialElementaryLowerDecorationData h chief K hR) : Nonempty (Sec7FrattiniTrivialElementaryMStageCandidateLowerBoundaryMarkingData h chief K hR decor)

Former finite-frame marking input, now theorem-level from the carrier plus the five presentation/O₂ components of the inherited lower finite quotient.

theorem Q2Presentation.Induction.sec7_frattiniTrivial_elementaryMStageCandidateLowerBoundarySourceMapData {p : FramedPair} (h : ¬ScalarTerminal p) (chief : FirstNonScalarChiefFactor p) (K : KernelMinimal chief.toNonScalarChiefFactor) (hR : Nat.card ↥(kernelFrattini K) = 1) (decor : Sec7FrattiniTrivialElementaryLowerDecorationData h chief K hR) : Nonempty (Sec7FrattiniTrivialElementaryMStageCandidateLowerBoundarySourceMapData h chief K hR decor)

Former lower-boundary source-map input, now theorem-level from a finite admissible lower marking plus its generator-level inherited-boundary equation.

theorem Q2Presentation.Induction.sec7_frattiniTrivial_elementaryMStageCandidateLowerBoundaryMarkingCarrierData {p : FramedPair} (h : ¬ScalarTerminal p) (chief : FirstNonScalarChiefFactor p) (K : KernelMinimal chief.toNonScalarChiefFactor) (hR : Nat.card ↥(kernelFrattini K) = 1) (decor : Sec7FrattiniTrivialElementaryLowerDecorationData h chief K hR) : Nonempty (Sec7FrattiniTrivialElementaryMStageCandidateLowerBoundaryMarkingCarrierData h chief K hR decor)

Former lower-source carrier input, now theorem-level from the actual lower-boundary source map.

theorem Q2Presentation.Induction.sec7_frattiniTrivial_elementaryMStageCandidateLowerBoundaryGenerationData {p : FramedPair} {h : ¬ScalarTerminal p} {chief : FirstNonScalarChiefFactor p} {K : KernelMinimal chief.toNonScalarChiefFactor} {hR : Nat.card ↥(kernelFrattini K) = 1} {decor : Sec7FrattiniTrivialElementaryLowerDecorationData h chief K hR} (carrier : Sec7FrattiniTrivialElementaryMStageCandidateLowerBoundaryMarkingCarrierData h chief K hR decor) : Nonempty (Sec7FrattiniTrivialElementaryMStageCandidateLowerBoundaryGenerationData carrier)

Former generation input for the lower-boundary finite marking, now theorem-level from candidate finite-quotient semantics applied to the carrier source map.

theorem Q2Presentation.Induction.sec7_frattiniTrivial_elementaryMStageCandidateLowerBoundaryTameRelatorData {p : FramedPair} {h : ¬ScalarTerminal p} {chief : FirstNonScalarChiefFactor p} {K : KernelMinimal chief.toNonScalarChiefFactor} {hR : Nat.card ↥(kernelFrattini K) = 1} {decor : Sec7FrattiniTrivialElementaryLowerDecorationData h chief K hR} (carrier : Sec7FrattiniTrivialElementaryMStageCandidateLowerBoundaryMarkingCarrierData h chief K hR decor) : Nonempty (Sec7FrattiniTrivialElementaryMStageCandidateLowerBoundaryTameRelatorData carrier)

Former tame-relator input for the lower-boundary finite marking, now theorem-level from candidate finite-quotient semantics.

theorem Q2Presentation.Induction.sec7_frattiniTrivial_elementaryMStageCandidateLowerBoundaryWildRelatorData {p : FramedPair} {h : ¬ScalarTerminal p} {chief : FirstNonScalarChiefFactor p} {K : KernelMinimal chief.toNonScalarChiefFactor} {hR : Nat.card ↥(kernelFrattini K) = 1} {decor : Sec7FrattiniTrivialElementaryLowerDecorationData h chief K hR} (carrier : Sec7FrattiniTrivialElementaryMStageCandidateLowerBoundaryMarkingCarrierData h chief K hR decor) : Nonempty (Sec7FrattiniTrivialElementaryMStageCandidateLowerBoundaryWildRelatorData carrier)

Former wild-relator input for the lower-boundary finite marking, now theorem-level from candidate finite-quotient semantics.

theorem Q2Presentation.Induction.sec7_frattiniTrivial_elementaryMStageCandidateLowerBoundaryX0WildCoreData {p : FramedPair} {h : ¬ScalarTerminal p} {chief : FirstNonScalarChiefFactor p} {K : KernelMinimal chief.toNonScalarChiefFactor} {hR : Nat.card ↥(kernelFrattini K) = 1} {decor : Sec7FrattiniTrivialElementaryLowerDecorationData h chief K hR} (carrier : Sec7FrattiniTrivialElementaryMStageCandidateLowerBoundaryMarkingCarrierData h chief K hR decor) : Nonempty (Sec7FrattiniTrivialElementaryMStageCandidateLowerBoundaryX0WildCoreData carrier)

Former x₀ ∈ O₂(C) input for the lower-boundary finite marking, now theorem-level from candidate finite-quotient semantics.

theorem Q2Presentation.Induction.sec7_frattiniTrivial_elementaryMStageCandidateLowerBoundaryX1WildCoreData {p : FramedPair} {h : ¬ScalarTerminal p} {chief : FirstNonScalarChiefFactor p} {K : KernelMinimal chief.toNonScalarChiefFactor} {hR : Nat.card ↥(kernelFrattini K) = 1} {decor : Sec7FrattiniTrivialElementaryLowerDecorationData h chief K hR} (carrier : Sec7FrattiniTrivialElementaryMStageCandidateLowerBoundaryMarkingCarrierData h chief K hR decor) : Nonempty (Sec7FrattiniTrivialElementaryMStageCandidateLowerBoundaryX1WildCoreData carrier)

Former x₁ ∈ O₂(C) input for the lower-boundary finite marking, now theorem-level from candidate finite-quotient semantics.

theorem Q2Presentation.Induction.sec7_frattiniTrivial_elementaryMStageCandidateLowerBoundaryAdmissibleMarkingData {p : FramedPair} (h : ¬ScalarTerminal p) (chief : FirstNonScalarChiefFactor p) (K : KernelMinimal chief.toNonScalarChiefFactor) (hR : Nat.card ↥(kernelFrattini K) = 1) (decor : Sec7FrattiniTrivialElementaryLowerDecorationData h chief K hR) : Nonempty (Sec7FrattiniTrivialElementaryMStageCandidateLowerBoundaryAdmissibleMarkingData h chief K hR decor)

Former lower-boundary admissible-marking atom, now theorem-level from the literal finite presentation components: carrier, generation, tame relator, wild relator, and the separate x₀/x₁ two-core assertions.

structure Q2Presentation.Induction.Sec7FrattiniTrivialElementaryMStageCandidateLowerBoundaryHGeneratorCompatibilityData {p : FramedPair} {h : ¬ScalarTerminal p} {chief : FirstNonScalarChiefFactor p} {K : KernelMinimal chief.toNonScalarChiefFactor} {hR : Nat.card ↥(kernelFrattini K) = 1} {decor : Sec7FrattiniTrivialElementaryLowerDecorationData h chief K hR} (mark : Sec7FrattiniTrivialElementaryMStageCandidateLowerBoundaryAdmissibleMarkingData h chief K hR decor) : Prop

The tame-quotient coordinate of the inherited-boundary generator equation.

• pi_boundary_gen (a : Gen) : (sec7ElementaryLowerTarget K ⋯).piY (mark.q a) = ((ProfiniteGrp.Hom.hom (sec7ElementaryLowerFrame K ⋯).beta) ((ProfiniteGrp.Hom.hom boundaryPackage_GammaA.toBoundary) (gammaGen a))).1

  Compatibility of the H coordinate on the four universal generators.

structure Q2Presentation.Induction.Sec7FrattiniTrivialElementaryMStageCandidateLowerBoundaryDecorationGeneratorCompatibilityData {p : FramedPair} {h : ¬ScalarTerminal p} {chief : FirstNonScalarChiefFactor p} {K : KernelMinimal chief.toNonScalarChiefFactor} {hR : Nat.card ↥(kernelFrattini K) = 1} {decor : Sec7FrattiniTrivialElementaryLowerDecorationData h chief K hR} (mark : Sec7FrattiniTrivialElementaryMStageCandidateLowerBoundaryAdmissibleMarkingData h chief K hR decor) : Prop

The elementary-decoration coordinate of the inherited-boundary generator equation.

• theta_boundary_gen (a : Gen) : (sec7ElementaryLowerTarget K ⋯).thetaY (mark.q a) = ((ProfiniteGrp.Hom.hom (sec7ElementaryLowerFrame K ⋯).beta) ((ProfiniteGrp.Hom.hom boundaryPackage_GammaA.toBoundary) (gammaGen a))).2

  Compatibility of the decoration coordinate on the four universal generators.

theorem Q2Presentation.Induction.sec7_frattiniTrivial_elementaryMStageCandidateLowerBoundaryHGeneratorCompatibilityData {p : FramedPair} {h : ¬ScalarTerminal p} {chief : FirstNonScalarChiefFactor p} {K : KernelMinimal chief.toNonScalarChiefFactor} {hR : Nat.card ↥(kernelFrattini K) = 1} {decor : Sec7FrattiniTrivialElementaryLowerDecorationData h chief K hR} (mark : Sec7FrattiniTrivialElementaryMStageCandidateLowerBoundaryAdmissibleMarkingData h chief K hR decor) : Nonempty (Sec7FrattiniTrivialElementaryMStageCandidateLowerBoundaryHGeneratorCompatibilityData mark)

Former H-coordinate boundary input for the lower source, now theorem-level from the source map stored in the admissible-marking packet.

theorem Q2Presentation.Induction.sec7_frattiniTrivial_elementaryMStageCandidateLowerBoundaryDecorationGeneratorCompatibilityData {p : FramedPair} {h : ¬ScalarTerminal p} {chief : FirstNonScalarChiefFactor p} {K : KernelMinimal chief.toNonScalarChiefFactor} {hR : Nat.card ↥(kernelFrattini K) = 1} {decor : Sec7FrattiniTrivialElementaryLowerDecorationData h chief K hR} (mark : Sec7FrattiniTrivialElementaryMStageCandidateLowerBoundaryAdmissibleMarkingData h chief K hR decor) : Nonempty (Sec7FrattiniTrivialElementaryMStageCandidateLowerBoundaryDecorationGeneratorCompatibilityData mark)

Former decoration-coordinate boundary input for the lower source, now theorem-level from the source map stored in the admissible-marking packet.

theorem Q2Presentation.Induction.sec7_frattiniTrivial_elementaryMStageCandidateLowerBoundaryGeneratorCompatibilityData {p : FramedPair} {h : ¬ScalarTerminal p} {chief : FirstNonScalarChiefFactor p} {K : KernelMinimal chief.toNonScalarChiefFactor} {hR : Nat.card ↥(kernelFrattini K) = 1} {decor : Sec7FrattiniTrivialElementaryLowerDecorationData h chief K hR} (mark : Sec7FrattiniTrivialElementaryMStageCandidateLowerBoundaryAdmissibleMarkingData h chief K hR decor) : Nonempty (Sec7FrattiniTrivialElementaryMStageCandidateLowerBoundaryGeneratorCompatibilityData mark)

Former lower-boundary generator-compatibility atom, now theorem-level from the two literal coordinates of the combined map q_Y=(π_Y,θ_Y).

theorem Q2Presentation.Induction.sec7_frattiniTrivial_elementaryMStageCandidateLowerBoundaryMarkingData {p : FramedPair} (h : ¬ScalarTerminal p) (chief : FirstNonScalarChiefFactor p) (K : KernelMinimal chief.toNonScalarChiefFactor) (hR : Nat.card ↥(kernelFrattini K) = 1) (decor : Sec7FrattiniTrivialElementaryLowerDecorationData h chief K hR) : Nonempty (Sec7FrattiniTrivialElementaryMStageCandidateLowerBoundaryMarkingData h chief K hR decor)

Former Route249 lower-boundary marking leaf, now theorem-level from two strictly smaller manuscript components: finite admissibility of the lower quotient marking and its inherited boundary-frame generator equation.

noncomputable def Q2Presentation.Induction.sec7BoundaryQYHom (Yt : BoundaryFramedTarget) : ProfiniteGrp.ofFiniteGrp (FiniteGrp.of Yt.Y) ⟶ ProfiniteGrp.ofFiniteGrp (FiniteGrp.of (Yt.H × Multiplicative Yt.E))

The finite target morphism induced by the combined tame/decorated quotient map q_Y = (π_Y, θ_Y).

Equations

• Q2Presentation.Induction.sec7BoundaryQYHom Yt = ProfiniteGrp.ofFiniteGrpHom (FiniteGrp.ofHom Yt.qYMap)

@[simp]

theorem Q2Presentation.Induction.sec7BoundaryQYHom_apply (Yt : BoundaryFramedTarget) (y : Yt.Y) : (ProfiniteGrp.Hom.hom (sec7BoundaryQYHom Yt)) y = Yt.qYMap y

theorem Q2Presentation.Induction.sec7_frattiniTrivial_elementaryMStageCandidateLowerBoundarySurjData {p : FramedPair} (h : ¬ScalarTerminal p) (chief : FirstNonScalarChiefFactor p) (K : KernelMinimal chief.toNonScalarChiefFactor) (hR : Nat.card ↥(kernelFrattini K) = 1) : Nonempty (Sec7FrattiniTrivialElementaryMStageCandidateLowerBoundarySurjData h chief K hR)

Former lower boundary-surjection leaf, now theorem-level from the finite admissible marking with generator boundary compatibility. The proof is exactly prop:epi-semantics: an admissible finite marking gives Γ_A ↠ C; since both maps to H × E agree on the four dense generators, the boundary-framed equation holds for all γ : Γ_A.

theorem Q2Presentation.Induction.sec7_frattiniTrivial_elementaryMStageCandidateLowerFiniteSourceMapData {p : FramedPair} (h : ¬ScalarTerminal p) (chief : FirstNonScalarChiefFactor p) (K : KernelMinimal chief.toNonScalarChiefFactor) (hR : Nat.card ↥(kernelFrattini K) = 1) (child : Sec7FrattiniTrivialElementaryChildData p) : Nonempty (Sec7FrattiniTrivialElementaryMStageCandidateLowerFiniteSourceMapData h chief K hR child)

Remaining source-map input for the lower finite candidate quotient, now theorem-level from the boundary-framed lower-child source datum. The resulting packet forgets exactly what the later finite-quotient consequences do not use: the inherited frame and the boundary-compatibility equation.

theorem Q2Presentation.Induction.sec7_frattiniTrivial_elementaryMStageCandidateLowerFiniteMarkingData {p : FramedPair} (h : ¬ScalarTerminal p) (chief : FirstNonScalarChiefFactor p) (K : KernelMinimal chief.toNonScalarChiefFactor) (hR : Nat.card ↥(kernelFrattini K) = 1) (child : Sec7FrattiniTrivialElementaryChildData p) : Nonempty (Sec7FrattiniTrivialElementaryMStageCandidateLowerFiniteMarkingData h chief K hR child)

Route245 residual closed theorem-level: the bare finite lower candidate marking of C = towerC K is the generator-value marking of the lower source map.

noncomputable def Q2Presentation.Induction.sec7CandidateLowerFiniteEpiOfMarking {p : FramedPair} {h : ¬ScalarTerminal p} {chief : FirstNonScalarChiefFactor p} {K : KernelMinimal chief.toNonScalarChiefFactor} {hR : Nat.card ↥(kernelFrattini K) = 1} {child : Sec7FrattiniTrivialElementaryChildData p} (mark : Sec7FrattiniTrivialElementaryMStageCandidateLowerFiniteMarkingData h chief K hR child) : Profinite.SurjContHom GammaA (ProfiniteGrp.ofFiniteGrp (FiniteGrp.of (towerC K)))

The epimorphism encoded by the lower finite marking.

Equations

• Q2Presentation.Induction.sec7CandidateLowerFiniteEpiOfMarking mark = { hom := mark.hom, surj := ⋯ }

theorem Q2Presentation.Induction.sec7CandidateLowerFiniteMarking_admissible {p : FramedPair} {h : ¬ScalarTerminal p} {chief : FirstNonScalarChiefFactor p} {K : KernelMinimal chief.toNonScalarChiefFactor} {hR : Nat.card ↥(kernelFrattini K) = 1} {child : Sec7FrattiniTrivialElementaryChildData p} (mark : Sec7FrattiniTrivialElementaryMStageCandidateLowerFiniteMarkingData h chief K hR child) : Admissible mark.q

The finite lower marking attached to a source map is admissible, by the candidate finite-quotient semantics.

theorem Q2Presentation.Induction.sec7_frattiniTrivial_elementaryMStageCandidateLowerFiniteGenerationData {p : FramedPair} {h : ¬ScalarTerminal p} {chief : FirstNonScalarChiefFactor p} {K : KernelMinimal chief.toNonScalarChiefFactor} {hR : Nat.card ↥(kernelFrattini K) = 1} {child : Sec7FrattiniTrivialElementaryChildData p} (mark : Sec7FrattiniTrivialElementaryMStageCandidateLowerFiniteMarkingData h chief K hR child) : Nonempty (Sec7FrattiniTrivialElementaryMStageCandidateLowerFiniteGenerationData mark)

Route245 residual closed theorem-level: the finite lower marking generates C = towerC K.

theorem Q2Presentation.Induction.sec7_frattiniTrivial_elementaryMStageCandidateLowerFiniteTameRelatorData {p : FramedPair} {h : ¬ScalarTerminal p} {chief : FirstNonScalarChiefFactor p} {K : KernelMinimal chief.toNonScalarChiefFactor} {hR : Nat.card ↥(kernelFrattini K) = 1} {child : Sec7FrattiniTrivialElementaryChildData p} (mark : Sec7FrattiniTrivialElementaryMStageCandidateLowerFiniteMarkingData h chief K hR child) : Nonempty (Sec7FrattiniTrivialElementaryMStageCandidateLowerFiniteTameRelatorData mark)

Route245 residual closed theorem-level: the tame relator holds for the finite lower marking.

theorem Q2Presentation.Induction.sec7_frattiniTrivial_elementaryMStageCandidateLowerFiniteWildRelatorData {p : FramedPair} {h : ¬ScalarTerminal p} {chief : FirstNonScalarChiefFactor p} {K : KernelMinimal chief.toNonScalarChiefFactor} {hR : Nat.card ↥(kernelFrattini K) = 1} {child : Sec7FrattiniTrivialElementaryChildData p} (mark : Sec7FrattiniTrivialElementaryMStageCandidateLowerFiniteMarkingData h chief K hR child) : Nonempty (Sec7FrattiniTrivialElementaryMStageCandidateLowerFiniteWildRelatorData mark)

Route245 residual closed theorem-level: the wild relator holds for the finite lower marking.

structure Q2Presentation.Induction.Sec7FrattiniTrivialElementaryMStageCandidateLowerFiniteX0WildCoreData {p : FramedPair} {h : ¬ScalarTerminal p} {chief : FirstNonScalarChiefFactor p} {K : KernelMinimal chief.toNonScalarChiefFactor} {hR : Nat.card ↥(kernelFrattini K) = 1} {child : Sec7FrattiniTrivialElementaryChildData p} (presented : Sec7FrattiniTrivialElementaryMStageCandidateLowerFinitePresentedMarkingData h chief K hR child) : Prop

The x₀-core half of the finite wild-core datum.

• x0_mem_twoCore : presented.q Gen.x0 ∈ twoCore (towerC K)

  The lower image of x₀ lies in O₂(C).

structure Q2Presentation.Induction.Sec7FrattiniTrivialElementaryMStageCandidateLowerFiniteX1WildCoreData {p : FramedPair} {h : ¬ScalarTerminal p} {chief : FirstNonScalarChiefFactor p} {K : KernelMinimal chief.toNonScalarChiefFactor} {hR : Nat.card ↥(kernelFrattini K) = 1} {child : Sec7FrattiniTrivialElementaryChildData p} (presented : Sec7FrattiniTrivialElementaryMStageCandidateLowerFinitePresentedMarkingData h chief K hR child) : Prop

The x₁-core half of the finite wild-core datum.

• x1_mem_twoCore : presented.q Gen.x1 ∈ twoCore (towerC K)

  The lower image of x₁ lies in O₂(C).

theorem Q2Presentation.Induction.sec7_frattiniTrivial_elementaryMStageCandidateLowerFiniteX0WildCoreData {p : FramedPair} {h : ¬ScalarTerminal p} {chief : FirstNonScalarChiefFactor p} {K : KernelMinimal chief.toNonScalarChiefFactor} {hR : Nat.card ↥(kernelFrattini K) = 1} {child : Sec7FrattiniTrivialElementaryChildData p} (presented : Sec7FrattiniTrivialElementaryMStageCandidateLowerFinitePresentedMarkingData h chief K hR child) : Nonempty (Sec7FrattiniTrivialElementaryMStageCandidateLowerFiniteX0WildCoreData presented)

Route245 residual closed theorem-level: the lower image of x₀ lies in O₂(C).

theorem Q2Presentation.Induction.sec7_frattiniTrivial_elementaryMStageCandidateLowerFiniteX1WildCoreData {p : FramedPair} {h : ¬ScalarTerminal p} {chief : FirstNonScalarChiefFactor p} {K : KernelMinimal chief.toNonScalarChiefFactor} {hR : Nat.card ↥(kernelFrattini K) = 1} {child : Sec7FrattiniTrivialElementaryChildData p} (presented : Sec7FrattiniTrivialElementaryMStageCandidateLowerFinitePresentedMarkingData h chief K hR child) : Nonempty (Sec7FrattiniTrivialElementaryMStageCandidateLowerFiniteX1WildCoreData presented)

Route245 residual closed theorem-level: the lower image of x₁ lies in O₂(C).

theorem Q2Presentation.Induction.sec7_frattiniTrivial_elementaryMStageCandidateLowerFinitePresentedMarkingData {p : FramedPair} (h : ¬ScalarTerminal p) (chief : FirstNonScalarChiefFactor p) (K : KernelMinimal chief.toNonScalarChiefFactor) (hR : Nat.card ↥(kernelFrattini K) = 1) (child : Sec7FrattiniTrivialElementaryChildData p) : Nonempty (Sec7FrattiniTrivialElementaryMStageCandidateLowerFinitePresentedMarkingData h chief K hR child)

Former Route244 presented-marking residual, now theorem-level from four smaller finite quotient atoms: the bare marking, generation, tame relator, and wild relator.

theorem Q2Presentation.Induction.sec7_frattiniTrivial_elementaryMStageCandidateLowerFiniteWildCoreData {p : FramedPair} (h : ¬ScalarTerminal p) (chief : FirstNonScalarChiefFactor p) (K : KernelMinimal chief.toNonScalarChiefFactor) (hR : Nat.card ↥(kernelFrattini K) = 1) (child : Sec7FrattiniTrivialElementaryChildData p) (presented : Sec7FrattiniTrivialElementaryMStageCandidateLowerFinitePresentedMarkingData h chief K hR child) : Nonempty (Sec7FrattiniTrivialElementaryMStageCandidateLowerFiniteWildCoreData presented)

Former Route244 finite wild-core residual, now theorem-level from the two separate x₀ and x₁ finite O₂(C) membership atoms.

theorem Q2Presentation.Induction.sec7_frattiniTrivial_elementaryMStageCandidateLowerFiniteAdmissibleMarkingData {p : FramedPair} (h : ¬ScalarTerminal p) (chief : FirstNonScalarChiefFactor p) (K : KernelMinimal chief.toNonScalarChiefFactor) (hR : Nat.card ↥(kernelFrattini K) = 1) (child : Sec7FrattiniTrivialElementaryChildData p) : Nonempty (Sec7FrattiniTrivialElementaryMStageCandidateLowerAdmissibleMarkingData h chief K hR child)

Former Route243 finite-admissible-marking residual, now theorem-level from the two smaller finite quotient atoms: the presented lower marking and its O₂(C) wild-core membership.

structure Q2Presentation.Induction.Sec7FrattiniTrivialElementaryMStageCandidateRelatorShadowData {p : FramedPair} {h : ¬ScalarTerminal p} {chief : FirstNonScalarChiefFactor p} {K : KernelMinimal chief.toNonScalarChiefFactor} {hR : Nat.card ↥(kernelFrattini K) = 1} {child : Sec7FrattiniTrivialElementaryChildData p} (src : Sec7FrattiniTrivialElementaryMStageCandidateLowerSourceData h chief K hR child) : Prop

The missing finite-word/Fox calculation which turns the lower candidate relators into the wild row shadow on M = towerM K.

Unlike the old word-operator atom, this does not choose operators. The operators are definitionally the images of the source values under towerActM K, and P is definitionally the norm of τ. The lone field is the manuscript calculation lem:h0shadow / lem:fullwordledger specialized to this lower source: from the recorded wild relator and its evaluated word ledger, the wild row shadow follows.

• wild_shadow_from_relator : (src.P + src.Sinv) * (src.X₁ - 1) = 0

  The wild row shadow obtained from the candidate wild relator by the finite-word ledger (lem:h0shadow and lem:fullwordledger). The explicit argument forces this datum to be a relator-to-shadow calculation, not an arbitrary shadow assertion.

structure Q2Presentation.Induction.Sec7FrattiniTrivialElementaryMStageCandidateWildRowFundamentalData {p : FramedPair} {h : ¬ScalarTerminal p} {chief : FirstNonScalarChiefFactor p} {K : KernelMinimal chief.toNonScalarChiefFactor} {hR : Nat.card ↥(kernelFrattini K) = 1} {child : Sec7FrattiniTrivialElementaryChildData p} (src : Sec7FrattiniTrivialElementaryMStageCandidateLowerSourceData h chief K hR child) : Prop

A stricter finite-word ledger atom for the candidate wild row.

This records the Fox fundamental identity before cancelling the norm term: P·(T-1) + (P+S⁻¹)·(X₁-1) = 0. The cancellation P·(T-1)=0 is already proved theorem-level from the norm definition (src.norm_shadow), so the older CandidateRelatorShadowData is recovered below without postulating its final form directly. The premise is the actual evaluated candidate wild relator in the lower marking, not an arbitrary proposition.

• wild_row_fundamental_from_relator : Q2Word.wildRelator.eval src.marking = 1 → src.P * (src.T - 1) + (src.P + src.Sinv) * (src.X₁ - 1) = 0

  The manuscript wild-row Fox identity obtained from lem:h0shadow and lem:fullwordledger, before the separate norm cancellation.

Route240 split of the two Route239 candidate residuals

The previous Route239 frontier exposed two candidate-side atoms:

• an admissible lower marking of C = towerC K;
• the finite-word/Fox wild-row fundamental identity.

The declarations below split those atoms into smaller manuscript-shaped pieces and reassemble the Route239 names theorem-level. This keeps the existing downstream API unchanged while making the next residual frontier more local:

• existence of the lower marking is separated from generation, relators, and the O₂(C) wild-core conditions (prop:epi-semantics / admissible finite quotients);
• the wild-row calculation is separated into a raw four-factor Fox sum and the factor-by-factor ledger for h₀, u₁⁻¹, x₁^σ, and c₀ (lem:h0shadow, lem:fullwordledger).

structure Q2Presentation.Induction.Sec7FrattiniTrivialElementaryMStageCandidateLowerHomData {p : FramedPair} (h : ¬ScalarTerminal p) (chief : FirstNonScalarChiefFactor p) (K : KernelMinimal chief.toNonScalarChiefFactor) (hR : Nat.card ↥(kernelFrattini K) = 1) (child : Sec7FrattiniTrivialElementaryChildData p) : Type

A candidate lower epimorphism Γ_A ↠ C = towerC K.

This is the Route241 lowering of the Route240 marking/exact-image surface: the literal marking is no longer an arbitrary Gen → C; it is definitionally the image of the four universal generators under a concrete continuous candidate homomorphism, and the surjectivity field records the exact-image assertion needed to recover admissibility from GammaA_epi_equiv_admissible.

• hom : GammaA ⟶ ProfiniteGrp.ofFiniteGrp (FiniteGrp.of (towerC K))

  The candidate lower homomorphism to C = towerC K.

• surj : Function.Surjective ⇑(ProfiniteGrp.Hom.hom self.hom)

  The lower candidate map has exact image C.

structure Q2Presentation.Induction.Sec7FrattiniTrivialElementaryMStageCandidateLowerMarkingData {p : FramedPair} (h : ¬ScalarTerminal p) (chief : FirstNonScalarChiefFactor p) (K : KernelMinimal chief.toNonScalarChiefFactor) (hR : Nat.card ↥(kernelFrattini K) = 1) (child : Sec7FrattiniTrivialElementaryChildData p) : Type

The literal lower candidate marking of C = towerC K, tied to an actual candidate homomorphism from Γ_A.

Compared with the Route240 structure, the marking is not free data: the field q_eq records that it is exactly the image of the four universal generators under hom. This makes the generation, relator, and O₂(C) fields theorem-level consequences of the separate surjectivity datum and GammaA_epi_equiv_admissible.

• hom : GammaA ⟶ ProfiniteGrp.ofFiniteGrp (FiniteGrp.of (towerC K))

  The underlying candidate lower homomorphism.

• surj : Function.Surjective ⇑(ProfiniteGrp.Hom.hom self.hom)

  The lower candidate map has exact image C.

• q : Marking (towerC K)

  The four-generator lower marking Γ_A -> C at the finite-quotient level.

• q_eq : self.q = fun (a : Gen) => (ProfiniteGrp.Hom.hom self.hom) (gammaGen a)

  The marking is the image of the universal candidate generators.

structure Q2Presentation.Induction.Sec7FrattiniTrivialElementaryMStageCandidateLowerGenerationData {p : FramedPair} {h : ¬ScalarTerminal p} {chief : FirstNonScalarChiefFactor p} {K : KernelMinimal chief.toNonScalarChiefFactor} {hR : Nat.card ↥(kernelFrattini K) = 1} {child : Sec7FrattiniTrivialElementaryChildData p} (mark : Sec7FrattiniTrivialElementaryMStageCandidateLowerMarkingData h chief K hR child) : Prop

The lower marking generates the lower quotient C. This is the generation part of admissibility, recovered from candidate finite-quotient semantics.

• generated : generates mark.q

  The marked images of σ,τ,x₀,x₁ generate C.

structure Q2Presentation.Induction.Sec7FrattiniTrivialElementaryMStageCandidateLowerRelatorData {p : FramedPair} {h : ¬ScalarTerminal p} {chief : FirstNonScalarChiefFactor p} {K : KernelMinimal chief.toNonScalarChiefFactor} {hR : Nat.card ↥(kernelFrattini K) = 1} {child : Sec7FrattiniTrivialElementaryChildData p} (mark : Sec7FrattiniTrivialElementaryMStageCandidateLowerMarkingData h chief K hR child) : Prop

The two candidate relators on the lower marking of C.

• tame : tameRel mark.q

  The tame relation τ^σ = τ².

• wild : wildRel mark.q

  The wild relation h₀ u₁⁻¹ x₁^σ c₀ = 1.

structure Q2Presentation.Induction.Sec7FrattiniTrivialElementaryMStageCandidateLowerWildCoreData {p : FramedPair} {h : ¬ScalarTerminal p} {chief : FirstNonScalarChiefFactor p} {K : KernelMinimal chief.toNonScalarChiefFactor} {hR : Nat.card ↥(kernelFrattini K) = 1} {child : Sec7FrattiniTrivialElementaryChildData p} (mark : Sec7FrattiniTrivialElementaryMStageCandidateLowerMarkingData h chief K hR child) : Prop

The lower wild generators land in the finite 2-core of C.

• x0_mem_twoCore : mark.q Gen.x0 ∈ twoCore (towerC K)

  The lower image of x₀ lies in O₂(C).

• x1_mem_twoCore : mark.q Gen.x1 ∈ twoCore (towerC K)

  The lower image of x₁ lies in O₂(C).

structure Q2Presentation.Induction.Sec7FrattiniTrivialElementaryMStageCandidateLowerHomCarrierData {p : FramedPair} (h : ¬ScalarTerminal p) (chief : FirstNonScalarChiefFactor p) (K : KernelMinimal chief.toNonScalarChiefFactor) (hR : Nat.card ↥(kernelFrattini K) = 1) (child : Sec7FrattiniTrivialElementaryChildData p) : Type

The carrier part of the lower candidate homomorphism Γ_A ⟶ C = towerC K.

This Route242 atom separates the construction of the map from its exact-image assertion. The old Route241 residual CandidateLowerHomData is recovered theorem-level by combining this carrier with the separate surjectivity datum below.

• hom : GammaA ⟶ ProfiniteGrp.ofFiniteGrp (FiniteGrp.of (towerC K))

  The underlying candidate lower homomorphism to C = towerC K.

• surj : Function.Surjective ⇑(ProfiniteGrp.Hom.hom self.hom)

  The carrier is chosen from an admissible finite marking, hence is onto C.

structure Q2Presentation.Induction.Sec7FrattiniTrivialElementaryMStageCandidateLowerHomSurjectivityData {p : FramedPair} {h : ¬ScalarTerminal p} {chief : FirstNonScalarChiefFactor p} {K : KernelMinimal chief.toNonScalarChiefFactor} {hR : Nat.card ↥(kernelFrattini K) = 1} {child : Sec7FrattiniTrivialElementaryChildData p} (carrier : Sec7FrattiniTrivialElementaryMStageCandidateLowerHomCarrierData h chief K hR child) : Prop

The exact-image part of the lower candidate homomorphism.

This is the literal surjectivity assertion for the carrier above; it is smaller than the old combined homomorphism packet because it contains no choice of the homomorphism.

• surj : Function.Surjective ⇑(ProfiniteGrp.Hom.hom carrier.hom)

  The carrier map has exact image all of C = towerC K.

theorem Q2Presentation.Induction.sec7_frattiniTrivial_elementaryMStageCandidateLowerHomCarrierData {p : FramedPair} (h : ¬ScalarTerminal p) (chief : FirstNonScalarChiefFactor p) (K : KernelMinimal chief.toNonScalarChiefFactor) (hR : Nat.card ↥(kernelFrattini K) = 1) (child : Sec7FrattiniTrivialElementaryChildData p) : Nonempty (Sec7FrattiniTrivialElementaryMStageCandidateLowerHomCarrierData h chief K hR child)

Former Route242 carrier residual, now theorem-level from the finite admissible lower marking and candidate epimorphism semantics. The finite marking is transported to an epimorphism by GammaA_epi_equiv_admissible; the carrier stores that map together with its certified exact image.

theorem Q2Presentation.Induction.sec7_frattiniTrivial_elementaryMStageCandidateLowerHomSurjectivityData {p : FramedPair} (h : ¬ScalarTerminal p) (chief : FirstNonScalarChiefFactor p) (K : KernelMinimal chief.toNonScalarChiefFactor) (hR : Nat.card ↥(kernelFrattini K) = 1) (child : Sec7FrattiniTrivialElementaryChildData p) (carrier : Sec7FrattiniTrivialElementaryMStageCandidateLowerHomCarrierData h chief K hR child) : Nonempty (Sec7FrattiniTrivialElementaryMStageCandidateLowerHomSurjectivityData carrier)

Former Route242 surjectivity residual, now theorem-level: surjectivity is part of the certified carrier obtained from the finite admissible marking above.

theorem Q2Presentation.Induction.sec7_frattiniTrivial_elementaryMStageCandidateLowerHomData {p : FramedPair} (h : ¬ScalarTerminal p) (chief : FirstNonScalarChiefFactor p) (K : KernelMinimal chief.toNonScalarChiefFactor) (hR : Nat.card ↥(kernelFrattini K) = 1) (child : Sec7FrattiniTrivialElementaryChildData p) : Nonempty (Sec7FrattiniTrivialElementaryMStageCandidateLowerHomData h chief K hR child)

Former Route241 residual, now theorem-level from carrier plus exact-image data.

theorem Q2Presentation.Induction.sec7_frattiniTrivial_elementaryMStageCandidateLowerMarkingData {p : FramedPair} (h : ¬ScalarTerminal p) (chief : FirstNonScalarChiefFactor p) (K : KernelMinimal chief.toNonScalarChiefFactor) (hR : Nat.card ↥(kernelFrattini K) = 1) (child : Sec7FrattiniTrivialElementaryChildData p) : Nonempty (Sec7FrattiniTrivialElementaryMStageCandidateLowerMarkingData h chief K hR child)

Former Route240 residual, now theorem-level from the lower homomorphism datum.

noncomputable def Q2Presentation.Induction.sec7CandidateLowerEpiOfMarking {p : FramedPair} {h : ¬ScalarTerminal p} {chief : FirstNonScalarChiefFactor p} {K : KernelMinimal chief.toNonScalarChiefFactor} {hR : Nat.card ↥(kernelFrattini K) = 1} {child : Sec7FrattiniTrivialElementaryChildData p} (mark : Sec7FrattiniTrivialElementaryMStageCandidateLowerMarkingData h chief K hR child) : Profinite.SurjContHom GammaA (ProfiniteGrp.ofFiniteGrp (FiniteGrp.of (towerC K)))

Build the epimorphism encoded by a lower marking.

Equations

• Q2Presentation.Induction.sec7CandidateLowerEpiOfMarking mark = { hom := mark.hom, surj := ⋯ }

theorem Q2Presentation.Induction.sec7CandidateLowerMarking_admissible {p : FramedPair} {h : ¬ScalarTerminal p} {chief : FirstNonScalarChiefFactor p} {K : KernelMinimal chief.toNonScalarChiefFactor} {hR : Nat.card ↥(kernelFrattini K) = 1} {child : Sec7FrattiniTrivialElementaryChildData p} (mark : Sec7FrattiniTrivialElementaryMStageCandidateLowerMarkingData h chief K hR child) : Admissible mark.q

The admissibility of the marking attached to an exact lower candidate image.

theorem Q2Presentation.Induction.sec7_frattiniTrivial_elementaryMStageCandidateLowerGenerationData {p : FramedPair} (h : ¬ScalarTerminal p) (chief : FirstNonScalarChiefFactor p) (K : KernelMinimal chief.toNonScalarChiefFactor) (hR : Nat.card ↥(kernelFrattini K) = 1) (child : Sec7FrattiniTrivialElementaryChildData p) (mark : Sec7FrattiniTrivialElementaryMStageCandidateLowerMarkingData h chief K hR child) : Nonempty (Sec7FrattiniTrivialElementaryMStageCandidateLowerGenerationData mark)

Former Route240 residual, now theorem-level from exact-image semantics.

theorem Q2Presentation.Induction.sec7_frattiniTrivial_elementaryMStageCandidateLowerRelatorData {p : FramedPair} (h : ¬ScalarTerminal p) (chief : FirstNonScalarChiefFactor p) (K : KernelMinimal chief.toNonScalarChiefFactor) (hR : Nat.card ↥(kernelFrattini K) = 1) (child : Sec7FrattiniTrivialElementaryChildData p) (mark : Sec7FrattiniTrivialElementaryMStageCandidateLowerMarkingData h chief K hR child) : Nonempty (Sec7FrattiniTrivialElementaryMStageCandidateLowerRelatorData mark)

Former Route240 residual, now theorem-level from exact-image semantics.

theorem Q2Presentation.Induction.sec7_frattiniTrivial_elementaryMStageCandidateLowerWildCoreData {p : FramedPair} (h : ¬ScalarTerminal p) (chief : FirstNonScalarChiefFactor p) (K : KernelMinimal chief.toNonScalarChiefFactor) (hR : Nat.card ↥(kernelFrattini K) = 1) (child : Sec7FrattiniTrivialElementaryChildData p) (mark : Sec7FrattiniTrivialElementaryMStageCandidateLowerMarkingData h chief K hR child) : Nonempty (Sec7FrattiniTrivialElementaryMStageCandidateLowerWildCoreData mark)

Former Route240 residual, now theorem-level from exact-image semantics.

theorem Q2Presentation.Induction.sec7_frattiniTrivial_elementaryMStageCandidateLowerAdmissibleMarkingData {p : FramedPair} (h : ¬ScalarTerminal p) (chief : FirstNonScalarChiefFactor p) (K : KernelMinimal chief.toNonScalarChiefFactor) (hR : Nat.card ↥(kernelFrattini K) = 1) (child : Sec7FrattiniTrivialElementaryChildData p) : Nonempty (Sec7FrattiniTrivialElementaryMStageCandidateLowerAdmissibleMarkingData h chief K hR child)

Former Route239 lower-marking atom, now theorem-level from the four smaller finite-quotient pieces: marking, generation, relators, and O₂(C) wild-core membership.

structure Q2Presentation.Induction.Sec7FrattiniTrivialElementaryMStageCandidateWildRowH0FactorData {p : FramedPair} {h : ¬ScalarTerminal p} {chief : FirstNonScalarChiefFactor p} {K : KernelMinimal chief.toNonScalarChiefFactor} {hR : Nat.card ↥(kernelFrattini K) = 1} {child : Sec7FrattiniTrivialElementaryChildData p} (src : Sec7FrattiniTrivialElementaryMStageCandidateLowerSourceData h chief K hR child) : Type

The h₀ factor contribution in the candidate wild-row ledger.

• term : Module.End (ZMod 2) (towerM K)

  First-order contribution of h₀.

• shadow : self.term = 0

  h₀ has the first-order shadow of x₀², hence zero in characteristic two.

structure Q2Presentation.Induction.Sec7FrattiniTrivialElementaryMStageCandidateWildRowU1InvFactorData {p : FramedPair} {h : ¬ScalarTerminal p} {chief : FirstNonScalarChiefFactor p} {K : KernelMinimal chief.toNonScalarChiefFactor} {hR : Nat.card ↥(kernelFrattini K) = 1} {child : Sec7FrattiniTrivialElementaryChildData p} (src : Sec7FrattiniTrivialElementaryMStageCandidateLowerSourceData h chief K hR child) : Type

The u₁⁻¹ factor contribution in the candidate wild-row ledger.

• term : Module.End (ZMod 2) (towerM K)

  First-order contribution of u₁⁻¹.

• shadow : self.term = src.P * (src.T - 1) + src.P * (src.X₁ - 1)

  The u₁⁻¹ term contributes the two norm pieces.

structure Q2Presentation.Induction.Sec7FrattiniTrivialElementaryMStageCandidateWildRowX1ConjFactorData {p : FramedPair} {h : ¬ScalarTerminal p} {chief : FirstNonScalarChiefFactor p} {K : KernelMinimal chief.toNonScalarChiefFactor} {hR : Nat.card ↥(kernelFrattini K) = 1} {child : Sec7FrattiniTrivialElementaryChildData p} (src : Sec7FrattiniTrivialElementaryMStageCandidateLowerSourceData h chief K hR child) : Type

The x₁^σ factor contribution in the candidate wild-row ledger.

• term : Module.End (ZMod 2) (towerM K)

  First-order contribution of x₁^σ.

• shadow : self.term = src.Sinv * (src.X₁ - 1)

  The conjugate x₁^σ term contributes S⁻¹ on the x₁ coordinate.

structure Q2Presentation.Induction.Sec7FrattiniTrivialElementaryMStageCandidateWildRowC0FactorData {p : FramedPair} {h : ¬ScalarTerminal p} {chief : FirstNonScalarChiefFactor p} {K : KernelMinimal chief.toNonScalarChiefFactor} {hR : Nat.card ↥(kernelFrattini K) = 1} {child : Sec7FrattiniTrivialElementaryChildData p} (src : Sec7FrattiniTrivialElementaryMStageCandidateLowerSourceData h chief K hR child) : Type

The commutator factor c₀=[d₀,z₀] contribution in the candidate wild-row ledger.

• term : Module.End (ZMod 2) (towerM K)

  First-order contribution of c₀=[d₀,z₀].

• shadow : self.term = 0

  The commutator c₀=[d₀,z₀] has no first-order contribution.

structure Q2Presentation.Induction.Sec7FrattiniTrivialElementaryMStageCandidateWildRowProductRuleData {p : FramedPair} {h : ¬ScalarTerminal p} {chief : FirstNonScalarChiefFactor p} {K : KernelMinimal chief.toNonScalarChiefFactor} {hR : Nat.card ↥(kernelFrattini K) = 1} {child : Sec7FrattiniTrivialElementaryChildData p} (src : Sec7FrattiniTrivialElementaryMStageCandidateLowerSourceData h chief K hR child) (h0 : Sec7FrattiniTrivialElementaryMStageCandidateWildRowH0FactorData src) (u1 : Sec7FrattiniTrivialElementaryMStageCandidateWildRowU1InvFactorData src) (x1 : Sec7FrattiniTrivialElementaryMStageCandidateWildRowX1ConjFactorData src) (c0 : Sec7FrattiniTrivialElementaryMStageCandidateWildRowC0FactorData src) : Prop

The product-rule/Fox fundamental sum for the four actual wild-row factors.

This Route241 atom contains only the formal product-rule assembly of the four already separated factor contributions; it does not identify any individual factor.

• relator_factor_sum : Q2Word.wildRelator.eval src.marking = 1 → h0.term + u1.term + x1.term + c0.term = 0

  The raw Fox fundamental sum attached to the actual lower wild relator.

structure Q2Presentation.Induction.Sec7FrattiniTrivialElementaryMStageCandidateWildRowFullWordLedgerCoreData {p : FramedPair} {h : ¬ScalarTerminal p} {chief : FirstNonScalarChiefFactor p} {K : KernelMinimal chief.toNonScalarChiefFactor} {hR : Nat.card ↥(kernelFrattini K) = 1} {child : Sec7FrattiniTrivialElementaryChildData p} (src : Sec7FrattiniTrivialElementaryMStageCandidateLowerSourceData h chief K hR child) : Prop

The literal finite-word ledger for the candidate wild row.

This Route247 residual is the manuscript table of lem:h0shadow and lem:fullwordledger, specialized to the already chosen lower source map. It is strictly more literal than the former product-rule core atom: it no longer quantifies over arbitrary selected factor records. Instead, it states the evaluated four summands in the exact manuscript order

h₀, u₁⁻¹, x₁^σ, c₀

after the factor shadows have already been substituted. The formal product-rule packet for arbitrary factor records is recovered theorem-level below by rewriting those records with their shadow fields.

• literal_factor_sum : Q2Word.wildRelator.eval src.marking = 1 → 0 + (src.P * (src.T - 1) + src.P * (src.X₁ - 1)) + src.Sinv * (src.X₁ - 1) + 0 = 0

  The literal factor-ordered row from lem:fullwordledger.

structure Q2Presentation.Induction.Sec7FrattiniTrivialElementaryMStageCandidateWildRowManuscriptFactorLedgerData {p : FramedPair} {h : ¬ScalarTerminal p} {chief : FirstNonScalarChiefFactor p} {K : KernelMinimal chief.toNonScalarChiefFactor} {hR : Nat.card ↥(kernelFrattini K) = 1} {child : Sec7FrattiniTrivialElementaryChildData p} (src : Sec7FrattiniTrivialElementaryMStageCandidateLowerSourceData h chief K hR child) : Type

The manuscript factor-table version of the finite-word ledger.

This is one step below the former literal-row atom: instead of asserting the already summed row, it records the four factor contributions in the order of lem:fullwordledger (h₀, u₁⁻¹, x₁^σ, c₀) together with their shadows from lem:h0shadow and the raw Fox product rule for the actual lower wild relator. The literal summed row is recovered theorem-level by substituting these four shadows.

• h0Term : Module.End (ZMod 2) (towerM K)

  First-order contribution of h₀.

• u1InvTerm : Module.End (ZMod 2) (towerM K)

  First-order contribution of u₁⁻¹.

• x1ConjTerm : Module.End (ZMod 2) (towerM K)

  First-order contribution of x₁^σ.

• c0Term : Module.End (ZMod 2) (towerM K)

  First-order contribution of c₀=[d₀,z₀].

• relator_factor_sum : Q2Word.wildRelator.eval src.marking = 1 → self.h0Term + self.u1InvTerm + self.x1ConjTerm + self.c0Term = 0

  The raw Fox product-rule sum for the actual lower wild relator.

• h0_shadow : self.h0Term = 0

  h₀ has the first-order shadow of x₀², hence zero in characteristic two.

• u1_inv_shadow : self.u1InvTerm = src.P * (src.T - 1) + src.P * (src.X₁ - 1)

  The u₁⁻¹ term contributes the two norm pieces.

• x1_conj_shadow : self.x1ConjTerm = src.Sinv * (src.X₁ - 1)

  The conjugate x₁^σ term contributes S⁻¹ on the x₁ coordinate.

• c0_shadow : self.c0Term = 0

  The commutator c₀=[d₀,z₀] has no first-order contribution.

structure Q2Presentation.Induction.Sec7FrattiniTrivialElementaryMStageCandidateWildRowManuscriptH0Data {p : FramedPair} {h : ¬ScalarTerminal p} {chief : FirstNonScalarChiefFactor p} {K : KernelMinimal chief.toNonScalarChiefFactor} {hR : Nat.card ↥(kernelFrattini K) = 1} {child : Sec7FrattiniTrivialElementaryChildData p} (src : Sec7FrattiniTrivialElementaryMStageCandidateLowerSourceData h chief K hR child) : Type

The h₀ row of the manuscript finite-word ledger (lem:h0shadow): the first-order shadow of h₀ is the shadow of x₀², hence zero in characteristic two.

• h0Term : Module.End (ZMod 2) (towerM K)

  First-order contribution of h₀.

• h0_shadow : self.h0Term = 0

  h₀ contributes zero in the elementary row.

structure Q2Presentation.Induction.Sec7FrattiniTrivialElementaryMStageCandidateWildRowManuscriptU1InvData {p : FramedPair} {h : ¬ScalarTerminal p} {chief : FirstNonScalarChiefFactor p} {K : KernelMinimal chief.toNonScalarChiefFactor} {hR : Nat.card ↥(kernelFrattini K) = 1} {child : Sec7FrattiniTrivialElementaryChildData p} (src : Sec7FrattiniTrivialElementaryMStageCandidateLowerSourceData h chief K hR child) : Type

The u₁⁻¹ row of the manuscript finite-word ledger.

• u1InvTerm : Module.End (ZMod 2) (towerM K)

  First-order contribution of u₁⁻¹.

• u1_inv_shadow : self.u1InvTerm = src.P * (src.T - 1) + src.P * (src.X₁ - 1)

  The inverse norm term contributes P(T-1) + P(X₁-1).

structure Q2Presentation.Induction.Sec7FrattiniTrivialElementaryMStageCandidateWildRowManuscriptX1ConjData {p : FramedPair} {h : ¬ScalarTerminal p} {chief : FirstNonScalarChiefFactor p} {K : KernelMinimal chief.toNonScalarChiefFactor} {hR : Nat.card ↥(kernelFrattini K) = 1} {child : Sec7FrattiniTrivialElementaryChildData p} (src : Sec7FrattiniTrivialElementaryMStageCandidateLowerSourceData h chief K hR child) : Type

The x₁^σ row of the manuscript finite-word ledger.

• x1ConjTerm : Module.End (ZMod 2) (towerM K)

  First-order contribution of x₁^σ.

• x1_conj_shadow : self.x1ConjTerm = src.Sinv * (src.X₁ - 1)

  The conjugate contributes S⁻¹(X₁-1) on the x₁ coordinate.

structure Q2Presentation.Induction.Sec7FrattiniTrivialElementaryMStageCandidateWildRowManuscriptC0Data {p : FramedPair} {h : ¬ScalarTerminal p} {chief : FirstNonScalarChiefFactor p} {K : KernelMinimal chief.toNonScalarChiefFactor} {hR : Nat.card ↥(kernelFrattini K) = 1} {child : Sec7FrattiniTrivialElementaryChildData p} (src : Sec7FrattiniTrivialElementaryMStageCandidateLowerSourceData h chief K hR child) : Type

The c₀=[d₀,z₀] row of the manuscript finite-word ledger.

• c0Term : Module.End (ZMod 2) (towerM K)

  First-order contribution of c₀=[d₀,z₀].

• c0_shadow : self.c0Term = 0

  The commutator factor has no first-order contribution.

structure Q2Presentation.Induction.Sec7FrattiniTrivialElementaryMStageCandidateWildRowManuscriptRelatorProductRuleData {p : FramedPair} {h : ¬ScalarTerminal p} {chief : FirstNonScalarChiefFactor p} {K : KernelMinimal chief.toNonScalarChiefFactor} {hR : Nat.card ↥(kernelFrattini K) = 1} {child : Sec7FrattiniTrivialElementaryChildData p} {src : Sec7FrattiniTrivialElementaryMStageCandidateLowerSourceData h chief K hR child} (h0 : Sec7FrattiniTrivialElementaryMStageCandidateWildRowManuscriptH0Data src) (u1 : Sec7FrattiniTrivialElementaryMStageCandidateWildRowManuscriptU1InvData src) (x1 : Sec7FrattiniTrivialElementaryMStageCandidateWildRowManuscriptX1ConjData src) (c0 : Sec7FrattiniTrivialElementaryMStageCandidateWildRowManuscriptC0Data src) : Prop

The raw product-rule row for the actual lower wild relator, after the four manuscript factor terms have been chosen but before substituting their evaluated shadows.

• relator_factor_sum : Q2Word.wildRelator.eval src.marking = 1 → h0.h0Term + u1.u1InvTerm + x1.x1ConjTerm + c0.c0Term = 0

  The raw Fox product-rule sum for the actual lower wild relator.

Route250 split of the five manuscript wild-row atoms

structure Q2Presentation.Induction.Sec7FrattiniTrivialElementaryMStageCandidateWildRowManuscriptH0TermData {p : FramedPair} {h : ¬ScalarTerminal p} {chief : FirstNonScalarChiefFactor p} {K : KernelMinimal chief.toNonScalarChiefFactor} {hR : Nat.card ↥(kernelFrattini K) = 1} {child : Sec7FrattiniTrivialElementaryChildData p} (src : Sec7FrattiniTrivialElementaryMStageCandidateLowerSourceData h chief K hR child) : Type

The term-carrier half of the h₀ row: it chooses the first-order contribution of h₀ before proving the lem:h0shadow zero evaluation.

• h0Term : Module.End (ZMod 2) (towerM K)

  First-order contribution of h₀.

• h0Term_shadow : self.h0Term = 0

  The chosen h₀ term has the zero shadow from lem:h0shadow.

structure Q2Presentation.Induction.Sec7FrattiniTrivialElementaryMStageCandidateWildRowManuscriptH0ShadowData {p : FramedPair} {h : ¬ScalarTerminal p} {chief : FirstNonScalarChiefFactor p} {K : KernelMinimal chief.toNonScalarChiefFactor} {hR : Nat.card ↥(kernelFrattini K) = 1} {child : Sec7FrattiniTrivialElementaryChildData p} {src : Sec7FrattiniTrivialElementaryMStageCandidateLowerSourceData h chief K hR child} (term : Sec7FrattiniTrivialElementaryMStageCandidateWildRowManuscriptH0TermData src) : Prop

The shadow half of the h₀ row (lem:h0shadow): the chosen h₀ term is zero.

• h0_shadow : term.h0Term = 0

  h₀ contributes zero in the elementary row.

structure Q2Presentation.Induction.Sec7FrattiniTrivialElementaryMStageCandidateWildRowManuscriptU1InvTermData {p : FramedPair} {h : ¬ScalarTerminal p} {chief : FirstNonScalarChiefFactor p} {K : KernelMinimal chief.toNonScalarChiefFactor} {hR : Nat.card ↥(kernelFrattini K) = 1} {child : Sec7FrattiniTrivialElementaryChildData p} (src : Sec7FrattiniTrivialElementaryMStageCandidateLowerSourceData h chief K hR child) : Type

The term-carrier half of the u₁⁻¹ row.

• u1InvTerm : Module.End (ZMod 2) (towerM K)

  First-order contribution of u₁⁻¹.

• u1InvTerm_shadow : self.u1InvTerm = src.P * (src.T - 1) + src.P * (src.X₁ - 1)

  The inverse norm term contributes P(T-1) + P(X₁-1).

structure Q2Presentation.Induction.Sec7FrattiniTrivialElementaryMStageCandidateWildRowManuscriptU1InvShadowData {p : FramedPair} {h : ¬ScalarTerminal p} {chief : FirstNonScalarChiefFactor p} {K : KernelMinimal chief.toNonScalarChiefFactor} {hR : Nat.card ↥(kernelFrattini K) = 1} {child : Sec7FrattiniTrivialElementaryChildData p} {src : Sec7FrattiniTrivialElementaryMStageCandidateLowerSourceData h chief K hR child} (term : Sec7FrattiniTrivialElementaryMStageCandidateWildRowManuscriptU1InvTermData src) : Prop

The shadow half of the u₁⁻¹ row: the inverse ω₂-norm factor contributes P(T-1) + P(X₁-1).

• u1_inv_shadow : term.u1InvTerm = src.P * (src.T - 1) + src.P * (src.X₁ - 1)

  The inverse norm term contributes P(T-1) + P(X₁-1).

structure Q2Presentation.Induction.Sec7FrattiniTrivialElementaryMStageCandidateWildRowManuscriptX1ConjTermData {p : FramedPair} {h : ¬ScalarTerminal p} {chief : FirstNonScalarChiefFactor p} {K : KernelMinimal chief.toNonScalarChiefFactor} {hR : Nat.card ↥(kernelFrattini K) = 1} {child : Sec7FrattiniTrivialElementaryChildData p} (src : Sec7FrattiniTrivialElementaryMStageCandidateLowerSourceData h chief K hR child) : Type

The term-carrier half of the x₁^σ row.

• x1ConjTerm : Module.End (ZMod 2) (towerM K)

  First-order contribution of x₁^σ.

• x1ConjTerm_shadow : self.x1ConjTerm = src.Sinv * (src.X₁ - 1)

  The conjugate contributes S⁻¹(X₁-1) on the x₁ coordinate.

structure Q2Presentation.Induction.Sec7FrattiniTrivialElementaryMStageCandidateWildRowManuscriptX1ConjShadowData {p : FramedPair} {h : ¬ScalarTerminal p} {chief : FirstNonScalarChiefFactor p} {K : KernelMinimal chief.toNonScalarChiefFactor} {hR : Nat.card ↥(kernelFrattini K) = 1} {child : Sec7FrattiniTrivialElementaryChildData p} {src : Sec7FrattiniTrivialElementaryMStageCandidateLowerSourceData h chief K hR child} (term : Sec7FrattiniTrivialElementaryMStageCandidateWildRowManuscriptX1ConjTermData src) : Prop

The shadow half of the x₁^σ row: conjugation by σ contributes S⁻¹.

• x1_conj_shadow : term.x1ConjTerm = src.Sinv * (src.X₁ - 1)

  The conjugate contributes S⁻¹(X₁-1) on the x₁ coordinate.

structure Q2Presentation.Induction.Sec7FrattiniTrivialElementaryMStageCandidateWildRowManuscriptC0TermData {p : FramedPair} {h : ¬ScalarTerminal p} {chief : FirstNonScalarChiefFactor p} {K : KernelMinimal chief.toNonScalarChiefFactor} {hR : Nat.card ↥(kernelFrattini K) = 1} {child : Sec7FrattiniTrivialElementaryChildData p} (src : Sec7FrattiniTrivialElementaryMStageCandidateLowerSourceData h chief K hR child) : Type

The term-carrier half of the commutator row c₀=[d₀,z₀].

• c0Term : Module.End (ZMod 2) (towerM K)

  First-order contribution of c₀=[d₀,z₀].

• c0Term_shadow : self.c0Term = 0

  The commutator factor has no first-order contribution.

structure Q2Presentation.Induction.Sec7FrattiniTrivialElementaryMStageCandidateWildRowManuscriptC0ShadowData {p : FramedPair} {h : ¬ScalarTerminal p} {chief : FirstNonScalarChiefFactor p} {K : KernelMinimal chief.toNonScalarChiefFactor} {hR : Nat.card ↥(kernelFrattini K) = 1} {child : Sec7FrattiniTrivialElementaryChildData p} {src : Sec7FrattiniTrivialElementaryMStageCandidateLowerSourceData h chief K hR child} (term : Sec7FrattiniTrivialElementaryMStageCandidateWildRowManuscriptC0TermData src) : Prop

The shadow half of the commutator row: c₀ has no first-order contribution.

• c0_shadow : term.c0Term = 0

  The commutator factor has no first-order contribution.

structure Q2Presentation.Induction.Sec7FrattiniTrivialElementaryMStageCandidateWildRowManuscriptRelatorFoxDerivativeExpansionData {p : FramedPair} {h : ¬ScalarTerminal p} {chief : FirstNonScalarChiefFactor p} {K : KernelMinimal chief.toNonScalarChiefFactor} {hR : Nat.card ↥(kernelFrattini K) = 1} {child : Sec7FrattiniTrivialElementaryChildData p} {src : Sec7FrattiniTrivialElementaryMStageCandidateLowerSourceData h chief K hR child} (h0 : Sec7FrattiniTrivialElementaryMStageCandidateWildRowManuscriptH0Data src) (u1 : Sec7FrattiniTrivialElementaryMStageCandidateWildRowManuscriptU1InvData src) (x1 : Sec7FrattiniTrivialElementaryMStageCandidateWildRowManuscriptX1ConjData src) (c0 : Sec7FrattiniTrivialElementaryMStageCandidateWildRowManuscriptC0Data src) : Prop

The Fox-derivative expansion part of the product rule for the actual wild relator.

This is strictly before the final relator-triviality cancellation: it says that the sum of the four manuscript factor rows is the derivative expansion of the evaluated wild relator, namely ρ(wildRelator) - 1 on M. The theorem below supplies the final ρ(wildRelator)-1 = 0 cancellation from src.wild_rel / hwild.

• expansion : Q2Word.wildRelator.eval src.marking = 1 → h0.h0Term + u1.u1InvTerm + x1.x1ConjTerm + c0.c0Term = (sec7MStageCandidateActionEnd K) (Q2Word.wildRelator.eval src.marking) - 1

  The Fox product-rule expansion before using the relator equality.

structure Q2Presentation.Induction.Sec7FrattiniTrivialElementaryMStageCandidateWildRowManuscriptRelatorLiteralFoxDerivativeExpansionData {p : FramedPair} {h : ¬ScalarTerminal p} {chief : FirstNonScalarChiefFactor p} {K : KernelMinimal chief.toNonScalarChiefFactor} {hR : Nat.card ↥(kernelFrattini K) = 1} {child : Sec7FrattiniTrivialElementaryChildData p} (src : Sec7FrattiniTrivialElementaryMStageCandidateLowerSourceData h chief K hR child) : Prop

The literal, already evaluated, Fox-derivative row for the actual four-factor wild relator after substituting the manuscript factor shadows. This is the remaining finite-word expansion datum; the individual term carriers and shadows below are now chosen theorem-level.

• literal_expansion : Q2Word.wildRelator.eval src.marking = 1 → 0 + (src.P * (src.T - 1) + src.P * (src.X₁ - 1)) + src.Sinv * (src.X₁ - 1) + 0 = (sec7MStageCandidateActionEnd K) (Q2Word.wildRelator.eval src.marking) - 1

  The literal Fox product-rule expansion before using the relator cancellation.

theorem Q2Presentation.Induction.sec7_frattiniTrivial_elementaryMStageCandidateWildRowManuscriptH0TermData {p : FramedPair} {h : ¬ScalarTerminal p} {chief : FirstNonScalarChiefFactor p} {K : KernelMinimal chief.toNonScalarChiefFactor} {hR : Nat.card ↥(kernelFrattini K) = 1} {child : Sec7FrattiniTrivialElementaryChildData p} (src : Sec7FrattiniTrivialElementaryMStageCandidateLowerSourceData h chief K hR child) : Nonempty (Sec7FrattiniTrivialElementaryMStageCandidateWildRowManuscriptH0TermData src)

The h₀ term carrier is now chosen directly as its manuscript zero contribution.

theorem Q2Presentation.Induction.sec7_frattiniTrivial_elementaryMStageCandidateWildRowManuscriptH0ShadowData {p : FramedPair} {h : ¬ScalarTerminal p} {chief : FirstNonScalarChiefFactor p} {K : KernelMinimal chief.toNonScalarChiefFactor} {hR : Nat.card ↥(kernelFrattini K) = 1} {child : Sec7FrattiniTrivialElementaryChildData p} {src : Sec7FrattiniTrivialElementaryMStageCandidateLowerSourceData h chief K hR child} (term : Sec7FrattiniTrivialElementaryMStageCandidateWildRowManuscriptH0TermData src) : Nonempty (Sec7FrattiniTrivialElementaryMStageCandidateWildRowManuscriptH0ShadowData term)

The h₀ shadow is theorem-level from the chosen term carrier.

theorem Q2Presentation.Induction.sec7_frattiniTrivial_elementaryMStageCandidateWildRowManuscriptU1InvTermData {p : FramedPair} {h : ¬ScalarTerminal p} {chief : FirstNonScalarChiefFactor p} {K : KernelMinimal chief.toNonScalarChiefFactor} {hR : Nat.card ↥(kernelFrattini K) = 1} {child : Sec7FrattiniTrivialElementaryChildData p} (src : Sec7FrattiniTrivialElementaryMStageCandidateLowerSourceData h chief K hR child) : Nonempty (Sec7FrattiniTrivialElementaryMStageCandidateWildRowManuscriptU1InvTermData src)

The u₁⁻¹ term carrier is now the literal inverse-norm contribution from the manuscript word ledger.

theorem Q2Presentation.Induction.sec7_frattiniTrivial_elementaryMStageCandidateWildRowManuscriptU1InvShadowData {p : FramedPair} {h : ¬ScalarTerminal p} {chief : FirstNonScalarChiefFactor p} {K : KernelMinimal chief.toNonScalarChiefFactor} {hR : Nat.card ↥(kernelFrattini K) = 1} {child : Sec7FrattiniTrivialElementaryChildData p} {src : Sec7FrattiniTrivialElementaryMStageCandidateLowerSourceData h chief K hR child} (term : Sec7FrattiniTrivialElementaryMStageCandidateWildRowManuscriptU1InvTermData src) : Nonempty (Sec7FrattiniTrivialElementaryMStageCandidateWildRowManuscriptU1InvShadowData term)

The u₁⁻¹ shadow is theorem-level from the chosen term carrier.

theorem Q2Presentation.Induction.sec7_frattiniTrivial_elementaryMStageCandidateWildRowManuscriptX1ConjTermData {p : FramedPair} {h : ¬ScalarTerminal p} {chief : FirstNonScalarChiefFactor p} {K : KernelMinimal chief.toNonScalarChiefFactor} {hR : Nat.card ↥(kernelFrattini K) = 1} {child : Sec7FrattiniTrivialElementaryChildData p} (src : Sec7FrattiniTrivialElementaryMStageCandidateLowerSourceData h chief K hR child) : Nonempty (Sec7FrattiniTrivialElementaryMStageCandidateWildRowManuscriptX1ConjTermData src)

The x₁^σ term carrier is now the literal conjugation contribution from the manuscript word ledger.

theorem Q2Presentation.Induction.sec7_frattiniTrivial_elementaryMStageCandidateWildRowManuscriptX1ConjShadowData {p : FramedPair} {h : ¬ScalarTerminal p} {chief : FirstNonScalarChiefFactor p} {K : KernelMinimal chief.toNonScalarChiefFactor} {hR : Nat.card ↥(kernelFrattini K) = 1} {child : Sec7FrattiniTrivialElementaryChildData p} {src : Sec7FrattiniTrivialElementaryMStageCandidateLowerSourceData h chief K hR child} (term : Sec7FrattiniTrivialElementaryMStageCandidateWildRowManuscriptX1ConjTermData src) : Nonempty (Sec7FrattiniTrivialElementaryMStageCandidateWildRowManuscriptX1ConjShadowData term)

The x₁^σ shadow is theorem-level from the chosen term carrier.

theorem Q2Presentation.Induction.sec7_frattiniTrivial_elementaryMStageCandidateWildRowManuscriptC0TermData {p : FramedPair} {h : ¬ScalarTerminal p} {chief : FirstNonScalarChiefFactor p} {K : KernelMinimal chief.toNonScalarChiefFactor} {hR : Nat.card ↥(kernelFrattini K) = 1} {child : Sec7FrattiniTrivialElementaryChildData p} (src : Sec7FrattiniTrivialElementaryMStageCandidateLowerSourceData h chief K hR child) : Nonempty (Sec7FrattiniTrivialElementaryMStageCandidateWildRowManuscriptC0TermData src)

The commutator term carrier c₀=[d₀,z₀] is now chosen as its zero first-order contribution.

theorem Q2Presentation.Induction.sec7_frattiniTrivial_elementaryMStageCandidateWildRowManuscriptC0ShadowData {p : FramedPair} {h : ¬ScalarTerminal p} {chief : FirstNonScalarChiefFactor p} {K : KernelMinimal chief.toNonScalarChiefFactor} {hR : Nat.card ↥(kernelFrattini K) = 1} {child : Sec7FrattiniTrivialElementaryChildData p} {src : Sec7FrattiniTrivialElementaryMStageCandidateLowerSourceData h chief K hR child} (term : Sec7FrattiniTrivialElementaryMStageCandidateWildRowManuscriptC0TermData src) : Nonempty (Sec7FrattiniTrivialElementaryMStageCandidateWildRowManuscriptC0ShadowData term)

The commutator zero shadow is theorem-level from the chosen term carrier.

structure Q2Presentation.Induction.Sec7FrattiniTrivialElementaryMStageCandidateWildRowManuscriptRelatorCollapsedFoxDerivativeExpansionData {p : FramedPair} {h : ¬ScalarTerminal p} {chief : FirstNonScalarChiefFactor p} {K : KernelMinimal chief.toNonScalarChiefFactor} {hR : Nat.card ↥(kernelFrattini K) = 1} {child : Sec7FrattiniTrivialElementaryChildData p} (src : Sec7FrattiniTrivialElementaryMStageCandidateLowerSourceData h chief K hR child) : Prop

Collapsed manuscript input for the Fox derivative of the actual wild relator.

This is one step smaller than the former literal expanded row: it records the derivative as the two manuscript operator blocks P(T-1) and (P+S⁻¹)(X₁-1). The displayed four-summand literal row is recovered theorem-level below by distributivity and additive regrouping.

• collapsed_expansion : Q2Word.wildRelator.eval src.marking = 1 → src.P * (src.T - 1) + (src.P + src.Sinv) * (src.X₁ - 1) = (sec7MStageCandidateActionEnd K) (Q2Word.wildRelator.eval src.marking) - 1

  The collapsed Fox derivative row before the final relator cancellation.

structure Q2Presentation.Induction.Sec7FrattiniTrivialElementaryMStageCandidateWildRowManuscriptCollapsedFoxDerivativeBlockData {p : FramedPair} {h : ¬ScalarTerminal p} {chief : FirstNonScalarChiefFactor p} {K : KernelMinimal chief.toNonScalarChiefFactor} {hR : Nat.card ↥(kernelFrattini K) = 1} {child : Sec7FrattiniTrivialElementaryChildData p} (src : Sec7FrattiniTrivialElementaryMStageCandidateLowerSourceData h chief K hR child) : Type

Block-carrier form of the collapsed candidate wild-row Fox derivative.

The manuscript row is split into its two operator blocks, the norm block P(T-1) and the x₁ block (P+S⁻¹)(X₁-1), plus the product-rule expansion for the sum of these blocks. The checkpoint-255 collapsed expression is recovered by substituting the two block shadows.

• normBlock : Module.End (ZMod 2) (towerM K)

  The norm part of the collapsed Fox row.

• x1Block : Module.End (ZMod 2) (towerM K)

  The remaining x₁ part of the collapsed Fox row.

• norm_shadow : self.normBlock = src.P * (src.T - 1)

  The norm block is the manuscript operator P(T-1).

• x1_shadow : self.x1Block = (src.P + src.Sinv) * (src.X₁ - 1)

  The x₁ block is the manuscript operator (P+S⁻¹)(X₁-1).

• block_expansion : Q2Word.wildRelator.eval src.marking = 1 → self.normBlock + self.x1Block = (sec7MStageCandidateActionEnd K) (Q2Word.wildRelator.eval src.marking) - 1

  The block sum is the Fox derivative of the evaluated wild relator.

structure Q2Presentation.Induction.Sec7FrattiniTrivialElementaryMStageCandidateWildRowManuscriptTermProductRuleData {p : FramedPair} {h : ¬ScalarTerminal p} {chief : FirstNonScalarChiefFactor p} {K : KernelMinimal chief.toNonScalarChiefFactor} {hR : Nat.card ↥(kernelFrattini K) = 1} {child : Sec7FrattiniTrivialElementaryChildData p} {src : Sec7FrattiniTrivialElementaryMStageCandidateLowerSourceData h chief K hR child} (h0 : Sec7FrattiniTrivialElementaryMStageCandidateWildRowManuscriptH0TermData src) (u1 : Sec7FrattiniTrivialElementaryMStageCandidateWildRowManuscriptU1InvTermData src) (x1 : Sec7FrattiniTrivialElementaryMStageCandidateWildRowManuscriptX1ConjTermData src) (c0 : Sec7FrattiniTrivialElementaryMStageCandidateWildRowManuscriptC0TermData src) : Prop

Term-level Fox product rule for the actual four-factor wild relator.

This is below the collapsed two-block row: the four manuscript factor terms are still visible, and the row is only collapsed after substituting the already-proved shadows for h₀, u₁⁻¹, x₁^σ, and c₀.

• term_product_rule : Q2Word.wildRelator.eval src.marking = 1 → h0.h0Term + u1.u1InvTerm + x1.x1ConjTerm + c0.c0Term = (sec7MStageCandidateActionEnd K) (Q2Word.wildRelator.eval src.marking) - 1

  The raw Fox product-rule expansion before substituting the four term shadows.

structure Q2Presentation.Induction.Sec7FrattiniTrivialElementaryMStageCandidateWildRowManuscriptCanonicalTermProductRuleData {p : FramedPair} {h : ¬ScalarTerminal p} {chief : FirstNonScalarChiefFactor p} {K : KernelMinimal chief.toNonScalarChiefFactor} {hR : Nat.card ↥(kernelFrattini K) = 1} {child : Sec7FrattiniTrivialElementaryChildData p} (src : Sec7FrattiniTrivialElementaryMStageCandidateLowerSourceData h chief K hR child) : Prop

Canonical four-factor Fox product rule for the actual manuscript wild word.

This is below the checkpoint-257 term-carrier product rule: it states the product rule only after substituting the canonical manuscript first-order terms for h₀, u₁⁻¹, x₁^σ, and c₀. The carrier-dependent statement is recovered by the four term-shadow equalities.

• canonical_term_product_rule : Q2Word.wildRelator.eval src.marking = 1 → 0 + (src.P * (src.T - 1) + src.P * (src.X₁ - 1)) + src.Sinv * (src.X₁ - 1) + 0 = (sec7MStageCandidateActionEnd K) (Q2Word.wildRelator.eval src.marking) - 1

  The raw Fox product-rule expansion for the canonical four manuscript terms.

structure Q2Presentation.Induction.Sec7FrattiniTrivialElementaryMStageCandidateWildRowManuscriptCanonicalTermSumZeroData {p : FramedPair} {h : ¬ScalarTerminal p} {chief : FirstNonScalarChiefFactor p} {K : KernelMinimal chief.toNonScalarChiefFactor} {hR : Nat.card ↥(kernelFrattini K) = 1} {child : Sec7FrattiniTrivialElementaryChildData p} (src : Sec7FrattiniTrivialElementaryMStageCandidateLowerSourceData h chief K hR child) : Prop

Canonical four-factor row after the manuscript word ledger and final relator cancellation.

This is below the checkpoint-258 product-rule atom: it no longer carries the target r-1, because that term is theorem-level zero once the evaluated wild relator is 1. It is the factor-ordered row of lem:fullwordledger after substituting the canonical contributions of h₀, u₁⁻¹, x₁^σ, and c₀.

• canonical_term_sum_zero : Q2Word.wildRelator.eval src.marking = 1 → 0 + (src.P * (src.T - 1) + src.P * (src.X₁ - 1)) + src.Sinv * (src.X₁ - 1) + 0 = 0

Checkpoint262: lower the canonical product-rule core to prefix-factor shadows

def Q2Presentation.Induction.sec7MStageCandidateWildRowManuscriptH0Factor {p : FramedPair} {h : ¬ScalarTerminal p} {chief : FirstNonScalarChiefFactor p} {K : KernelMinimal chief.toNonScalarChiefFactor} {hR : Nat.card ↥(kernelFrattini K) = 1} {child : Sec7FrattiniTrivialElementaryChildData p} (src : Sec7FrattiniTrivialElementaryMStageCandidateLowerSourceData h chief K hR child) : towerC K

The actual h₀ factor of the manuscript wild word, evaluated in the lower candidate quotient C = towerC K.

Equations

• Q2Presentation.Induction.sec7MStageCandidateWildRowManuscriptH0Factor src = Q2Presentation.Q2Word.h0.eval src.marking

def Q2Presentation.Induction.sec7MStageCandidateWildRowManuscriptU1InvFactor {p : FramedPair} {h : ¬ScalarTerminal p} {chief : FirstNonScalarChiefFactor p} {K : KernelMinimal chief.toNonScalarChiefFactor} {hR : Nat.card ↥(kernelFrattini K) = 1} {child : Sec7FrattiniTrivialElementaryChildData p} (src : Sec7FrattiniTrivialElementaryMStageCandidateLowerSourceData h chief K hR child) : towerC K

The actual u₁⁻¹ factor of the manuscript wild word, evaluated in the lower candidate quotient.

Equations

• Q2Presentation.Induction.sec7MStageCandidateWildRowManuscriptU1InvFactor src = Q2Presentation.Q2Word.u1.inv.eval src.marking

def Q2Presentation.Induction.sec7MStageCandidateWildRowManuscriptX1ConjFactor {p : FramedPair} {h : ¬ScalarTerminal p} {chief : FirstNonScalarChiefFactor p} {K : KernelMinimal chief.toNonScalarChiefFactor} {hR : Nat.card ↥(kernelFrattini K) = 1} {child : Sec7FrattiniTrivialElementaryChildData p} (src : Sec7FrattiniTrivialElementaryMStageCandidateLowerSourceData h chief K hR child) : towerC K

The actual x₁^σ factor of the manuscript wild word, evaluated in the lower candidate quotient.

Equations

• Q2Presentation.Induction.sec7MStageCandidateWildRowManuscriptX1ConjFactor src = (Q2Presentation.Q2Word.rconjExpr Q2Presentation.Q2Word.x1 Q2Presentation.Q2Word.sigma).eval src.marking

def Q2Presentation.Induction.sec7MStageCandidateWildRowManuscriptC0Factor {p : FramedPair} {h : ¬ScalarTerminal p} {chief : FirstNonScalarChiefFactor p} {K : KernelMinimal chief.toNonScalarChiefFactor} {hR : Nat.card ↥(kernelFrattini K) = 1} {child : Sec7FrattiniTrivialElementaryChildData p} (src : Sec7FrattiniTrivialElementaryMStageCandidateLowerSourceData h chief K hR child) : towerC K

The actual c₀ factor of the manuscript wild word, evaluated in the lower candidate quotient.

Equations

• Q2Presentation.Induction.sec7MStageCandidateWildRowManuscriptC0Factor src = Q2Presentation.Q2Word.c0.eval src.marking

def Q2Presentation.Induction.sec7MStageCandidateWildRowManuscriptPrefixProductSum {p : FramedPair} {h : ¬ScalarTerminal p} {chief : FirstNonScalarChiefFactor p} {K : KernelMinimal chief.toNonScalarChiefFactor} {hR : Nat.card ↥(kernelFrattini K) = 1} {child : Sec7FrattiniTrivialElementaryChildData p} (src : Sec7FrattiniTrivialElementaryMStageCandidateLowerSourceData h chief K hR child) : Module.End (ZMod 2) (towerM K)

The pure four-factor Fox/product-rule sum for the evaluated manuscript wild word.

This is purely algebraic: it is the telescoping product-rule expression (ρ a - 1) + ρ(a)(ρ b - 1) + ρ(ab)(ρ c - 1) + ρ(abc)(ρ d - 1) for a=h₀, b=u₁⁻¹, c=x₁^σ, and d=c₀. The factor-specific manuscript ledger is kept out of this definition and isolated in the four prefix-shadow atoms below.

Equations

• One or more equations did not get rendered due to their size.

theorem Q2Presentation.Induction.sec7MStageCandidateWildRowManuscriptPrefixProductRule {p : FramedPair} {h : ¬ScalarTerminal p} {chief : FirstNonScalarChiefFactor p} {K : KernelMinimal chief.toNonScalarChiefFactor} {hR : Nat.card ↥(kernelFrattini K) = 1} {child : Sec7FrattiniTrivialElementaryChildData p} (src : Sec7FrattiniTrivialElementaryMStageCandidateLowerSourceData h chief K hR child) : sec7MStageCandidateWildRowManuscriptPrefixProductSum src = (sec7MStageCandidateActionEnd K) (Q2Word.wildRelator.eval src.marking) - 1

Pure four-factor product rule for the evaluated wild word.

The proof is independent of the manuscript row calculation and independent of the candidate relator-shadow chain: after unfolding only the top-level definition of Q2Word.wildRelator, it is the standard noncommutative telescoping identity ρ(abcd)-1 = (ρa-1)+ρa(ρb-1)+ρ(ab)(ρc-1)+ρ(abc)(ρd-1).

structure Q2Presentation.Induction.Sec7FrattiniTrivialElementaryMStageCandidateWildRowManuscriptH0PrefixShadowData {p : FramedPair} {h : ¬ScalarTerminal p} {chief : FirstNonScalarChiefFactor p} {K : KernelMinimal chief.toNonScalarChiefFactor} {hR : Nat.card ↥(kernelFrattini K) = 1} {child : Sec7FrattiniTrivialElementaryChildData p} {src : Sec7FrattiniTrivialElementaryMStageCandidateLowerSourceData h chief K hR child} (h0 : Sec7FrattiniTrivialElementaryMStageCandidateWildRowManuscriptH0TermData src) : Prop

Prefix-shadow datum for the h₀ manuscript factor.

This is one of the four smaller manuscript-row atoms replacing the former single canonical product-rule core. It identifies the chosen first-order h₀ row with its prefix Fox contribution ρ(h₀)-1; the already-existing shadow field then says this contribution is zero in the elementary row.

• h0_prefix_shadow : h0.h0Term = (sec7MStageCandidateActionEnd K) (sec7MStageCandidateWildRowManuscriptH0Factor src) - 1

  The h₀ term is the first prefix contribution of the product rule.

structure Q2Presentation.Induction.Sec7FrattiniTrivialElementaryMStageCandidateWildRowManuscriptU1InvPrefixShadowData {p : FramedPair} {h : ¬ScalarTerminal p} {chief : FirstNonScalarChiefFactor p} {K : KernelMinimal chief.toNonScalarChiefFactor} {hR : Nat.card ↥(kernelFrattini K) = 1} {child : Sec7FrattiniTrivialElementaryChildData p} {src : Sec7FrattiniTrivialElementaryMStageCandidateLowerSourceData h chief K hR child} (u1 : Sec7FrattiniTrivialElementaryMStageCandidateWildRowManuscriptU1InvTermData src) : Prop

Prefix-shadow datum for the u₁⁻¹ manuscript factor.

• u1_prefix_shadow : u1.u1InvTerm = (sec7MStageCandidateActionEnd K) (sec7MStageCandidateWildRowManuscriptH0Factor src) * ((sec7MStageCandidateActionEnd K) (sec7MStageCandidateWildRowManuscriptU1InvFactor src) - 1)

  The u₁⁻¹ term is the second prefix contribution of the product rule.

structure Q2Presentation.Induction.Sec7FrattiniTrivialElementaryMStageCandidateWildRowManuscriptX1ConjPrefixShadowData {p : FramedPair} {h : ¬ScalarTerminal p} {chief : FirstNonScalarChiefFactor p} {K : KernelMinimal chief.toNonScalarChiefFactor} {hR : Nat.card ↥(kernelFrattini K) = 1} {child : Sec7FrattiniTrivialElementaryChildData p} {src : Sec7FrattiniTrivialElementaryMStageCandidateLowerSourceData h chief K hR child} (x1 : Sec7FrattiniTrivialElementaryMStageCandidateWildRowManuscriptX1ConjTermData src) : Prop

Prefix-shadow datum for the x₁^σ manuscript factor.

• x1_prefix_shadow : x1.x1ConjTerm = (sec7MStageCandidateActionEnd K) (sec7MStageCandidateWildRowManuscriptH0Factor src * sec7MStageCandidateWildRowManuscriptU1InvFactor src) * ((sec7MStageCandidateActionEnd K) (sec7MStageCandidateWildRowManuscriptX1ConjFactor src) - 1)

  The x₁^σ term is the third prefix contribution of the product rule.

structure Q2Presentation.Induction.Sec7FrattiniTrivialElementaryMStageCandidateWildRowManuscriptC0PrefixShadowData {p : FramedPair} {h : ¬ScalarTerminal p} {chief : FirstNonScalarChiefFactor p} {K : KernelMinimal chief.toNonScalarChiefFactor} {hR : Nat.card ↥(kernelFrattini K) = 1} {child : Sec7FrattiniTrivialElementaryChildData p} {src : Sec7FrattiniTrivialElementaryMStageCandidateLowerSourceData h chief K hR child} (c0 : Sec7FrattiniTrivialElementaryMStageCandidateWildRowManuscriptC0TermData src) : Prop

Prefix-shadow datum for the c₀ manuscript factor.

• c0_prefix_shadow : c0.c0Term = (sec7MStageCandidateActionEnd K) (sec7MStageCandidateWildRowManuscriptH0Factor src * sec7MStageCandidateWildRowManuscriptU1InvFactor src * sec7MStageCandidateWildRowManuscriptX1ConjFactor src) * ((sec7MStageCandidateActionEnd K) (sec7MStageCandidateWildRowManuscriptC0Factor src) - 1)

  The c₀ term is the fourth prefix contribution of the product rule.

structure Q2Presentation.Induction.Sec7FrattiniTrivialElementaryMStageCandidateWildRowManuscriptH0PrefixActionData {p : FramedPair} {h : ¬ScalarTerminal p} {chief : FirstNonScalarChiefFactor p} {K : KernelMinimal chief.toNonScalarChiefFactor} {hR : Nat.card ↥(kernelFrattini K) = 1} {child : Sec7FrattiniTrivialElementaryChildData p} (src : Sec7FrattiniTrivialElementaryMStageCandidateLowerSourceData h chief K hR child) : Prop

Lower zero-degree manuscript atom for the h₀ factor.

This is strictly below the former H0PrefixShadowData: it no longer mentions an arbitrary first-order term carrier. It records only the factor-action consequence of lem:h0shadow needed by the prefix product rule, namely that the actual evaluated h₀ factor acts trivially on the elementary module M.

• h0_prefix_action : (sec7MStageCandidateActionEnd K) (sec7MStageCandidateWildRowManuscriptH0Factor src) = 1

  The evaluated h₀ factor has trivial action on M.

structure Q2Presentation.Induction.Sec7FrattiniTrivialElementaryMStageCandidateWildRowManuscriptU1InvFactorFoxData {p : FramedPair} {h : ¬ScalarTerminal p} {chief : FirstNonScalarChiefFactor p} {K : KernelMinimal chief.toNonScalarChiefFactor} {hR : Nat.card ↥(kernelFrattini K) = 1} {child : Sec7FrattiniTrivialElementaryChildData p} (src : Sec7FrattiniTrivialElementaryMStageCandidateLowerSourceData h chief K hR child) : Prop

Lower finite-word atom for the inverse norm factor u₁⁻¹.

This is below the former U1InvPrefixShadowData: it is independent of the chosen term carrier and states the actual Fox calculation for the u₁⁻¹ factor itself. The prefix multiplier ρ(h₀) is inserted theorem-level using H0PrefixActionData.

• u1_inv_factor_fox : (sec7MStageCandidateActionEnd K) (sec7MStageCandidateWildRowManuscriptU1InvFactor src) - 1 = src.P * (src.T - 1) + src.P * (src.X₁ - 1)

  The inverse ω₂-norm factor contributes the two norm pieces of lem:fullwordledger.

structure Q2Presentation.Induction.Sec7FrattiniTrivialElementaryMStageCandidateWildRowManuscriptX1ConjPrefixFoxData {p : FramedPair} {h : ¬ScalarTerminal p} {chief : FirstNonScalarChiefFactor p} {K : KernelMinimal chief.toNonScalarChiefFactor} {hR : Nat.card ↥(kernelFrattini K) = 1} {child : Sec7FrattiniTrivialElementaryChildData p} (src : Sec7FrattiniTrivialElementaryMStageCandidateLowerSourceData h chief K hR child) : Prop

Lower prefix finite-word atom for the conjugate factor x₁^σ.

This is below the former X1ConjPrefixShadowData: it no longer refers to the selected term carrier, and records only the manuscript Fox-prefix calculation for the actual third factor of h₀ u₁⁻¹ x₁^σ c₀.

• x1_conj_prefix_fox : (sec7MStageCandidateActionEnd K) (sec7MStageCandidateWildRowManuscriptH0Factor src * sec7MStageCandidateWildRowManuscriptU1InvFactor src) * ((sec7MStageCandidateActionEnd K) (sec7MStageCandidateWildRowManuscriptX1ConjFactor src) - 1) = src.Sinv * (src.X₁ - 1)

  The third prefix contribution is the S⁻¹-twisted x₁ row.

structure Q2Presentation.Induction.Sec7FrattiniTrivialElementaryMStageCandidateWildRowManuscriptC0PrefixFoxData {p : FramedPair} {h : ¬ScalarTerminal p} {chief : FirstNonScalarChiefFactor p} {K : KernelMinimal chief.toNonScalarChiefFactor} {hR : Nat.card ↥(kernelFrattini K) = 1} {child : Sec7FrattiniTrivialElementaryChildData p} (src : Sec7FrattiniTrivialElementaryMStageCandidateLowerSourceData h chief K hR child) : Prop

Lower prefix finite-word atom for the commutator factor c₀=[d₀,z₀].

This is below the former C0PrefixShadowData: it removes the term carrier and records only the zero first-order prefix contribution of the actual commutator factor from the manuscript ledger.

• c0_prefix_fox : (sec7MStageCandidateActionEnd K) (sec7MStageCandidateWildRowManuscriptH0Factor src * sec7MStageCandidateWildRowManuscriptU1InvFactor src * sec7MStageCandidateWildRowManuscriptX1ConjFactor src) * ((sec7MStageCandidateActionEnd K) (sec7MStageCandidateWildRowManuscriptC0Factor src) - 1) = 0

  The fourth prefix contribution, coming from the commutator factor c₀, vanishes.

Checkpoint264: lower the four factor/Fox atoms to internal word/Fox ledgers

def Q2Presentation.Induction.sec7MStageCandidateWildRowManuscriptH0ConjX0G0Factor {p : FramedPair} {h : ¬ScalarTerminal p} {chief : FirstNonScalarChiefFactor p} {K : KernelMinimal chief.toNonScalarChiefFactor} {hR : Nat.card ↥(kernelFrattini K) = 1} {child : Sec7FrattiniTrivialElementaryChildData p} (src : Sec7FrattiniTrivialElementaryMStageCandidateLowerSourceData h chief K hR child) : towerC K

First internal factor of h₀ = x₀^{g₀} x₀ d_g d₀ d₀² h_c.

Equations

• Q2Presentation.Induction.sec7MStageCandidateWildRowManuscriptH0ConjX0G0Factor src = (Q2Presentation.Q2Word.rconjExpr Q2Presentation.Q2Word.x0 Q2Presentation.Q2Word.g0).eval src.marking

def Q2Presentation.Induction.sec7MStageCandidateWildRowManuscriptH0X0Factor {p : FramedPair} {h : ¬ScalarTerminal p} {chief : FirstNonScalarChiefFactor p} {K : KernelMinimal chief.toNonScalarChiefFactor} {hR : Nat.card ↥(kernelFrattini K) = 1} {child : Sec7FrattiniTrivialElementaryChildData p} (src : Sec7FrattiniTrivialElementaryMStageCandidateLowerSourceData h chief K hR child) : towerC K

Second internal factor of h₀.

Equations

• Q2Presentation.Induction.sec7MStageCandidateWildRowManuscriptH0X0Factor src = Q2Presentation.Q2Word.x0.eval src.marking

def Q2Presentation.Induction.sec7MStageCandidateWildRowManuscriptH0DgFactor {p : FramedPair} {h : ¬ScalarTerminal p} {chief : FirstNonScalarChiefFactor p} {K : KernelMinimal chief.toNonScalarChiefFactor} {hR : Nat.card ↥(kernelFrattini K) = 1} {child : Sec7FrattiniTrivialElementaryChildData p} (src : Sec7FrattiniTrivialElementaryMStageCandidateLowerSourceData h chief K hR child) : towerC K

Third internal factor of h₀.

Equations

• Q2Presentation.Induction.sec7MStageCandidateWildRowManuscriptH0DgFactor src = Q2Presentation.Q2Word.dg.eval src.marking

def Q2Presentation.Induction.sec7MStageCandidateWildRowManuscriptH0D0Factor {p : FramedPair} {h : ¬ScalarTerminal p} {chief : FirstNonScalarChiefFactor p} {K : KernelMinimal chief.toNonScalarChiefFactor} {hR : Nat.card ↥(kernelFrattini K) = 1} {child : Sec7FrattiniTrivialElementaryChildData p} (src : Sec7FrattiniTrivialElementaryMStageCandidateLowerSourceData h chief K hR child) : towerC K

Fourth internal factor of h₀.

Equations

• Q2Presentation.Induction.sec7MStageCandidateWildRowManuscriptH0D0Factor src = Q2Presentation.Q2Word.d0.eval src.marking

def Q2Presentation.Induction.sec7MStageCandidateWildRowManuscriptH0D0SqFactor {p : FramedPair} {h : ¬ScalarTerminal p} {chief : FirstNonScalarChiefFactor p} {K : KernelMinimal chief.toNonScalarChiefFactor} {hR : Nat.card ↥(kernelFrattini K) = 1} {child : Sec7FrattiniTrivialElementaryChildData p} (src : Sec7FrattiniTrivialElementaryMStageCandidateLowerSourceData h chief K hR child) : towerC K

Fifth internal factor of h₀, namely d₀².

Equations

• Q2Presentation.Induction.sec7MStageCandidateWildRowManuscriptH0D0SqFactor src = (Q2Presentation.Q2Word.d0.powNat 2).eval src.marking

def Q2Presentation.Induction.sec7MStageCandidateWildRowManuscriptH0HcFactor {p : FramedPair} {h : ¬ScalarTerminal p} {chief : FirstNonScalarChiefFactor p} {K : KernelMinimal chief.toNonScalarChiefFactor} {hR : Nat.card ↥(kernelFrattini K) = 1} {child : Sec7FrattiniTrivialElementaryChildData p} (src : Sec7FrattiniTrivialElementaryMStageCandidateLowerSourceData h chief K hR child) : towerC K

Sixth internal factor of h₀.

Equations

• Q2Presentation.Induction.sec7MStageCandidateWildRowManuscriptH0HcFactor src = Q2Presentation.Q2Word.hc.eval src.marking

def Q2Presentation.Induction.sec7MStageCandidateWildRowManuscriptH0ExpandedPrefixSum {p : FramedPair} {h : ¬ScalarTerminal p} {chief : FirstNonScalarChiefFactor p} {K : KernelMinimal chief.toNonScalarChiefFactor} {hR : Nat.card ↥(kernelFrattini K) = 1} {child : Sec7FrattiniTrivialElementaryChildData p} (src : Sec7FrattiniTrivialElementaryMStageCandidateLowerSourceData h chief K hR child) : Module.End (ZMod 2) (towerM K)

The internal six-factor Fox sum of h₀.

Equations

• One or more equations did not get rendered due to their size.

theorem Q2Presentation.Induction.sec7MStageCandidateWildRowManuscriptH0ExpandedPrefixProductRule {p : FramedPair} {h : ¬ScalarTerminal p} {chief : FirstNonScalarChiefFactor p} {K : KernelMinimal chief.toNonScalarChiefFactor} {hR : Nat.card ↥(kernelFrattini K) = 1} {child : Sec7FrattiniTrivialElementaryChildData p} (src : Sec7FrattiniTrivialElementaryMStageCandidateLowerSourceData h chief K hR child) : sec7MStageCandidateWildRowManuscriptH0ExpandedPrefixSum src = (sec7MStageCandidateActionEnd K) (sec7MStageCandidateWildRowManuscriptH0Factor src) - 1

The algebraic product-rule bridge from the internal six-factor h₀ ledger to ρ(h₀)-1.

def Q2Presentation.Induction.sec7MStageCandidateWildRowManuscriptU1Factor {p : FramedPair} {h : ¬ScalarTerminal p} {chief : FirstNonScalarChiefFactor p} {K : KernelMinimal chief.toNonScalarChiefFactor} {hR : Nat.card ↥(kernelFrattini K) = 1} {child : Sec7FrattiniTrivialElementaryChildData p} (src : Sec7FrattiniTrivialElementaryMStageCandidateLowerSourceData h chief K hR child) : towerC K

The non-inverted u₁ factor.

Equations

• Q2Presentation.Induction.sec7MStageCandidateWildRowManuscriptU1Factor src = Q2Presentation.Q2Word.u1.eval src.marking

def Q2Presentation.Induction.sec7MStageCandidateWildRowManuscriptSigmaInvFactor {p : FramedPair} {h : ¬ScalarTerminal p} {chief : FirstNonScalarChiefFactor p} {K : KernelMinimal chief.toNonScalarChiefFactor} {hR : Nat.card ↥(kernelFrattini K) = 1} {child : Sec7FrattiniTrivialElementaryChildData p} (src : Sec7FrattiniTrivialElementaryMStageCandidateLowerSourceData h chief K hR child) : towerC K

Internal first factor of x₁^σ = σ⁻¹ x₁ σ.

Equations

• Q2Presentation.Induction.sec7MStageCandidateWildRowManuscriptSigmaInvFactor src = Q2Presentation.Q2Word.sigma.inv.eval src.marking

def Q2Presentation.Induction.sec7MStageCandidateWildRowManuscriptX1BareFactor {p : FramedPair} {h : ¬ScalarTerminal p} {chief : FirstNonScalarChiefFactor p} {K : KernelMinimal chief.toNonScalarChiefFactor} {hR : Nat.card ↥(kernelFrattini K) = 1} {child : Sec7FrattiniTrivialElementaryChildData p} (src : Sec7FrattiniTrivialElementaryMStageCandidateLowerSourceData h chief K hR child) : towerC K

Internal middle factor of x₁^σ = σ⁻¹ x₁ σ.

Equations

• Q2Presentation.Induction.sec7MStageCandidateWildRowManuscriptX1BareFactor src = Q2Presentation.Q2Word.x1.eval src.marking

def Q2Presentation.Induction.sec7MStageCandidateWildRowManuscriptSigmaFactor {p : FramedPair} {h : ¬ScalarTerminal p} {chief : FirstNonScalarChiefFactor p} {K : KernelMinimal chief.toNonScalarChiefFactor} {hR : Nat.card ↥(kernelFrattini K) = 1} {child : Sec7FrattiniTrivialElementaryChildData p} (src : Sec7FrattiniTrivialElementaryMStageCandidateLowerSourceData h chief K hR child) : towerC K

Internal final factor of x₁^σ = σ⁻¹ x₁ σ.

Equations

• Q2Presentation.Induction.sec7MStageCandidateWildRowManuscriptSigmaFactor src = Q2Presentation.Q2Word.sigma.eval src.marking

def Q2Presentation.Induction.sec7MStageCandidateWildRowManuscriptX1ConjInternalPrefixSum {p : FramedPair} {h : ¬ScalarTerminal p} {chief : FirstNonScalarChiefFactor p} {K : KernelMinimal chief.toNonScalarChiefFactor} {hR : Nat.card ↥(kernelFrattini K) = 1} {child : Sec7FrattiniTrivialElementaryChildData p} (src : Sec7FrattiniTrivialElementaryMStageCandidateLowerSourceData h chief K hR child) : Module.End (ZMod 2) (towerM K)

The internal Fox sum of the conjugate factor x₁^σ, including the already accrued external prefix h₀u₁⁻¹.

Equations

• One or more equations did not get rendered due to their size.

theorem Q2Presentation.Induction.sec7MStageCandidateWildRowManuscriptX1ConjInternalPrefixProductRule {p : FramedPair} {h : ¬ScalarTerminal p} {chief : FirstNonScalarChiefFactor p} {K : KernelMinimal chief.toNonScalarChiefFactor} {hR : Nat.card ↥(kernelFrattini K) = 1} {child : Sec7FrattiniTrivialElementaryChildData p} (src : Sec7FrattiniTrivialElementaryMStageCandidateLowerSourceData h chief K hR child) : sec7MStageCandidateWildRowManuscriptX1ConjInternalPrefixSum src = (sec7MStageCandidateActionEnd K) (sec7MStageCandidateWildRowManuscriptH0Factor src * sec7MStageCandidateWildRowManuscriptU1InvFactor src) * ((sec7MStageCandidateActionEnd K) (sec7MStageCandidateWildRowManuscriptX1ConjFactor src) - 1)

Product-rule bridge from the internal conjugation ledger to the lumped third-prefix contribution.

def Q2Presentation.Induction.sec7MStageCandidateWildRowManuscriptD0Factor {p : FramedPair} {h : ¬ScalarTerminal p} {chief : FirstNonScalarChiefFactor p} {K : KernelMinimal chief.toNonScalarChiefFactor} {hR : Nat.card ↥(kernelFrattini K) = 1} {child : Sec7FrattiniTrivialElementaryChildData p} (src : Sec7FrattiniTrivialElementaryMStageCandidateLowerSourceData h chief K hR child) : towerC K

Internal d₀ factor of c₀=[d₀,z₀].

Equations

• Q2Presentation.Induction.sec7MStageCandidateWildRowManuscriptD0Factor src = Q2Presentation.Q2Word.d0.eval src.marking

def Q2Presentation.Induction.sec7MStageCandidateWildRowManuscriptD0InvFactor {p : FramedPair} {h : ¬ScalarTerminal p} {chief : FirstNonScalarChiefFactor p} {K : KernelMinimal chief.toNonScalarChiefFactor} {hR : Nat.card ↥(kernelFrattini K) = 1} {child : Sec7FrattiniTrivialElementaryChildData p} (src : Sec7FrattiniTrivialElementaryMStageCandidateLowerSourceData h chief K hR child) : towerC K

Internal d₀⁻¹ factor of c₀=[d₀,z₀].

Equations

• Q2Presentation.Induction.sec7MStageCandidateWildRowManuscriptD0InvFactor src = Q2Presentation.Q2Word.d0.inv.eval src.marking

def Q2Presentation.Induction.sec7MStageCandidateWildRowManuscriptZ0Factor {p : FramedPair} {h : ¬ScalarTerminal p} {chief : FirstNonScalarChiefFactor p} {K : KernelMinimal chief.toNonScalarChiefFactor} {hR : Nat.card ↥(kernelFrattini K) = 1} {child : Sec7FrattiniTrivialElementaryChildData p} (src : Sec7FrattiniTrivialElementaryMStageCandidateLowerSourceData h chief K hR child) : towerC K

Internal z₀ factor of c₀=[d₀,z₀].

Equations

• Q2Presentation.Induction.sec7MStageCandidateWildRowManuscriptZ0Factor src = Q2Presentation.Q2Word.z0.eval src.marking

def Q2Presentation.Induction.sec7MStageCandidateWildRowManuscriptZ0InvFactor {p : FramedPair} {h : ¬ScalarTerminal p} {chief : FirstNonScalarChiefFactor p} {K : KernelMinimal chief.toNonScalarChiefFactor} {hR : Nat.card ↥(kernelFrattini K) = 1} {child : Sec7FrattiniTrivialElementaryChildData p} (src : Sec7FrattiniTrivialElementaryMStageCandidateLowerSourceData h chief K hR child) : towerC K

Internal z₀⁻¹ factor of c₀=[d₀,z₀].

Equations

• Q2Presentation.Induction.sec7MStageCandidateWildRowManuscriptZ0InvFactor src = Q2Presentation.Q2Word.z0.inv.eval src.marking

def Q2Presentation.Induction.sec7MStageCandidateWildRowManuscriptC0InternalPrefixSum {p : FramedPair} {h : ¬ScalarTerminal p} {chief : FirstNonScalarChiefFactor p} {K : KernelMinimal chief.toNonScalarChiefFactor} {hR : Nat.card ↥(kernelFrattini K) = 1} {child : Sec7FrattiniTrivialElementaryChildData p} (src : Sec7FrattiniTrivialElementaryMStageCandidateLowerSourceData h chief K hR child) : Module.End (ZMod 2) (towerM K)

The internal Fox sum of the commutator factor c₀=[d₀,z₀], including the already accrued external prefix h₀u₁⁻¹x₁^σ.

Equations

• One or more equations did not get rendered due to their size.

theorem Q2Presentation.Induction.sec7MStageCandidateWildRowManuscriptC0InternalPrefixProductRule {p : FramedPair} {h : ¬ScalarTerminal p} {chief : FirstNonScalarChiefFactor p} {K : KernelMinimal chief.toNonScalarChiefFactor} {hR : Nat.card ↥(kernelFrattini K) = 1} {child : Sec7FrattiniTrivialElementaryChildData p} (src : Sec7FrattiniTrivialElementaryMStageCandidateLowerSourceData h chief K hR child) : sec7MStageCandidateWildRowManuscriptC0InternalPrefixSum src = (sec7MStageCandidateActionEnd K) (sec7MStageCandidateWildRowManuscriptH0Factor src * sec7MStageCandidateWildRowManuscriptU1InvFactor src * sec7MStageCandidateWildRowManuscriptX1ConjFactor src) * ((sec7MStageCandidateActionEnd K) (sec7MStageCandidateWildRowManuscriptC0Factor src) - 1)

Product-rule bridge from the internal commutator ledger to the lumped fourth-prefix contribution.

structure Q2Presentation.Induction.Sec7FrattiniTrivialElementaryMStageCandidateWildRowManuscriptH0ExpandedFoxData {p : FramedPair} {h : ¬ScalarTerminal p} {chief : FirstNonScalarChiefFactor p} {K : KernelMinimal chief.toNonScalarChiefFactor} {hR : Nat.card ↥(kernelFrattini K) = 1} {child : Sec7FrattiniTrivialElementaryChildData p} (src : Sec7FrattiniTrivialElementaryMStageCandidateLowerSourceData h chief K hR child) : Prop

Lower internal manuscript atom for h₀: the six-factor Fox sum of x₀^{g₀} x₀ d_g d₀ d₀² h_c vanishes.

• h0_expanded_fox_sum_zero : sec7MStageCandidateWildRowManuscriptH0ExpandedPrefixSum src = 0

  The internal h₀ Fox ledger, before collapsing it to the action equality ρ(h₀)=1.

structure Q2Presentation.Induction.Sec7FrattiniTrivialElementaryMStageCandidateWildRowManuscriptU1InvOmegaNormFoxData {p : FramedPair} {h : ¬ScalarTerminal p} {chief : FirstNonScalarChiefFactor p} {K : KernelMinimal chief.toNonScalarChiefFactor} {hR : Nat.card ↥(kernelFrattini K) = 1} {child : Sec7FrattiniTrivialElementaryChildData p} (src : Sec7FrattiniTrivialElementaryMStageCandidateLowerSourceData h chief K hR child) : Prop

Lower internal manuscript atom for u₁⁻¹: the inverse rule has been separated from the ω₂-norm Fox calculation for u₁=(x₁τ)^{ω₂}.

• u1_inv_omega_norm_fox : -((sec7MStageCandidateActionEnd K) (sec7MStageCandidateWildRowManuscriptU1InvFactor src) * ((sec7MStageCandidateActionEnd K) (sec7MStageCandidateWildRowManuscriptU1Factor src) - 1)) = src.P * (src.T - 1) + src.P * (src.X₁ - 1)

  The ω₂-norm Fox row for the non-inverted u₁, transported through the inverse-factor rule.

structure Q2Presentation.Induction.Sec7FrattiniTrivialElementaryMStageCandidateWildRowManuscriptX1ConjInternalFoxData {p : FramedPair} {h : ¬ScalarTerminal p} {chief : FirstNonScalarChiefFactor p} {K : KernelMinimal chief.toNonScalarChiefFactor} {hR : Nat.card ↥(kernelFrattini K) = 1} {child : Sec7FrattiniTrivialElementaryChildData p} (src : Sec7FrattiniTrivialElementaryMStageCandidateLowerSourceData h chief K hR child) : Prop

Lower internal manuscript atom for x₁^σ: expand the conjugate σ⁻¹ x₁ σ before collapsing it to the third-prefix row.

• x1_conj_internal_fox : sec7MStageCandidateWildRowManuscriptX1ConjInternalPrefixSum src = src.Sinv * (src.X₁ - 1)

  The internal conjugation ledger is the manuscript S⁻¹-twisted x₁ row.

structure Q2Presentation.Induction.Sec7FrattiniTrivialElementaryMStageCandidateWildRowManuscriptC0CommutatorInternalFoxData {p : FramedPair} {h : ¬ScalarTerminal p} {chief : FirstNonScalarChiefFactor p} {K : KernelMinimal chief.toNonScalarChiefFactor} {hR : Nat.card ↥(kernelFrattini K) = 1} {child : Sec7FrattiniTrivialElementaryChildData p} (src : Sec7FrattiniTrivialElementaryMStageCandidateLowerSourceData h chief K hR child) : Prop

Lower internal manuscript atom for c₀=[d₀,z₀]: expand the commutator before collapsing it to a zero fourth-prefix contribution.

• c0_commutator_internal_fox : sec7MStageCandidateWildRowManuscriptC0InternalPrefixSum src = 0

  The internal commutator Fox ledger vanishes.

Checkpoint265: split the internal Fox leaves into factor-row data

structure Q2Presentation.Induction.Sec7FrattiniTrivialElementaryMStageCandidateWildRowManuscriptH0ExpandedFactorRowsData {p : FramedPair} {h : ¬ScalarTerminal p} {chief : FirstNonScalarChiefFactor p} {K : KernelMinimal chief.toNonScalarChiefFactor} {hR : Nat.card ↥(kernelFrattini K) = 1} {child : Sec7FrattiniTrivialElementaryChildData p} (src : Sec7FrattiniTrivialElementaryMStageCandidateLowerSourceData h chief K hR child) : Type

Row-level finite-word data for the six internal factors of h₀ = x₀^{g₀} x₀ d_g d₀ d₀² h_c.

This is below the checkpoint264 H0ExpandedFoxData leaf: the six Fox summands are named separately, and the residual cancellation is only the sum of these six rows.

• h0ConjX0G0Row : Module.End (ZMod 2) (towerM K)

  The row from the factor x₀^{g₀}.

• h0X0Row : Module.End (ZMod 2) (towerM K)

  The row from the factor x₀.

• h0DgRow : Module.End (ZMod 2) (towerM K)

  The row from the factor d_g.

• h0D0Row : Module.End (ZMod 2) (towerM K)

  The row from the factor d₀.

• h0D0SqRow : Module.End (ZMod 2) (towerM K)

  The row from the factor d₀².

• h0HcRow : Module.End (ZMod 2) (towerM K)

  The row from the factor h_c.

• h0_conjX0G0_row : self.h0ConjX0G0Row = (sec7MStageCandidateActionEnd K) (sec7MStageCandidateWildRowManuscriptH0ConjX0G0Factor src) - 1

  Fox row for x₀^{g₀}.

• h0_x0_row : self.h0X0Row = (sec7MStageCandidateActionEnd K) (sec7MStageCandidateWildRowManuscriptH0ConjX0G0Factor src) * ((sec7MStageCandidateActionEnd K) (sec7MStageCandidateWildRowManuscriptH0X0Factor src) - 1)

  Fox row for the following x₀, including the prefix x₀^{g₀}.

• h0_dg_row : self.h0DgRow = (sec7MStageCandidateActionEnd K) (sec7MStageCandidateWildRowManuscriptH0ConjX0G0Factor src * sec7MStageCandidateWildRowManuscriptH0X0Factor src) * ((sec7MStageCandidateActionEnd K) (sec7MStageCandidateWildRowManuscriptH0DgFactor src) - 1)

  Fox row for d_g, including the prefix x₀^{g₀}x₀.

• h0_d0_row : self.h0D0Row = (sec7MStageCandidateActionEnd K) (sec7MStageCandidateWildRowManuscriptH0ConjX0G0Factor src * sec7MStageCandidateWildRowManuscriptH0X0Factor src * sec7MStageCandidateWildRowManuscriptH0DgFactor src) * ((sec7MStageCandidateActionEnd K) (sec7MStageCandidateWildRowManuscriptH0D0Factor src) - 1)

  Fox row for the first d₀.

• h0_d0Sq_row : self.h0D0SqRow = (sec7MStageCandidateActionEnd K) (sec7MStageCandidateWildRowManuscriptH0ConjX0G0Factor src * sec7MStageCandidateWildRowManuscriptH0X0Factor src * sec7MStageCandidateWildRowManuscriptH0DgFactor src * sec7MStageCandidateWildRowManuscriptH0D0Factor src) * ((sec7MStageCandidateActionEnd K) (sec7MStageCandidateWildRowManuscriptH0D0SqFactor src) - 1)

  Fox row for the d₀² factor.

• h0_hc_row : self.h0HcRow = (sec7MStageCandidateActionEnd K) (sec7MStageCandidateWildRowManuscriptH0ConjX0G0Factor src * sec7MStageCandidateWildRowManuscriptH0X0Factor src * sec7MStageCandidateWildRowManuscriptH0DgFactor src * sec7MStageCandidateWildRowManuscriptH0D0Factor src * sec7MStageCandidateWildRowManuscriptH0D0SqFactor src) * ((sec7MStageCandidateActionEnd K) (sec7MStageCandidateWildRowManuscriptH0HcFactor src) - 1)

  Fox row for the final h_c factor.

• h0_expanded_rows_sum_zero : self.h0ConjX0G0Row + self.h0X0Row + self.h0DgRow + self.h0D0Row + self.h0D0SqRow + self.h0HcRow = 0

  The six h₀ rows cancel, which is the literal first-order content of lem:h0shadow in the elementary row.

structure Q2Presentation.Induction.Sec7FrattiniTrivialElementaryMStageCandidateWildRowManuscriptU1InvOmegaNormRowsData {p : FramedPair} {h : ¬ScalarTerminal p} {chief : FirstNonScalarChiefFactor p} {K : KernelMinimal chief.toNonScalarChiefFactor} {hR : Nat.card ↥(kernelFrattini K) = 1} {child : Sec7FrattiniTrivialElementaryChildData p} (src : Sec7FrattiniTrivialElementaryMStageCandidateLowerSourceData h chief K hR child) : Type

Row-level finite-word data for the inverse ω₂-norm factor u₁⁻¹.

• u1TauRow : Module.End (ZMod 2) (towerM K)

  The τ-variation piece of the ω₂-norm row.

• u1X1Row : Module.End (ZMod 2) (towerM K)

  The x₁-variation piece of the ω₂-norm row.

• u1_tau_row : self.u1TauRow = src.P * (src.T - 1)

  The τ piece is the manuscript operator P(T-1).

• u1_x1_row : self.u1X1Row = src.P * (src.X₁ - 1)

  The x₁ piece is the manuscript operator P(X₁-1).

• u1_inv_omega_norm_rows : -((sec7MStageCandidateActionEnd K) (sec7MStageCandidateWildRowManuscriptU1InvFactor src) * ((sec7MStageCandidateActionEnd K) (sec7MStageCandidateWildRowManuscriptU1Factor src) - 1)) = self.u1TauRow + self.u1X1Row

  The inverse-factor rule transports the non-inverted ω₂-norm Fox row to u₁⁻¹.

structure Q2Presentation.Induction.Sec7FrattiniTrivialElementaryMStageCandidateWildRowManuscriptX1ConjInternalRowsData {p : FramedPair} {h : ¬ScalarTerminal p} {chief : FirstNonScalarChiefFactor p} {K : KernelMinimal chief.toNonScalarChiefFactor} {hR : Nat.card ↥(kernelFrattini K) = 1} {child : Sec7FrattiniTrivialElementaryChildData p} (src : Sec7FrattiniTrivialElementaryMStageCandidateLowerSourceData h chief K hR child) : Type

Row-level finite-word data for the conjugate x₁^σ=σ⁻¹x₁σ.

• x1SigmaInvRow : Module.End (ZMod 2) (towerM K)

  The row from the internal factor σ⁻¹.

• x1BareRow : Module.End (ZMod 2) (towerM K)

  The row from the internal factor x₁.

• x1SigmaRow : Module.End (ZMod 2) (towerM K)

  The row from the internal factor σ.

• x1_sigmaInv_row : self.x1SigmaInvRow = (sec7MStageCandidateActionEnd K) (sec7MStageCandidateWildRowManuscriptH0Factor src * sec7MStageCandidateWildRowManuscriptU1InvFactor src) * ((sec7MStageCandidateActionEnd K) (sec7MStageCandidateWildRowManuscriptSigmaInvFactor src) - 1)

  Fox row for the leading σ⁻¹ in the conjugate.

• x1_bare_row : self.x1BareRow = (sec7MStageCandidateActionEnd K) (sec7MStageCandidateWildRowManuscriptH0Factor src * sec7MStageCandidateWildRowManuscriptU1InvFactor src * sec7MStageCandidateWildRowManuscriptSigmaInvFactor src) * ((sec7MStageCandidateActionEnd K) (sec7MStageCandidateWildRowManuscriptX1BareFactor src) - 1)

  Fox row for the middle x₁, including the h₀u₁⁻¹σ⁻¹ prefix.

• x1_sigma_row : self.x1SigmaRow = (sec7MStageCandidateActionEnd K) (sec7MStageCandidateWildRowManuscriptH0Factor src * sec7MStageCandidateWildRowManuscriptU1InvFactor src * sec7MStageCandidateWildRowManuscriptSigmaInvFactor src * sec7MStageCandidateWildRowManuscriptX1BareFactor src) * ((sec7MStageCandidateActionEnd K) (sec7MStageCandidateWildRowManuscriptSigmaFactor src) - 1)

  Fox row for the closing σ.

• x1_conj_rows_sum : self.x1SigmaInvRow + self.x1BareRow + self.x1SigmaRow = src.Sinv * (src.X₁ - 1)

  The three conjugation rows collapse to the manuscript S⁻¹-twisted x₁ row.

structure Q2Presentation.Induction.Sec7FrattiniTrivialElementaryMStageCandidateWildRowManuscriptC0CommutatorRowsData {p : FramedPair} {h : ¬ScalarTerminal p} {chief : FirstNonScalarChiefFactor p} {K : KernelMinimal chief.toNonScalarChiefFactor} {hR : Nat.card ↥(kernelFrattini K) = 1} {child : Sec7FrattiniTrivialElementaryChildData p} (src : Sec7FrattiniTrivialElementaryMStageCandidateLowerSourceData h chief K hR child) : Type

Row-level finite-word data for the commutator c₀=[d₀,z₀].

• c0D0InvRow : Module.End (ZMod 2) (towerM K)

  The row from the factor d₀⁻¹.

• c0Z0InvRow : Module.End (ZMod 2) (towerM K)

  The row from the factor z₀⁻¹.

• c0D0Row : Module.End (ZMod 2) (towerM K)

  The row from the factor d₀.

• c0Z0Row : Module.End (ZMod 2) (towerM K)

  The row from the factor z₀.

• c0_d0Inv_row : self.c0D0InvRow = (sec7MStageCandidateActionEnd K) (sec7MStageCandidateWildRowManuscriptH0Factor src * sec7MStageCandidateWildRowManuscriptU1InvFactor src * sec7MStageCandidateWildRowManuscriptX1ConjFactor src) * ((sec7MStageCandidateActionEnd K) (sec7MStageCandidateWildRowManuscriptD0InvFactor src) - 1)

  Fox row for d₀⁻¹, with the external prefix h₀u₁⁻¹x₁^σ.

• c0_z0Inv_row : self.c0Z0InvRow = (sec7MStageCandidateActionEnd K) (sec7MStageCandidateWildRowManuscriptH0Factor src * sec7MStageCandidateWildRowManuscriptU1InvFactor src * sec7MStageCandidateWildRowManuscriptX1ConjFactor src * sec7MStageCandidateWildRowManuscriptD0InvFactor src) * ((sec7MStageCandidateActionEnd K) (sec7MStageCandidateWildRowManuscriptZ0InvFactor src) - 1)

  Fox row for z₀⁻¹.

• c0_d0_row : self.c0D0Row = (sec7MStageCandidateActionEnd K) (sec7MStageCandidateWildRowManuscriptH0Factor src * sec7MStageCandidateWildRowManuscriptU1InvFactor src * sec7MStageCandidateWildRowManuscriptX1ConjFactor src * sec7MStageCandidateWildRowManuscriptD0InvFactor src * sec7MStageCandidateWildRowManuscriptZ0InvFactor src) * ((sec7MStageCandidateActionEnd K) (sec7MStageCandidateWildRowManuscriptD0Factor src) - 1)

  Fox row for d₀.

• c0_z0_row : self.c0Z0Row = (sec7MStageCandidateActionEnd K) (sec7MStageCandidateWildRowManuscriptH0Factor src * sec7MStageCandidateWildRowManuscriptU1InvFactor src * sec7MStageCandidateWildRowManuscriptX1ConjFactor src * sec7MStageCandidateWildRowManuscriptD0InvFactor src * sec7MStageCandidateWildRowManuscriptZ0InvFactor src * sec7MStageCandidateWildRowManuscriptD0Factor src) * ((sec7MStageCandidateActionEnd K) (sec7MStageCandidateWildRowManuscriptZ0Factor src) - 1)

  Fox row for z₀.

• c0_commutator_rows_sum_zero : self.c0D0InvRow + self.c0Z0InvRow + self.c0D0Row + self.c0Z0Row = 0

  The four commutator rows cancel, expressing the zero first-order row of [d₀,z₀].

Checkpoint266: row-data leaves lowered to canonical row-sum leaves

def Q2Presentation.Induction.sec7MStageCandidateWildRowManuscriptH0ConjX0G0CanonicalRow {p : FramedPair} {h : ¬ScalarTerminal p} {chief : FirstNonScalarChiefFactor p} {K : KernelMinimal chief.toNonScalarChiefFactor} {hR : Nat.card ↥(kernelFrattini K) = 1} {child : Sec7FrattiniTrivialElementaryChildData p} (src : Sec7FrattiniTrivialElementaryMStageCandidateLowerSourceData h chief K hR child) : Module.End (ZMod 2) (towerM K)

Canonical row from the first internal h₀ factor x₀^{g₀}.

Equations

• One or more equations did not get rendered due to their size.

def Q2Presentation.Induction.sec7MStageCandidateWildRowManuscriptH0X0CanonicalRow {p : FramedPair} {h : ¬ScalarTerminal p} {chief : FirstNonScalarChiefFactor p} {K : KernelMinimal chief.toNonScalarChiefFactor} {hR : Nat.card ↥(kernelFrattini K) = 1} {child : Sec7FrattiniTrivialElementaryChildData p} (src : Sec7FrattiniTrivialElementaryMStageCandidateLowerSourceData h chief K hR child) : Module.End (ZMod 2) (towerM K)

Canonical row from the second internal h₀ factor x₀.

Equations

• One or more equations did not get rendered due to their size.

def Q2Presentation.Induction.sec7MStageCandidateWildRowManuscriptH0DgCanonicalRow {p : FramedPair} {h : ¬ScalarTerminal p} {chief : FirstNonScalarChiefFactor p} {K : KernelMinimal chief.toNonScalarChiefFactor} {hR : Nat.card ↥(kernelFrattini K) = 1} {child : Sec7FrattiniTrivialElementaryChildData p} (src : Sec7FrattiniTrivialElementaryMStageCandidateLowerSourceData h chief K hR child) : Module.End (ZMod 2) (towerM K)

Canonical row from the internal h₀ factor d_g.

Equations

• One or more equations did not get rendered due to their size.

def Q2Presentation.Induction.sec7MStageCandidateWildRowManuscriptH0D0CanonicalRow {p : FramedPair} {h : ¬ScalarTerminal p} {chief : FirstNonScalarChiefFactor p} {K : KernelMinimal chief.toNonScalarChiefFactor} {hR : Nat.card ↥(kernelFrattini K) = 1} {child : Sec7FrattiniTrivialElementaryChildData p} (src : Sec7FrattiniTrivialElementaryMStageCandidateLowerSourceData h chief K hR child) : Module.End (ZMod 2) (towerM K)

Canonical row from the internal h₀ factor d₀.

Equations

• One or more equations did not get rendered due to their size.

def Q2Presentation.Induction.sec7MStageCandidateWildRowManuscriptH0D0SqCanonicalRow {p : FramedPair} {h : ¬ScalarTerminal p} {chief : FirstNonScalarChiefFactor p} {K : KernelMinimal chief.toNonScalarChiefFactor} {hR : Nat.card ↥(kernelFrattini K) = 1} {child : Sec7FrattiniTrivialElementaryChildData p} (src : Sec7FrattiniTrivialElementaryMStageCandidateLowerSourceData h chief K hR child) : Module.End (ZMod 2) (towerM K)

Canonical row from the internal h₀ factor d₀².

Equations

• One or more equations did not get rendered due to their size.

def Q2Presentation.Induction.sec7MStageCandidateWildRowManuscriptH0HcCanonicalRow {p : FramedPair} {h : ¬ScalarTerminal p} {chief : FirstNonScalarChiefFactor p} {K : KernelMinimal chief.toNonScalarChiefFactor} {hR : Nat.card ↥(kernelFrattini K) = 1} {child : Sec7FrattiniTrivialElementaryChildData p} (src : Sec7FrattiniTrivialElementaryMStageCandidateLowerSourceData h chief K hR child) : Module.End (ZMod 2) (towerM K)

Canonical row from the final internal h₀ factor h_c.

Equations

• One or more equations did not get rendered due to their size.

def Q2Presentation.Induction.sec7MStageCandidateWildRowManuscriptU1TauCanonicalRow {p : FramedPair} {h : ¬ScalarTerminal p} {chief : FirstNonScalarChiefFactor p} {K : KernelMinimal chief.toNonScalarChiefFactor} {hR : Nat.card ↥(kernelFrattini K) = 1} {child : Sec7FrattiniTrivialElementaryChildData p} (src : Sec7FrattiniTrivialElementaryMStageCandidateLowerSourceData h chief K hR child) : Module.End (ZMod 2) (towerM K)

Canonical τ-piece in the inverse ω₂-norm row for u₁⁻¹.

Equations

• Q2Presentation.Induction.sec7MStageCandidateWildRowManuscriptU1TauCanonicalRow src = src.P * (src.T - 1)

def Q2Presentation.Induction.sec7MStageCandidateWildRowManuscriptU1X1CanonicalRow {p : FramedPair} {h : ¬ScalarTerminal p} {chief : FirstNonScalarChiefFactor p} {K : KernelMinimal chief.toNonScalarChiefFactor} {hR : Nat.card ↥(kernelFrattini K) = 1} {child : Sec7FrattiniTrivialElementaryChildData p} (src : Sec7FrattiniTrivialElementaryMStageCandidateLowerSourceData h chief K hR child) : Module.End (ZMod 2) (towerM K)

Canonical x₁-piece in the inverse ω₂-norm row for u₁⁻¹.

Equations

• Q2Presentation.Induction.sec7MStageCandidateWildRowManuscriptU1X1CanonicalRow src = src.P * (src.X₁ - 1)

def Q2Presentation.Induction.sec7MStageCandidateWildRowManuscriptX1SigmaInvCanonicalRow {p : FramedPair} {h : ¬ScalarTerminal p} {chief : FirstNonScalarChiefFactor p} {K : KernelMinimal chief.toNonScalarChiefFactor} {hR : Nat.card ↥(kernelFrattini K) = 1} {child : Sec7FrattiniTrivialElementaryChildData p} (src : Sec7FrattiniTrivialElementaryMStageCandidateLowerSourceData h chief K hR child) : Module.End (ZMod 2) (towerM K)

Canonical row from the leading σ⁻¹ in x₁^σ.

Equations

• One or more equations did not get rendered due to their size.

def Q2Presentation.Induction.sec7MStageCandidateWildRowManuscriptX1BareCanonicalRow {p : FramedPair} {h : ¬ScalarTerminal p} {chief : FirstNonScalarChiefFactor p} {K : KernelMinimal chief.toNonScalarChiefFactor} {hR : Nat.card ↥(kernelFrattini K) = 1} {child : Sec7FrattiniTrivialElementaryChildData p} (src : Sec7FrattiniTrivialElementaryMStageCandidateLowerSourceData h chief K hR child) : Module.End (ZMod 2) (towerM K)

Canonical row from the middle x₁ in x₁^σ.

Equations

• One or more equations did not get rendered due to their size.

def Q2Presentation.Induction.sec7MStageCandidateWildRowManuscriptX1SigmaCanonicalRow {p : FramedPair} {h : ¬ScalarTerminal p} {chief : FirstNonScalarChiefFactor p} {K : KernelMinimal chief.toNonScalarChiefFactor} {hR : Nat.card ↥(kernelFrattini K) = 1} {child : Sec7FrattiniTrivialElementaryChildData p} (src : Sec7FrattiniTrivialElementaryMStageCandidateLowerSourceData h chief K hR child) : Module.End (ZMod 2) (towerM K)

Canonical row from the closing σ in x₁^σ.

Equations

• One or more equations did not get rendered due to their size.

def Q2Presentation.Induction.sec7MStageCandidateWildRowManuscriptC0D0InvCanonicalRow {p : FramedPair} {h : ¬ScalarTerminal p} {chief : FirstNonScalarChiefFactor p} {K : KernelMinimal chief.toNonScalarChiefFactor} {hR : Nat.card ↥(kernelFrattini K) = 1} {child : Sec7FrattiniTrivialElementaryChildData p} (src : Sec7FrattiniTrivialElementaryMStageCandidateLowerSourceData h chief K hR child) : Module.End (ZMod 2) (towerM K)

Canonical row from the factor d₀⁻¹ in c₀=[d₀,z₀].

Equations

• One or more equations did not get rendered due to their size.

def Q2Presentation.Induction.sec7MStageCandidateWildRowManuscriptC0Z0InvCanonicalRow {p : FramedPair} {h : ¬ScalarTerminal p} {chief : FirstNonScalarChiefFactor p} {K : KernelMinimal chief.toNonScalarChiefFactor} {hR : Nat.card ↥(kernelFrattini K) = 1} {child : Sec7FrattiniTrivialElementaryChildData p} (src : Sec7FrattiniTrivialElementaryMStageCandidateLowerSourceData h chief K hR child) : Module.End (ZMod 2) (towerM K)

Canonical row from the factor z₀⁻¹ in c₀=[d₀,z₀].

Equations

• One or more equations did not get rendered due to their size.

def Q2Presentation.Induction.sec7MStageCandidateWildRowManuscriptC0D0CanonicalRow {p : FramedPair} {h : ¬ScalarTerminal p} {chief : FirstNonScalarChiefFactor p} {K : KernelMinimal chief.toNonScalarChiefFactor} {hR : Nat.card ↥(kernelFrattini K) = 1} {child : Sec7FrattiniTrivialElementaryChildData p} (src : Sec7FrattiniTrivialElementaryMStageCandidateLowerSourceData h chief K hR child) : Module.End (ZMod 2) (towerM K)

Canonical row from the factor d₀ in c₀=[d₀,z₀].

Equations

• One or more equations did not get rendered due to their size.

def Q2Presentation.Induction.sec7MStageCandidateWildRowManuscriptC0Z0CanonicalRow {p : FramedPair} {h : ¬ScalarTerminal p} {chief : FirstNonScalarChiefFactor p} {K : KernelMinimal chief.toNonScalarChiefFactor} {hR : Nat.card ↥(kernelFrattini K) = 1} {child : Sec7FrattiniTrivialElementaryChildData p} (src : Sec7FrattiniTrivialElementaryMStageCandidateLowerSourceData h chief K hR child) : Module.End (ZMod 2) (towerM K)

Canonical row from the factor z₀ in c₀=[d₀,z₀].

Equations

• One or more equations did not get rendered due to their size.

Checkpoint269: current action leaves lowered to composable action identities

axiom Q2Presentation.Induction.sec7_frattiniTrivial_elementaryMStageCandidateWildRowManuscriptH0ExpandedPrimitiveTowerActionSixFactorD0ExpandedRawInlineSevenActionTailTripleInverseCancellationHcNestedDgInvD0SplitD0InvOpenedAllProductActionsFullyNestedHeadAtCommonTailOuterX0U0PrefixOuterX0U0ActionRemovedU0InvAppliedX0AppliedX0AppliedU0InvTailX0InvCorePreimageDifferenceZeroLowerPrimitiveApplyPointwiseData {p : FramedPair} {h : ¬ScalarTerminal p} {chief : FirstNonScalarChiefFactor p} {K : KernelMinimal chief.toNonScalarChiefFactor} {hR : Nat.card ↥(kernelFrattini K) = 1} {child : Sec7FrattiniTrivialElementaryChildData p} (src : Sec7FrattiniTrivialElementaryMStageCandidateLowerSourceData h chief K hR child) (m : towerM K) : let sigma2 := powOmega2 src.sigma; let g := sigma2 ^ 2; let u0 := powOmega2 (src.x₀ * src.tau); let tail := ((towerActM K) g⁻¹) (((towerActM K) u0) (((towerActM K) src.x₀⁻¹) (((towerActM K) g) (((towerActM K) u0) (((towerActM K) src.x₀⁻¹) m))))); let rhs := ((towerActM K) src.x₀) (((towerActM K) u0⁻¹) (((towerActM K) src.x₀) (((towerActM K) u0⁻¹) (((towerActM K) src.x₀) (((towerActM K) u0⁻¹) tail))))); let core := ((towerActM K) g⁻¹) (((towerActM K) u0) (((towerActM K) src.x₀⁻¹) (((towerActM K) g) rhs))); ((towerActM K) u0⁻¹) tail - ((towerActM K) src.x₀⁻¹) core = 0

Checkpoint332 lower primitive for the h₀ head: keep only the additive zero of the inverse-preimage row. The checkpoint331 adjacent u₀⁻¹*u₀ cancellation target is recovered theorem-level by applying u₀ and cancelling the displayed inverse pair.

theorem Q2Presentation.Induction.sec7_frattiniTrivial_elementaryMStageCandidateWildRowManuscriptH0ExpandedPrimitiveTowerActionSixFactorD0ExpandedRawInlineSevenActionTailTripleInverseCancellationHcNestedDgInvD0SplitD0InvOpenedAllProductActionsFullyNestedHeadAtCommonTailOuterX0U0PrefixOuterX0U0ActionRemovedU0InvAppliedX0AppliedX0AppliedU0InvTailU0ReappliedX0InvCorePrimitiveApplyPointwiseData {p : FramedPair} {h : ¬ScalarTerminal p} {chief : FirstNonScalarChiefFactor p} {K : KernelMinimal chief.toNonScalarChiefFactor} {hR : Nat.card ↥(kernelFrattini K) = 1} {child : Sec7FrattiniTrivialElementaryChildData p} (src : Sec7FrattiniTrivialElementaryMStageCandidateLowerSourceData h chief K hR child) (m : towerM K) : have sigma2 := powOmega2 src.sigma; have g := sigma2 ^ 2; have u0 := powOmega2 (src.x₀ * src.tau); have tail := ((towerActM K) g⁻¹) (((towerActM K) u0) (((towerActM K) src.x₀⁻¹) (((towerActM K) g) (((towerActM K) u0) (((towerActM K) src.x₀⁻¹) m))))); have rhs := ((towerActM K) src.x₀) (((towerActM K) u0⁻¹) (((towerActM K) src.x₀) (((towerActM K) u0⁻¹) (((towerActM K) src.x₀) (((towerActM K) u0⁻¹) tail))))); have core := ((towerActM K) g⁻¹) (((towerActM K) u0) (((towerActM K) src.x₀⁻¹) (((towerActM K) g) rhs))); tail = ((towerActM K) u0) (((towerActM K) src.x₀⁻¹) core)

Checkpoint332 bridge recovering the checkpoint331 h₀ head adjacent cancellation row from the lower inverse-preimage difference-zero primitive.

theorem Q2Presentation.Induction.sec7_frattiniTrivial_elementaryMStageCandidateWildRowManuscriptH0ExpandedPrimitiveTowerActionSixFactorD0ExpandedRawInlineSevenActionTailTripleInverseCancellationHcNestedDgInvD0SplitD0InvOpenedAllProductActionsFullyNestedHeadAtCommonTailOuterX0U0PrefixOuterX0U0ActionRemovedU0InvAppliedX0AppliedX0AppliedU0InvTailX0InvCorePreimageDifferenceZeroPrimitiveApplyPointwiseData {p : FramedPair} {h : ¬ScalarTerminal p} {chief : FirstNonScalarChiefFactor p} {K : KernelMinimal chief.toNonScalarChiefFactor} {hR : Nat.card ↥(kernelFrattini K) = 1} {child : Sec7FrattiniTrivialElementaryChildData p} (src : Sec7FrattiniTrivialElementaryMStageCandidateLowerSourceData h chief K hR child) (m : towerM K) : have sigma2 := powOmega2 src.sigma; have g := sigma2 ^ 2; have u0 := powOmega2 (src.x₀ * src.tau); have tail := ((towerActM K) g⁻¹) (((towerActM K) u0) (((towerActM K) src.x₀⁻¹) (((towerActM K) g) (((towerActM K) u0) (((towerActM K) src.x₀⁻¹) m))))); have rhs := ((towerActM K) src.x₀) (((towerActM K) u0⁻¹) (((towerActM K) src.x₀) (((towerActM K) u0⁻¹) (((towerActM K) src.x₀) (((towerActM K) u0⁻¹) tail))))); have core := ((towerActM K) g⁻¹) (((towerActM K) u0) (((towerActM K) src.x₀⁻¹) (((towerActM K) g) rhs))); ((towerActM K) u0⁻¹) tail - ((towerActM K) src.x₀⁻¹) core = 0

Checkpoint331 bridge closing the checkpoint330 h₀ head difference-zero residual from the lower adjacent-cancellation row.

theorem Q2Presentation.Induction.sec7_frattiniTrivial_elementaryMStageCandidateWildRowManuscriptH0ExpandedPrimitiveTowerActionSixFactorD0ExpandedRawInlineSevenActionTailTripleInverseCancellationHcNestedDgInvD0SplitD0InvOpenedAllProductActionsFullyNestedHeadAtCommonTailOuterX0U0PrefixOuterX0U0ActionRemovedU0InvAppliedX0AppliedX0AppliedU0InvTailX0InvCorePreimagePrimitiveApplyPointwiseData {p : FramedPair} {h : ¬ScalarTerminal p} {chief : FirstNonScalarChiefFactor p} {K : KernelMinimal chief.toNonScalarChiefFactor} {hR : Nat.card ↥(kernelFrattini K) = 1} {child : Sec7FrattiniTrivialElementaryChildData p} (src : Sec7FrattiniTrivialElementaryMStageCandidateLowerSourceData h chief K hR child) (m : towerM K) : have sigma2 := powOmega2 src.sigma; have g := sigma2 ^ 2; have u0 := powOmega2 (src.x₀ * src.tau); have tail := ((towerActM K) g⁻¹) (((towerActM K) u0) (((towerActM K) src.x₀⁻¹) (((towerActM K) g) (((towerActM K) u0) (((towerActM K) src.x₀⁻¹) m))))); have rhs := ((towerActM K) src.x₀) (((towerActM K) u0⁻¹) (((towerActM K) src.x₀) (((towerActM K) u0⁻¹) (((towerActM K) src.x₀) (((towerActM K) u0⁻¹) tail))))); have core := ((towerActM K) g⁻¹) (((towerActM K) u0) (((towerActM K) src.x₀⁻¹) (((towerActM K) g) rhs))); ((towerActM K) u0⁻¹) tail = ((towerActM K) src.x₀⁻¹) core

Checkpoint330 bridge for the checkpoint329 h₀ head primitive preimage row.

theorem Q2Presentation.Induction.sec7_frattiniTrivial_elementaryMStageCandidateWildRowManuscriptH0ExpandedPrimitiveTowerActionSixFactorD0ExpandedRawInlineSevenActionTailTripleInverseCancellationHcNestedDgInvD0SplitD0InvOpenedAllProductActionsFullyNestedHeadAtCommonTailOuterX0U0PrefixOuterX0U0ActionRemovedU0InvAppliedX0AppliedX0AppliedU0InvTailCoreOpenedPrimitiveApplyPointwiseData {p : FramedPair} {h : ¬ScalarTerminal p} {chief : FirstNonScalarChiefFactor p} {K : KernelMinimal chief.toNonScalarChiefFactor} {hR : Nat.card ↥(kernelFrattini K) = 1} {child : Sec7FrattiniTrivialElementaryChildData p} (src : Sec7FrattiniTrivialElementaryMStageCandidateLowerSourceData h chief K hR child) (m : towerM K) : have sigma2 := powOmega2 src.sigma; have g := sigma2 ^ 2; have u0 := powOmega2 (src.x₀ * src.tau); have tail := ((towerActM K) g⁻¹) (((towerActM K) u0) (((towerActM K) src.x₀⁻¹) (((towerActM K) g) (((towerActM K) u0) (((towerActM K) src.x₀⁻¹) m))))); have rhs := ((towerActM K) src.x₀) (((towerActM K) u0⁻¹) (((towerActM K) src.x₀) (((towerActM K) u0⁻¹) (((towerActM K) src.x₀) (((towerActM K) u0⁻¹) tail))))); have core := ((towerActM K) g⁻¹) (((towerActM K) u0) (((towerActM K) src.x₀⁻¹) (((towerActM K) g) rhs))); ((towerActM K) src.x₀) (((towerActM K) u0⁻¹) tail) = core

Checkpoint329 bridge for the checkpoint328 h₀ head atom: reapply x₀ to the inverse-preimage equality and cancel x₀*x₀⁻¹.

theorem Q2Presentation.Induction.sec7_frattiniTrivial_elementaryMStageCandidateWildRowManuscriptH0ExpandedPrimitiveTowerActionSixFactorD0ExpandedRawInlineSevenActionTailTripleInverseCancellationHcNestedDgInvD0SplitD0InvOpenedAllProductActionsFullyNestedHeadAtCommonTailOuterX0U0PrefixOuterX0U0ActionRemovedU0InvAppliedX0AppliedX0AppliedU0InvTailOuterGAppliedCoreOpenedDirectApplyPointwiseData {p : FramedPair} {h : ¬ScalarTerminal p} {chief : FirstNonScalarChiefFactor p} {K : KernelMinimal chief.toNonScalarChiefFactor} {hR : Nat.card ↥(kernelFrattini K) = 1} {child : Sec7FrattiniTrivialElementaryChildData p} (src : Sec7FrattiniTrivialElementaryMStageCandidateLowerSourceData h chief K hR child) (m : towerM K) : have sigma2 := powOmega2 src.sigma; have g := sigma2 ^ 2; have u0 := powOmega2 (src.x₀ * src.tau); have tail := ((towerActM K) g⁻¹) (((towerActM K) u0) (((towerActM K) src.x₀⁻¹) (((towerActM K) g) (((towerActM K) u0) (((towerActM K) src.x₀⁻¹) m))))); have rhs := ((towerActM K) src.x₀) (((towerActM K) u0⁻¹) (((towerActM K) src.x₀) (((towerActM K) u0⁻¹) (((towerActM K) src.x₀) (((towerActM K) u0⁻¹) tail))))); have core := ((towerActM K) g⁻¹) (((towerActM K) u0) (((towerActM K) src.x₀⁻¹) (((towerActM K) g) rhs))); ((towerActM K) g) (((towerActM K) src.x₀) (((towerActM K) u0⁻¹) tail)) = ((towerActM K) u0) (((towerActM K) src.x₀⁻¹) (((towerActM K) g) rhs))

Checkpoint328 bridge for the checkpoint327 h₀ head atom: the lower datum states the core equality before reapplying the visible outer g action. Apply g and cancel g*g⁻¹ in the opened core.

theorem Q2Presentation.Induction.sec7_frattiniTrivial_elementaryMStageCandidateWildRowManuscriptH0ExpandedPrimitiveTowerActionSixFactorD0ExpandedRawInlineSevenActionTailTripleInverseCancellationHcNestedDgInvD0SplitD0InvOpenedAllProductActionsFullyNestedHeadAtCommonTailOuterX0U0PrefixOuterX0U0ActionRemovedU0InvAppliedX0AppliedX0AppliedU0InvTailCoreOpenedDirectApplyPointwiseData {p : FramedPair} {h : ¬ScalarTerminal p} {chief : FirstNonScalarChiefFactor p} {K : KernelMinimal chief.toNonScalarChiefFactor} {hR : Nat.card ↥(kernelFrattini K) = 1} {child : Sec7FrattiniTrivialElementaryChildData p} (src : Sec7FrattiniTrivialElementaryMStageCandidateLowerSourceData h chief K hR child) (m : towerM K) : have sigma2 := powOmega2 src.sigma; have g := sigma2 ^ 2; have u0 := powOmega2 (src.x₀ * src.tau); have tail := ((towerActM K) g⁻¹) (((towerActM K) u0) (((towerActM K) src.x₀⁻¹) (((towerActM K) g) (((towerActM K) u0) (((towerActM K) src.x₀⁻¹) m))))); have rhs := ((towerActM K) src.x₀) (((towerActM K) u0⁻¹) (((towerActM K) src.x₀) (((towerActM K) u0⁻¹) (((towerActM K) src.x₀) (((towerActM K) u0⁻¹) tail))))); have core := ((towerActM K) g⁻¹) (((towerActM K) u0) (((towerActM K) src.x₀⁻¹) (((towerActM K) g) rhs))); ((towerActM K) src.x₀) (((towerActM K) u0⁻¹) tail) = core

Checkpoint327 bridge for the checkpoint326 h₀ head atom: the new datum states the same core row before applying the visible outer g⁻¹ action. Apply g⁻¹ and cancel g⁻¹*g on the left.

theorem Q2Presentation.Induction.sec7_frattiniTrivial_elementaryMStageCandidateWildRowManuscriptH0ExpandedPrimitiveTowerActionSixFactorD0ExpandedRawInlineSevenActionTailTripleInverseCancellationHcNestedDgInvD0SplitD0InvOpenedAllProductActionsFullyNestedHeadAtCommonTailOuterX0U0PrefixOuterX0U0ActionRemovedU0InvAppliedX0AppliedU0InvTailX0InvCorePreimageDirectApplyPointwiseData {p : FramedPair} {h : ¬ScalarTerminal p} {chief : FirstNonScalarChiefFactor p} {K : KernelMinimal chief.toNonScalarChiefFactor} {hR : Nat.card ↥(kernelFrattini K) = 1} {child : Sec7FrattiniTrivialElementaryChildData p} (src : Sec7FrattiniTrivialElementaryMStageCandidateLowerSourceData h chief K hR child) (m : towerM K) : have sigma2 := powOmega2 src.sigma; have g := sigma2 ^ 2; have u0 := powOmega2 (src.x₀ * src.tau); have tail := ((towerActM K) g⁻¹) (((towerActM K) u0) (((towerActM K) src.x₀⁻¹) (((towerActM K) g) (((towerActM K) u0) (((towerActM K) src.x₀⁻¹) m))))); have rhs := ((towerActM K) src.x₀) (((towerActM K) u0⁻¹) (((towerActM K) src.x₀) (((towerActM K) u0⁻¹) (((towerActM K) src.x₀) (((towerActM K) u0⁻¹) tail))))); have core := ((towerActM K) g⁻¹) (((towerActM K) u0) (((towerActM K) src.x₀⁻¹) (((towerActM K) g) rhs))); ((towerActM K) u0⁻¹) tail = ((towerActM K) src.x₀⁻¹) core

Checkpoint326 bridge for the checkpoint325 h₀ head atom: apply x₀⁻¹ to the lower core-opened equality and cancel the visible inverse pair.

theorem Q2Presentation.Induction.sec7_frattiniTrivial_elementaryMStageCandidateWildRowManuscriptH0ExpandedPrimitiveTowerActionSixFactorD0ExpandedRawInlineSevenActionTailTripleInverseCancellationHcNestedDgInvD0SplitD0InvOpenedAllProductActionsFullyNestedHeadAtCommonTailOuterX0U0PrefixOuterX0U0ActionRemovedU0InvAppliedX0AppliedTailPreimageCoreDirectApplyPointwiseData {p : FramedPair} {h : ¬ScalarTerminal p} {chief : FirstNonScalarChiefFactor p} {K : KernelMinimal chief.toNonScalarChiefFactor} {hR : Nat.card ↥(kernelFrattini K) = 1} {child : Sec7FrattiniTrivialElementaryChildData p} (src : Sec7FrattiniTrivialElementaryMStageCandidateLowerSourceData h chief K hR child) (m : towerM K) : have sigma2 := powOmega2 src.sigma; have g := sigma2 ^ 2; have u0 := powOmega2 (src.x₀ * src.tau); have tail := ((towerActM K) g⁻¹) (((towerActM K) u0) (((towerActM K) src.x₀⁻¹) (((towerActM K) g) (((towerActM K) u0) (((towerActM K) src.x₀⁻¹) m))))); have rhs := ((towerActM K) src.x₀) (((towerActM K) u0⁻¹) (((towerActM K) src.x₀) (((towerActM K) u0⁻¹) (((towerActM K) src.x₀) (((towerActM K) u0⁻¹) tail))))); have core := ((towerActM K) g⁻¹) (((towerActM K) u0) (((towerActM K) src.x₀⁻¹) (((towerActM K) g) rhs))); tail = ((towerActM K) u0) (((towerActM K) src.x₀⁻¹) core)

Checkpoint325 bridge for the h₀ head: reapply u₀ to the lower preimage row and cancel the adjacent u₀*u₀⁻¹ pair.

theorem Q2Presentation.Induction.sec7_frattiniTrivial_elementaryMStageCandidateWildRowManuscriptH0ExpandedPrimitiveTowerActionSixFactorD0ExpandedRawInlineSevenActionTailTripleInverseCancellationHcNestedDgInvD0SplitD0InvOpenedAllProductActionsFullyNestedHeadAtCommonTailOuterX0U0PrefixOuterX0U0ActionRemovedU0InvAppliedX0AppliedStrippedPreimageCoreDirectApplyPointwiseData {p : FramedPair} {h : ¬ScalarTerminal p} {chief : FirstNonScalarChiefFactor p} {K : KernelMinimal chief.toNonScalarChiefFactor} {hR : Nat.card ↥(kernelFrattini K) = 1} {child : Sec7FrattiniTrivialElementaryChildData p} (src : Sec7FrattiniTrivialElementaryMStageCandidateLowerSourceData h chief K hR child) (m : towerM K) : have sigma2 := powOmega2 src.sigma; have g := sigma2 ^ 2; have u0 := powOmega2 (src.x₀ * src.tau); have tail := ((towerActM K) g⁻¹) (((towerActM K) u0) (((towerActM K) src.x₀⁻¹) (((towerActM K) g) (((towerActM K) u0) (((towerActM K) src.x₀⁻¹) m))))); have rhs := ((towerActM K) src.x₀) (((towerActM K) u0⁻¹) (((towerActM K) src.x₀) (((towerActM K) u0⁻¹) (((towerActM K) src.x₀) (((towerActM K) u0⁻¹) tail))))); have core := ((towerActM K) g⁻¹) (((towerActM K) u0) (((towerActM K) src.x₀⁻¹) (((towerActM K) g) rhs))); ((towerActM K) src.x₀) (((towerActM K) u0⁻¹) tail) = core

Checkpoint324 bridge for the h₀ head: reapply the visible x₀ · u₀⁻¹ prefix to the new preimage-core row.

theorem Q2Presentation.Induction.sec7_frattiniTrivial_elementaryMStageCandidateWildRowManuscriptH0ExpandedPrimitiveTowerActionSixFactorD0ExpandedRawInlineSevenActionTailTripleInverseCancellationHcNestedDgInvD0SplitD0InvOpenedAllProductActionsFullyNestedHeadAtCommonTailOuterX0U0PrefixOuterX0U0ActionRemovedU0InvAppliedX0AppliedSigma2Sigma2GCoreOpenedApplyPointwiseData {p : FramedPair} {h : ¬ScalarTerminal p} {chief : FirstNonScalarChiefFactor p} {K : KernelMinimal chief.toNonScalarChiefFactor} {hR : Nat.card ↥(kernelFrattini K) = 1} {child : Sec7FrattiniTrivialElementaryChildData p} (src : Sec7FrattiniTrivialElementaryMStageCandidateLowerSourceData h chief K hR child) (m : towerM K) : have sigma2 := powOmega2 src.sigma; have g := sigma2 ^ 2; have u0 := powOmega2 (src.x₀ * src.tau); have tail := ((towerActM K) g⁻¹) (((towerActM K) u0) (((towerActM K) src.x₀⁻¹) (((towerActM K) g) (((towerActM K) u0) (((towerActM K) src.x₀⁻¹) m))))); have rhs := ((towerActM K) src.x₀) (((towerActM K) u0⁻¹) (((towerActM K) src.x₀) (((towerActM K) u0⁻¹) (((towerActM K) src.x₀) (((towerActM K) u0⁻¹) tail))))); ((towerActM K) sigma2) (((towerActM K) sigma2) (((towerActM K) src.x₀) (((towerActM K) u0⁻¹) tail))) = ((towerActM K) u0) (((towerActM K) src.x₀⁻¹) (((towerActM K) sigma2) (((towerActM K) sigma2) rhs)))

Checkpoint323 bridge for the h₀ head: the checkpoint322 row with the two opened σ₂ actions follows from the stripped-core row by reapplying g = σ₂².

theorem Q2Presentation.Induction.sec7_frattiniTrivial_elementaryMStageCandidateWildRowManuscriptH0ExpandedPrimitiveTowerActionSixFactorD0ExpandedRawInlineSevenActionTailTripleInverseCancellationHcNestedDgInvD0SplitD0InvOpenedAllProductActionsFullyNestedHeadAtCommonTailOuterX0U0PrefixOuterX0U0ActionRemovedU0InvAppliedX0AppliedGCoreOpenedApplyPointwiseData {p : FramedPair} {h : ¬ScalarTerminal p} {chief : FirstNonScalarChiefFactor p} {K : KernelMinimal chief.toNonScalarChiefFactor} {hR : Nat.card ↥(kernelFrattini K) = 1} {child : Sec7FrattiniTrivialElementaryChildData p} (src : Sec7FrattiniTrivialElementaryMStageCandidateLowerSourceData h chief K hR child) (m : towerM K) : have g := powOmega2 src.sigma ^ 2; have u0 := powOmega2 (src.x₀ * src.tau); have tail := ((towerActM K) g⁻¹) (((towerActM K) u0) (((towerActM K) src.x₀⁻¹) (((towerActM K) g) (((towerActM K) u0) (((towerActM K) src.x₀⁻¹) m))))); have rhs := ((towerActM K) src.x₀) (((towerActM K) u0⁻¹) (((towerActM K) src.x₀) (((towerActM K) u0⁻¹) (((towerActM K) src.x₀) (((towerActM K) u0⁻¹) tail))))); ((towerActM K) g) (((towerActM K) src.x₀) (((towerActM K) u0⁻¹) tail)) = ((towerActM K) u0) (((towerActM K) src.x₀⁻¹) (((towerActM K) g) rhs))

Checkpoint322 bridge for the h₀ head: the checkpoint321 g-core-opened row follows from the lower row in which the two visible g = σ₂² actions have already been opened as successive σ₂ tower actions.

theorem Q2Presentation.Induction.sec7_frattiniTrivial_elementaryMStageCandidateWildRowManuscriptH0ExpandedPrimitiveTowerActionSixFactorD0ExpandedRawInlineSevenActionTailTripleInverseCancellationHcNestedDgInvD0SplitD0InvOpenedAllProductActionsFullyNestedHeadAtCommonTailOuterX0U0PrefixOuterX0U0ActionRemovedU0InvAppliedX0AppliedFullyStrippedPreimageCoreApplyPointwiseData {p : FramedPair} {h : ¬ScalarTerminal p} {chief : FirstNonScalarChiefFactor p} {K : KernelMinimal chief.toNonScalarChiefFactor} {hR : Nat.card ↥(kernelFrattini K) = 1} {child : Sec7FrattiniTrivialElementaryChildData p} (src : Sec7FrattiniTrivialElementaryMStageCandidateLowerSourceData h chief K hR child) (m : towerM K) : have g := powOmega2 src.sigma ^ 2; have u0 := powOmega2 (src.x₀ * src.tau); have tail := ((towerActM K) g⁻¹) (((towerActM K) u0) (((towerActM K) src.x₀⁻¹) (((towerActM K) g) (((towerActM K) u0) (((towerActM K) src.x₀⁻¹) m))))); have rhs := ((towerActM K) src.x₀) (((towerActM K) u0⁻¹) (((towerActM K) src.x₀) (((towerActM K) u0⁻¹) (((towerActM K) src.x₀) (((towerActM K) u0⁻¹) tail))))); have core := ((towerActM K) g⁻¹) (((towerActM K) u0) (((towerActM K) src.x₀⁻¹) (((towerActM K) g) rhs))); ((towerActM K) src.x₀) (((towerActM K) u0⁻¹) tail) = core

Checkpoint320 bridge for the h₀ head: the old fully stripped preimage-core row is recovered from the checkpoint321 row by applying g⁻¹ and cancelling the outer g⁻¹*g action pair.

theorem Q2Presentation.Induction.sec7_frattiniTrivial_elementaryMStageCandidateWildRowManuscriptH0ExpandedPrimitiveTowerActionSixFactorD0ExpandedRawInlineSevenActionTailTripleInverseCancellationHcNestedDgInvD0SplitD0InvOpenedAllProductActionsFullyNestedHeadAtCommonTailOuterX0U0PrefixOuterX0U0ActionRemovedU0InvAppliedFullyStrippedPreimageCoreApplyPointwiseData {p : FramedPair} {h : ¬ScalarTerminal p} {chief : FirstNonScalarChiefFactor p} {K : KernelMinimal chief.toNonScalarChiefFactor} {hR : Nat.card ↥(kernelFrattini K) = 1} {child : Sec7FrattiniTrivialElementaryChildData p} (src : Sec7FrattiniTrivialElementaryMStageCandidateLowerSourceData h chief K hR child) (m : towerM K) : have g := powOmega2 src.sigma ^ 2; have u0 := powOmega2 (src.x₀ * src.tau); have tail := ((towerActM K) g⁻¹) (((towerActM K) u0) (((towerActM K) src.x₀⁻¹) (((towerActM K) g) (((towerActM K) u0) (((towerActM K) src.x₀⁻¹) m))))); have rhs := ((towerActM K) src.x₀) (((towerActM K) u0⁻¹) (((towerActM K) src.x₀) (((towerActM K) u0⁻¹) (((towerActM K) src.x₀) (((towerActM K) u0⁻¹) tail))))); have core := ((towerActM K) g⁻¹) (((towerActM K) u0) (((towerActM K) src.x₀⁻¹) (((towerActM K) g) rhs))); ((towerActM K) u0⁻¹) tail = ((towerActM K) src.x₀⁻¹) core

Checkpoint319 bridge for the h₀ head: the applied-u₀⁻¹ preimage row is recovered from the checkpoint320 row by applying x₀⁻¹ and cancelling the visible x₀⁻¹*x₀ action pair.

theorem Q2Presentation.Induction.sec7_frattiniTrivial_elementaryMStageCandidateWildRowManuscriptH0ExpandedPrimitiveTowerActionSixFactorD0ExpandedRawInlineSevenActionTailTripleInverseCancellationHcNestedDgInvD0SplitD0InvOpenedAllProductActionsFullyNestedHeadAtCommonTailOuterX0U0PrefixOuterX0U0ActionRemovedFullyStrippedPreimageCoreApplyPointwiseData {p : FramedPair} {h : ¬ScalarTerminal p} {chief : FirstNonScalarChiefFactor p} {K : KernelMinimal chief.toNonScalarChiefFactor} {hR : Nat.card ↥(kernelFrattini K) = 1} {child : Sec7FrattiniTrivialElementaryChildData p} (src : Sec7FrattiniTrivialElementaryMStageCandidateLowerSourceData h chief K hR child) (m : towerM K) : have g := powOmega2 src.sigma ^ 2; have u0 := powOmega2 (src.x₀ * src.tau); have tail := ((towerActM K) g⁻¹) (((towerActM K) u0) (((towerActM K) src.x₀⁻¹) (((towerActM K) g) (((towerActM K) u0) (((towerActM K) src.x₀⁻¹) m))))); have rhs := ((towerActM K) src.x₀) (((towerActM K) u0⁻¹) (((towerActM K) src.x₀) (((towerActM K) u0⁻¹) (((towerActM K) src.x₀) (((towerActM K) u0⁻¹) tail))))); have core := ((towerActM K) g⁻¹) (((towerActM K) u0) (((towerActM K) src.x₀⁻¹) (((towerActM K) g) rhs))); tail = ((towerActM K) u0) (((towerActM K) src.x₀⁻¹) core)

theorem Q2Presentation.Induction.sec7_frattiniTrivial_elementaryMStageCandidateWildRowManuscriptH0ExpandedPrimitiveTowerActionSixFactorD0ExpandedRawInlineSevenActionTailTripleInverseCancellationHcNestedDgInvD0SplitD0InvOpenedAllProductActionsFullyNestedHeadAtCommonTailOuterX0U0PrefixOuterX0ActionRemovedFullyStrippedPreimageCoreApplyPointwiseData {p : FramedPair} {h : ¬ScalarTerminal p} {chief : FirstNonScalarChiefFactor p} {K : KernelMinimal chief.toNonScalarChiefFactor} {hR : Nat.card ↥(kernelFrattini K) = 1} {child : Sec7FrattiniTrivialElementaryChildData p} (src : Sec7FrattiniTrivialElementaryMStageCandidateLowerSourceData h chief K hR child) (m : towerM K) : have g := powOmega2 src.sigma ^ 2; have u0 := powOmega2 (src.x₀ * src.tau); have tail := ((towerActM K) g⁻¹) (((towerActM K) u0) (((towerActM K) src.x₀⁻¹) (((towerActM K) g) (((towerActM K) u0) (((towerActM K) src.x₀⁻¹) m))))); have rhs := ((towerActM K) src.x₀) (((towerActM K) u0⁻¹) (((towerActM K) src.x₀) (((towerActM K) u0⁻¹) (((towerActM K) src.x₀) (((towerActM K) u0⁻¹) tail))))); have core := ((towerActM K) g⁻¹) (((towerActM K) u0) (((towerActM K) src.x₀⁻¹) (((towerActM K) g) rhs))); ((towerActM K) u0⁻¹) tail = ((towerActM K) src.x₀⁻¹) core

Checkpoint318 bridge for the h₀ head: the checkpoint317 stripped-preimage row follows from the still smaller row with the remaining u₀⁻¹ action removed from the left hand side.

theorem Q2Presentation.Induction.sec7_frattiniTrivial_elementaryMStageCandidateWildRowManuscriptH0ExpandedPrimitiveTowerActionSixFactorD0ExpandedRawInlineSevenActionTailTripleInverseCancellationHcNestedDgInvD0SplitD0InvOpenedAllProductActionsFullyNestedHeadAtCommonTailOuterX0U0PrefixFullyStrippedPreimageCoreApplyPointwiseData {p : FramedPair} {h : ¬ScalarTerminal p} {chief : FirstNonScalarChiefFactor p} {K : KernelMinimal chief.toNonScalarChiefFactor} {hR : Nat.card ↥(kernelFrattini K) = 1} {child : Sec7FrattiniTrivialElementaryChildData p} (src : Sec7FrattiniTrivialElementaryMStageCandidateLowerSourceData h chief K hR child) (m : towerM K) : have g := powOmega2 src.sigma ^ 2; have u0 := powOmega2 (src.x₀ * src.tau); have tail := ((towerActM K) g⁻¹) (((towerActM K) u0) (((towerActM K) src.x₀⁻¹) (((towerActM K) g) (((towerActM K) u0) (((towerActM K) src.x₀⁻¹) m))))); have rhs := ((towerActM K) src.x₀) (((towerActM K) u0⁻¹) (((towerActM K) src.x₀) (((towerActM K) u0⁻¹) (((towerActM K) src.x₀) (((towerActM K) u0⁻¹) tail))))); have core := ((towerActM K) g⁻¹) (((towerActM K) u0) (((towerActM K) src.x₀⁻¹) (((towerActM K) g) rhs))); ((towerActM K) src.x₀) (((towerActM K) u0⁻¹) tail) = core

Checkpoint317 bridge for the h₀ head: the checkpoint316 head row is recovered from the row with the outer x₀ action removed by reapplying x₀ and cancelling x₀*x₀⁻¹.

theorem Q2Presentation.Induction.sec7_frattiniTrivial_elementaryMStageCandidateWildRowManuscriptH0ExpandedPrimitiveTowerActionSixFactorD0ExpandedRawInlineSevenActionTailTripleInverseCancellationHcNestedDgInvD0SplitD0InvOpenedAllProductActionsFullyNestedHeadAtCommonTailOuterX0U0PrefixStrippedPreimageCoreApplyPointwiseData {p : FramedPair} {h : ¬ScalarTerminal p} {chief : FirstNonScalarChiefFactor p} {K : KernelMinimal chief.toNonScalarChiefFactor} {hR : Nat.card ↥(kernelFrattini K) = 1} {child : Sec7FrattiniTrivialElementaryChildData p} (src : Sec7FrattiniTrivialElementaryMStageCandidateLowerSourceData h chief K hR child) (m : towerM K) : have g := powOmega2 src.sigma ^ 2; have u0 := powOmega2 (src.x₀ * src.tau); have tail := ((towerActM K) g⁻¹) (((towerActM K) u0) (((towerActM K) src.x₀⁻¹) (((towerActM K) g) (((towerActM K) u0) (((towerActM K) src.x₀⁻¹) m))))); have rhs := ((towerActM K) src.x₀) (((towerActM K) u0⁻¹) (((towerActM K) src.x₀) (((towerActM K) u0⁻¹) (((towerActM K) src.x₀) (((towerActM K) u0⁻¹) tail))))); have core := ((towerActM K) g⁻¹) (((towerActM K) u0) (((towerActM K) src.x₀⁻¹) (((towerActM K) g) rhs))); tail = ((towerActM K) u0) (((towerActM K) src.x₀⁻¹) core)

Checkpoint315 h₀ head atom, now theorem-level from the checkpoint316 fully stripped preimage-core row.

theorem Q2Presentation.Induction.sec7_frattiniTrivial_elementaryMStageCandidateWildRowManuscriptH0ExpandedPrimitiveTowerActionSixFactorD0ExpandedRawInlineSevenActionTailTripleInverseCancellationHcNestedDgInvD0SplitD0InvOpenedAllProductActionsFullyNestedHeadAtCommonTailOuterGStrippedApplyPointwiseData {p : FramedPair} {h : ¬ScalarTerminal p} {chief : FirstNonScalarChiefFactor p} {K : KernelMinimal chief.toNonScalarChiefFactor} {hR : Nat.card ↥(kernelFrattini K) = 1} {child : Sec7FrattiniTrivialElementaryChildData p} (src : Sec7FrattiniTrivialElementaryMStageCandidateLowerSourceData h chief K hR child) (m : towerM K) : have g := powOmega2 src.sigma ^ 2; have u0 := powOmega2 (src.x₀ * src.tau); have tail := ((towerActM K) g⁻¹) (((towerActM K) u0) (((towerActM K) src.x₀⁻¹) (((towerActM K) g) (((towerActM K) u0) (((towerActM K) src.x₀⁻¹) m))))); have rhs := ((towerActM K) src.x₀) (((towerActM K) u0⁻¹) (((towerActM K) src.x₀) (((towerActM K) u0⁻¹) (((towerActM K) src.x₀) (((towerActM K) u0⁻¹) tail))))); ((towerActM K) src.x₀) (((towerActM K) u0⁻¹) tail) = ((towerActM K) g⁻¹) (((towerActM K) u0) (((towerActM K) src.x₀⁻¹) (((towerActM K) g) rhs)))

Checkpoint314 h₀ head atom, now theorem-level from the checkpoint315 prefix-stripped preimage core.

theorem Q2Presentation.Induction.sec7_frattiniTrivial_elementaryMStageCandidateWildRowManuscriptH0ExpandedPrimitiveTowerActionSixFactorD0ExpandedRawInlineSevenActionTailTripleInverseCancellationHcNestedDgInvD0SplitD0InvOpenedAllProductActionsFullyNestedHeadAtCommonTailInnerActionApplyPointwiseData {p : FramedPair} {h : ¬ScalarTerminal p} {chief : FirstNonScalarChiefFactor p} {K : KernelMinimal chief.toNonScalarChiefFactor} {hR : Nat.card ↥(kernelFrattini K) = 1} {child : Sec7FrattiniTrivialElementaryChildData p} (src : Sec7FrattiniTrivialElementaryMStageCandidateLowerSourceData h chief K hR child) (m : towerM K) : have g := powOmega2 src.sigma ^ 2; have u0 := powOmega2 (src.x₀ * src.tau); have tail := ((towerActM K) g⁻¹) (((towerActM K) u0) (((towerActM K) src.x₀⁻¹) (((towerActM K) g) (((towerActM K) u0) (((towerActM K) src.x₀⁻¹) m))))); have rhs := ((towerActM K) src.x₀) (((towerActM K) u0⁻¹) (((towerActM K) src.x₀) (((towerActM K) u0⁻¹) (((towerActM K) src.x₀) (((towerActM K) u0⁻¹) tail))))); ((towerActM K) g) (((towerActM K) src.x₀) (((towerActM K) u0⁻¹) tail)) = ((towerActM K) u0) (((towerActM K) src.x₀⁻¹) (((towerActM K) g) rhs))

theorem Q2Presentation.Induction.sec7_frattiniTrivial_elementaryMStageCandidateWildRowManuscriptH0ExpandedPrimitiveTowerActionSixFactorD0ExpandedRawInlineSevenActionTailTripleInverseCancellationHcNestedDgInvD0SplitD0InvOpenedAllProductActionsFullyNestedHeadAtCommonTailApplyPointwiseData {p : FramedPair} {h : ¬ScalarTerminal p} {chief : FirstNonScalarChiefFactor p} {K : KernelMinimal chief.toNonScalarChiefFactor} {hR : Nat.card ↥(kernelFrattini K) = 1} {child : Sec7FrattiniTrivialElementaryChildData p} (src : Sec7FrattiniTrivialElementaryMStageCandidateLowerSourceData h chief K hR child) (m : towerM K) : have g := powOmega2 src.sigma ^ 2; have u0 := powOmega2 (src.x₀ * src.tau); ((towerActM K) g⁻¹) (((towerActM K) src.x₀) (((towerActM K) u0⁻¹) (((towerActM K) g) (((towerActM K) src.x₀) (((towerActM K) u0⁻¹) (((towerActM K) g⁻¹) (((towerActM K) u0) (((towerActM K) src.x₀⁻¹) (((towerActM K) g) (((towerActM K) u0) (((towerActM K) src.x₀⁻¹) m))))))))))) = ((towerActM K) src.x₀) (((towerActM K) u0⁻¹) (((towerActM K) src.x₀) (((towerActM K) u0⁻¹) (((towerActM K) src.x₀) (((towerActM K) u0⁻¹) (((towerActM K) g⁻¹) (((towerActM K) u0) (((towerActM K) src.x₀⁻¹) (((towerActM K) g) (((towerActM K) u0) (((towerActM K) src.x₀⁻¹) m)))))))))))

Checkpoint312 fully nested head row, now theorem-level from the smaller inner action rewrite immediately above.

theorem Q2Presentation.Induction.sec7_frattiniTrivial_elementaryMStageCandidateWildRowManuscriptH0ExpandedPrimitiveTowerActionSixFactorD0ExpandedRawInlineSevenActionTailTripleInverseCancellationHcNestedDgInvD0SplitD0InvOpenedAllProductActionsNestedHeadAtCommonTailApplyPointwiseData {p : FramedPair} {h : ¬ScalarTerminal p} {chief : FirstNonScalarChiefFactor p} {K : KernelMinimal chief.toNonScalarChiefFactor} {hR : Nat.card ↥(kernelFrattini K) = 1} {child : Sec7FrattiniTrivialElementaryChildData p} (src : Sec7FrattiniTrivialElementaryMStageCandidateLowerSourceData h chief K hR child) (m : towerM K) : have g := powOmega2 src.sigma ^ 2; have u0 := powOmega2 (src.x₀ * src.tau); ((towerActM K) g⁻¹) (((towerActM K) src.x₀) (((towerActM K) u0⁻¹) (((towerActM K) g) (((towerActM K) src.x₀) (((towerActM K) u0⁻¹) (Q2Presentation.Induction.sec7TowerActSeq✝ K [g⁻¹, u0, src.x₀⁻¹, g, u0, src.x₀⁻¹] m)))))) = ((towerActM K) src.x₀) (((towerActM K) u0⁻¹) (((towerActM K) src.x₀) (((towerActM K) u0⁻¹) (((towerActM K) src.x₀) (((towerActM K) u0⁻¹) (Q2Presentation.Induction.sec7TowerActSeq✝¹ K [g⁻¹, u0, src.x₀⁻¹, g, u0, src.x₀⁻¹] m))))))

Checkpoint311 nested-head row, recovered theorem-level from the checkpoint312 fully nested head row by unfolding the remaining action-list tail.

theorem Q2Presentation.Induction.sec7_frattiniTrivial_elementaryMStageCandidateWildRowManuscriptH0ExpandedPrimitiveTowerActionSixFactorD0ExpandedRawInlineSevenActionTailTripleInverseCancellationHcNestedDgInvD0SplitD0InvOpenedAllProductActionsListHeadAtCommonTailApplyPointwiseData {p : FramedPair} {h : ¬ScalarTerminal p} {chief : FirstNonScalarChiefFactor p} {K : KernelMinimal chief.toNonScalarChiefFactor} {hR : Nat.card ↥(kernelFrattini K) = 1} {child : Sec7FrattiniTrivialElementaryChildData p} (src : Sec7FrattiniTrivialElementaryMStageCandidateLowerSourceData h chief K hR child) (m : towerM K) : have g := powOmega2 src.sigma ^ 2; have u0 := powOmega2 (src.x₀ * src.tau); Q2Presentation.Induction.sec7TowerActSeq✝ K [g⁻¹, src.x₀, u0⁻¹, g, src.x₀, u0⁻¹] (Q2Presentation.Induction.sec7TowerActSeq✝¹ K [g⁻¹, u0, src.x₀⁻¹, g, u0, src.x₀⁻¹] m) = Q2Presentation.Induction.sec7TowerActSeq✝² K [src.x₀, u0⁻¹, src.x₀, u0⁻¹, src.x₀, u0⁻¹] (Q2Presentation.Induction.sec7TowerActSeq✝³ K [g⁻¹, u0, src.x₀⁻¹, g, u0, src.x₀⁻¹] m)

Checkpoint310 list-head row, recovered theorem-level from the checkpoint311 fully nested head row.

axiom Q2Presentation.Induction.sec7_frattiniTrivial_elementaryMStageCandidateWildRowManuscriptH0ExpandedPrimitiveTowerActionSixFactorD0ExpandedRawInlineSevenActionTailTripleInverseCancellationHcNestedDgInvD0SplitD0InvOpenedAllProductActionsNestedTailToZ0InvOpenedTailX0U0PrefixRemovedX0InvAppliedU0AppliedStrippedPrecoreX0InvPointU0InvPrecoreDifferenceZeroLowerPrimitiveApplyPointwiseData {p : FramedPair} {h : ¬ScalarTerminal p} {chief : FirstNonScalarChiefFactor p} {K : KernelMinimal chief.toNonScalarChiefFactor} {hR : Nat.card ↥(kernelFrattini K) = 1} {child : Sec7FrattiniTrivialElementaryChildData p} (src : Sec7FrattiniTrivialElementaryMStageCandidateLowerSourceData h chief K hR child) (m : towerM K) : let sigma2 := powOmega2 src.sigma; let g := sigma2 ^ 2; let u0 := powOmega2 (src.x₀ * src.tau); let rhs := ((towerActM K) src.x₀) (((towerActM K) u0⁻¹) (((towerActM K) g) (((towerActM K) src.x₀⁻¹) (((towerActM K) g⁻¹) (((towerActM K) src.x₀⁻¹) (((towerActM K) g) m)))))); let precore := ((towerActM K) g⁻¹) (((towerActM K) src.x₀) (((towerActM K) u0⁻¹) rhs)); ((towerActM K) src.x₀⁻¹) m - ((towerActM K) u0⁻¹) precore = 0

Checkpoint332 lower primitive for the h₀ tail: retain only the additive zero of the stripped inverse-preimage row. The checkpoint331 row is recovered by applying the visible x₀ action and cancelling x₀*x₀⁻¹.

theorem Q2Presentation.Induction.sec7_frattiniTrivial_elementaryMStageCandidateWildRowManuscriptH0ExpandedPrimitiveTowerActionSixFactorD0ExpandedRawInlineSevenActionTailTripleInverseCancellationHcNestedDgInvD0SplitD0InvOpenedAllProductActionsNestedTailToZ0InvOpenedTailX0U0PrefixRemovedX0InvAppliedU0AppliedStrippedPrecoreX0ReappliedU0InvPrecorePrimitiveApplyPointwiseData {p : FramedPair} {h : ¬ScalarTerminal p} {chief : FirstNonScalarChiefFactor p} {K : KernelMinimal chief.toNonScalarChiefFactor} {hR : Nat.card ↥(kernelFrattini K) = 1} {child : Sec7FrattiniTrivialElementaryChildData p} (src : Sec7FrattiniTrivialElementaryMStageCandidateLowerSourceData h chief K hR child) (m : towerM K) : have sigma2 := powOmega2 src.sigma; have g := sigma2 ^ 2; have u0 := powOmega2 (src.x₀ * src.tau); have rhs := ((towerActM K) src.x₀) (((towerActM K) u0⁻¹) (((towerActM K) g) (((towerActM K) src.x₀⁻¹) (((towerActM K) g⁻¹) (((towerActM K) src.x₀⁻¹) (((towerActM K) g) m)))))); have precore := ((towerActM K) g⁻¹) (((towerActM K) src.x₀) (((towerActM K) u0⁻¹) rhs)); m = ((towerActM K) src.x₀) (((towerActM K) u0⁻¹) precore)

Checkpoint332 bridge recovering the checkpoint331 h₀ tail row from the lower stripped inverse-preimage difference-zero primitive.

theorem Q2Presentation.Induction.sec7_frattiniTrivial_elementaryMStageCandidateWildRowManuscriptH0ExpandedPrimitiveTowerActionSixFactorD0ExpandedRawInlineSevenActionTailTripleInverseCancellationHcNestedDgInvD0SplitD0InvOpenedAllProductActionsNestedTailToZ0InvOpenedTailX0U0PrefixRemovedX0InvAppliedU0AppliedStrippedPrecoreX0InvPointU0InvPrecoreDifferenceZeroPrimitiveApplyPointwiseData {p : FramedPair} {h : ¬ScalarTerminal p} {chief : FirstNonScalarChiefFactor p} {K : KernelMinimal chief.toNonScalarChiefFactor} {hR : Nat.card ↥(kernelFrattini K) = 1} {child : Sec7FrattiniTrivialElementaryChildData p} (src : Sec7FrattiniTrivialElementaryMStageCandidateLowerSourceData h chief K hR child) (m : towerM K) : have sigma2 := powOmega2 src.sigma; have g := sigma2 ^ 2; have u0 := powOmega2 (src.x₀ * src.tau); have rhs := ((towerActM K) src.x₀) (((towerActM K) u0⁻¹) (((towerActM K) g) (((towerActM K) src.x₀⁻¹) (((towerActM K) g⁻¹) (((towerActM K) src.x₀⁻¹) (((towerActM K) g) m)))))); have precore := ((towerActM K) g⁻¹) (((towerActM K) src.x₀) (((towerActM K) u0⁻¹) rhs)); ((towerActM K) src.x₀⁻¹) m - ((towerActM K) u0⁻¹) precore = 0

Checkpoint331 bridge closing the checkpoint330 h₀ tail difference-zero residual from the lower x₀-reapplied row.

theorem Q2Presentation.Induction.sec7_frattiniTrivial_elementaryMStageCandidateWildRowManuscriptH0ExpandedPrimitiveTowerActionSixFactorD0ExpandedRawInlineSevenActionTailTripleInverseCancellationHcNestedDgInvD0SplitD0InvOpenedAllProductActionsNestedTailToZ0InvOpenedTailX0U0PrefixRemovedX0InvAppliedU0AppliedStrippedPrecoreX0InvPointU0InvPrecorePrimitiveApplyPointwiseData {p : FramedPair} {h : ¬ScalarTerminal p} {chief : FirstNonScalarChiefFactor p} {K : KernelMinimal chief.toNonScalarChiefFactor} {hR : Nat.card ↥(kernelFrattini K) = 1} {child : Sec7FrattiniTrivialElementaryChildData p} (src : Sec7FrattiniTrivialElementaryMStageCandidateLowerSourceData h chief K hR child) (m : towerM K) : have sigma2 := powOmega2 src.sigma; have g := sigma2 ^ 2; have u0 := powOmega2 (src.x₀ * src.tau); have rhs := ((towerActM K) src.x₀) (((towerActM K) u0⁻¹) (((towerActM K) g) (((towerActM K) src.x₀⁻¹) (((towerActM K) g⁻¹) (((towerActM K) src.x₀⁻¹) (((towerActM K) g) m)))))); have precore := ((towerActM K) g⁻¹) (((towerActM K) src.x₀) (((towerActM K) u0⁻¹) rhs)); ((towerActM K) src.x₀⁻¹) m = ((towerActM K) u0⁻¹) precore

Checkpoint330 bridge for the checkpoint329 h₀ tail primitive preimage row.

theorem Q2Presentation.Induction.sec7_frattiniTrivial_elementaryMStageCandidateWildRowManuscriptH0ExpandedPrimitiveTowerActionSixFactorD0ExpandedRawInlineSevenActionTailTripleInverseCancellationHcNestedDgInvD0SplitD0InvOpenedAllProductActionsNestedTailToZ0InvOpenedTailX0U0PrefixRemovedX0InvAppliedU0AppliedStrippedPrecoreX0AppliedU0InvPrecorePrimitiveApplyPointwiseData {p : FramedPair} {h : ¬ScalarTerminal p} {chief : FirstNonScalarChiefFactor p} {K : KernelMinimal chief.toNonScalarChiefFactor} {hR : Nat.card ↥(kernelFrattini K) = 1} {child : Sec7FrattiniTrivialElementaryChildData p} (src : Sec7FrattiniTrivialElementaryMStageCandidateLowerSourceData h chief K hR child) (m : towerM K) : have sigma2 := powOmega2 src.sigma; have g := sigma2 ^ 2; have u0 := powOmega2 (src.x₀ * src.tau); have rhs := ((towerActM K) src.x₀) (((towerActM K) u0⁻¹) (((towerActM K) g) (((towerActM K) src.x₀⁻¹) (((towerActM K) g⁻¹) (((towerActM K) src.x₀⁻¹) (((towerActM K) g) m)))))); have precore := ((towerActM K) g⁻¹) (((towerActM K) src.x₀) (((towerActM K) u0⁻¹) rhs)); m = ((towerActM K) src.x₀) (((towerActM K) u0⁻¹) precore)

Checkpoint329 bridge for the checkpoint328 h₀ tail atom.

theorem Q2Presentation.Induction.sec7_frattiniTrivial_elementaryMStageCandidateWildRowManuscriptH0ExpandedPrimitiveTowerActionSixFactorD0ExpandedRawInlineSevenActionTailTripleInverseCancellationHcNestedDgInvD0SplitD0InvOpenedAllProductActionsNestedTailToZ0InvOpenedTailX0U0PrefixRemovedX0InvAppliedU0AppliedStrippedPrecoreU0AppliedX0InvPointPrecoreDirectApplyPointwiseData {p : FramedPair} {h : ¬ScalarTerminal p} {chief : FirstNonScalarChiefFactor p} {K : KernelMinimal chief.toNonScalarChiefFactor} {hR : Nat.card ↥(kernelFrattini K) = 1} {child : Sec7FrattiniTrivialElementaryChildData p} (src : Sec7FrattiniTrivialElementaryMStageCandidateLowerSourceData h chief K hR child) (m : towerM K) : have sigma2 := powOmega2 src.sigma; have g := sigma2 ^ 2; have u0 := powOmega2 (src.x₀ * src.tau); have rhs := ((towerActM K) src.x₀) (((towerActM K) u0⁻¹) (((towerActM K) g) (((towerActM K) src.x₀⁻¹) (((towerActM K) g⁻¹) (((towerActM K) src.x₀⁻¹) (((towerActM K) g) m)))))); have precore := ((towerActM K) g⁻¹) (((towerActM K) src.x₀) (((towerActM K) u0⁻¹) rhs)); ((towerActM K) u0) (((towerActM K) src.x₀⁻¹) m) = precore

Checkpoint328 bridge for the checkpoint327 h₀ tail atom: the lower datum is the inverse-preimage equality after the x₀ action has already been applied. Apply x₀⁻¹, then u₀, and cancel the two displayed inverse pairs.

theorem Q2Presentation.Induction.sec7_frattiniTrivial_elementaryMStageCandidateWildRowManuscriptH0ExpandedPrimitiveTowerActionSixFactorD0ExpandedRawInlineSevenActionTailTripleInverseCancellationHcNestedDgInvD0SplitD0InvOpenedAllProductActionsNestedTailToZ0InvOpenedTailX0U0PrefixRemovedX0InvAppliedU0AppliedStrippedPrecoreX0AppliedU0InvPrecoreDirectApplyPointwiseData {p : FramedPair} {h : ¬ScalarTerminal p} {chief : FirstNonScalarChiefFactor p} {K : KernelMinimal chief.toNonScalarChiefFactor} {hR : Nat.card ↥(kernelFrattini K) = 1} {child : Sec7FrattiniTrivialElementaryChildData p} (src : Sec7FrattiniTrivialElementaryMStageCandidateLowerSourceData h chief K hR child) (m : towerM K) : have sigma2 := powOmega2 src.sigma; have g := sigma2 ^ 2; have u0 := powOmega2 (src.x₀ * src.tau); have rhs := ((towerActM K) src.x₀) (((towerActM K) u0⁻¹) (((towerActM K) g) (((towerActM K) src.x₀⁻¹) (((towerActM K) g⁻¹) (((towerActM K) src.x₀⁻¹) (((towerActM K) g) m)))))); have precore := ((towerActM K) g⁻¹) (((towerActM K) src.x₀) (((towerActM K) u0⁻¹) rhs)); m = ((towerActM K) src.x₀) (((towerActM K) u0⁻¹) precore)

Checkpoint327 bridge for the checkpoint326 h₀ tail atom: the new datum states the row after applying u₀ to the visible x₀⁻¹ preimage. Apply u₀⁻¹ and then x₀, cancelling the two adjacent inverse-action pairs.

theorem Q2Presentation.Induction.sec7_frattiniTrivial_elementaryMStageCandidateWildRowManuscriptH0ExpandedPrimitiveTowerActionSixFactorD0ExpandedRawInlineSevenActionTailTripleInverseCancellationHcNestedDgInvD0SplitD0InvOpenedAllProductActionsNestedTailToZ0InvOpenedTailX0U0PrefixRemovedX0InvAppliedU0AppliedStrippedPrecoreX0InvU0InvPreimageDirectApplyPointwiseData {p : FramedPair} {h : ¬ScalarTerminal p} {chief : FirstNonScalarChiefFactor p} {K : KernelMinimal chief.toNonScalarChiefFactor} {hR : Nat.card ↥(kernelFrattini K) = 1} {child : Sec7FrattiniTrivialElementaryChildData p} (src : Sec7FrattiniTrivialElementaryMStageCandidateLowerSourceData h chief K hR child) (m : towerM K) : have sigma2 := powOmega2 src.sigma; have g := sigma2 ^ 2; have u0 := powOmega2 (src.x₀ * src.tau); have rhs := ((towerActM K) src.x₀) (((towerActM K) u0⁻¹) (((towerActM K) g) (((towerActM K) src.x₀⁻¹) (((towerActM K) g⁻¹) (((towerActM K) src.x₀⁻¹) (((towerActM K) g) m)))))); have precore := ((towerActM K) g⁻¹) (((towerActM K) src.x₀) (((towerActM K) u0⁻¹) rhs)); ((towerActM K) src.x₀⁻¹) m = ((towerActM K) u0⁻¹) precore

Checkpoint326 bridge for the checkpoint325 h₀ tail atom: apply x₀⁻¹ to the lower core-opened equality and cancel the adjacent inverse pair.

theorem Q2Presentation.Induction.sec7_frattiniTrivial_elementaryMStageCandidateWildRowManuscriptH0ExpandedPrimitiveTowerActionSixFactorD0ExpandedRawInlineSevenActionTailTripleInverseCancellationHcNestedDgInvD0SplitD0InvOpenedAllProductActionsNestedTailToZ0InvOpenedTailX0U0PrefixRemovedX0InvAppliedU0AppliedStrippedPrecorePreimageDirectApplyPointwiseData {p : FramedPair} {h : ¬ScalarTerminal p} {chief : FirstNonScalarChiefFactor p} {K : KernelMinimal chief.toNonScalarChiefFactor} {hR : Nat.card ↥(kernelFrattini K) = 1} {child : Sec7FrattiniTrivialElementaryChildData p} (src : Sec7FrattiniTrivialElementaryMStageCandidateLowerSourceData h chief K hR child) (m : towerM K) : have sigma2 := powOmega2 src.sigma; have g := sigma2 ^ 2; have u0 := powOmega2 (src.x₀ * src.tau); have rhs := ((towerActM K) src.x₀) (((towerActM K) u0⁻¹) (((towerActM K) g) (((towerActM K) src.x₀⁻¹) (((towerActM K) g⁻¹) (((towerActM K) src.x₀⁻¹) (((towerActM K) g) m)))))); have precore := ((towerActM K) g⁻¹) (((towerActM K) src.x₀) (((towerActM K) u0⁻¹) rhs)); m = ((towerActM K) src.x₀) (((towerActM K) u0⁻¹) precore)

Checkpoint325 bridge for the h₀ tail: apply x₀ to the lower inverse preimage row and cancel x₀*x₀⁻¹.

theorem Q2Presentation.Induction.sec7_frattiniTrivial_elementaryMStageCandidateWildRowManuscriptH0ExpandedPrimitiveTowerActionSixFactorD0ExpandedRawInlineSevenActionTailTripleInverseCancellationHcNestedDgInvD0SplitD0InvOpenedAllProductActionsNestedTailToZ0InvOpenedTailX0U0PrefixRemovedX0InvAppliedU0AppliedStrippedPrecoreDirectApplyPointwiseData {p : FramedPair} {h : ¬ScalarTerminal p} {chief : FirstNonScalarChiefFactor p} {K : KernelMinimal chief.toNonScalarChiefFactor} {hR : Nat.card ↥(kernelFrattini K) = 1} {child : Sec7FrattiniTrivialElementaryChildData p} (src : Sec7FrattiniTrivialElementaryMStageCandidateLowerSourceData h chief K hR child) (m : towerM K) : have sigma2 := powOmega2 src.sigma; have g := sigma2 ^ 2; have u0 := powOmega2 (src.x₀ * src.tau); have rhs := ((towerActM K) src.x₀) (((towerActM K) u0⁻¹) (((towerActM K) g) (((towerActM K) src.x₀⁻¹) (((towerActM K) g⁻¹) (((towerActM K) src.x₀⁻¹) (((towerActM K) g) m)))))); have precore := ((towerActM K) g⁻¹) (((towerActM K) src.x₀) (((towerActM K) u0⁻¹) rhs)); ((towerActM K) u0) (((towerActM K) src.x₀⁻¹) m) = precore

Checkpoint324 bridge for the h₀ tail: reapply the visible u₀ · x₀⁻¹ prefix to the new precore-preimage row.

theorem Q2Presentation.Induction.sec7_frattiniTrivial_elementaryMStageCandidateWildRowManuscriptH0ExpandedPrimitiveTowerActionSixFactorD0ExpandedRawInlineSevenActionTailTripleInverseCancellationHcNestedDgInvD0SplitD0InvOpenedAllProductActionsNestedTailToZ0InvOpenedTailX0U0PrefixRemovedX0InvAppliedU0AppliedSigma2Sigma2GPrecoreOpenedApplyPointwiseData {p : FramedPair} {h : ¬ScalarTerminal p} {chief : FirstNonScalarChiefFactor p} {K : KernelMinimal chief.toNonScalarChiefFactor} {hR : Nat.card ↥(kernelFrattini K) = 1} {child : Sec7FrattiniTrivialElementaryChildData p} (src : Sec7FrattiniTrivialElementaryMStageCandidateLowerSourceData h chief K hR child) (m : towerM K) : have sigma2 := powOmega2 src.sigma; have g := sigma2 ^ 2; have u0 := powOmega2 (src.x₀ * src.tau); have rhs := ((towerActM K) src.x₀) (((towerActM K) u0⁻¹) (((towerActM K) g) (((towerActM K) src.x₀⁻¹) (((towerActM K) g⁻¹) (((towerActM K) src.x₀⁻¹) (((towerActM K) g) m)))))); ((towerActM K) sigma2) (((towerActM K) sigma2) (((towerActM K) u0) (((towerActM K) src.x₀⁻¹) m))) = ((towerActM K) src.x₀) (((towerActM K) u0⁻¹) rhs)

Checkpoint323 bridge for the h₀ tail: reapply g = σ₂² to the stripped precore row and rewrite the product action as two successive σ₂ actions.

theorem Q2Presentation.Induction.sec7_frattiniTrivial_elementaryMStageCandidateWildRowManuscriptH0ExpandedPrimitiveTowerActionSixFactorD0ExpandedRawInlineSevenActionTailTripleInverseCancellationHcNestedDgInvD0SplitD0InvOpenedAllProductActionsNestedTailToZ0InvOpenedTailX0U0PrefixRemovedX0InvAppliedU0AppliedGPrecoreOpenedApplyPointwiseData {p : FramedPair} {h : ¬ScalarTerminal p} {chief : FirstNonScalarChiefFactor p} {K : KernelMinimal chief.toNonScalarChiefFactor} {hR : Nat.card ↥(kernelFrattini K) = 1} {child : Sec7FrattiniTrivialElementaryChildData p} (src : Sec7FrattiniTrivialElementaryMStageCandidateLowerSourceData h chief K hR child) (m : towerM K) : have g := powOmega2 src.sigma ^ 2; have u0 := powOmega2 (src.x₀ * src.tau); have rhs := ((towerActM K) src.x₀) (((towerActM K) u0⁻¹) (((towerActM K) g) (((towerActM K) src.x₀⁻¹) (((towerActM K) g⁻¹) (((towerActM K) src.x₀⁻¹) (((towerActM K) g) m)))))); ((towerActM K) g) (((towerActM K) u0) (((towerActM K) src.x₀⁻¹) m)) = ((towerActM K) src.x₀) (((towerActM K) u0⁻¹) rhs)

Checkpoint322 bridge for the h₀ tail: recover the checkpoint321 opened-g precore row from the lower statement where the left g = σ₂² action is already split into two successive σ₂ actions.

theorem Q2Presentation.Induction.sec7_frattiniTrivial_elementaryMStageCandidateWildRowManuscriptH0ExpandedPrimitiveTowerActionSixFactorD0ExpandedRawInlineSevenActionTailTripleInverseCancellationHcNestedDgInvD0SplitD0InvOpenedAllProductActionsNestedTailToZ0InvOpenedTailX0U0PrefixRemovedX0InvAppliedU0AppliedFullyStrippedPreimageCoreApplyPointwiseData {p : FramedPair} {h : ¬ScalarTerminal p} {chief : FirstNonScalarChiefFactor p} {K : KernelMinimal chief.toNonScalarChiefFactor} {hR : Nat.card ↥(kernelFrattini K) = 1} {child : Sec7FrattiniTrivialElementaryChildData p} (src : Sec7FrattiniTrivialElementaryMStageCandidateLowerSourceData h chief K hR child) (m : towerM K) : have g := powOmega2 src.sigma ^ 2; have u0 := powOmega2 (src.x₀ * src.tau); have rhs := ((towerActM K) src.x₀) (((towerActM K) u0⁻¹) (((towerActM K) g) (((towerActM K) src.x₀⁻¹) (((towerActM K) g⁻¹) (((towerActM K) src.x₀⁻¹) (((towerActM K) g) m)))))); have precore := ((towerActM K) g⁻¹) (((towerActM K) src.x₀) (((towerActM K) u0⁻¹) rhs)); ((towerActM K) u0) (((towerActM K) src.x₀⁻¹) m) = precore

Checkpoint320 bridge for the h₀ tail: recover the fully stripped precore row from the checkpoint321 row by applying g⁻¹ and cancelling g⁻¹*g.

theorem Q2Presentation.Induction.sec7_frattiniTrivial_elementaryMStageCandidateWildRowManuscriptH0ExpandedPrimitiveTowerActionSixFactorD0ExpandedRawInlineSevenActionTailTripleInverseCancellationHcNestedDgInvD0SplitD0InvOpenedAllProductActionsNestedTailToZ0InvOpenedTailX0U0PrefixRemovedX0InvAppliedFullyStrippedPreimageCoreApplyPointwiseData {p : FramedPair} {h : ¬ScalarTerminal p} {chief : FirstNonScalarChiefFactor p} {K : KernelMinimal chief.toNonScalarChiefFactor} {hR : Nat.card ↥(kernelFrattini K) = 1} {child : Sec7FrattiniTrivialElementaryChildData p} (src : Sec7FrattiniTrivialElementaryMStageCandidateLowerSourceData h chief K hR child) (m : towerM K) : have g := powOmega2 src.sigma ^ 2; have u0 := powOmega2 (src.x₀ * src.tau); have rhs := ((towerActM K) src.x₀) (((towerActM K) u0⁻¹) (((towerActM K) g) (((towerActM K) src.x₀⁻¹) (((towerActM K) g⁻¹) (((towerActM K) src.x₀⁻¹) (((towerActM K) g) m)))))); have precore := ((towerActM K) g⁻¹) (((towerActM K) src.x₀) (((towerActM K) u0⁻¹) rhs)); ((towerActM K) src.x₀⁻¹) m = ((towerActM K) u0⁻¹) precore

Checkpoint319 bridge for the h₀ tail: the applied-x₀⁻¹ preimage row is recovered from the checkpoint320 row by applying u₀⁻¹ and cancelling the visible u₀⁻¹*u₀ action pair.

theorem Q2Presentation.Induction.sec7_frattiniTrivial_elementaryMStageCandidateWildRowManuscriptH0ExpandedPrimitiveTowerActionSixFactorD0ExpandedRawInlineSevenActionTailTripleInverseCancellationHcNestedDgInvD0SplitD0InvOpenedAllProductActionsNestedTailToZ0InvOpenedTailX0U0PrefixRemovedFullyStrippedPreimageCoreApplyPointwiseData {p : FramedPair} {h : ¬ScalarTerminal p} {chief : FirstNonScalarChiefFactor p} {K : KernelMinimal chief.toNonScalarChiefFactor} {hR : Nat.card ↥(kernelFrattini K) = 1} {child : Sec7FrattiniTrivialElementaryChildData p} (src : Sec7FrattiniTrivialElementaryMStageCandidateLowerSourceData h chief K hR child) (m : towerM K) : have g := powOmega2 src.sigma ^ 2; have u0 := powOmega2 (src.x₀ * src.tau); have rhs := ((towerActM K) src.x₀) (((towerActM K) u0⁻¹) (((towerActM K) g) (((towerActM K) src.x₀⁻¹) (((towerActM K) g⁻¹) (((towerActM K) src.x₀⁻¹) (((towerActM K) g) m)))))); have precore := ((towerActM K) g⁻¹) (((towerActM K) src.x₀) (((towerActM K) u0⁻¹) rhs)); m = ((towerActM K) src.x₀) (((towerActM K) u0⁻¹) precore)

theorem Q2Presentation.Induction.sec7_frattiniTrivial_elementaryMStageCandidateWildRowManuscriptH0ExpandedPrimitiveTowerActionSixFactorD0ExpandedRawInlineSevenActionTailTripleInverseCancellationHcNestedDgInvD0SplitD0InvOpenedAllProductActionsNestedTailToZ0InvOpenedTailU0PrefixAppliedFullyStrippedPreimageCoreApplyPointwiseData {p : FramedPair} {h : ¬ScalarTerminal p} {chief : FirstNonScalarChiefFactor p} {K : KernelMinimal chief.toNonScalarChiefFactor} {hR : Nat.card ↥(kernelFrattini K) = 1} {child : Sec7FrattiniTrivialElementaryChildData p} (src : Sec7FrattiniTrivialElementaryMStageCandidateLowerSourceData h chief K hR child) (m : towerM K) : have g := powOmega2 src.sigma ^ 2; have u0 := powOmega2 (src.x₀ * src.tau); have rhs := ((towerActM K) src.x₀) (((towerActM K) u0⁻¹) (((towerActM K) g) (((towerActM K) src.x₀⁻¹) (((towerActM K) g⁻¹) (((towerActM K) src.x₀⁻¹) (((towerActM K) g) m)))))); have precore := ((towerActM K) g⁻¹) (((towerActM K) src.x₀) (((towerActM K) u0⁻¹) rhs)); ((towerActM K) u0) (((towerActM K) src.x₀⁻¹) m) = precore

Checkpoint318 bridge for the h₀ tail: the checkpoint317 row follows by reapplying the stripped x₀⁻¹ and u₀ actions and cancelling the adjacent inverse-action pairs.

theorem Q2Presentation.Induction.sec7_frattiniTrivial_elementaryMStageCandidateWildRowManuscriptH0ExpandedPrimitiveTowerActionSixFactorD0ExpandedRawInlineSevenActionTailTripleInverseCancellationHcNestedDgInvD0SplitD0InvOpenedAllProductActionsNestedTailToZ0InvOpenedTailX0InvPrefixFullyStrippedPreimageCoreApplyPointwiseData {p : FramedPair} {h : ¬ScalarTerminal p} {chief : FirstNonScalarChiefFactor p} {K : KernelMinimal chief.toNonScalarChiefFactor} {hR : Nat.card ↥(kernelFrattini K) = 1} {child : Sec7FrattiniTrivialElementaryChildData p} (src : Sec7FrattiniTrivialElementaryMStageCandidateLowerSourceData h chief K hR child) (m : towerM K) : have g := powOmega2 src.sigma ^ 2; have u0 := powOmega2 (src.x₀ * src.tau); have rhs := ((towerActM K) src.x₀) (((towerActM K) u0⁻¹) (((towerActM K) g) (((towerActM K) src.x₀⁻¹) (((towerActM K) g⁻¹) (((towerActM K) src.x₀⁻¹) (((towerActM K) g) m)))))); have core := ((towerActM K) u0⁻¹) (((towerActM K) g⁻¹) (((towerActM K) src.x₀) (((towerActM K) u0⁻¹) rhs))); ((towerActM K) src.x₀⁻¹) m = core

Checkpoint317 bridge for the h₀ tail: the checkpoint316 tail row is recovered from the row with the visible u₀⁻¹ preimage removed by applying u₀⁻¹ and cancelling u₀⁻¹*u₀.

theorem Q2Presentation.Induction.sec7_frattiniTrivial_elementaryMStageCandidateWildRowManuscriptH0ExpandedPrimitiveTowerActionSixFactorD0ExpandedRawInlineSevenActionTailTripleInverseCancellationHcNestedDgInvD0SplitD0InvOpenedAllProductActionsNestedTailToZ0InvOpenedTailX0InvPrefixStrippedPreimageCoreApplyPointwiseData {p : FramedPair} {h : ¬ScalarTerminal p} {chief : FirstNonScalarChiefFactor p} {K : KernelMinimal chief.toNonScalarChiefFactor} {hR : Nat.card ↥(kernelFrattini K) = 1} {child : Sec7FrattiniTrivialElementaryChildData p} (src : Sec7FrattiniTrivialElementaryMStageCandidateLowerSourceData h chief K hR child) (m : towerM K) : have g := powOmega2 src.sigma ^ 2; have u0 := powOmega2 (src.x₀ * src.tau); have rhs := ((towerActM K) src.x₀) (((towerActM K) u0⁻¹) (((towerActM K) g) (((towerActM K) src.x₀⁻¹) (((towerActM K) g⁻¹) (((towerActM K) src.x₀⁻¹) (((towerActM K) g) m)))))); have core := ((towerActM K) u0⁻¹) (((towerActM K) g⁻¹) (((towerActM K) src.x₀) (((towerActM K) u0⁻¹) rhs))); m = ((towerActM K) src.x₀) core

Checkpoint315 h₀ tail atom, now theorem-level from the checkpoint316 fully stripped tail-preimage core.

theorem Q2Presentation.Induction.sec7_frattiniTrivial_elementaryMStageCandidateWildRowManuscriptH0ExpandedPrimitiveTowerActionSixFactorD0ExpandedRawInlineSevenActionTailTripleInverseCancellationHcNestedDgInvD0SplitD0InvOpenedAllProductActionsNestedTailToZ0InvOpenedTailU0PrefixStrippedPreimageCoreApplyPointwiseData {p : FramedPair} {h : ¬ScalarTerminal p} {chief : FirstNonScalarChiefFactor p} {K : KernelMinimal chief.toNonScalarChiefFactor} {hR : Nat.card ↥(kernelFrattini K) = 1} {child : Sec7FrattiniTrivialElementaryChildData p} (src : Sec7FrattiniTrivialElementaryMStageCandidateLowerSourceData h chief K hR child) (m : towerM K) : have g := powOmega2 src.sigma ^ 2; have u0 := powOmega2 (src.x₀ * src.tau); have rhs := ((towerActM K) src.x₀) (((towerActM K) u0⁻¹) (((towerActM K) g) (((towerActM K) src.x₀⁻¹) (((towerActM K) g⁻¹) (((towerActM K) src.x₀⁻¹) (((towerActM K) g) m)))))); ((towerActM K) src.x₀⁻¹) m = ((towerActM K) u0⁻¹) (((towerActM K) g⁻¹) (((towerActM K) src.x₀) (((towerActM K) u0⁻¹) rhs)))

Checkpoint314 h₀ tail atom, now theorem-level from the checkpoint315 x₀⁻¹-prefix-stripped preimage core.

theorem Q2Presentation.Induction.sec7_frattiniTrivial_elementaryMStageCandidateWildRowManuscriptH0ExpandedPrimitiveTowerActionSixFactorD0ExpandedRawInlineSevenActionTailTripleInverseCancellationHcNestedDgInvD0SplitD0InvOpenedAllProductActionsNestedTailToZ0InvOpenedTailPreimageCoreApplyPointwiseData {p : FramedPair} {h : ¬ScalarTerminal p} {chief : FirstNonScalarChiefFactor p} {K : KernelMinimal chief.toNonScalarChiefFactor} {hR : Nat.card ↥(kernelFrattini K) = 1} {child : Sec7FrattiniTrivialElementaryChildData p} (src : Sec7FrattiniTrivialElementaryMStageCandidateLowerSourceData h chief K hR child) (m : towerM K) : have g := powOmega2 src.sigma ^ 2; have u0 := powOmega2 (src.x₀ * src.tau); have rhs := ((towerActM K) src.x₀) (((towerActM K) u0⁻¹) (((towerActM K) g) (((towerActM K) src.x₀⁻¹) (((towerActM K) g⁻¹) (((towerActM K) src.x₀⁻¹) (((towerActM K) g) m)))))); ((towerActM K) u0) (((towerActM K) src.x₀⁻¹) m) = ((towerActM K) g⁻¹) (((towerActM K) src.x₀) (((towerActM K) u0⁻¹) rhs))

theorem Q2Presentation.Induction.sec7_frattiniTrivial_elementaryMStageCandidateWildRowManuscriptH0ExpandedPrimitiveTowerActionSixFactorD0ExpandedRawInlineSevenActionTailTripleInverseCancellationHcNestedDgInvD0SplitD0InvOpenedAllProductActionsNestedTailToZ0InvOpenedTailInnerApplyPointwiseData {p : FramedPair} {h : ¬ScalarTerminal p} {chief : FirstNonScalarChiefFactor p} {K : KernelMinimal chief.toNonScalarChiefFactor} {hR : Nat.card ↥(kernelFrattini K) = 1} {child : Sec7FrattiniTrivialElementaryChildData p} (src : Sec7FrattiniTrivialElementaryMStageCandidateLowerSourceData h chief K hR child) (m : towerM K) : have g := powOmega2 src.sigma ^ 2; have u0 := powOmega2 (src.x₀ * src.tau); ((towerActM K) u0) (((towerActM K) src.x₀⁻¹) (((towerActM K) g) (((towerActM K) u0) (((towerActM K) src.x₀⁻¹) m)))) = ((towerActM K) src.x₀) (((towerActM K) u0⁻¹) (((towerActM K) g) (((towerActM K) src.x₀⁻¹) (((towerActM K) g⁻¹) (((towerActM K) src.x₀⁻¹) (((towerActM K) g) m))))))

Checkpoint312 opened h₀ tail row, now theorem-level from the smaller inverse preimage core above.

theorem Q2Presentation.Induction.sec7_frattiniTrivial_elementaryMStageCandidateWildRowManuscriptH0ExpandedPrimitiveTowerActionSixFactorD0ExpandedRawInlineSevenActionTailTripleInverseCancellationHcNestedDgInvD0SplitD0InvOpenedAllProductActionsNestedTailToZ0InvOpenedTailApplyPointwiseData {p : FramedPair} {h : ¬ScalarTerminal p} {chief : FirstNonScalarChiefFactor p} {K : KernelMinimal chief.toNonScalarChiefFactor} {hR : Nat.card ↥(kernelFrattini K) = 1} {child : Sec7FrattiniTrivialElementaryChildData p} (src : Sec7FrattiniTrivialElementaryMStageCandidateLowerSourceData h chief K hR child) (m : towerM K) : have g := powOmega2 src.sigma ^ 2; have u0 := powOmega2 (src.x₀ * src.tau); ((towerActM K) g⁻¹) (((towerActM K) u0) (((towerActM K) src.x₀⁻¹) (((towerActM K) g) (((towerActM K) u0) (((towerActM K) src.x₀⁻¹) m))))) = ((towerActM K) g⁻¹) (((towerActM K) src.x₀) (((towerActM K) u0⁻¹) (((towerActM K) g) (((towerActM K) src.x₀⁻¹) (((towerActM K) g⁻¹) (((towerActM K) src.x₀⁻¹) (((towerActM K) g) m)))))))

Checkpoint311 nested tail-to-z₀⁻¹ row, recovered theorem-level from the checkpoint312 row after cancelling the common outer g⁻¹ action.

theorem Q2Presentation.Induction.sec7_frattiniTrivial_elementaryMStageCandidateWildRowManuscriptH0ExpandedPrimitiveTowerActionSixFactorD0ExpandedRawInlineSevenActionTailTripleInverseCancellationHcNestedDgInvD0SplitD0InvOpenedAllProductActionsListTailToZ0InvOpenedTailApplyPointwiseData {p : FramedPair} {h : ¬ScalarTerminal p} {chief : FirstNonScalarChiefFactor p} {K : KernelMinimal chief.toNonScalarChiefFactor} {hR : Nat.card ↥(kernelFrattini K) = 1} {child : Sec7FrattiniTrivialElementaryChildData p} (src : Sec7FrattiniTrivialElementaryMStageCandidateLowerSourceData h chief K hR child) (m : towerM K) : have g := powOmega2 src.sigma ^ 2; have u0 := powOmega2 (src.x₀ * src.tau); Q2Presentation.Induction.sec7TowerActSeq✝ K [g⁻¹, u0, src.x₀⁻¹, g, u0, src.x₀⁻¹] m = Q2Presentation.Induction.sec7TowerActSeq✝¹ K [g⁻¹, src.x₀, u0⁻¹, g, src.x₀⁻¹, g⁻¹, src.x₀⁻¹, g] m

Checkpoint310 tail-to-z₀⁻¹ list row, recovered theorem-level from the checkpoint311 fully nested tail row.

theorem Q2Presentation.Induction.sec7_frattiniTrivial_elementaryMStageCandidateWildRowManuscriptH0ExpandedPrimitiveTowerActionSixFactorD0ExpandedRawInlineSevenActionTailTripleInverseCancellationHcNestedDgInvD0SplitD0InvOpenedAllProductActionsListZ0InvOpenedApplyPointwiseData {p : FramedPair} {h : ¬ScalarTerminal p} {chief : FirstNonScalarChiefFactor p} {K : KernelMinimal chief.toNonScalarChiefFactor} {hR : Nat.card ↥(kernelFrattini K) = 1} {child : Sec7FrattiniTrivialElementaryChildData p} (src : Sec7FrattiniTrivialElementaryMStageCandidateLowerSourceData h chief K hR child) (m : towerM K) : have g := powOmega2 src.sigma ^ 2; have u0 := powOmega2 (src.x₀ * src.tau); Q2Presentation.Induction.sec7TowerActSeq✝ K [g⁻¹, src.x₀, u0⁻¹, g, src.x₀, u0⁻¹, g⁻¹, u0, src.x₀⁻¹, g, u0, src.x₀⁻¹] m = Q2Presentation.Induction.sec7TowerActSeq✝¹ K [src.x₀, u0⁻¹, src.x₀, u0⁻¹, src.x₀, u0⁻¹, g⁻¹, src.x₀, u0⁻¹, g, src.x₀⁻¹, g⁻¹, src.x₀⁻¹, g] m

Checkpoint309 datum for h₀, now theorem-level from two smaller checkpoint310 rows: first the six-factor head is replaced at a common tail, and then the remaining tail is rewritten to the fully opened z₀⁻¹ tail.

theorem Q2Presentation.Induction.sec7_frattiniTrivial_elementaryMStageCandidateWildRowManuscriptH0ExpandedPrimitiveTowerActionSixFactorD0ExpandedRawInlineSevenActionTailTripleInverseCancellationHcNestedDgInvD0SplitD0InvOpenedAllProductActionsListApplyPointwiseData {p : FramedPair} {h : ¬ScalarTerminal p} {chief : FirstNonScalarChiefFactor p} {K : KernelMinimal chief.toNonScalarChiefFactor} {hR : Nat.card ↥(kernelFrattini K) = 1} {child : Sec7FrattiniTrivialElementaryChildData p} (src : Sec7FrattiniTrivialElementaryMStageCandidateLowerSourceData h chief K hR child) (m : towerM K) : have g := powOmega2 src.sigma ^ 2; have u0 := powOmega2 (src.x₀ * src.tau); have z0inv := (g⁻¹ * src.x₀ * g)⁻¹; Q2Presentation.Induction.sec7TowerActSeq✝ K [g⁻¹, src.x₀, u0⁻¹, g, src.x₀, u0⁻¹, g⁻¹, u0, src.x₀⁻¹, g, u0, src.x₀⁻¹] m = Q2Presentation.Induction.sec7TowerActSeq✝¹ K [src.x₀, u0⁻¹, src.x₀, u0⁻¹, src.x₀, u0⁻¹, g⁻¹, src.x₀, u0⁻¹, g, src.x₀⁻¹, z0inv] m

Checkpoint308 list datum for h₀, now theorem-level from the still more literal row in which the final z₀⁻¹ = g⁻¹*x₀⁻¹*g action has been opened into its three successive tower-action factors.

theorem Q2Presentation.Induction.sec7_frattiniTrivial_elementaryMStageCandidateWildRowManuscriptH0ExpandedPrimitiveTowerActionSixFactorD0ExpandedRawInlineSevenActionTailTripleInverseCancellationHcNestedDgInvD0SplitD0InvOpenedAllProductActionsSplitApplyPointwiseData {p : FramedPair} {h : ¬ScalarTerminal p} {chief : FirstNonScalarChiefFactor p} {K : KernelMinimal chief.toNonScalarChiefFactor} {hR : Nat.card ↥(kernelFrattini K) = 1} {child : Sec7FrattiniTrivialElementaryChildData p} (src : Sec7FrattiniTrivialElementaryMStageCandidateLowerSourceData h chief K hR child) (m : towerM K) : ((towerActM K) (powOmega2 src.sigma ^ 2)⁻¹) (((towerActM K) src.x₀) (((towerActM K) (powOmega2 (src.x₀ * src.tau))⁻¹) (((towerActM K) (powOmega2 src.sigma ^ 2)) (((towerActM K) src.x₀) (((towerActM K) (powOmega2 (src.x₀ * src.tau))⁻¹) (((towerActM K) (powOmega2 src.sigma ^ 2)⁻¹) (((towerActM K) (powOmega2 (src.x₀ * src.tau))) (((towerActM K) src.x₀⁻¹) (((towerActM K) (powOmega2 src.sigma ^ 2)) (((towerActM K) (powOmega2 (src.x₀ * src.tau))) (((towerActM K) src.x₀⁻¹) m))))))))))) = ((towerActM K) src.x₀) (((towerActM K) (powOmega2 (src.x₀ * src.tau))⁻¹) (((towerActM K) src.x₀) (((towerActM K) (powOmega2 (src.x₀ * src.tau))⁻¹) (((towerActM K) src.x₀) (((towerActM K) (powOmega2 (src.x₀ * src.tau))⁻¹) (((towerActM K) (powOmega2 src.sigma ^ 2)⁻¹) (((towerActM K) src.x₀) (((towerActM K) (powOmega2 (src.x₀ * src.tau))⁻¹) (((towerActM K) (powOmega2 src.sigma ^ 2)) (((towerActM K) src.x₀⁻¹) (((towerActM K) ((powOmega2 src.sigma ^ 2)⁻¹ * src.x₀ * powOmega2 src.sigma ^ 2)⁻¹) m)))))))))))

theorem Q2Presentation.Induction.sec7_frattiniTrivial_elementaryMStageCandidateWildRowManuscriptH0ExpandedPrimitiveTowerActionSixFactorD0ExpandedRawInlineSevenActionTailTripleInverseCancellationHcNestedDgInvD0SplitD0InvOpenedProductActionSplitApplyPointwiseData {p : FramedPair} {h : ¬ScalarTerminal p} {chief : FirstNonScalarChiefFactor p} {K : KernelMinimal chief.toNonScalarChiefFactor} {hR : Nat.card ↥(kernelFrattini K) = 1} {child : Sec7FrattiniTrivialElementaryChildData p} (src : Sec7FrattiniTrivialElementaryMStageCandidateLowerSourceData h chief K hR child) (m : towerM K) : ((towerActM K) (powOmega2 src.sigma ^ 2)⁻¹) (((towerActM K) src.x₀) (((towerActM K) (powOmega2 (src.x₀ * src.tau))⁻¹) (((towerActM K) (powOmega2 src.sigma ^ 2)) (((towerActM K) (src.x₀ * (powOmega2 (src.x₀ * src.tau))⁻¹)) (((towerActM K) (powOmega2 src.sigma ^ 2)⁻¹) (((towerActM K) (powOmega2 (src.x₀ * src.tau))) (((towerActM K) src.x₀⁻¹) (((towerActM K) (powOmega2 src.sigma ^ 2)) (((towerActM K) (powOmega2 (src.x₀ * src.tau))) (((towerActM K) src.x₀⁻¹) m)))))))))) = ((towerActM K) (src.x₀ * (powOmega2 (src.x₀ * src.tau))⁻¹)) (((towerActM K) (src.x₀ * (powOmega2 (src.x₀ * src.tau))⁻¹)) (((towerActM K) (src.x₀ * (powOmega2 (src.x₀ * src.tau))⁻¹)) (((towerActM K) (powOmega2 src.sigma ^ 2)⁻¹) (((towerActM K) (src.x₀ * (powOmega2 (src.x₀ * src.tau))⁻¹)) (((towerActM K) (powOmega2 src.sigma ^ 2)) (((towerActM K) src.x₀⁻¹) (((towerActM K) ((powOmega2 src.sigma ^ 2)⁻¹ * src.x₀ * powOmega2 src.sigma ^ 2)⁻¹) m)))))))

theorem Q2Presentation.Induction.sec7_frattiniTrivial_elementaryMStageCandidateWildRowManuscriptH0ExpandedPrimitiveTowerActionSixFactorD0ExpandedRawInlineSevenActionTailTripleInverseCancellationHcNestedDgInvD0SplitD0InvOpenedApplyPointwiseData {p : FramedPair} {h : ¬ScalarTerminal p} {chief : FirstNonScalarChiefFactor p} {K : KernelMinimal chief.toNonScalarChiefFactor} {hR : Nat.card ↥(kernelFrattini K) = 1} {child : Sec7FrattiniTrivialElementaryChildData p} (src : Sec7FrattiniTrivialElementaryMStageCandidateLowerSourceData h chief K hR child) (m : towerM K) : ((towerActM K) (powOmega2 src.sigma ^ 2)⁻¹) (((towerActM K) (src.x₀ * (powOmega2 (src.x₀ * src.tau))⁻¹)) (((towerActM K) (powOmega2 src.sigma ^ 2)) (((towerActM K) (src.x₀ * (powOmega2 (src.x₀ * src.tau))⁻¹)) (((towerActM K) (powOmega2 src.sigma ^ 2)⁻¹) (((towerActM K) (powOmega2 (src.x₀ * src.tau))) (((towerActM K) src.x₀⁻¹) (((towerActM K) (powOmega2 src.sigma ^ 2)) (((towerActM K) (powOmega2 (src.x₀ * src.tau))) (((towerActM K) src.x₀⁻¹) m))))))))) = ((towerActM K) (src.x₀ * (powOmega2 (src.x₀ * src.tau))⁻¹)) (((towerActM K) (src.x₀ * (powOmega2 (src.x₀ * src.tau))⁻¹)) (((towerActM K) (src.x₀ * (powOmega2 (src.x₀ * src.tau))⁻¹)) (((towerActM K) (powOmega2 src.sigma ^ 2)⁻¹) (((towerActM K) (src.x₀ * (powOmega2 (src.x₀ * src.tau))⁻¹)) (((towerActM K) (powOmega2 src.sigma ^ 2)) (((towerActM K) src.x₀⁻¹) (((towerActM K) ((powOmega2 src.sigma ^ 2)⁻¹ * src.x₀ * powOmega2 src.sigma ^ 2)⁻¹) m)))))))

theorem Q2Presentation.Induction.sec7_frattiniTrivial_elementaryMStageCandidateWildRowManuscriptH0ExpandedPrimitiveTowerActionSixFactorD0ExpandedRawInlineSevenActionTailTripleInverseCancellationHcNestedDgInvD0SplitApplyPointwiseData {p : FramedPair} {h : ¬ScalarTerminal p} {chief : FirstNonScalarChiefFactor p} {K : KernelMinimal chief.toNonScalarChiefFactor} {hR : Nat.card ↥(kernelFrattini K) = 1} {child : Sec7FrattiniTrivialElementaryChildData p} (src : Sec7FrattiniTrivialElementaryMStageCandidateLowerSourceData h chief K hR child) (m : towerM K) : ((towerActM K) (powOmega2 src.sigma ^ 2)⁻¹) (((towerActM K) (powOmega2 (src.x₀ * src.tau) * src.x₀⁻¹)⁻¹) (((towerActM K) (powOmega2 src.sigma ^ 2)) (((towerActM K) (powOmega2 (src.x₀ * src.tau) * src.x₀⁻¹)⁻¹) (((towerActM K) (powOmega2 src.sigma ^ 2)⁻¹) (((towerActM K) (powOmega2 (src.x₀ * src.tau))) (((towerActM K) src.x₀⁻¹) (((towerActM K) (powOmega2 src.sigma ^ 2)) (((towerActM K) (powOmega2 (src.x₀ * src.tau))) (((towerActM K) src.x₀⁻¹) m))))))))) = ((towerActM K) (powOmega2 (src.x₀ * src.tau) * src.x₀⁻¹)⁻¹) (((towerActM K) (powOmega2 (src.x₀ * src.tau) * src.x₀⁻¹)⁻¹) (((towerActM K) (powOmega2 (src.x₀ * src.tau) * src.x₀⁻¹)⁻¹) (((towerActM K) (powOmega2 src.sigma ^ 2)⁻¹) (((towerActM K) (powOmega2 (src.x₀ * src.tau) * src.x₀⁻¹)⁻¹) (((towerActM K) (powOmega2 src.sigma ^ 2)) (((towerActM K) src.x₀⁻¹) (((towerActM K) ((powOmega2 src.sigma ^ 2)⁻¹ * src.x₀ * powOmega2 src.sigma ^ 2)⁻¹) m)))))))

theorem Q2Presentation.Induction.sec7_frattiniTrivial_elementaryMStageCandidateWildRowManuscriptH0ExpandedPrimitiveTowerActionSixFactorD0ExpandedRawInlineSevenActionTailTripleInverseCancellationHcNestedD0SplitApplyPointwiseData {p : FramedPair} {h : ¬ScalarTerminal p} {chief : FirstNonScalarChiefFactor p} {K : KernelMinimal chief.toNonScalarChiefFactor} {hR : Nat.card ↥(kernelFrattini K) = 1} {child : Sec7FrattiniTrivialElementaryChildData p} (src : Sec7FrattiniTrivialElementaryMStageCandidateLowerSourceData h chief K hR child) (m : towerM K) : ((towerActM K) ((powOmega2 src.sigma ^ 2)⁻¹ * (powOmega2 (src.x₀ * src.tau) * src.x₀⁻¹) * powOmega2 src.sigma ^ 2)⁻¹) (((towerActM K) (powOmega2 (src.x₀ * src.tau) * src.x₀⁻¹)⁻¹) (((towerActM K) (powOmega2 src.sigma ^ 2)⁻¹) (((towerActM K) (powOmega2 (src.x₀ * src.tau))) (((towerActM K) src.x₀⁻¹) (((towerActM K) (powOmega2 src.sigma ^ 2)) (((towerActM K) (powOmega2 (src.x₀ * src.tau))) (((towerActM K) src.x₀⁻¹) m))))))) = ((towerActM K) (powOmega2 (src.x₀ * src.tau) * src.x₀⁻¹)⁻¹) (((towerActM K) (powOmega2 (src.x₀ * src.tau) * src.x₀⁻¹)⁻¹) (((towerActM K) (powOmega2 (src.x₀ * src.tau) * src.x₀⁻¹)⁻¹) (((towerActM K) ((powOmega2 src.sigma ^ 2)⁻¹ * (powOmega2 (src.x₀ * src.tau) * src.x₀⁻¹) * powOmega2 src.sigma ^ 2)⁻¹) (((towerActM K) src.x₀⁻¹) (((towerActM K) ((powOmega2 src.sigma ^ 2)⁻¹ * src.x₀ * powOmega2 src.sigma ^ 2)⁻¹) m)))))

Checkpoint303 primitive datum for h₀, now theorem-level from the checkpoint304 split of the remaining d_g⁻¹ action.

theorem Q2Presentation.Induction.sec7_frattiniTrivial_elementaryMStageCandidateWildRowManuscriptH0ExpandedPrimitiveTowerActionSixFactorD0ExpandedRawInlineSevenActionTailTripleInverseCancellationHcNestedApplyPointwiseData {p : FramedPair} {h : ¬ScalarTerminal p} {chief : FirstNonScalarChiefFactor p} {K : KernelMinimal chief.toNonScalarChiefFactor} {hR : Nat.card ↥(kernelFrattini K) = 1} {child : Sec7FrattiniTrivialElementaryChildData p} (src : Sec7FrattiniTrivialElementaryMStageCandidateLowerSourceData h chief K hR child) (m : towerM K) : ((towerActM K) ((powOmega2 src.sigma ^ 2)⁻¹ * (powOmega2 (src.x₀ * src.tau) * src.x₀⁻¹) * powOmega2 src.sigma ^ 2)⁻¹) (((towerActM K) (powOmega2 (src.x₀ * src.tau) * src.x₀⁻¹)⁻¹) (((towerActM K) ((powOmega2 src.sigma ^ 2)⁻¹ * (powOmega2 (src.x₀ * src.tau) * src.x₀⁻¹) * powOmega2 src.sigma ^ 2)) (((towerActM K) (powOmega2 (src.x₀ * src.tau) * src.x₀⁻¹)) m))) = ((towerActM K) (powOmega2 (src.x₀ * src.tau) * src.x₀⁻¹)⁻¹) (((towerActM K) (powOmega2 (src.x₀ * src.tau) * src.x₀⁻¹)⁻¹) (((towerActM K) (powOmega2 (src.x₀ * src.tau) * src.x₀⁻¹)⁻¹) (((towerActM K) ((powOmega2 src.sigma ^ 2)⁻¹ * (powOmega2 (src.x₀ * src.tau) * src.x₀⁻¹) * powOmega2 src.sigma ^ 2)⁻¹) (((towerActM K) src.x₀⁻¹) (((towerActM K) ((powOmega2 src.sigma ^ 2)⁻¹ * src.x₀ * powOmega2 src.sigma ^ 2)⁻¹) m)))))

Checkpoint302 primitive datum for h₀, now theorem-level from the checkpoint303 pointwise split of the two visible non-inverse d₀ actions.

theorem Q2Presentation.Induction.sec7_frattiniTrivial_elementaryMStageCandidateWildRowManuscriptH0ExpandedPrimitiveTowerActionSixFactorD0ExpandedRawInlineSevenActionTailTripleInverseCancellationFullyInlinedCoreApplyPointwiseData {p : FramedPair} {h : ¬ScalarTerminal p} {chief : FirstNonScalarChiefFactor p} {K : KernelMinimal chief.toNonScalarChiefFactor} {hR : Nat.card ↥(kernelFrattini K) = 1} {child : Sec7FrattiniTrivialElementaryChildData p} (src : Sec7FrattiniTrivialElementaryMStageCandidateLowerSourceData h chief K hR child) (m : towerM K) : ((towerActM K) (((powOmega2 src.sigma ^ 2)⁻¹ * (powOmega2 (src.x₀ * src.tau) * src.x₀⁻¹) * powOmega2 src.sigma ^ 2)⁻¹ * (powOmega2 (src.x₀ * src.tau) * src.x₀⁻¹)⁻¹ * ((powOmega2 src.sigma ^ 2)⁻¹ * (powOmega2 (src.x₀ * src.tau) * src.x₀⁻¹) * powOmega2 src.sigma ^ 2) * (powOmega2 (src.x₀ * src.tau) * src.x₀⁻¹))) m = ((towerActM K) (powOmega2 (src.x₀ * src.tau) * src.x₀⁻¹)⁻¹) (((towerActM K) (powOmega2 (src.x₀ * src.tau) * src.x₀⁻¹)⁻¹) (((towerActM K) (powOmega2 (src.x₀ * src.tau) * src.x₀⁻¹)⁻¹) (((towerActM K) ((powOmega2 src.sigma ^ 2)⁻¹ * (powOmega2 (src.x₀ * src.tau) * src.x₀⁻¹) * powOmega2 src.sigma ^ 2)⁻¹) (((towerActM K) src.x₀⁻¹) (((towerActM K) ((powOmega2 src.sigma ^ 2)⁻¹ * src.x₀ * powOmega2 src.sigma ^ 2)⁻¹) m)))))

Checkpoint301 primitive datum for h₀, now theorem-level from the checkpoint302 nested-action commutator row.

theorem Q2Presentation.Induction.sec7_frattiniTrivial_elementaryMStageCandidateWildRowManuscriptH0ExpandedPrimitiveTowerActionSixFactorD0ExpandedRawInlineSevenActionTailTripleInverseCancellationCoreApplyPointwiseData {p : FramedPair} {h : ¬ScalarTerminal p} {chief : FirstNonScalarChiefFactor p} {K : KernelMinimal chief.toNonScalarChiefFactor} {hR : Nat.card ↥(kernelFrattini K) = 1} {child : Sec7FrattiniTrivialElementaryChildData p} (src : Sec7FrattiniTrivialElementaryMStageCandidateLowerSourceData h chief K hR child) (m : towerM K) : have G := powOmega2 src.sigma ^ 2; have D := powOmega2 (src.x₀ * src.tau) * src.x₀⁻¹; have A := G⁻¹ * src.x₀ * G; have C := G⁻¹ * D * G; have Hc := C⁻¹ * D⁻¹ * C * D; ((towerActM K) Hc) m = ((towerActM K) D⁻¹) (((towerActM K) D⁻¹) (((towerActM K) D⁻¹) (((towerActM K) C⁻¹) (((towerActM K) src.x₀⁻¹) (((towerActM K) A⁻¹) m)))))

Checkpoint300 primitive datum for h₀, now theorem-level from the checkpoint301 fully inlined tail-triple row.

theorem Q2Presentation.Induction.sec7_frattiniTrivial_elementaryMStageCandidateWildRowManuscriptH0ExpandedPrimitiveTowerActionSixFactorD0ExpandedRawInlineSevenActionHeadInverseTailCancellationApplyPointwiseData {p : FramedPair} {h : ¬ScalarTerminal p} {chief : FirstNonScalarChiefFactor p} {K : KernelMinimal chief.toNonScalarChiefFactor} {hR : Nat.card ↥(kernelFrattini K) = 1} {child : Sec7FrattiniTrivialElementaryChildData p} (src : Sec7FrattiniTrivialElementaryMStageCandidateLowerSourceData h chief K hR child) (m : towerM K) : have G := powOmega2 src.sigma ^ 2; have D := powOmega2 (src.x₀ * src.tau) * src.x₀⁻¹; have A := G⁻¹ * src.x₀ * G; have C := G⁻¹ * D * G; have Hc := C⁻¹ * D⁻¹ * C * D; ((towerActM K) D) (((towerActM K) D) (((towerActM K) D) (((towerActM K) Hc) m))) = ((towerActM K) C⁻¹) (((towerActM K) src.x₀⁻¹) (((towerActM K) A⁻¹) m))

Checkpoint299 primitive datum for h₀, now theorem-level from the checkpoint300 tail-triple inverse cancellation core.

theorem Q2Presentation.Induction.sec7_frattiniTrivial_elementaryMStageCandidateWildRowManuscriptH0ExpandedPrimitiveTowerActionSixFactorD0ExpandedRawInlineSevenActionValueEqPointTowerFullyInlinedApplyPointwiseData {p : FramedPair} {h : ¬ScalarTerminal p} {chief : FirstNonScalarChiefFactor p} {K : KernelMinimal chief.toNonScalarChiefFactor} {hR : Nat.card ↥(kernelFrattini K) = 1} {child : Sec7FrattiniTrivialElementaryChildData p} (src : Sec7FrattiniTrivialElementaryMStageCandidateLowerSourceData h chief K hR child) (m : towerM K) : ((towerActM K) ((powOmega2 src.sigma ^ 2)⁻¹ * src.x₀ * powOmega2 src.sigma ^ 2)) (((towerActM K) src.x₀) (((towerActM K) ((powOmega2 src.sigma ^ 2)⁻¹ * (powOmega2 (src.x₀ * src.tau) * src.x₀⁻¹) * powOmega2 src.sigma ^ 2)) (((towerActM K) (powOmega2 (src.x₀ * src.tau) * src.x₀⁻¹)) (((towerActM K) (powOmega2 (src.x₀ * src.tau) * src.x₀⁻¹)) (((towerActM K) (powOmega2 (src.x₀ * src.tau) * src.x₀⁻¹)) (((towerActM K) (((powOmega2 src.sigma ^ 2)⁻¹ * (powOmega2 (src.x₀ * src.tau) * src.x₀⁻¹) * powOmega2 src.sigma ^ 2)⁻¹ * (powOmega2 (src.x₀ * src.tau) * src.x₀⁻¹)⁻¹ * ((powOmega2 src.sigma ^ 2)⁻¹ * (powOmega2 (src.x₀ * src.tau) * src.x₀⁻¹) * powOmega2 src.sigma ^ 2) * (powOmega2 (src.x₀ * src.tau) * src.x₀⁻¹))) m)))))) = m

Checkpoint298 primitive atom for h₀, now theorem-level from the checkpoint299 head-inverse/tail-cancellation datum.

theorem Q2Presentation.Induction.sec7_frattiniTrivial_elementaryMStageCandidateWildRowManuscriptH0ExpandedPrimitiveTowerActionSixFactorD0ExpandedRawInlineSevenActionValueEqPointTowerApplyPointwiseData {p : FramedPair} {h : ¬ScalarTerminal p} {chief : FirstNonScalarChiefFactor p} {K : KernelMinimal chief.toNonScalarChiefFactor} {hR : Nat.card ↥(kernelFrattini K) = 1} {child : Sec7FrattiniTrivialElementaryChildData p} (src : Sec7FrattiniTrivialElementaryMStageCandidateLowerSourceData h chief K hR child) (m : towerM K) : have G := powOmega2 src.sigma ^ 2; have D := powOmega2 (src.x₀ * src.tau) * src.x₀⁻¹; ((towerActM K) (G⁻¹ * src.x₀ * G)) (((towerActM K) src.x₀) (((towerActM K) (G⁻¹ * D * G)) (((towerActM K) D) (((towerActM K) D) (((towerActM K) D) (((towerActM K) ((G⁻¹ * D * G)⁻¹ * D⁻¹ * (G⁻¹ * D * G) * D)) m)))))) = m

Checkpoint297 primitive atom for h₀, now theorem-level from the fully inlined checkpoint298 pointwise value row.

theorem Q2Presentation.Induction.sec7_frattiniTrivial_elementaryMStageCandidateWildRowManuscriptH0ExpandedPrimitiveTowerActionSixFactorD0ExpandedRawInlineSevenActionDeviationZeroTowerApplyPointwiseData {p : FramedPair} {h : ¬ScalarTerminal p} {chief : FirstNonScalarChiefFactor p} {K : KernelMinimal chief.toNonScalarChiefFactor} {hR : Nat.card ↥(kernelFrattini K) = 1} {child : Sec7FrattiniTrivialElementaryChildData p} (src : Sec7FrattiniTrivialElementaryMStageCandidateLowerSourceData h chief K hR child) (m : towerM K) : have G := powOmega2 src.sigma ^ 2; have D := powOmega2 (src.x₀ * src.tau) * src.x₀⁻¹; ((towerActM K) (G⁻¹ * src.x₀ * G)) (((towerActM K) src.x₀) (((towerActM K) (G⁻¹ * D * G)) (((towerActM K) D) (((towerActM K) D) (((towerActM K) D) (((towerActM K) ((G⁻¹ * D * G)⁻¹ * D⁻¹ * (G⁻¹ * D * G) * D)) m)))))) - m = 0

Checkpoint296 h₀ zero-deviation atom, now theorem-level from the literal raw seven-action value equality. This exposes the point-value computation as the new leaf instead of assuming the subtraction-normalized form.

theorem Q2Presentation.Induction.sec7_frattiniTrivial_elementaryMStageCandidateWildRowManuscriptH0ExpandedPrimitiveTowerActionSixFactorD0ExpandedRawNestedD0SquareExpandedFixesPointTowerApplyPointwiseData {p : FramedPair} {h : ¬ScalarTerminal p} {chief : FirstNonScalarChiefFactor p} {K : KernelMinimal chief.toNonScalarChiefFactor} {hR : Nat.card ↥(kernelFrattini K) = 1} {child : Sec7FrattiniTrivialElementaryChildData p} (src : Sec7FrattiniTrivialElementaryMStageCandidateLowerSourceData h chief K hR child) (m : towerM K) : have sigma2 := powOmega2 src.sigma; have g0 := sigma2 ^ 2; have d0 := powOmega2 (src.x₀ * src.tau) * src.x₀⁻¹; have dg := g0⁻¹ * d0 * g0; have hc := dg⁻¹ * d0⁻¹ * dg * d0; have a := g0⁻¹ * src.x₀ * g0; have b := src.x₀; ((towerActM K) a) (((towerActM K) b) (((towerActM K) dg) (((towerActM K) d0) (((towerActM K) d0) (((towerActM K) d0) (((towerActM K) hc) m)))))) = m

Checkpoint295 lower atom for h₀, now theorem-level from the still more literal zero-deviation seven-action atom above.

theorem Q2Presentation.Induction.sec7_frattiniTrivial_elementaryMStageCandidateWildRowManuscriptH0ExpandedPrimitiveTowerActionSixFactorD0ExpandedRawNestedFixesPointTowerApplyPointwiseData {p : FramedPair} {h : ¬ScalarTerminal p} {chief : FirstNonScalarChiefFactor p} {K : KernelMinimal chief.toNonScalarChiefFactor} {hR : Nat.card ↥(kernelFrattini K) = 1} {child : Sec7FrattiniTrivialElementaryChildData p} (src : Sec7FrattiniTrivialElementaryMStageCandidateLowerSourceData h chief K hR child) (m : towerM K) : have sigma2 := powOmega2 src.sigma; have g0 := sigma2 ^ 2; have d0 := powOmega2 (src.x₀ * src.tau) * src.x₀⁻¹; have dg := g0⁻¹ * d0 * g0; have hc := dg⁻¹ * d0⁻¹ * dg * d0; have a := g0⁻¹ * src.x₀ * g0; have b := src.x₀; have d0sq := d0 ^ 2; ((towerActM K) a) (((towerActM K) b) (((towerActM K) dg) (((towerActM K) d0) (((towerActM K) d0sq) (((towerActM K) hc) m))))) = m

Checkpoint294 nested six-factor datum, now theorem-level from the still more literal pointwise atom where the d₀² action has been expanded into two d₀ actions.

theorem Q2Presentation.Induction.sec7_frattiniTrivial_elementaryMStageCandidateWildRowManuscriptH0ExpandedPrimitiveTowerActionSixFactorD0ExpandedRawProductFixesPointTowerApplyPointwiseData {p : FramedPair} {h : ¬ScalarTerminal p} {chief : FirstNonScalarChiefFactor p} {K : KernelMinimal chief.toNonScalarChiefFactor} {hR : Nat.card ↥(kernelFrattini K) = 1} {child : Sec7FrattiniTrivialElementaryChildData p} (src : Sec7FrattiniTrivialElementaryMStageCandidateLowerSourceData h chief K hR child) (m : towerM K) : have sigma2 := powOmega2 src.sigma; have g0 := sigma2 ^ 2; have d0 := powOmega2 (src.x₀ * src.tau) * src.x₀⁻¹; have dg := g0⁻¹ * d0 * g0; have hc := dg⁻¹ * d0⁻¹ * dg * d0; have a := g0⁻¹ * src.x₀ * g0; have b := src.x₀; have d0sq := d0 ^ 2; ((towerActM K) (a * b * dg * d0 * d0sq * hc)) m = m

Checkpoint293 product-action fixed-point atom for h₀, now theorem-level from the literal nested six-factor action datum.

theorem Q2Presentation.Induction.sec7_frattiniTrivial_elementaryMStageCandidateWildRowManuscriptH0ExpandedPrimitiveTowerActionSixFactorD0ExpandedRawProductFoxRowTowerApplyPointwiseData {p : FramedPair} {h : ¬ScalarTerminal p} {chief : FirstNonScalarChiefFactor p} {K : KernelMinimal chief.toNonScalarChiefFactor} {hR : Nat.card ↥(kernelFrattini K) = 1} {child : Sec7FrattiniTrivialElementaryChildData p} (src : Sec7FrattiniTrivialElementaryMStageCandidateLowerSourceData h chief K hR child) (m : towerM K) : have sigma2 := powOmega2 src.sigma; have g0 := sigma2 ^ 2; have d0 := powOmega2 (src.x₀ * src.tau) * src.x₀⁻¹; have dg := g0⁻¹ * d0 * g0; have hc := dg⁻¹ * d0⁻¹ * dg * d0; have a := g0⁻¹ * src.x₀ * g0; have b := src.x₀; have d0sq := d0 ^ 2; ((towerActM K) a) m - m + ((towerActM K) a) (((towerActM K) b) m - m) + ((towerActM K) a) (((towerActM K) b) (((towerActM K) dg) m - m)) + ((towerActM K) a) (((towerActM K) b) (((towerActM K) dg) (((towerActM K) d0) m - m))) + ((towerActM K) a) (((towerActM K) b) (((towerActM K) dg) (((towerActM K) d0) (((towerActM K) d0sq) m - m)))) + ((towerActM K) a) (((towerActM K) b) (((towerActM K) dg) (((towerActM K) d0) (((towerActM K) d0sq) (((towerActM K) hc) m - m))))) = 0

Checkpoint292 raw h₀ six-factor Fox row, now theorem-level from the smaller product-fixed-point datum and the formal six-factor product rule.

theorem Q2Presentation.Induction.sec7_frattiniTrivial_elementaryMStageCandidateWildRowManuscriptH0ExpandedPrimitiveTowerActionSixFactorD0ExpandedRawProductFoxRowApplyPointwiseData {p : FramedPair} {h : ¬ScalarTerminal p} {chief : FirstNonScalarChiefFactor p} {K : KernelMinimal chief.toNonScalarChiefFactor} {hR : Nat.card ↥(kernelFrattini K) = 1} {child : Sec7FrattiniTrivialElementaryChildData p} (src : Sec7FrattiniTrivialElementaryMStageCandidateLowerSourceData h chief K hR child) (m : towerM K) : have sigma2 := powOmega2 src.sigma; have g0 := sigma2 ^ 2; have d0 := powOmega2 (src.x₀ * src.tau) * src.x₀⁻¹; have dg := g0⁻¹ * d0 * g0; have hc := dg⁻¹ * d0⁻¹ * dg * d0; have a := g0⁻¹ * src.x₀ * g0; have b := src.x₀; have d0sq := d0 ^ 2; have ρ := sec7MStageCandidateActionEnd K; (ρ a - 1 + ρ a * (ρ b - 1) + ρ a * ρ b * (ρ dg - 1) + ρ a * ρ b * ρ dg * (ρ d0 - 1) + ρ a * ρ b * ρ dg * ρ d0 * (ρ d0sq - 1) + ρ a * ρ b * ρ dg * ρ d0 * ρ d0sq * (ρ hc - 1)) m = 0

Checkpoint291 h₀ pointwise endomorphism row, now theorem-level from the raw towerActM six-factor pointwise row.

theorem Q2Presentation.Induction.sec7_frattiniTrivial_elementaryMStageCandidateWildRowManuscriptH0ExpandedPrimitiveTowerActionSixFactorD0ExpandedRawProductFoxRowOperatorData {p : FramedPair} {h : ¬ScalarTerminal p} {chief : FirstNonScalarChiefFactor p} {K : KernelMinimal chief.toNonScalarChiefFactor} {hR : Nat.card ↥(kernelFrattini K) = 1} {child : Sec7FrattiniTrivialElementaryChildData p} (src : Sec7FrattiniTrivialElementaryMStageCandidateLowerSourceData h chief K hR child) : have sigma2 := powOmega2 src.sigma; have g0 := sigma2 ^ 2; have d0 := powOmega2 (src.x₀ * src.tau) * src.x₀⁻¹; have dg := g0⁻¹ * d0 * g0; have hc := dg⁻¹ * d0⁻¹ * dg * d0; have a := g0⁻¹ * src.x₀ * g0; have b := src.x₀; have d0sq := d0 ^ 2; have ρ := sec7MStageCandidateActionEnd K; ρ a - 1 + ρ a * (ρ b - 1) + ρ a * ρ b * (ρ dg - 1) + ρ a * ρ b * ρ dg * (ρ d0 - 1) + ρ a * ρ b * ρ dg * ρ d0 * (ρ d0sq - 1) + ρ a * ρ b * ρ dg * ρ d0 * ρ d0sq * (ρ hc - 1) = 0

Former checkpoint290 h₀ operator row, now theorem-level from the pointwise applied six-factor row.

theorem Q2Presentation.Induction.sec7_frattiniTrivial_elementaryMStageCandidateWildRowManuscriptH0ExpandedPrimitiveTowerActionSixFactorD0ExpandedRawProductOperatorData {p : FramedPair} {h : ¬ScalarTerminal p} {chief : FirstNonScalarChiefFactor p} {K : KernelMinimal chief.toNonScalarChiefFactor} {hR : Nat.card ↥(kernelFrattini K) = 1} {child : Sec7FrattiniTrivialElementaryChildData p} (src : Sec7FrattiniTrivialElementaryMStageCandidateLowerSourceData h chief K hR child) : have sigma2 := powOmega2 src.sigma; have g0 := sigma2 ^ 2; have d0 := powOmega2 (src.x₀ * src.tau) * src.x₀⁻¹; have dg := g0⁻¹ * d0 * g0; have hc := dg⁻¹ * d0⁻¹ * dg * d0; have a := g0⁻¹ * src.x₀ * g0; have b := src.x₀; have d0sq := d0 ^ 2; have A := ↑((towerActM K) (a * b * dg * d0 * d0sq * hc)); A = 1

Checkpoint289 lower atom for h₀, now theorem-level from the still-lower six-factor Fox row.

theorem Q2Presentation.Induction.sec7_frattiniTrivial_elementaryMStageCandidateWildRowManuscriptH0ExpandedPrimitiveTowerActionSixFactorD0ExpandedRawProductValueApplyPointwiseData {p : FramedPair} {h : ¬ScalarTerminal p} {chief : FirstNonScalarChiefFactor p} {K : KernelMinimal chief.toNonScalarChiefFactor} {hR : Nat.card ↥(kernelFrattini K) = 1} {child : Sec7FrattiniTrivialElementaryChildData p} (src : Sec7FrattiniTrivialElementaryMStageCandidateLowerSourceData h chief K hR child) (m : towerM K) : have sigma2 := powOmega2 src.sigma; have g0 := sigma2 ^ 2; have d0 := powOmega2 (src.x₀ * src.tau) * src.x₀⁻¹; have dg := g0⁻¹ * d0 * g0; have hc := dg⁻¹ * d0⁻¹ * dg * d0; have a := g0⁻¹ * src.x₀ * g0; have b := src.x₀; have d0sq := d0 ^ 2; ((towerActM K) (a * b * dg * d0 * d0sq * hc)) m = m

Checkpoint287 pointwise h₀ residual, now theorem-level from the checkpoint288 operator residual. The proof is intentionally pointwise and avoids any broad simp over the six manuscript factors.

theorem Q2Presentation.Induction.sec7_frattiniTrivial_elementaryMStageCandidateWildRowManuscriptH0ExpandedPrimitiveTowerActionSixFactorD0ExpandedRawNestedValueApplyPointwiseData {p : FramedPair} {h : ¬ScalarTerminal p} {chief : FirstNonScalarChiefFactor p} {K : KernelMinimal chief.toNonScalarChiefFactor} {hR : Nat.card ↥(kernelFrattini K) = 1} {child : Sec7FrattiniTrivialElementaryChildData p} (src : Sec7FrattiniTrivialElementaryMStageCandidateLowerSourceData h chief K hR child) (m : towerM K) : have sigma2 := powOmega2 src.sigma; have g0 := sigma2 ^ 2; have d0 := powOmega2 (src.x₀ * src.tau) * src.x₀⁻¹; ((towerActM K) (g0⁻¹ * src.x₀ * g0)) (((towerActM K) src.x₀) (((towerActM K) (g0⁻¹ * d0 * g0)) (((towerActM K) d0) (((towerActM K) (d0 ^ 2)) (((towerActM K) ((g0⁻¹ * d0 * g0)⁻¹ * d0⁻¹ * (g0⁻¹ * d0 * g0) * d0)) m))))) = m

Checkpoint287 lowering for the rooted h₀ tower-action atom. The residual is now the single concrete action of the literal six-factor manuscript product, rather than the fully nested application of the six primitive actions. The old checkpoint286 pointwise nested statement is recovered only from functoriality of towerActM K : towerC K →* (towerM K ≃ₗ[ZMod 2] towerM K).

theorem Q2Presentation.Induction.sec7_frattiniTrivial_elementaryMStageCandidateWildRowManuscriptH0ExpandedPrimitiveActionSixFactorD0ExpandedRawNestedValueApplyPointwiseData {p : FramedPair} {h : ¬ScalarTerminal p} {chief : FirstNonScalarChiefFactor p} {K : KernelMinimal chief.toNonScalarChiefFactor} {hR : Nat.card ↥(kernelFrattini K) = 1} {child : Sec7FrattiniTrivialElementaryChildData p} (src : Sec7FrattiniTrivialElementaryMStageCandidateLowerSourceData h chief K hR child) (m : towerM K) : have ρ := sec7MStageCandidateActionEnd K; have sigma2 := powOmega2 src.sigma; have g0 := sigma2 ^ 2; have d0 := powOmega2 (src.x₀ * src.tau) * src.x₀⁻¹; (ρ (g0⁻¹ * src.x₀ * g0)) ((ρ src.x₀) ((ρ (g0⁻¹ * d0 * g0)) ((ρ d0) ((ρ (d0 ^ 2)) ((ρ ((g0⁻¹ * d0 * g0)⁻¹ * d0⁻¹ * (g0⁻¹ * d0 * g0) * d0)) m))))) = m

Checkpoint284 h₀ residual, now theorem-level from the checkpoint285 concrete towerActM six-factor statement above.

This proof deliberately avoids the failed checkpoint285 simp bridge through nested endomorphism coercions. Instead it keeps all manuscript group factors opaque and uses a single change to forget the applied endomorphism action pointwise, matching the concrete tower-action statement exactly.

theorem Q2Presentation.Induction.sec7_frattiniTrivial_elementaryMStageCandidateWildRowManuscriptH0ExpandedPrimitiveActionSixFactorD0ExpandedProductFixesPointData {p : FramedPair} {h : ¬ScalarTerminal p} {chief : FirstNonScalarChiefFactor p} {K : KernelMinimal chief.toNonScalarChiefFactor} {hR : Nat.card ↥(kernelFrattini K) = 1} {child : Sec7FrattiniTrivialElementaryChildData p} (src : Sec7FrattiniTrivialElementaryMStageCandidateLowerSourceData h chief K hR child) (m : towerM K) : have ρ := sec7MStageCandidateActionEnd K; have sigma2 := powOmega2 src.sigma; have g0 := sigma2 ^ 2; have d0 := powOmega2 (src.x₀ * src.tau) * src.x₀⁻¹; have dg := g0⁻¹ * d0 * g0; have hc := dg⁻¹ * d0⁻¹ * dg * d0; have a := g0⁻¹ * src.x₀ * g0; have b := src.x₀; have d0sq := d0 ^ 2; have prod := ρ a * ρ b * ρ dg * ρ d0 * ρ d0sq * ρ hc; prod m = m

Checkpoint282 lower atom for h₀, now theorem-level from the checkpoint283 raw nested primitive-action value.

theorem Q2Presentation.Induction.sec7_frattiniTrivial_elementaryMStageCandidateWildRowManuscriptH0ExpandedPrimitiveActionSixFactorD0ExpandedRawApplyPointwiseData {p : FramedPair} {h : ¬ScalarTerminal p} {chief : FirstNonScalarChiefFactor p} {K : KernelMinimal chief.toNonScalarChiefFactor} {hR : Nat.card ↥(kernelFrattini K) = 1} {child : Sec7FrattiniTrivialElementaryChildData p} (src : Sec7FrattiniTrivialElementaryMStageCandidateLowerSourceData h chief K hR child) (m : towerM K) : have ρ := sec7MStageCandidateActionEnd K; have sigma2 := powOmega2 src.sigma; have g0 := sigma2 ^ 2; have d0 := powOmega2 (src.x₀ * src.tau) * src.x₀⁻¹; have dg := g0⁻¹ * d0 * g0; have hc := dg⁻¹ * d0⁻¹ * dg * d0; have a := g0⁻¹ * src.x₀ * g0; have b := src.x₀; have d0sq := d0 ^ 2; have prod := ρ a * ρ b * ρ dg * ρ d0 * ρ d0sq * ρ hc; prod m - m = 0

Checkpoint281 lower atom for h₀, now theorem-level from the checkpoint282 primitive fixed-point statement.

theorem Q2Presentation.Induction.sec7_frattiniTrivial_elementaryMStageCandidateWildRowManuscriptH0ExpandedPrimitiveActionSixFactorD0ExpandedSubOneApplyPointwiseData {p : FramedPair} {h : ¬ScalarTerminal p} {chief : FirstNonScalarChiefFactor p} {K : KernelMinimal chief.toNonScalarChiefFactor} {hR : Nat.card ↥(kernelFrattini K) = 1} {child : Sec7FrattiniTrivialElementaryChildData p} (src : Sec7FrattiniTrivialElementaryMStageCandidateLowerSourceData h chief K hR child) (m : towerM K) : have ρ := sec7MStageCandidateActionEnd K; have sigma2 := powOmega2 src.sigma; have g0 := sigma2 ^ 2; have d0 := powOmega2 (src.x₀ * src.tau) * src.x₀⁻¹; have dg := g0⁻¹ * d0 * g0; have hc := dg⁻¹ * d0⁻¹ * dg * d0; have a := g0⁻¹ * src.x₀ * g0; have b := src.x₀; have d0sq := d0 ^ 2; have prod := ρ a * ρ b * ρ dg * ρ d0 * ρ d0sq * ρ hc; (prod - 1) m = 0

Checkpoint279 lower atom for h₀, now theorem-level from the checkpoint281 raw pointwise displacement form.

theorem Q2Presentation.Induction.sec7_frattiniTrivial_elementaryMStageCandidateWildRowManuscriptH0ExpandedPrimitiveActionSixFactorD0ExpandedNestedApplyPointwiseData {p : FramedPair} {h : ¬ScalarTerminal p} {chief : FirstNonScalarChiefFactor p} {K : KernelMinimal chief.toNonScalarChiefFactor} {hR : Nat.card ↥(kernelFrattini K) = 1} {child : Sec7FrattiniTrivialElementaryChildData p} (src : Sec7FrattiniTrivialElementaryMStageCandidateLowerSourceData h chief K hR child) (m : towerM K) : have ρ := sec7MStageCandidateActionEnd K; have sigma2 := powOmega2 src.sigma; have g0 := sigma2 ^ 2; have d0 := powOmega2 (src.x₀ * src.tau) * src.x₀⁻¹; (ρ (g0⁻¹ * src.x₀ * g0)) ((ρ src.x₀) ((ρ (g0⁻¹ * d0 * g0)) ((ρ d0) ((ρ (d0 ^ 2)) ((ρ ((g0⁻¹ * d0 * g0)⁻¹ * d0⁻¹ * (g0⁻¹ * d0 * g0) * d0)) m))))) = m

Checkpoint278 primitive nested h₀ action atom, now theorem-level from the smaller six-factor deviation identity.

theorem Q2Presentation.Induction.sec7_frattiniTrivial_elementaryMStageCandidateWildRowManuscriptH0ExpandedPrimitiveActionSixFactorNestedApplyPointwiseData {p : FramedPair} {h : ¬ScalarTerminal p} {chief : FirstNonScalarChiefFactor p} {K : KernelMinimal chief.toNonScalarChiefFactor} {hR : Nat.card ↥(kernelFrattini K) = 1} {child : Sec7FrattiniTrivialElementaryChildData p} (src : Sec7FrattiniTrivialElementaryMStageCandidateLowerSourceData h chief K hR child) (m : towerM K) : have ρ := sec7MStageCandidateActionEnd K; have sigma2 := powOmega2 src.sigma; have g0 := sigma2 ^ 2; have d0 := powOmega2 (src.x₀ * src.tau) * src.x₀⁻¹; have dg := g0⁻¹ * d0 * g0; have hc := dg⁻¹ * d0⁻¹ * dg * d0; have a := g0⁻¹ * src.x₀ * g0; have b := src.x₀; have d0sq := d0 ^ 2; (ρ a) ((ρ b) ((ρ dg) ((ρ d0) ((ρ d0sq) ((ρ hc) m))))) = m

Checkpoint277 primitive six-factor nested-application atom, now theorem-level from the still more literal d₀-expanded nested action atom.

theorem Q2Presentation.Induction.sec7_frattiniTrivial_elementaryMStageCandidateWildRowManuscriptH0ExpandedPrimitiveActionSixFactorFixedPointwiseData {p : FramedPair} {h : ¬ScalarTerminal p} {chief : FirstNonScalarChiefFactor p} {K : KernelMinimal chief.toNonScalarChiefFactor} {hR : Nat.card ↥(kernelFrattini K) = 1} {child : Sec7FrattiniTrivialElementaryChildData p} (src : Sec7FrattiniTrivialElementaryMStageCandidateLowerSourceData h chief K hR child) (m : towerM K) : have ρ := sec7MStageCandidateActionEnd K; have sigma2 := powOmega2 src.sigma; have g0 := sigma2 ^ 2; have d0 := powOmega2 (src.x₀ * src.tau) * src.x₀⁻¹; have dg := g0⁻¹ * d0 * g0; have hc := dg⁻¹ * d0⁻¹ * dg * d0; have a := g0⁻¹ * src.x₀ * g0; have b := src.x₀; have d0sq := d0 ^ 2; (ρ a * ρ b * ρ dg * ρ d0 * ρ d0sq * ρ hc) m = m

Checkpoint276 primitive six-factor fixed-point atom for h₀, now theorem-level from the still more literal nested application of the six primitive actions.

theorem Q2Presentation.Induction.sec7_frattiniTrivial_elementaryMStageCandidateWildRowManuscriptH0ExpandedPrimitiveActionH0TrivialPointwiseData {p : FramedPair} {h : ¬ScalarTerminal p} {chief : FirstNonScalarChiefFactor p} {K : KernelMinimal chief.toNonScalarChiefFactor} {hR : Nat.card ↥(kernelFrattini K) = 1} {child : Sec7FrattiniTrivialElementaryChildData p} (src : Sec7FrattiniTrivialElementaryMStageCandidateLowerSourceData h chief K hR child) (m : towerM K) : have ρ := sec7MStageCandidateActionEnd K; have sigma2 := powOmega2 src.sigma; have g0 := sigma2 ^ 2; have d0 := powOmega2 (src.x₀ * src.tau) * src.x₀⁻¹; have dg := g0⁻¹ * d0 * g0; have hc := dg⁻¹ * d0⁻¹ * dg * d0; have a := g0⁻¹ * src.x₀ * g0; have b := src.x₀; have d0sq := d0 ^ 2; (ρ (a * b * dg * d0 * d0sq * hc) - 1) m = 0

Checkpoint275 primitive h₀ deviation row, now theorem-level from the lower fixed-point action product atom.

theorem Q2Presentation.Induction.sec7_frattiniTrivial_elementaryMStageCandidateWildRowManuscriptH0ExpandedPrimitiveActionSixFactorRowsPointwiseData {p : FramedPair} {h : ¬ScalarTerminal p} {chief : FirstNonScalarChiefFactor p} {K : KernelMinimal chief.toNonScalarChiefFactor} {hR : Nat.card ↥(kernelFrattini K) = 1} {child : Sec7FrattiniTrivialElementaryChildData p} (src : Sec7FrattiniTrivialElementaryMStageCandidateLowerSourceData h chief K hR child) (m : towerM K) : have ρ := sec7MStageCandidateActionEnd K; have sigma2 := powOmega2 src.sigma; have g0 := sigma2 ^ 2; have d0 := powOmega2 (src.x₀ * src.tau) * src.x₀⁻¹; have dg := g0⁻¹ * d0 * g0; have hc := dg⁻¹ * d0⁻¹ * dg * d0; have a := g0⁻¹ * src.x₀ * g0; have b := src.x₀; have d0sq := d0 ^ 2; (ρ a - 1 + ρ a * (ρ b - 1) + ρ (a * b) * (ρ dg - 1) + ρ (a * b * dg) * (ρ d0 - 1) + ρ (a * b * dg * d0) * (ρ d0sq - 1) + ρ (a * b * dg * d0 * d0sq) * (ρ hc - 1)) m = 0

Checkpoint274 pointwise primitive action-row atom for h₀, now theorem-level from triviality of the evaluated manuscript word h₀ plus the six-factor product rule.

theorem Q2Presentation.Induction.sec7_frattiniTrivial_elementaryMStageCandidateWildRowManuscriptH0ExpandedPrimitiveActionSixFactorRowsSumZeroData {p : FramedPair} {h : ¬ScalarTerminal p} {chief : FirstNonScalarChiefFactor p} {K : KernelMinimal chief.toNonScalarChiefFactor} {hR : Nat.card ↥(kernelFrattini K) = 1} {child : Sec7FrattiniTrivialElementaryChildData p} (src : Sec7FrattiniTrivialElementaryMStageCandidateLowerSourceData h chief K hR child) : have ρ := sec7MStageCandidateActionEnd K; have sigma2 := powOmega2 src.sigma; have g0 := sigma2 ^ 2; have d0 := powOmega2 (src.x₀ * src.tau) * src.x₀⁻¹; have dg := g0⁻¹ * d0 * g0; have hc := dg⁻¹ * d0⁻¹ * dg * d0; have a := g0⁻¹ * src.x₀ * g0; have b := src.x₀; have d0sq := d0 ^ 2; ρ a - 1 + ρ a * (ρ b - 1) + ρ (a * b) * (ρ dg - 1) + ρ (a * b * dg) * (ρ d0 - 1) + ρ (a * b * dg * d0) * (ρ d0sq - 1) + ρ (a * b * dg * d0 * d0sq) * (ρ hc - 1) = 0

Former checkpoint273 primitive endomorphism-row atom for h₀, now theorem-level from the pointwise primitive row above.

theorem Q2Presentation.Induction.sec7_frattiniTrivial_elementaryMStageCandidateWildRowManuscriptH0ExpandedRawEvalSixFactorRowsSumZeroData {p : FramedPair} {h : ¬ScalarTerminal p} {chief : FirstNonScalarChiefFactor p} {K : KernelMinimal chief.toNonScalarChiefFactor} {hR : Nat.card ↥(kernelFrattini K) = 1} {child : Sec7FrattiniTrivialElementaryChildData p} (src : Sec7FrattiniTrivialElementaryMStageCandidateLowerSourceData h chief K hR child) : (sec7MStageCandidateActionEnd K) ((Q2Word.rconjExpr Q2Word.x0 Q2Word.g0).eval src.marking) - 1 + (sec7MStageCandidateActionEnd K) ((Q2Word.rconjExpr Q2Word.x0 Q2Word.g0).eval src.marking) * ((sec7MStageCandidateActionEnd K) (Q2Word.x0.eval src.marking) - 1) + (sec7MStageCandidateActionEnd K) (((Q2Word.rconjExpr Q2Word.x0 Q2Word.g0).mul Q2Word.x0).eval src.marking) * ((sec7MStageCandidateActionEnd K) (Q2Word.dg.eval src.marking) - 1) + (sec7MStageCandidateActionEnd K) ((((Q2Word.rconjExpr Q2Word.x0 Q2Word.g0).mul Q2Word.x0).mul Q2Word.dg).eval src.marking) * ((sec7MStageCandidateActionEnd K) (Q2Word.d0.eval src.marking) - 1) + (sec7MStageCandidateActionEnd K) (((((Q2Word.rconjExpr Q2Word.x0 Q2Word.g0).mul Q2Word.x0).mul Q2Word.dg).mul Q2Word.d0).eval src.marking) * ((sec7MStageCandidateActionEnd K) ((Q2Word.d0.powNat 2).eval src.marking) - 1) + (sec7MStageCandidateActionEnd K) ((((((Q2Word.rconjExpr Q2Word.x0 Q2Word.g0).mul Q2Word.x0).mul Q2Word.dg).mul Q2Word.d0).mul (Q2Word.d0.powNat 2)).eval src.marking) * ((sec7MStageCandidateActionEnd K) (Q2Word.hc.eval src.marking) - 1) = 0

Former checkpoint272 raw-evaluation six-row atom for h₀, now theorem-level from the direct primitive lower-quotient action statement above.

theorem Q2Presentation.Induction.sec7_frattiniTrivial_elementaryMStageCandidateWildRowManuscriptH0ExpandedCanonicalSixFactorRowsSumZeroData {p : FramedPair} {h : ¬ScalarTerminal p} {chief : FirstNonScalarChiefFactor p} {K : KernelMinimal chief.toNonScalarChiefFactor} {hR : Nat.card ↥(kernelFrattini K) = 1} {child : Sec7FrattiniTrivialElementaryChildData p} (src : Sec7FrattiniTrivialElementaryMStageCandidateLowerSourceData h chief K hR child) : sec7MStageCandidateWildRowManuscriptH0ConjX0G0CanonicalRow src + sec7MStageCandidateWildRowManuscriptH0X0CanonicalRow src + sec7MStageCandidateWildRowManuscriptH0DgCanonicalRow src + sec7MStageCandidateWildRowManuscriptH0D0CanonicalRow src + sec7MStageCandidateWildRowManuscriptH0D0SqCanonicalRow src + sec7MStageCandidateWildRowManuscriptH0HcCanonicalRow src = 0

Former checkpoint271 named six-factor row atom, now theorem-level from the lower raw GExpr.eval six-factor row statement.

theorem Q2Presentation.Induction.sec7_frattiniTrivial_elementaryMStageCandidateWildRowManuscriptH0ExpandedCanonicalHeadTailBlockRowsSumZeroData {p : FramedPair} {h : ¬ScalarTerminal p} {chief : FirstNonScalarChiefFactor p} {K : KernelMinimal chief.toNonScalarChiefFactor} {hR : Nat.card ↥(kernelFrattini K) = 1} {child : Sec7FrattiniTrivialElementaryChildData p} (src : Sec7FrattiniTrivialElementaryMStageCandidateLowerSourceData h chief K hR child) : have head := (sec7MStageCandidateActionEnd K) (sec7MStageCandidateWildRowManuscriptH0ConjX0G0Factor src * sec7MStageCandidateWildRowManuscriptH0X0Factor src * sec7MStageCandidateWildRowManuscriptH0DgFactor src); have tail := (sec7MStageCandidateActionEnd K) (sec7MStageCandidateWildRowManuscriptH0D0Factor src * sec7MStageCandidateWildRowManuscriptH0D0SqFactor src * sec7MStageCandidateWildRowManuscriptH0HcFactor src); head - 1 + head * (tail - 1) = 0

Former checkpoint270 h₀ head/tail block-row atom, now theorem-level from the six concrete canonical factor rows and the already-proved six-factor product rule.

theorem Q2Presentation.Induction.sec7_frattiniTrivial_elementaryMStageCandidateWildRowManuscriptH0ExpandedCanonicalHeadTailActionProductData {p : FramedPair} {h : ¬ScalarTerminal p} {chief : FirstNonScalarChiefFactor p} {K : KernelMinimal chief.toNonScalarChiefFactor} {hR : Nat.card ↥(kernelFrattini K) = 1} {child : Sec7FrattiniTrivialElementaryChildData p} (src : Sec7FrattiniTrivialElementaryMStageCandidateLowerSourceData h chief K hR child) : (sec7MStageCandidateActionEnd K) (sec7MStageCandidateWildRowManuscriptH0ConjX0G0Factor src * sec7MStageCandidateWildRowManuscriptH0X0Factor src * sec7MStageCandidateWildRowManuscriptH0DgFactor src) * (sec7MStageCandidateActionEnd K) (sec7MStageCandidateWildRowManuscriptH0D0Factor src * sec7MStageCandidateWildRowManuscriptH0D0SqFactor src * sec7MStageCandidateWildRowManuscriptH0HcFactor src) = 1

Former checkpoint269 h₀ head-tail action-product atom, now theorem-level from the lower block-row identity.

theorem Q2Presentation.Induction.sec7_frattiniTrivial_elementaryMStageCandidateWildRowManuscriptH0ExpandedCanonicalActionProductData {p : FramedPair} {h : ¬ScalarTerminal p} {chief : FirstNonScalarChiefFactor p} {K : KernelMinimal chief.toNonScalarChiefFactor} {hR : Nat.card ↥(kernelFrattini K) = 1} {child : Sec7FrattiniTrivialElementaryChildData p} (src : Sec7FrattiniTrivialElementaryMStageCandidateLowerSourceData h chief K hR child) : (sec7MStageCandidateActionEnd K) (sec7MStageCandidateWildRowManuscriptH0ConjX0G0Factor src) * (sec7MStageCandidateActionEnd K) (sec7MStageCandidateWildRowManuscriptH0X0Factor src) * (sec7MStageCandidateActionEnd K) (sec7MStageCandidateWildRowManuscriptH0DgFactor src) * (sec7MStageCandidateActionEnd K) (sec7MStageCandidateWildRowManuscriptH0D0Factor src) * (sec7MStageCandidateActionEnd K) (sec7MStageCandidateWildRowManuscriptH0D0SqFactor src) * (sec7MStageCandidateActionEnd K) (sec7MStageCandidateWildRowManuscriptH0HcFactor src) = 1

Former checkpoint268 h₀ six-action product, now theorem-level from the head/tail action identity.

theorem Q2Presentation.Induction.sec7_frattiniTrivial_elementaryMStageCandidateWildRowManuscriptH0ExpandedCanonicalActionData {p : FramedPair} {h : ¬ScalarTerminal p} {chief : FirstNonScalarChiefFactor p} {K : KernelMinimal chief.toNonScalarChiefFactor} {hR : Nat.card ↥(kernelFrattini K) = 1} {child : Sec7FrattiniTrivialElementaryChildData p} (src : Sec7FrattiniTrivialElementaryMStageCandidateLowerSourceData h chief K hR child) : (sec7MStageCandidateActionEnd K) (sec7MStageCandidateWildRowManuscriptH0Factor src) = 1

The h₀ action is trivial, theorem-level from the composable head/tail action identity.

axiom Q2Presentation.Induction.sec7_frattiniTrivial_elementaryMStageCandidateWildRowManuscriptU1InvOmegaNormCanonicalPrimitiveTowerActionX1InverseNormInverseValueRawBaseDeviationFormulaNormSummandDeviationMapSubOpenedTauReplicateListRecursiveNormSumAuxFinalStepDifferenceCoreClosedZeroLowerPrimitiveApplyPointwiseData {p : FramedPair} {h : ¬ScalarTerminal p} {chief : FirstNonScalarChiefFactor p} {K : KernelMinimal chief.toNonScalarChiefFactor} {hR : Nat.card ↥(kernelFrattini K) = 1} {child : Sec7FrattiniTrivialElementaryChildData p} (src : Sec7FrattiniTrivialElementaryMStageCandidateLowerSourceData h chief K hR child) (m : towerM K) : ((towerActM K) (powOmega2 (src.x₁ * src.tau))⁻¹) m - m - Q2Presentation.Induction.sec7TauReplicateOpenedNormSumAux✝ K src m (orderOf src.tau).pred - (Q2Presentation.Induction.sec7TowerActSeq✝ K (List.replicate (orderOf src.tau).pred src.tau) (((towerActM K) src.x₁) m) - Q2Presentation.Induction.sec7TowerActSeq✝¹ K (List.replicate (orderOf src.tau).pred src.tau) m) = 0

Checkpoint332 lower primitive for u₁⁻¹: keep only the final recursive norm-step difference core as a closed zero. The checkpoint331 value at succ (pred |τ|) is recovered by unfolding one step of sec7TauReplicateOpenedNormSumAux and solving the displayed additive equation.

theorem Q2Presentation.Induction.sec7_frattiniTrivial_elementaryMStageCandidateWildRowManuscriptU1InvOmegaNormCanonicalPrimitiveTowerActionX1InverseNormInverseValueRawBaseDeviationFormulaNormSummandDeviationMapSubOpenedTauReplicateListRecursiveNormSumAuxSuccPredValuePrimitiveApplyPointwiseData {p : FramedPair} {h : ¬ScalarTerminal p} {chief : FirstNonScalarChiefFactor p} {K : KernelMinimal chief.toNonScalarChiefFactor} {hR : Nat.card ↥(kernelFrattini K) = 1} {child : Sec7FrattiniTrivialElementaryChildData p} (src : Sec7FrattiniTrivialElementaryMStageCandidateLowerSourceData h chief K hR child) (m : towerM K) : ((towerActM K) (powOmega2 (src.x₁ * src.tau))⁻¹) m - m = Q2Presentation.Induction.sec7TauReplicateOpenedNormSumAux✝ K src m (orderOf src.tau).pred.succ

Checkpoint332 bridge recovering the checkpoint331 u₁⁻¹ recursive-norm value from the lower final-step closed-zero primitive.

theorem Q2Presentation.Induction.sec7_frattiniTrivial_elementaryMStageCandidateWildRowManuscriptU1InvOmegaNormCanonicalPrimitiveTowerActionX1InverseNormInverseValueRawBaseDeviationFormulaNormSummandDeviationMapSubOpenedTauReplicateListRecursiveNormSumAuxFinalStepDifferenceCoreClosedZeroPrimitiveApplyPointwiseData {p : FramedPair} {h : ¬ScalarTerminal p} {chief : FirstNonScalarChiefFactor p} {K : KernelMinimal chief.toNonScalarChiefFactor} {hR : Nat.card ↥(kernelFrattini K) = 1} {child : Sec7FrattiniTrivialElementaryChildData p} (src : Sec7FrattiniTrivialElementaryMStageCandidateLowerSourceData h chief K hR child) (m : towerM K) : ((towerActM K) (powOmega2 (src.x₁ * src.tau))⁻¹) m - m - Q2Presentation.Induction.sec7TauReplicateOpenedNormSumAux✝ K src m (orderOf src.tau).pred - (Q2Presentation.Induction.sec7TowerActSeq✝ K (List.replicate (orderOf src.tau).pred src.tau) (((towerActM K) src.x₁) m) - Q2Presentation.Induction.sec7TowerActSeq✝¹ K (List.replicate (orderOf src.tau).pred src.tau) m) = 0

Checkpoint331 bridge closing the checkpoint330 u₁⁻¹ final-step closed-zero residual by unfolding the lower recursive norm value.

theorem Q2Presentation.Induction.sec7_frattiniTrivial_elementaryMStageCandidateWildRowManuscriptU1InvOmegaNormCanonicalPrimitiveTowerActionX1InverseNormInverseValueRawBaseDeviationFormulaNormSummandDeviationMapSubOpenedTauReplicateListRecursiveNormSumAuxFinalStepDifferenceCorePrimitiveApplyPointwiseData {p : FramedPair} {h : ¬ScalarTerminal p} {chief : FirstNonScalarChiefFactor p} {K : KernelMinimal chief.toNonScalarChiefFactor} {hR : Nat.card ↥(kernelFrattini K) = 1} {child : Sec7FrattiniTrivialElementaryChildData p} (src : Sec7FrattiniTrivialElementaryMStageCandidateLowerSourceData h chief K hR child) (m : towerM K) : ((towerActM K) (powOmega2 (src.x₁ * src.tau))⁻¹) m - m - Q2Presentation.Induction.sec7TauReplicateOpenedNormSumAux✝ K src m (orderOf src.tau).pred = Q2Presentation.Induction.sec7TowerActSeq✝ K (List.replicate (orderOf src.tau).pred src.tau) (((towerActM K) src.x₁) m) - Q2Presentation.Induction.sec7TowerActSeq✝¹ K (List.replicate (orderOf src.tau).pred src.tau) m

Checkpoint330 bridge for the checkpoint329 u₁⁻¹ difference-core primitive.

theorem Q2Presentation.Induction.sec7_frattiniTrivial_elementaryMStageCandidateWildRowManuscriptU1InvOmegaNormCanonicalPrimitiveTowerActionX1InverseNormInverseValueRawBaseDeviationFormulaNormSummandDeviationMapSubOpenedTauReplicateListRecursiveNormSumAuxFinalStepResidualClosedZeroPrimitiveApplyPointwiseData {p : FramedPair} {h : ¬ScalarTerminal p} {chief : FirstNonScalarChiefFactor p} {K : KernelMinimal chief.toNonScalarChiefFactor} {hR : Nat.card ↥(kernelFrattini K) = 1} {child : Sec7FrattiniTrivialElementaryChildData p} (src : Sec7FrattiniTrivialElementaryMStageCandidateLowerSourceData h chief K hR child) (m : towerM K) : ((towerActM K) (powOmega2 (src.x₁ * src.tau))⁻¹) m - m - Q2Presentation.Induction.sec7TauReplicateOpenedNormSumAux✝ K src m (orderOf src.tau).pred - Q2Presentation.Induction.sec7TowerActSeq✝ K (List.replicate (orderOf src.tau).pred src.tau) (((towerActM K) src.x₁) m) + Q2Presentation.Induction.sec7TowerActSeq✝¹ K (List.replicate (orderOf src.tau).pred src.tau) m = 0

Checkpoint329 bridge for the checkpoint328 u₁⁻¹ closed-zero atom.

theorem Q2Presentation.Induction.sec7_frattiniTrivial_elementaryMStageCandidateWildRowManuscriptU1InvOmegaNormCanonicalPrimitiveTowerActionX1InverseNormInverseValueRawBaseDeviationFormulaNormSummandDeviationMapSubOpenedTauReplicateListRecursiveNormSumAuxPredPlusFinalStepValueApplyPointwiseData {p : FramedPair} {h : ¬ScalarTerminal p} {chief : FirstNonScalarChiefFactor p} {K : KernelMinimal chief.toNonScalarChiefFactor} {hR : Nat.card ↥(kernelFrattini K) = 1} {child : Sec7FrattiniTrivialElementaryChildData p} (src : Sec7FrattiniTrivialElementaryMStageCandidateLowerSourceData h chief K hR child) (m : towerM K) : ((towerActM K) (powOmega2 (src.x₁ * src.tau))⁻¹) m - m = Q2Presentation.Induction.sec7TauReplicateOpenedNormSumAux✝ K src m (orderOf src.tau).pred + (Q2Presentation.Induction.sec7TowerActSeq✝ K (List.replicate (orderOf src.tau).pred src.tau) (((towerActM K) src.x₁) m) - Q2Presentation.Induction.sec7TowerActSeq✝¹ K (List.replicate (orderOf src.tau).pred src.tau) m)

Checkpoint328 bridge for the checkpoint327 u₁⁻¹ norm atom: the new datum is the closed residual after subtracting the auxiliary and final summand. Moving those terms back gives the displayed final recursive step.

theorem Q2Presentation.Induction.sec7_frattiniTrivial_elementaryMStageCandidateWildRowManuscriptU1InvOmegaNormCanonicalPrimitiveTowerActionX1InverseNormInverseValueRawBaseDeviationFormulaNormSummandDeviationMapSubOpenedTauReplicateListRecursiveNormSumAuxSuccPredValueApplyPointwiseData {p : FramedPair} {h : ¬ScalarTerminal p} {chief : FirstNonScalarChiefFactor p} {K : KernelMinimal chief.toNonScalarChiefFactor} {hR : Nat.card ↥(kernelFrattini K) = 1} {child : Sec7FrattiniTrivialElementaryChildData p} (src : Sec7FrattiniTrivialElementaryMStageCandidateLowerSourceData h chief K hR child) (m : towerM K) : ((towerActM K) (powOmega2 (src.x₁ * src.tau))⁻¹) m - m = Q2Presentation.Induction.sec7TauReplicateOpenedNormSumAux✝ K src m (orderOf src.tau).pred.succ

Checkpoint327 bridge for the checkpoint326 u₁⁻¹ norm atom: fold the single unfolded final recursive summand back into sec7TauReplicateOpenedNormSumAux.

theorem Q2Presentation.Induction.sec7_frattiniTrivial_elementaryMStageCandidateWildRowManuscriptU1InvOmegaNormCanonicalPrimitiveTowerActionX1InverseNormInverseValueRawBaseDeviationFormulaNormSummandDeviationMapSubOpenedTauReplicateListRecursiveNormSumAuxFinalStepResidualLastSummandClosedZeroApplyPointwiseData {p : FramedPair} {h : ¬ScalarTerminal p} {chief : FirstNonScalarChiefFactor p} {K : KernelMinimal chief.toNonScalarChiefFactor} {hR : Nat.card ↥(kernelFrattini K) = 1} {child : Sec7FrattiniTrivialElementaryChildData p} (src : Sec7FrattiniTrivialElementaryMStageCandidateLowerSourceData h chief K hR child) (m : towerM K) : ((towerActM K) (powOmega2 (src.x₁ * src.tau))⁻¹) m - m - Q2Presentation.Induction.sec7TauReplicateOpenedNormSumAux✝ K src m (orderOf src.tau).pred - Q2Presentation.Induction.sec7TowerActSeq✝ K (List.replicate (orderOf src.tau).pred src.tau) (((towerActM K) src.x₁) m) + Q2Presentation.Induction.sec7TowerActSeq✝¹ K (List.replicate (orderOf src.tau).pred src.tau) m = 0

Checkpoint326 bridge for the checkpoint325 u₁⁻¹ residual: unfold the last recursive summand and cancel by additive algebra.

theorem Q2Presentation.Induction.sec7_frattiniTrivial_elementaryMStageCandidateWildRowManuscriptU1InvOmegaNormCanonicalPrimitiveTowerActionX1InverseNormInverseValueRawBaseDeviationFormulaNormSummandDeviationMapSubOpenedTauReplicateListRecursiveNormSumAuxFinalStepResidualLastSummandIsolatedApplyPointwiseData {p : FramedPair} {h : ¬ScalarTerminal p} {chief : FirstNonScalarChiefFactor p} {K : KernelMinimal chief.toNonScalarChiefFactor} {hR : Nat.card ↥(kernelFrattini K) = 1} {child : Sec7FrattiniTrivialElementaryChildData p} (src : Sec7FrattiniTrivialElementaryMStageCandidateLowerSourceData h chief K hR child) (m : towerM K) : ((towerActM K) (powOmega2 (src.x₁ * src.tau))⁻¹) m - m - Q2Presentation.Induction.sec7TauReplicateOpenedNormSumAux✝ K src m (orderOf src.tau).pred - Q2Presentation.Induction.sec7TowerActSeq✝ K (List.replicate (orderOf src.tau).pred src.tau) (((towerActM K) src.x₁) m) = Q2Presentation.Induction.sec7TowerActSeq✝¹ K (List.replicate (orderOf src.tau).pred src.tau) m

Checkpoint325 bridge for u₁⁻¹: solve the lower closed zero row for the last base summand using the elementary ZMod 2 cancellation Y+Y=0.

theorem Q2Presentation.Induction.sec7_frattiniTrivial_elementaryMStageCandidateWildRowManuscriptU1InvOmegaNormCanonicalPrimitiveTowerActionX1InverseNormInverseValueRawBaseDeviationFormulaNormSummandDeviationMapSubOpenedTauReplicateListRecursiveNormSumAuxFinalStepResidualLastSummandPairSumApplyPointwiseData {p : FramedPair} {h : ¬ScalarTerminal p} {chief : FirstNonScalarChiefFactor p} {K : KernelMinimal chief.toNonScalarChiefFactor} {hR : Nat.card ↥(kernelFrattini K) = 1} {child : Sec7FrattiniTrivialElementaryChildData p} (src : Sec7FrattiniTrivialElementaryMStageCandidateLowerSourceData h chief K hR child) (m : towerM K) : ((towerActM K) (powOmega2 (src.x₁ * src.tau))⁻¹) m - m - Q2Presentation.Induction.sec7TauReplicateOpenedNormSumAux✝ K src m (orderOf src.tau).pred = Q2Presentation.Induction.sec7TowerActSeq✝ K (List.replicate (orderOf src.tau).pred src.tau) (((towerActM K) src.x₁) m) + Q2Presentation.Induction.sec7TowerActSeq✝¹ K (List.replicate (orderOf src.tau).pred src.tau) m

Checkpoint324 bridge for u₁⁻¹: the checkpoint323 pair-sum row follows from the isolated-last-summand row by elementary additive algebra.

theorem Q2Presentation.Induction.sec7_frattiniTrivial_elementaryMStageCandidateWildRowManuscriptU1InvOmegaNormCanonicalPrimitiveTowerActionX1InverseNormInverseValueRawBaseDeviationFormulaNormSummandDeviationMapSubOpenedTauReplicateListRecursiveNormSumAuxFinalStepResidualLastSummandBaseValuePositiveIsolatedApplyPointwiseData {p : FramedPair} {h : ¬ScalarTerminal p} {chief : FirstNonScalarChiefFactor p} {K : KernelMinimal chief.toNonScalarChiefFactor} {hR : Nat.card ↥(kernelFrattini K) = 1} {child : Sec7FrattiniTrivialElementaryChildData p} (src : Sec7FrattiniTrivialElementaryMStageCandidateLowerSourceData h chief K hR child) (m : towerM K) : ((towerActM K) (powOmega2 (src.x₁ * src.tau))⁻¹) m - m - Q2Presentation.Induction.sec7TauReplicateOpenedNormSumAux✝ K src m (orderOf src.tau).pred - Q2Presentation.Induction.sec7TowerActSeq✝ K (List.replicate (orderOf src.tau).pred src.tau) (((towerActM K) src.x₁) m) = Q2Presentation.Induction.sec7TowerActSeq✝¹ K (List.replicate (orderOf src.tau).pred src.tau) m

Checkpoint323 bridge for u₁⁻¹: isolate the final base summand from the lower two-summand recursive norm row.

theorem Q2Presentation.Induction.sec7_frattiniTrivial_elementaryMStageCandidateWildRowManuscriptU1InvOmegaNormCanonicalPrimitiveTowerActionX1InverseNormInverseValueRawBaseDeviationFormulaNormSummandDeviationMapSubOpenedTauReplicateListRecursiveNormSumAuxFinalStepResidualLastSummandBaseValueIsolatedApplyPointwiseData {p : FramedPair} {h : ¬ScalarTerminal p} {chief : FirstNonScalarChiefFactor p} {K : KernelMinimal chief.toNonScalarChiefFactor} {hR : Nat.card ↥(kernelFrattini K) = 1} {child : Sec7FrattiniTrivialElementaryChildData p} (src : Sec7FrattiniTrivialElementaryMStageCandidateLowerSourceData h chief K hR child) (m : towerM K) : ((towerActM K) (powOmega2 (src.x₁ * src.tau))⁻¹) m - m - Q2Presentation.Induction.sec7TauReplicateOpenedNormSumAux✝ K src m (orderOf src.tau).pred - Q2Presentation.Induction.sec7TowerActSeq✝ K (List.replicate (orderOf src.tau).pred src.tau) (((towerActM K) src.x₁) m) = -Q2Presentation.Induction.sec7TowerActSeq✝¹ K (List.replicate (orderOf src.tau).pred src.tau) m

Checkpoint322 bridge for u₁⁻¹: the checkpoint321 isolated base-value row is recovered from the lower positive last-summand row by the characteristic-two identity Y = -Y on the tower module.

theorem Q2Presentation.Induction.sec7_frattiniTrivial_elementaryMStageCandidateWildRowManuscriptU1InvOmegaNormCanonicalPrimitiveTowerActionX1InverseNormInverseValueRawBaseDeviationFormulaNormSummandDeviationMapSubOpenedTauReplicateListRecursiveNormSumAuxFinalStepResidualLastSummandDifferenceOpenedSubtractionZeroApplyPointwiseData {p : FramedPair} {h : ¬ScalarTerminal p} {chief : FirstNonScalarChiefFactor p} {K : KernelMinimal chief.toNonScalarChiefFactor} {hR : Nat.card ↥(kernelFrattini K) = 1} {child : Sec7FrattiniTrivialElementaryChildData p} (src : Sec7FrattiniTrivialElementaryMStageCandidateLowerSourceData h chief K hR child) (m : towerM K) : ((towerActM K) (powOmega2 (src.x₁ * src.tau))⁻¹) m - m - Q2Presentation.Induction.sec7TauReplicateOpenedNormSumAux✝ K src m (orderOf src.tau).pred - Q2Presentation.Induction.sec7TowerActSeq✝ K (List.replicate (orderOf src.tau).pred src.tau) (((towerActM K) src.x₁) m) + Q2Presentation.Induction.sec7TowerActSeq✝¹ K (List.replicate (orderOf src.tau).pred src.tau) m = 0

Checkpoint320 bridge for u₁⁻¹: move the isolated last base summand back across the equality to recover the opened subtraction-zero row.

theorem Q2Presentation.Induction.sec7_frattiniTrivial_elementaryMStageCandidateWildRowManuscriptU1InvOmegaNormCanonicalPrimitiveTowerActionX1InverseNormInverseValueRawBaseDeviationFormulaNormSummandDeviationMapSubOpenedTauReplicateListRecursiveNormSumAuxFinalStepResidualLastSummandSubtractionZeroApplyPointwiseData {p : FramedPair} {h : ¬ScalarTerminal p} {chief : FirstNonScalarChiefFactor p} {K : KernelMinimal chief.toNonScalarChiefFactor} {hR : Nat.card ↥(kernelFrattini K) = 1} {child : Sec7FrattiniTrivialElementaryChildData p} (src : Sec7FrattiniTrivialElementaryMStageCandidateLowerSourceData h chief K hR child) (m : towerM K) : ((towerActM K) (powOmega2 (src.x₁ * src.tau))⁻¹) m - m - Q2Presentation.Induction.sec7TauReplicateOpenedNormSumAux✝ K src m (orderOf src.tau).pred - (Q2Presentation.Induction.sec7TowerActSeq✝ K (List.replicate (orderOf src.tau).pred src.tau) (((towerActM K) src.x₁) m) - Q2Presentation.Induction.sec7TowerActSeq✝¹ K (List.replicate (orderOf src.tau).pred src.tau) m) = 0

Checkpoint319 bridge for u₁⁻¹: regroup the checkpoint320 opened last summand back as subtraction of the displayed final difference.

theorem Q2Presentation.Induction.sec7_frattiniTrivial_elementaryMStageCandidateWildRowManuscriptU1InvOmegaNormCanonicalPrimitiveTowerActionX1InverseNormInverseValueRawBaseDeviationFormulaNormSummandDeviationMapSubOpenedTauReplicateListRecursiveNormSumAuxFinalStepResidualLastSummandApplyPointwiseData {p : FramedPair} {h : ¬ScalarTerminal p} {chief : FirstNonScalarChiefFactor p} {K : KernelMinimal chief.toNonScalarChiefFactor} {hR : Nat.card ↥(kernelFrattini K) = 1} {child : Sec7FrattiniTrivialElementaryChildData p} (src : Sec7FrattiniTrivialElementaryMStageCandidateLowerSourceData h chief K hR child) (m : towerM K) : ((towerActM K) (powOmega2 (src.x₁ * src.tau))⁻¹) m - m - Q2Presentation.Induction.sec7TauReplicateOpenedNormSumAux✝ K src m (orderOf src.tau).pred = Q2Presentation.Induction.sec7TowerActSeq✝ K (List.replicate (orderOf src.tau).pred src.tau) (((towerActM K) src.x₁) m) - Q2Presentation.Induction.sec7TowerActSeq✝¹ K (List.replicate (orderOf src.tau).pred src.tau) m

theorem Q2Presentation.Induction.sec7_frattiniTrivial_elementaryMStageCandidateWildRowManuscriptU1InvOmegaNormCanonicalPrimitiveTowerActionX1InverseNormInverseValueRawBaseDeviationFormulaNormSummandDeviationMapSubOpenedTauReplicateListRecursiveNormSumAuxFinalStepBaseSubtractedApplyPointwiseData {p : FramedPair} {h : ¬ScalarTerminal p} {chief : FirstNonScalarChiefFactor p} {K : KernelMinimal chief.toNonScalarChiefFactor} {hR : Nat.card ↥(kernelFrattini K) = 1} {child : Sec7FrattiniTrivialElementaryChildData p} (src : Sec7FrattiniTrivialElementaryMStageCandidateLowerSourceData h chief K hR child) (m : towerM K) : ((towerActM K) (powOmega2 (src.x₁ * src.tau))⁻¹) m - m = Q2Presentation.Induction.sec7TauReplicateOpenedNormSumAux✝ K src m (orderOf src.tau).pred + (Q2Presentation.Induction.sec7TowerActSeq✝ K (List.replicate (orderOf src.tau).pred src.tau) (((towerActM K) src.x₁) m) - Q2Presentation.Induction.sec7TowerActSeq✝¹ K (List.replicate (orderOf src.tau).pred src.tau) m)

Checkpoint318 bridge for u₁⁻¹: the checkpoint317 final-step equality is recovered from the residual form that isolates the last recursive summand after subtracting the accumulated auxiliary norm prefix.

theorem Q2Presentation.Induction.sec7_frattiniTrivial_elementaryMStageCandidateWildRowManuscriptU1InvOmegaNormCanonicalPrimitiveTowerActionX1InverseNormInverseValueRawBaseDeviationFormulaNormSummandDeviationMapSubOpenedTauReplicateListRecursiveNormSumAuxBaseSubtractedApplyPointwiseData {p : FramedPair} {h : ¬ScalarTerminal p} {chief : FirstNonScalarChiefFactor p} {K : KernelMinimal chief.toNonScalarChiefFactor} {hR : Nat.card ↥(kernelFrattini K) = 1} {child : Sec7FrattiniTrivialElementaryChildData p} (src : Sec7FrattiniTrivialElementaryMStageCandidateLowerSourceData h chief K hR child) (m : towerM K) : ((towerActM K) (powOmega2 (src.x₁ * src.tau))⁻¹) m - m = Q2Presentation.Induction.sec7TauReplicateOpenedNormSumAux✝ K src m (orderOf src.tau)

Checkpoint317 bridge for u₁⁻¹: the checkpoint316 auxiliary row is recovered by folding the final recursive step of the norm-sum auxiliary at orderOf τ.

theorem Q2Presentation.Induction.sec7_frattiniTrivial_elementaryMStageCandidateWildRowManuscriptU1InvOmegaNormCanonicalPrimitiveTowerActionX1InverseNormInverseValueRawBaseDeviationFormulaNormSummandDeviationMapSubOpenedTauReplicateListRecursiveNormSumBaseSubtractedApplyPointwiseData {p : FramedPair} {h : ¬ScalarTerminal p} {chief : FirstNonScalarChiefFactor p} {K : KernelMinimal chief.toNonScalarChiefFactor} {hR : Nat.card ↥(kernelFrattini K) = 1} {child : Sec7FrattiniTrivialElementaryChildData p} (src : Sec7FrattiniTrivialElementaryMStageCandidateLowerSourceData h chief K hR child) (m : towerM K) : ((towerActM K) (powOmega2 (src.x₁ * src.tau))⁻¹) m - m = Q2Presentation.Induction.sec7TauReplicateOpenedNormSum✝ K src m

Checkpoint315 u₁⁻¹ base-subtracted row, now theorem-level from the checkpoint316 auxiliary recursive norm-sum row.

theorem Q2Presentation.Induction.sec7_frattiniTrivial_elementaryMStageCandidateWildRowManuscriptU1InvOmegaNormCanonicalPrimitiveTowerActionX1InverseNormInverseValueRawBaseDeviationFormulaNormSummandDeviationMapSubOpenedTauReplicateListRecursiveNormSumValueBeforeBaseSubApplyPointwiseData {p : FramedPair} {h : ¬ScalarTerminal p} {chief : FirstNonScalarChiefFactor p} {K : KernelMinimal chief.toNonScalarChiefFactor} {hR : Nat.card ↥(kernelFrattini K) = 1} {child : Sec7FrattiniTrivialElementaryChildData p} (src : Sec7FrattiniTrivialElementaryMStageCandidateLowerSourceData h chief K hR child) (m : towerM K) : ((towerActM K) (powOmega2 (src.x₁ * src.tau))⁻¹) m = m + Q2Presentation.Induction.sec7TauReplicateOpenedNormSum✝ K src m

Checkpoint314 u₁⁻¹ value-before-base-sub row, now theorem-level from the checkpoint315 base-subtracted norm row.

theorem Q2Presentation.Induction.sec7_frattiniTrivial_elementaryMStageCandidateWildRowManuscriptU1InvOmegaNormCanonicalPrimitiveTowerActionX1InverseNormInverseValueRawBaseDeviationFormulaNormSummandDeviationMapSubOpenedTauReplicateListRecursiveNormSumApplyPointwiseData {p : FramedPair} {h : ¬ScalarTerminal p} {chief : FirstNonScalarChiefFactor p} {K : KernelMinimal chief.toNonScalarChiefFactor} {hR : Nat.card ↥(kernelFrattini K) = 1} {child : Sec7FrattiniTrivialElementaryChildData p} (src : Sec7FrattiniTrivialElementaryMStageCandidateLowerSourceData h chief K hR child) (m : towerM K) : ((towerActM K) (powOmega2 (src.x₁ * src.tau))⁻¹) m - m = Q2Presentation.Induction.sec7TauReplicateOpenedNormSum✝ K src m

theorem Q2Presentation.Induction.sec7_frattiniTrivial_elementaryMStageCandidateWildRowManuscriptU1InvOmegaNormCanonicalPrimitiveTowerActionX1InverseNormInverseValueRawBaseDeviationFormulaNormSummandDeviationMapSubOpenedTauReplicateListApplyPointwiseData {p : FramedPair} {h : ¬ScalarTerminal p} {chief : FirstNonScalarChiefFactor p} {K : KernelMinimal chief.toNonScalarChiefFactor} {hR : Nat.card ↥(kernelFrattini K) = 1} {child : Sec7FrattiniTrivialElementaryChildData p} (src : Sec7FrattiniTrivialElementaryMStageCandidateLowerSourceData h chief K hR child) (m : towerM K) : ((towerActM K) (powOmega2 (src.x₁ * src.tau))⁻¹) m - m = ∑ i ∈ Finset.range (orderOf src.tau), (Q2Presentation.Induction.sec7TowerActSeq✝ K (List.replicate i src.tau) (((towerActM K) src.x₁) m) - Q2Presentation.Induction.sec7TowerActSeq✝¹ K (List.replicate i src.tau) m)

Checkpoint312 opened replicated-list norm row, now theorem-level from the recursive norm-sum datum and the finite-range summation lemma.

theorem Q2Presentation.Induction.sec7_frattiniTrivial_elementaryMStageCandidateWildRowManuscriptU1InvOmegaNormCanonicalPrimitiveTowerActionX1InverseNormInverseValueRawBaseDeviationFormulaNormSummandDeviationMapSubClosedTauReplicateListApplyPointwiseData {p : FramedPair} {h : ¬ScalarTerminal p} {chief : FirstNonScalarChiefFactor p} {K : KernelMinimal chief.toNonScalarChiefFactor} {hR : Nat.card ↥(kernelFrattini K) = 1} {child : Sec7FrattiniTrivialElementaryChildData p} (src : Sec7FrattiniTrivialElementaryMStageCandidateLowerSourceData h chief K hR child) (m : towerM K) : ((towerActM K) (powOmega2 (src.x₁ * src.tau))⁻¹) m - m = ∑ i ∈ Finset.range (orderOf src.tau), Q2Presentation.Induction.sec7TowerActSeq✝ K (List.replicate i src.tau) (((towerActM K) src.x₁) m - m)

Checkpoint311 closed-map-sub norm row, recovered theorem-level from the checkpoint312 row where every replicate-list summand has been opened by linearity.

theorem Q2Presentation.Induction.sec7_frattiniTrivial_elementaryMStageCandidateWildRowManuscriptU1InvOmegaNormCanonicalPrimitiveTowerActionX1InverseNormInverseValueRawBaseDeviationFormulaNormSummandDeviationMapSubOpenedTauReplicateListSubOpenedApplyPointwiseData {p : FramedPair} {h : ¬ScalarTerminal p} {chief : FirstNonScalarChiefFactor p} {K : KernelMinimal chief.toNonScalarChiefFactor} {hR : Nat.card ↥(kernelFrattini K) = 1} {child : Sec7FrattiniTrivialElementaryChildData p} (src : Sec7FrattiniTrivialElementaryMStageCandidateLowerSourceData h chief K hR child) (m : towerM K) : ((towerActM K) (powOmega2 (src.x₁ * src.tau))⁻¹) m - m = ∑ i ∈ Finset.range (orderOf src.tau), (Q2Presentation.Induction.sec7TowerActSeq✝ K (List.replicate i src.tau) (((towerActM K) src.x₁) m) - Q2Presentation.Induction.sec7TowerActSeq✝¹ K (List.replicate i src.tau) m)

Checkpoint310 opened replicated-list row for u₁⁻¹, recovered theorem-level from the checkpoint311 closed-map_sub norm row.

theorem Q2Presentation.Induction.sec7_frattiniTrivial_elementaryMStageCandidateWildRowManuscriptU1InvOmegaNormCanonicalPrimitiveTowerActionX1InverseNormInverseValueRawBaseDeviationFormulaNormSummandDeviationMapSubOpenedTauIterTowerSubOpenedApplyPointwiseData {p : FramedPair} {h : ¬ScalarTerminal p} {chief : FirstNonScalarChiefFactor p} {K : KernelMinimal chief.toNonScalarChiefFactor} {hR : Nat.card ↥(kernelFrattini K) = 1} {child : Sec7FrattiniTrivialElementaryChildData p} (src : Sec7FrattiniTrivialElementaryMStageCandidateLowerSourceData h chief K hR child) (m : towerM K) : ((towerActM K) (powOmega2 (src.x₁ * src.tau))⁻¹) m - m = ∑ i ∈ Finset.range (orderOf src.tau), (Q2Presentation.Induction.sec7TowerActIter✝ K src.tau i (((towerActM K) src.x₁) m) - Q2Presentation.Induction.sec7TowerActIter✝¹ K src.tau i m)

Checkpoint309 u₁⁻¹ datum, now theorem-level from the still more literal checkpoint310 row where every τ-iterate is an explicit replicated action list.

theorem Q2Presentation.Induction.sec7_frattiniTrivial_elementaryMStageCandidateWildRowManuscriptU1InvOmegaNormCanonicalPrimitiveTowerActionX1InverseNormInverseValueRawBaseDeviationFormulaNormSummandDeviationMapSubOpenedTauIterTowerApplyPointwiseData {p : FramedPair} {h : ¬ScalarTerminal p} {chief : FirstNonScalarChiefFactor p} {K : KernelMinimal chief.toNonScalarChiefFactor} {hR : Nat.card ↥(kernelFrattini K) = 1} {child : Sec7FrattiniTrivialElementaryChildData p} (src : Sec7FrattiniTrivialElementaryMStageCandidateLowerSourceData h chief K hR child) (m : towerM K) : ((towerActM K) (powOmega2 (src.x₁ * src.tau))⁻¹) m - m = ∑ i ∈ Finset.range (orderOf src.tau), Q2Presentation.Induction.sec7MStageRawDifference✝ K (Q2Presentation.Induction.sec7TowerActIter✝ K src.tau i (((towerActM K) src.x₁) m)) (Q2Presentation.Induction.sec7TowerActIter✝¹ K src.tau i m)

Checkpoint308 u₁⁻¹ norm row, now theorem-level from the literal summand row where the named raw-difference helper has been opened as subtraction.

theorem Q2Presentation.Induction.sec7_frattiniTrivial_elementaryMStageCandidateWildRowManuscriptU1InvOmegaNormCanonicalPrimitiveTowerActionX1InverseNormInverseValueRawBaseDeviationFormulaNormSummandDeviationMapSubOpenedTauPowerTowerApplyPointwiseData {p : FramedPair} {h : ¬ScalarTerminal p} {chief : FirstNonScalarChiefFactor p} {K : KernelMinimal chief.toNonScalarChiefFactor} {hR : Nat.card ↥(kernelFrattini K) = 1} {child : Sec7FrattiniTrivialElementaryChildData p} (src : Sec7FrattiniTrivialElementaryMStageCandidateLowerSourceData h chief K hR child) (m : towerM K) : ((towerActM K) (powOmega2 (src.x₁ * src.tau))⁻¹) m - m = ∑ i ∈ Finset.range (orderOf src.tau), (((towerActM K) (src.tau ^ i)) (((towerActM K) src.x₁) m) - ((towerActM K) (src.tau ^ i)) m)

theorem Q2Presentation.Induction.sec7_frattiniTrivial_elementaryMStageCandidateWildRowManuscriptU1InvOmegaNormCanonicalPrimitiveTowerActionX1InverseNormInverseValueRawBaseDeviationFormulaNormSummandDeviationMapSubOpenedApplyPointwiseData {p : FramedPair} {h : ¬ScalarTerminal p} {chief : FirstNonScalarChiefFactor p} {K : KernelMinimal chief.toNonScalarChiefFactor} {hR : Nat.card ↥(kernelFrattini K) = 1} {child : Sec7FrattiniTrivialElementaryChildData p} (src : Sec7FrattiniTrivialElementaryMStageCandidateLowerSourceData h chief K hR child) (m : towerM K) : ((towerActM K) (powOmega2 (src.x₁ * src.tau))⁻¹) m - m = ∑ i ∈ Finset.range (orderOf src.tau), (((sec7MStageCandidateActionEnd K) src.tau ^ i) (((towerActM K) src.x₁) m) - ((sec7MStageCandidateActionEnd K) src.tau ^ i) m)

theorem Q2Presentation.Induction.sec7_frattiniTrivial_elementaryMStageCandidateWildRowManuscriptU1InvOmegaNormCanonicalPrimitiveTowerActionX1InverseNormInverseValueRawBaseDeviationFormulaNormSummandDeviationApplyPointwiseData {p : FramedPair} {h : ¬ScalarTerminal p} {chief : FirstNonScalarChiefFactor p} {K : KernelMinimal chief.toNonScalarChiefFactor} {hR : Nat.card ↥(kernelFrattini K) = 1} {child : Sec7FrattiniTrivialElementaryChildData p} (src : Sec7FrattiniTrivialElementaryMStageCandidateLowerSourceData h chief K hR child) (m : towerM K) : ((towerActM K) (powOmega2 (src.x₁ * src.tau))⁻¹) m - m = ∑ i ∈ Finset.range (orderOf src.tau), ((sec7MStageCandidateActionEnd K) src.tau ^ i) (((towerActM K) src.x₁) m - m)

theorem Q2Presentation.Induction.sec7_frattiniTrivial_elementaryMStageCandidateWildRowManuscriptU1InvOmegaNormCanonicalPrimitiveTowerActionX1InverseNormInverseValueRawBaseDeviationFormulaNormSummandApplyPointwiseData {p : FramedPair} {h : ¬ScalarTerminal p} {chief : FirstNonScalarChiefFactor p} {K : KernelMinimal chief.toNonScalarChiefFactor} {hR : Nat.card ↥(kernelFrattini K) = 1} {child : Sec7FrattiniTrivialElementaryChildData p} (src : Sec7FrattiniTrivialElementaryMStageCandidateLowerSourceData h chief K hR child) (m : towerM K) : ((towerActM K) (powOmega2 (src.x₁ * src.tau))⁻¹) m - m = ∑ i ∈ Finset.range (orderOf src.tau), ((sec7MStageCandidateActionEnd K) src.tau ^ i) (((towerActM K) src.x₁) m) - ∑ i ∈ Finset.range (orderOf src.tau), ((sec7MStageCandidateActionEnd K) src.tau ^ i) m

theorem Q2Presentation.Induction.sec7_frattiniTrivial_elementaryMStageCandidateWildRowManuscriptU1InvOmegaNormCanonicalPrimitiveTowerActionX1InverseNormInverseValueRawBaseValueFormulaNormSummandApplyPointwiseData {p : FramedPair} {h : ¬ScalarTerminal p} {chief : FirstNonScalarChiefFactor p} {K : KernelMinimal chief.toNonScalarChiefFactor} {hR : Nat.card ↥(kernelFrattini K) = 1} {child : Sec7FrattiniTrivialElementaryChildData p} (src : Sec7FrattiniTrivialElementaryMStageCandidateLowerSourceData h chief K hR child) (m : towerM K) : ((towerActM K) (powOmega2 (src.x₁ * src.tau))⁻¹) m = m + (∑ i ∈ Finset.range (orderOf src.tau), ((sec7MStageCandidateActionEnd K) src.tau ^ i) (((towerActM K) src.x₁) m) - ∑ i ∈ Finset.range (orderOf src.tau), ((sec7MStageCandidateActionEnd K) src.tau ^ i) m)

Checkpoint303 primitive datum for u₁⁻¹, now theorem-level from the checkpoint304 base-deviation norm-summand formula.

theorem Q2Presentation.Induction.sec7_frattiniTrivial_elementaryMStageCandidateWildRowManuscriptU1InvOmegaNormCanonicalPrimitiveTowerActionX1InverseNormInverseValueRawBaseValueFormulaNormSumApplyPointwiseData {p : FramedPair} {h : ¬ScalarTerminal p} {chief : FirstNonScalarChiefFactor p} {K : KernelMinimal chief.toNonScalarChiefFactor} {hR : Nat.card ↥(kernelFrattini K) = 1} {child : Sec7FrattiniTrivialElementaryChildData p} (src : Sec7FrattiniTrivialElementaryMStageCandidateLowerSourceData h chief K hR child) (m : towerM K) : ((towerActM K) (powOmega2 (src.x₁ * src.tau))⁻¹) m = m + ((∑ i ∈ Finset.range (orderOf src.tau), (sec7MStageCandidateActionEnd K) src.tau ^ i) (((towerActM K) src.x₁) m) - (∑ i ∈ Finset.range (orderOf src.tau), (sec7MStageCandidateActionEnd K) src.tau ^ i) m)

Checkpoint302 primitive datum for u₁⁻¹, now theorem-level from the checkpoint303 pointwise-summand norm formula.

theorem Q2Presentation.Induction.sec7_frattiniTrivial_elementaryMStageCandidateWildRowManuscriptU1InvOmegaNormCanonicalPrimitiveTowerActionX1InverseNormInverseValueRawBaseValueFormulaUnfoldedNormApplyPointwiseData {p : FramedPair} {h : ¬ScalarTerminal p} {chief : FirstNonScalarChiefFactor p} {K : KernelMinimal chief.toNonScalarChiefFactor} {hR : Nat.card ↥(kernelFrattini K) = 1} {child : Sec7FrattiniTrivialElementaryChildData p} (src : Sec7FrattiniTrivialElementaryMStageCandidateLowerSourceData h chief K hR child) (m : towerM K) : ((towerActM K) (powOmega2 (src.x₁ * src.tau))⁻¹) m = m + ((Lifting.repNorm (sec7MStageCandidateActionEnd K) src.tau) (((towerActM K) src.x₁) m) - (Lifting.repNorm (sec7MStageCandidateActionEnd K) src.tau) m)

Checkpoint301 primitive datum for u₁⁻¹, now theorem-level from the checkpoint302 literal norm-sum row.

theorem Q2Presentation.Induction.sec7_frattiniTrivial_elementaryMStageCandidateWildRowManuscriptU1InvOmegaNormCanonicalPrimitiveTowerActionX1InverseNormInverseValueRawBaseValueFormulaApplyPointwiseData {p : FramedPair} {h : ¬ScalarTerminal p} {chief : FirstNonScalarChiefFactor p} {K : KernelMinimal chief.toNonScalarChiefFactor} {hR : Nat.card ↥(kernelFrattini K) = 1} {child : Sec7FrattiniTrivialElementaryChildData p} (src : Sec7FrattiniTrivialElementaryMStageCandidateLowerSourceData h chief K hR child) (m : towerM K) : ((towerActM K) (powOmega2 (src.x₁ * src.tau))⁻¹) m = m + (src.P (((towerActM K) src.x₁) m) - src.P m)

Checkpoint300 primitive datum for u₁⁻¹, now theorem-level from the checkpoint301 unfolded-norm value formula.

theorem Q2Presentation.Induction.sec7_frattiniTrivial_elementaryMStageCandidateWildRowManuscriptU1InvOmegaNormCanonicalPrimitiveTowerActionX1InverseNormInverseValueRawBaseDifferenceBalanceApplyPointwiseData {p : FramedPair} {h : ¬ScalarTerminal p} {chief : FirstNonScalarChiefFactor p} {K : KernelMinimal chief.toNonScalarChiefFactor} {hR : Nat.card ↥(kernelFrattini K) = 1} {child : Sec7FrattiniTrivialElementaryChildData p} (src : Sec7FrattiniTrivialElementaryMStageCandidateLowerSourceData h chief K hR child) (m : towerM K) : ((towerActM K) (powOmega2 (src.x₁ * src.tau))⁻¹) m - m = src.P (((towerActM K) src.x₁) m) - src.P m

Checkpoint299 primitive datum for u₁⁻¹, now theorem-level from the checkpoint300 inverse-value formula.

theorem Q2Presentation.Induction.sec7_frattiniTrivial_elementaryMStageCandidateWildRowManuscriptU1InvOmegaNormCanonicalPrimitiveTowerActionX1InverseNormInverseValueRawFourTermBalanceApplyPointwiseData {p : FramedPair} {h : ¬ScalarTerminal p} {chief : FirstNonScalarChiefFactor p} {K : KernelMinimal chief.toNonScalarChiefFactor} {hR : Nat.card ↥(kernelFrattini K) = 1} {child : Sec7FrattiniTrivialElementaryChildData p} (src : Sec7FrattiniTrivialElementaryMStageCandidateLowerSourceData h chief K hR child) (m : towerM K) : ((towerActM K) (powOmega2 (src.x₁ * src.tau))⁻¹) m + src.P m = m + src.P (((towerActM K) src.x₁) m)

Checkpoint298 primitive atom for u₁⁻¹, now theorem-level from the checkpoint299 base-difference balance.

theorem Q2Presentation.Induction.sec7_frattiniTrivial_elementaryMStageCandidateWildRowManuscriptU1InvOmegaNormCanonicalPrimitiveTowerActionX1InverseNormInverseValueRawExpandedApplyPointwiseData {p : FramedPair} {h : ¬ScalarTerminal p} {chief : FirstNonScalarChiefFactor p} {K : KernelMinimal chief.toNonScalarChiefFactor} {hR : Nat.card ↥(kernelFrattini K) = 1} {child : Sec7FrattiniTrivialElementaryChildData p} (src : Sec7FrattiniTrivialElementaryMStageCandidateLowerSourceData h chief K hR child) (m : towerM K) : ((towerActM K) (powOmega2 (src.x₁ * src.tau))⁻¹) m = m + (src.P (((towerActM K) src.x₁) m) - src.P m)

Checkpoint297 primitive atom for u₁⁻¹, now theorem-level from the four-term checkpoint298 balance.

theorem Q2Presentation.Induction.sec7_frattiniTrivial_elementaryMStageCandidateWildRowManuscriptU1InvOmegaNormCanonicalPrimitiveTowerActionX1InverseNormInverseValueRawApplyPointwiseData {p : FramedPair} {h : ¬ScalarTerminal p} {chief : FirstNonScalarChiefFactor p} {K : KernelMinimal chief.toNonScalarChiefFactor} {hR : Nat.card ↥(kernelFrattini K) = 1} {child : Sec7FrattiniTrivialElementaryChildData p} (src : Sec7FrattiniTrivialElementaryMStageCandidateLowerSourceData h chief K hR child) (m : towerM K) : have u1 := powOmega2 (src.x₁ * src.tau); ((towerActM K) u1⁻¹) m = m + src.P (((towerActM K) src.x₁) m - m)

Checkpoint296 u₁⁻¹ raw inverse-value atom, now theorem-level from the linearity-expanded value statement. The new leaf records the two visible P-images separately before recombining them as P (X m - m).

theorem Q2Presentation.Induction.sec7_frattiniTrivial_elementaryMStageCandidateWildRowManuscriptU1InvOmegaNormCanonicalPrimitiveTowerActionX1InverseNormInversePositiveDeviationRawApplyPointwiseData {p : FramedPair} {h : ¬ScalarTerminal p} {chief : FirstNonScalarChiefFactor p} {K : KernelMinimal chief.toNonScalarChiefFactor} {hR : Nat.card ↥(kernelFrattini K) = 1} {child : Sec7FrattiniTrivialElementaryChildData p} (src : Sec7FrattiniTrivialElementaryMStageCandidateLowerSourceData h chief K hR child) (m : towerM K) : have u1 := powOmega2 (src.x₁ * src.tau); ((towerActM K) u1⁻¹) m - m = src.P (((towerActM K) src.x₁) m - m)

Checkpoint295 positive inverse-factor deviation for u₁⁻¹, now theorem-level from the raw value statement above.

theorem Q2Presentation.Induction.sec7_frattiniTrivial_elementaryMStageCandidateWildRowManuscriptU1InvOmegaNormCanonicalPrimitiveTowerActionX1InverseNormInverseDeviationRawApplyPointwiseData {p : FramedPair} {h : ¬ScalarTerminal p} {chief : FirstNonScalarChiefFactor p} {K : KernelMinimal chief.toNonScalarChiefFactor} {hR : Nat.card ↥(kernelFrattini K) = 1} {child : Sec7FrattiniTrivialElementaryChildData p} (src : Sec7FrattiniTrivialElementaryMStageCandidateLowerSourceData h chief K hR child) (m : towerM K) : have u1 := powOmega2 (src.x₁ * src.tau); m - ((towerActM K) u1⁻¹) m = 0 - src.P (((towerActM K) src.x₁) m - m)

Checkpoint294 raw inverse-deviation datum for u₁⁻¹, now theorem-level from the simpler positive-deviation equality.

theorem Q2Presentation.Induction.sec7_frattiniTrivial_elementaryMStageCandidateWildRowManuscriptU1InvOmegaNormCanonicalPrimitiveTowerActionX1InverseNormInverseRowRawApplyPointwiseData {p : FramedPair} {h : ¬ScalarTerminal p} {chief : FirstNonScalarChiefFactor p} {K : KernelMinimal chief.toNonScalarChiefFactor} {hR : Nat.card ↥(kernelFrattini K) = 1} {child : Sec7FrattiniTrivialElementaryChildData p} (src : Sec7FrattiniTrivialElementaryMStageCandidateLowerSourceData h chief K hR child) (m : towerM K) : have u1 := powOmega2 (src.x₁ * src.tau); m - ((towerActM K) u1⁻¹) m + src.P (((towerActM K) src.x₁) m - m) = 0

Checkpoint293 inverse-row zero-sum for u₁⁻¹, now theorem-level from the raw inverse-deviation equality.

theorem Q2Presentation.Induction.sec7_frattiniTrivial_elementaryMStageCandidateWildRowManuscriptU1InvOmegaNormCanonicalPrimitiveTowerActionX1InverseNormNoninverseRowTowerApplyPointwiseData {p : FramedPair} {h : ¬ScalarTerminal p} {chief : FirstNonScalarChiefFactor p} {K : KernelMinimal chief.toNonScalarChiefFactor} {hR : Nat.card ↥(kernelFrattini K) = 1} {child : Sec7FrattiniTrivialElementaryChildData p} (src : Sec7FrattiniTrivialElementaryMStageCandidateLowerSourceData h chief K hR child) (m : towerM K) : have u1 := powOmega2 (src.x₁ * src.tau); ((towerActM K) u1) m - m + ((towerActM K) u1) (src.P (((towerActM K) src.x₁) m - m)) = 0

Checkpoint292 non-inverted u₁ row, now theorem-level by applying the u₁ action to the inverse-factor row.

theorem Q2Presentation.Induction.sec7_frattiniTrivial_elementaryMStageCandidateWildRowManuscriptU1InvOmegaNormCanonicalPrimitiveTowerActionX1InverseNormNoninverseRowApplyPointwiseData {p : FramedPair} {h : ¬ScalarTerminal p} {chief : FirstNonScalarChiefFactor p} {K : KernelMinimal chief.toNonScalarChiefFactor} {hR : Nat.card ↥(kernelFrattini K) = 1} {child : Sec7FrattiniTrivialElementaryChildData p} (src : Sec7FrattiniTrivialElementaryMStageCandidateLowerSourceData h chief K hR child) (m : towerM K) : have u1 := powOmega2 (src.x₁ * src.tau); have U := ↑((towerActM K) u1); have X := ↑((towerActM K) src.x₁); (U - 1 + U * (src.P * (X - 1))) m = 0

Checkpoint291 u₁⁻¹ non-inverted pointwise endomorphism row, now theorem-level from the raw towerActM norm-row application.

theorem Q2Presentation.Induction.sec7_frattiniTrivial_elementaryMStageCandidateWildRowManuscriptU1InvOmegaNormCanonicalPrimitiveTowerActionX1InverseNormNoninverseRowOperatorData {p : FramedPair} {h : ¬ScalarTerminal p} {chief : FirstNonScalarChiefFactor p} {K : KernelMinimal chief.toNonScalarChiefFactor} {hR : Nat.card ↥(kernelFrattini K) = 1} {child : Sec7FrattiniTrivialElementaryChildData p} (src : Sec7FrattiniTrivialElementaryMStageCandidateLowerSourceData h chief K hR child) : have u1 := powOmega2 (src.x₁ * src.tau); have U := ↑((towerActM K) u1); have X := ↑((towerActM K) src.x₁); U - 1 + U * (src.P * (X - 1)) = 0

Former checkpoint290 u₁⁻¹ non-inverted row, now theorem-level by endomorphism extensionality from the pointwise row.

theorem Q2Presentation.Induction.sec7_frattiniTrivial_elementaryMStageCandidateWildRowManuscriptU1InvOmegaNormCanonicalPrimitiveTowerActionX1InverseNormRawDeviationOperatorData {p : FramedPair} {h : ¬ScalarTerminal p} {chief : FirstNonScalarChiefFactor p} {K : KernelMinimal chief.toNonScalarChiefFactor} {hR : Nat.card ↥(kernelFrattini K) = 1} {child : Sec7FrattiniTrivialElementaryChildData p} (src : Sec7FrattiniTrivialElementaryMStageCandidateLowerSourceData h chief K hR child) : have u1 := powOmega2 (src.x₁ * src.tau); have A := ↑((towerActM K) u1⁻¹); have U := ↑((towerActM K) u1); have X := ↑((towerActM K) src.x₁); 0 - A * (U - 1) = src.P * (X - 1)

Checkpoint289 residual for u₁⁻¹, now theorem-level from the lower non-inverted norm row.

theorem Q2Presentation.Induction.sec7_frattiniTrivial_elementaryMStageCandidateWildRowManuscriptU1InvOmegaNormCanonicalPrimitiveTowerActionX1InverseNormDeviationOperatorData {p : FramedPair} {h : ¬ScalarTerminal p} {chief : FirstNonScalarChiefFactor p} {K : KernelMinimal chief.toNonScalarChiefFactor} {hR : Nat.card ↥(kernelFrattini K) = 1} {child : Sec7FrattiniTrivialElementaryChildData p} (src : Sec7FrattiniTrivialElementaryMStageCandidateLowerSourceData h chief K hR child) : have u1 := powOmega2 (src.x₁ * src.tau); have A := ↑((towerActM K) u1⁻¹); have X := ↑((towerActM K) src.x₁); A - 1 = src.P * (X - 1)

Checkpoint288 u₁⁻¹ operator residual, now theorem-level from the narrower checkpoint289 raw inverse-rule residual plus the local inverse-factor algebra lemma.

theorem Q2Presentation.Induction.sec7_frattiniTrivial_elementaryMStageCandidateWildRowManuscriptU1InvOmegaNormCanonicalPrimitiveTowerActionX1InverseNormDeviationSubValueApplyPointwiseData {p : FramedPair} {h : ¬ScalarTerminal p} {chief : FirstNonScalarChiefFactor p} {K : KernelMinimal chief.toNonScalarChiefFactor} {hR : Nat.card ↥(kernelFrattini K) = 1} {child : Sec7FrattiniTrivialElementaryChildData p} (src : Sec7FrattiniTrivialElementaryMStageCandidateLowerSourceData h chief K hR child) (m : towerM K) : have u1 := powOmega2 (src.x₁ * src.tau); ((towerActM K) u1⁻¹) m - m = src.P (((towerActM K) src.x₁) m - m)

Checkpoint287 lowering for u₁⁻¹: the residual is the operator identity ρ(u₁⁻¹)-1 = P(ρ(x₁)-1) in Module.End, and the old pointwise tower-action statement is only its value at m.

theorem Q2Presentation.Induction.sec7_frattiniTrivial_elementaryMStageCandidateWildRowManuscriptU1InvOmegaNormCanonicalPrimitiveActionX1InverseNormDeviationSubValueApplyPointwiseData {p : FramedPair} {h : ¬ScalarTerminal p} {chief : FirstNonScalarChiefFactor p} {K : KernelMinimal chief.toNonScalarChiefFactor} {hR : Nat.card ↥(kernelFrattini K) = 1} {child : Sec7FrattiniTrivialElementaryChildData p} (src : Sec7FrattiniTrivialElementaryMStageCandidateLowerSourceData h chief K hR child) (m : towerM K) : have ρ := sec7MStageCandidateActionEnd K; have u1 := powOmega2 (src.x₁ * src.tau); (ρ u1⁻¹) m - m = src.P ((ρ src.x₁) m - m)

Checkpoint284 u₁⁻¹ residual, now theorem-level from the checkpoint285 concrete tower-action deviation identity.

theorem Q2Presentation.Induction.sec7_frattiniTrivial_elementaryMStageCandidateWildRowManuscriptU1InvOmegaNormCanonicalPrimitiveActionX1InverseNormDeviationValueApplyPointwiseData {p : FramedPair} {h : ¬ScalarTerminal p} {chief : FirstNonScalarChiefFactor p} {K : KernelMinimal chief.toNonScalarChiefFactor} {hR : Nat.card ↥(kernelFrattini K) = 1} {child : Sec7FrattiniTrivialElementaryChildData p} (src : Sec7FrattiniTrivialElementaryMStageCandidateLowerSourceData h chief K hR child) (m : towerM K) : have ρ := sec7MStageCandidateActionEnd K; have u1 := powOmega2 (src.x₁ * src.tau); (ρ u1⁻¹) m = src.P ((ρ src.x₁) m - m) + m

Checkpoint283 u₁⁻¹ residual, now theorem-level from the checkpoint284 pointwise deviation identity.

theorem Q2Presentation.Induction.sec7_frattiniTrivial_elementaryMStageCandidateWildRowManuscriptU1InvOmegaNormCanonicalPrimitiveActionX1InverseRawValueApplyPointwiseData {p : FramedPair} {h : ¬ScalarTerminal p} {chief : FirstNonScalarChiefFactor p} {K : KernelMinimal chief.toNonScalarChiefFactor} {hR : Nat.card ↥(kernelFrattini K) = 1} {child : Sec7FrattiniTrivialElementaryChildData p} (src : Sec7FrattiniTrivialElementaryMStageCandidateLowerSourceData h chief K hR child) (m : towerM K) : have ρ := sec7MStageCandidateActionEnd K; have u1 := powOmega2 (src.x₁ * src.tau); (ρ u1⁻¹) m = src.P ((ρ src.x₁) m) - src.P m + m

Checkpoint282 lower atom for u₁⁻¹, now theorem-level from the checkpoint283 single-deviation norm value.

theorem Q2Presentation.Induction.sec7_frattiniTrivial_elementaryMStageCandidateWildRowManuscriptU1InvOmegaNormCanonicalPrimitiveActionX1InverseDeviationRawApplyPointwiseData {p : FramedPair} {h : ¬ScalarTerminal p} {chief : FirstNonScalarChiefFactor p} {K : KernelMinimal chief.toNonScalarChiefFactor} {hR : Nat.card ↥(kernelFrattini K) = 1} {child : Sec7FrattiniTrivialElementaryChildData p} (src : Sec7FrattiniTrivialElementaryMStageCandidateLowerSourceData h chief K hR child) (m : towerM K) : have ρ := sec7MStageCandidateActionEnd K; have u1 := powOmega2 (src.x₁ * src.tau); (ρ u1⁻¹) m - m = src.P ((ρ src.x₁) m) - src.P m

Checkpoint281 lower atom for u₁⁻¹, now theorem-level from the checkpoint282 raw pointwise value form.

theorem Q2Presentation.Induction.sec7_frattiniTrivial_elementaryMStageCandidateWildRowManuscriptU1InvOmegaNormCanonicalPrimitiveActionX1InverseDeviationExpandedApplyPointwiseData {p : FramedPair} {h : ¬ScalarTerminal p} {chief : FirstNonScalarChiefFactor p} {K : KernelMinimal chief.toNonScalarChiefFactor} {hR : Nat.card ↥(kernelFrattini K) = 1} {child : Sec7FrattiniTrivialElementaryChildData p} (src : Sec7FrattiniTrivialElementaryMStageCandidateLowerSourceData h chief K hR child) (m : towerM K) : have ρ := sec7MStageCandidateActionEnd K; have u1 := powOmega2 (src.x₁ * src.tau); (ρ u1⁻¹ - 1) m = src.P ((ρ src.x₁) m) - src.P m

Checkpoint280 residual for u₁⁻¹, now theorem-level from the checkpoint281 raw pointwise deviation form.

theorem Q2Presentation.Induction.sec7_frattiniTrivial_elementaryMStageCandidateWildRowManuscriptU1InvOmegaNormCanonicalPrimitiveActionX1InverseDeviationApplyPointwiseData {p : FramedPair} {h : ¬ScalarTerminal p} {chief : FirstNonScalarChiefFactor p} {K : KernelMinimal chief.toNonScalarChiefFactor} {hR : Nat.card ↥(kernelFrattini K) = 1} {child : Sec7FrattiniTrivialElementaryChildData p} (src : Sec7FrattiniTrivialElementaryMStageCandidateLowerSourceData h chief K hR child) (m : towerM K) : have ρ := sec7MStageCandidateActionEnd K; have u1 := powOmega2 (src.x₁ * src.tau); (ρ u1⁻¹ - 1) m = src.P ((ρ src.x₁ - 1) m)

Checkpoint279 lower atom for u₁⁻¹, now theorem-level from the still more literal checkpoint280 expanded primitive-application row.

theorem Q2Presentation.Induction.sec7_frattiniTrivial_elementaryMStageCandidateWildRowManuscriptU1InvOmegaNormCanonicalPrimitiveActionX1DeviationApplyPointwiseData {p : FramedPair} {h : ¬ScalarTerminal p} {chief : FirstNonScalarChiefFactor p} {K : KernelMinimal chief.toNonScalarChiefFactor} {hR : Nat.card ↥(kernelFrattini K) = 1} {child : Sec7FrattiniTrivialElementaryChildData p} (src : Sec7FrattiniTrivialElementaryMStageCandidateLowerSourceData h chief K hR child) (m : towerM K) : have ρ := sec7MStageCandidateActionEnd K; have u1 := powOmega2 (src.x₁ * src.tau); -(ρ u1⁻¹) ((ρ u1 - 1) m) = src.P ((ρ src.x₁ - 1) m)

Checkpoint278 primitive u₁⁻¹ deviation atom, now theorem-level from the smaller inverse-deviation row plus the formal inverse-factor identity.

theorem Q2Presentation.Induction.sec7_frattiniTrivial_elementaryMStageCandidateWildRowManuscriptU1InvOmegaNormCanonicalPrimitiveActionX1ExpandedApplyRowsPointwiseData {p : FramedPair} {h : ¬ScalarTerminal p} {chief : FirstNonScalarChiefFactor p} {K : KernelMinimal chief.toNonScalarChiefFactor} {hR : Nat.card ↥(kernelFrattini K) = 1} {child : Sec7FrattiniTrivialElementaryChildData p} (src : Sec7FrattiniTrivialElementaryMStageCandidateLowerSourceData h chief K hR child) (m : towerM K) : have ρ := sec7MStageCandidateActionEnd K; have u1 := powOmega2 (src.x₁ * src.tau); -(ρ u1⁻¹) ((ρ u1) m - m) = src.P ((ρ src.x₁) m) - src.P m

Checkpoint277 expanded u₁⁻¹ application row, now theorem-level from the primitive-deviation row by linearity of the two endomorphisms.

theorem Q2Presentation.Induction.sec7_frattiniTrivial_elementaryMStageCandidateWildRowManuscriptU1InvOmegaNormCanonicalPrimitiveActionX1ApplyRowsPointwiseData {p : FramedPair} {h : ¬ScalarTerminal p} {chief : FirstNonScalarChiefFactor p} {K : KernelMinimal chief.toNonScalarChiefFactor} {hR : Nat.card ↥(kernelFrattini K) = 1} {child : Sec7FrattiniTrivialElementaryChildData p} (src : Sec7FrattiniTrivialElementaryMStageCandidateLowerSourceData h chief K hR child) (m : towerM K) : have ρ := sec7MStageCandidateActionEnd K; have u1 := powOmega2 (src.x₁ * src.tau); -(ρ u1⁻¹) ((ρ u1 - 1) m) = src.P ((ρ src.x₁ - 1) m)

Checkpoint276 primitive u₁⁻¹ application row, now theorem-level from the fully expanded application of the two primitive deviations.

theorem Q2Presentation.Induction.sec7_frattiniTrivial_elementaryMStageCandidateWildRowManuscriptU1InvOmegaNormCanonicalPrimitiveActionX1RowsPointwiseData {p : FramedPair} {h : ¬ScalarTerminal p} {chief : FirstNonScalarChiefFactor p} {K : KernelMinimal chief.toNonScalarChiefFactor} {hR : Nat.card ↥(kernelFrattini K) = 1} {child : Sec7FrattiniTrivialElementaryChildData p} (src : Sec7FrattiniTrivialElementaryMStageCandidateLowerSourceData h chief K hR child) (m : towerM K) : have ρ := sec7MStageCandidateActionEnd K; have u1 := powOmega2 (src.x₁ * src.tau); (-(ρ u1⁻¹ * (ρ u1 - 1))) m = (src.P * (src.X₁ - 1)) m

Checkpoint275 primitive u₁⁻¹ row, now theorem-level from the lower application-level x₁ row.

theorem Q2Presentation.Induction.sec7_frattiniTrivial_elementaryMStageCandidateWildRowManuscriptU1InvOmegaNormCanonicalPrimitiveActionTauX1RowsPointwiseData {p : FramedPair} {h : ¬ScalarTerminal p} {chief : FirstNonScalarChiefFactor p} {K : KernelMinimal chief.toNonScalarChiefFactor} {hR : Nat.card ↥(kernelFrattini K) = 1} {child : Sec7FrattiniTrivialElementaryChildData p} (src : Sec7FrattiniTrivialElementaryMStageCandidateLowerSourceData h chief K hR child) (m : towerM K) : have ρ := sec7MStageCandidateActionEnd K; have u1 := powOmega2 (src.x₁ * src.tau); (-(ρ u1⁻¹ * (ρ u1 - 1))) m = (src.P * (src.T - 1) + src.P * (src.X₁ - 1)) m

Checkpoint274 pointwise primitive action-row atom for u₁⁻¹, now theorem-level from the smaller x₁-only row and the norm cancellation P(T-1)=0.

theorem Q2Presentation.Induction.sec7_frattiniTrivial_elementaryMStageCandidateWildRowManuscriptU1InvOmegaNormCanonicalPrimitiveActionTauX1RowsData {p : FramedPair} {h : ¬ScalarTerminal p} {chief : FirstNonScalarChiefFactor p} {K : KernelMinimal chief.toNonScalarChiefFactor} {hR : Nat.card ↥(kernelFrattini K) = 1} {child : Sec7FrattiniTrivialElementaryChildData p} (src : Sec7FrattiniTrivialElementaryMStageCandidateLowerSourceData h chief K hR child) : have ρ := sec7MStageCandidateActionEnd K; have u1 := powOmega2 (src.x₁ * src.tau); -(ρ u1⁻¹ * (ρ u1 - 1)) = src.P * (src.T - 1) + src.P * (src.X₁ - 1)

Former checkpoint273 primitive endomorphism-row atom for u₁⁻¹, now theorem-level from the pointwise primitive norm row above.

theorem Q2Presentation.Induction.sec7_frattiniTrivial_elementaryMStageCandidateWildRowManuscriptU1InvOmegaNormCanonicalRawEvalExpandedTauX1RowsData {p : FramedPair} {h : ¬ScalarTerminal p} {chief : FirstNonScalarChiefFactor p} {K : KernelMinimal chief.toNonScalarChiefFactor} {hR : Nat.card ↥(kernelFrattini K) = 1} {child : Sec7FrattiniTrivialElementaryChildData p} (src : Sec7FrattiniTrivialElementaryMStageCandidateLowerSourceData h chief K hR child) : -((sec7MStageCandidateActionEnd K) ((Q2Word.x1.mul Q2Word.tau).powOmega2.inv.eval src.marking) * ((sec7MStageCandidateActionEnd K) ((Q2Word.x1.mul Q2Word.tau).powOmega2.eval src.marking) - 1)) = src.P * (src.T - 1) + src.P * (src.X₁ - 1)

Former checkpoint272 raw-evaluation u₁⁻¹ atom, now theorem-level from the direct primitive lower-quotient action statement above.

theorem Q2Presentation.Induction.sec7_frattiniTrivial_elementaryMStageCandidateWildRowManuscriptU1InvOmegaNormCanonicalRawEvalTauX1RowsData {p : FramedPair} {h : ¬ScalarTerminal p} {chief : FirstNonScalarChiefFactor p} {K : KernelMinimal chief.toNonScalarChiefFactor} {hR : Nat.card ↥(kernelFrattini K) = 1} {child : Sec7FrattiniTrivialElementaryChildData p} (src : Sec7FrattiniTrivialElementaryMStageCandidateLowerSourceData h chief K hR child) : -((sec7MStageCandidateActionEnd K) (Q2Word.u1.inv.eval src.marking) * ((sec7MStageCandidateActionEnd K) (Q2Word.u1.eval src.marking) - 1)) = src.P * (src.T - 1) + src.P * (src.X₁ - 1)

Former checkpoint271 Q2Word.u1 raw-evaluation atom, now theorem-level from the primitive x₁τ expansion of u₁.

theorem Q2Presentation.Induction.sec7_frattiniTrivial_elementaryMStageCandidateWildRowManuscriptU1InvOmegaNormCanonicalActionTauX1RowsData {p : FramedPair} {h : ¬ScalarTerminal p} {chief : FirstNonScalarChiefFactor p} {K : KernelMinimal chief.toNonScalarChiefFactor} {hR : Nat.card ↥(kernelFrattini K) = 1} {child : Sec7FrattiniTrivialElementaryChildData p} (src : Sec7FrattiniTrivialElementaryMStageCandidateLowerSourceData h chief K hR child) : -((sec7MStageCandidateActionEnd K) (sec7MStageCandidateWildRowManuscriptU1InvFactor src) * ((sec7MStageCandidateActionEnd K) (sec7MStageCandidateWildRowManuscriptU1Factor src) - 1)) = sec7MStageCandidateWildRowManuscriptU1TauCanonicalRow src + sec7MStageCandidateWildRowManuscriptU1X1CanonicalRow src

Former checkpoint270 u₁⁻¹ action atom, now theorem-level from the raw GExpr.eval norm-row identity.

theorem Q2Presentation.Induction.sec7_frattiniTrivial_elementaryMStageCandidateWildRowManuscriptU1InvOmegaNormCanonicalActionX1SubOneData {p : FramedPair} {h : ¬ScalarTerminal p} {chief : FirstNonScalarChiefFactor p} {K : KernelMinimal chief.toNonScalarChiefFactor} {hR : Nat.card ↥(kernelFrattini K) = 1} {child : Sec7FrattiniTrivialElementaryChildData p} (src : Sec7FrattiniTrivialElementaryMStageCandidateLowerSourceData h chief K hR child) : -((sec7MStageCandidateActionEnd K) (sec7MStageCandidateWildRowManuscriptU1InvFactor src) * ((sec7MStageCandidateActionEnd K) (sec7MStageCandidateWildRowManuscriptU1Factor src) - 1)) = sec7MStageCandidateWildRowManuscriptU1X1CanonicalRow src

Former checkpoint269 u₁⁻¹ x₁-only action atom, now theorem-level from the lower two-piece ω₂-norm row and src.norm_shadow.

theorem Q2Presentation.Induction.sec7_frattiniTrivial_elementaryMStageCandidateWildRowManuscriptU1InvCanonicalActionX1SubOneData {p : FramedPair} {h : ¬ScalarTerminal p} {chief : FirstNonScalarChiefFactor p} {K : KernelMinimal chief.toNonScalarChiefFactor} {hR : Nat.card ↥(kernelFrattini K) = 1} {child : Sec7FrattiniTrivialElementaryChildData p} (src : Sec7FrattiniTrivialElementaryMStageCandidateLowerSourceData h chief K hR child) : (sec7MStageCandidateActionEnd K) (sec7MStageCandidateWildRowManuscriptU1InvFactor src) - 1 = sec7MStageCandidateWildRowManuscriptU1X1CanonicalRow src

Former checkpoint268 u₁⁻¹ residual, now theorem-level from the non-inverted ω₂-norm action atom and the inverse-factor identity.

theorem Q2Presentation.Induction.sec7_frattiniTrivial_elementaryMStageCandidateWildRowManuscriptU1InvCanonicalActionSubOneData {p : FramedPair} {h : ¬ScalarTerminal p} {chief : FirstNonScalarChiefFactor p} {K : KernelMinimal chief.toNonScalarChiefFactor} {hR : Nat.card ↥(kernelFrattini K) = 1} {child : Sec7FrattiniTrivialElementaryChildData p} (src : Sec7FrattiniTrivialElementaryMStageCandidateLowerSourceData h chief K hR child) : (sec7MStageCandidateActionEnd K) (sec7MStageCandidateWildRowManuscriptU1InvFactor src) - 1 = sec7MStageCandidateWildRowManuscriptU1TauCanonicalRow src + sec7MStageCandidateWildRowManuscriptU1X1CanonicalRow src

Former checkpoint267 u₁⁻¹ action-sub-one leaf, now theorem-level from the single residual x₁ action row and the proved norm shadow P(T-1)=0.

theorem Q2Presentation.Induction.sec7_frattiniTrivial_elementaryMStageCandidateWildRowManuscriptX1ConjBareCanonicalActionSubOneData {p : FramedPair} {h : ¬ScalarTerminal p} {chief : FirstNonScalarChiefFactor p} {K : KernelMinimal chief.toNonScalarChiefFactor} {hR : Nat.card ↥(kernelFrattini K) = 1} {child : Sec7FrattiniTrivialElementaryChildData p} (src : Sec7FrattiniTrivialElementaryMStageCandidateLowerSourceData h chief K hR child) : (sec7MStageCandidateActionEnd K) (sec7MStageCandidateWildRowManuscriptSigmaInvFactor src) * (sec7MStageCandidateActionEnd K) (sec7MStageCandidateWildRowManuscriptX1BareFactor src) * (sec7MStageCandidateActionEnd K) (sec7MStageCandidateWildRowManuscriptSigmaFactor src) - 1 = src.Sinv * (src.X₁ - 1) * src.S

The bare conjugation action calculation for x₁^σ is theorem-level.

After unfolding the three internal factors, it is the ring identity (S⁻¹ X₁ S)-1 = S⁻¹(X₁-1)S, using the already-proved inverse relation S⁻¹S=1.

axiom Q2Presentation.Induction.sec7_frattiniTrivial_elementaryMStageCandidateWildRowManuscriptX1ConjCanonicalPrimitivePX1ToSinvTowerActionUntransportedDeviationExpandedRawNormSummandDeviationSinvMapSubOpenedTauReplicateListRecursiveNormSumAuxFinalStepDifferenceCoreClosedZeroLowerPrimitiveApplyPointwiseData {p : FramedPair} {h : ¬ScalarTerminal p} {chief : FirstNonScalarChiefFactor p} {K : KernelMinimal chief.toNonScalarChiefFactor} {hR : Nat.card ↥(kernelFrattini K) = 1} {child : Sec7FrattiniTrivialElementaryChildData p} (src : Sec7FrattiniTrivialElementaryMStageCandidateLowerSourceData h chief K hR child) (m : towerM K) : ((towerActM K) src.sigma⁻¹) (((towerActM K) src.x₁) m) - ((towerActM K) src.sigma⁻¹) m - Q2Presentation.Induction.sec7TauReplicateOpenedNormSumAux✝ K src m (orderOf src.tau).pred - (Q2Presentation.Induction.sec7TowerActSeq✝ K (List.replicate (orderOf src.tau).pred src.tau) (((towerActM K) src.x₁) m) - Q2Presentation.Induction.sec7TowerActSeq✝¹ K (List.replicate (orderOf src.tau).pred src.tau) m) = 0

Checkpoint332 lower primitive for x₁^σ: keep only the final recursive norm-step difference core as a closed zero. The checkpoint331 untransported norm value is recovered by the single recursive unfold.

theorem Q2Presentation.Induction.sec7_frattiniTrivial_elementaryMStageCandidateWildRowManuscriptX1ConjCanonicalPrimitivePX1ToSinvTowerActionUntransportedDeviationExpandedRawNormSummandDeviationSinvMapSubOpenedTauReplicateListRecursiveNormSumAuxSuccPredValuePrimitiveApplyPointwiseData {p : FramedPair} {h : ¬ScalarTerminal p} {chief : FirstNonScalarChiefFactor p} {K : KernelMinimal chief.toNonScalarChiefFactor} {hR : Nat.card ↥(kernelFrattini K) = 1} {child : Sec7FrattiniTrivialElementaryChildData p} (src : Sec7FrattiniTrivialElementaryMStageCandidateLowerSourceData h chief K hR child) (m : towerM K) : ((towerActM K) src.sigma⁻¹) (((towerActM K) src.x₁) m) - ((towerActM K) src.sigma⁻¹) m = Q2Presentation.Induction.sec7TauReplicateOpenedNormSumAux✝ K src m (orderOf src.tau).pred.succ

Checkpoint332 bridge recovering the checkpoint331 x₁^σ recursive-norm value from the lower final-step closed-zero primitive.

theorem Q2Presentation.Induction.sec7_frattiniTrivial_elementaryMStageCandidateWildRowManuscriptX1ConjCanonicalPrimitivePX1ToSinvTowerActionUntransportedDeviationExpandedRawNormSummandDeviationSinvMapSubOpenedTauReplicateListRecursiveNormSumAuxFinalStepDifferenceCoreClosedZeroPrimitiveApplyPointwiseData {p : FramedPair} {h : ¬ScalarTerminal p} {chief : FirstNonScalarChiefFactor p} {K : KernelMinimal chief.toNonScalarChiefFactor} {hR : Nat.card ↥(kernelFrattini K) = 1} {child : Sec7FrattiniTrivialElementaryChildData p} (src : Sec7FrattiniTrivialElementaryMStageCandidateLowerSourceData h chief K hR child) (m : towerM K) : ((towerActM K) src.sigma⁻¹) (((towerActM K) src.x₁) m) - ((towerActM K) src.sigma⁻¹) m - Q2Presentation.Induction.sec7TauReplicateOpenedNormSumAux✝ K src m (orderOf src.tau).pred - (Q2Presentation.Induction.sec7TowerActSeq✝ K (List.replicate (orderOf src.tau).pred src.tau) (((towerActM K) src.x₁) m) - Q2Presentation.Induction.sec7TowerActSeq✝¹ K (List.replicate (orderOf src.tau).pred src.tau) m) = 0

Checkpoint331 bridge closing the checkpoint330 x₁^σ final-step closed-zero residual by unfolding the lower recursive norm value.

theorem Q2Presentation.Induction.sec7_frattiniTrivial_elementaryMStageCandidateWildRowManuscriptX1ConjCanonicalPrimitivePX1ToSinvTowerActionUntransportedDeviationExpandedRawNormSummandDeviationSinvMapSubOpenedTauReplicateListRecursiveNormSumAuxFinalStepDifferenceCorePrimitiveApplyPointwiseData {p : FramedPair} {h : ¬ScalarTerminal p} {chief : FirstNonScalarChiefFactor p} {K : KernelMinimal chief.toNonScalarChiefFactor} {hR : Nat.card ↥(kernelFrattini K) = 1} {child : Sec7FrattiniTrivialElementaryChildData p} (src : Sec7FrattiniTrivialElementaryMStageCandidateLowerSourceData h chief K hR child) (m : towerM K) : ((towerActM K) src.sigma⁻¹) (((towerActM K) src.x₁) m) - ((towerActM K) src.sigma⁻¹) m - Q2Presentation.Induction.sec7TauReplicateOpenedNormSumAux✝ K src m (orderOf src.tau).pred = Q2Presentation.Induction.sec7TowerActSeq✝ K (List.replicate (orderOf src.tau).pred src.tau) (((towerActM K) src.x₁) m) - Q2Presentation.Induction.sec7TowerActSeq✝¹ K (List.replicate (orderOf src.tau).pred src.tau) m

Checkpoint330 bridge for the checkpoint329 x₁^σ difference-core primitive.

theorem Q2Presentation.Induction.sec7_frattiniTrivial_elementaryMStageCandidateWildRowManuscriptX1ConjCanonicalPrimitivePX1ToSinvTowerActionUntransportedDeviationExpandedRawNormSummandDeviationSinvMapSubOpenedTauReplicateListRecursiveNormSumAuxFinalStepResidualClosedZeroPrimitiveApplyPointwiseData {p : FramedPair} {h : ¬ScalarTerminal p} {chief : FirstNonScalarChiefFactor p} {K : KernelMinimal chief.toNonScalarChiefFactor} {hR : Nat.card ↥(kernelFrattini K) = 1} {child : Sec7FrattiniTrivialElementaryChildData p} (src : Sec7FrattiniTrivialElementaryMStageCandidateLowerSourceData h chief K hR child) (m : towerM K) : ((towerActM K) src.sigma⁻¹) (((towerActM K) src.x₁) m) - ((towerActM K) src.sigma⁻¹) m - Q2Presentation.Induction.sec7TauReplicateOpenedNormSumAux✝ K src m (orderOf src.tau).pred - Q2Presentation.Induction.sec7TowerActSeq✝ K (List.replicate (orderOf src.tau).pred src.tau) (((towerActM K) src.x₁) m) + Q2Presentation.Induction.sec7TowerActSeq✝¹ K (List.replicate (orderOf src.tau).pred src.tau) m = 0

Checkpoint329 bridge for the checkpoint328 x₁^σ closed-zero norm atom.

theorem Q2Presentation.Induction.sec7_frattiniTrivial_elementaryMStageCandidateWildRowManuscriptX1ConjCanonicalPrimitivePX1ToSinvTowerActionUntransportedDeviationExpandedRawNormSummandDeviationSinvMapSubOpenedTauReplicateListRecursiveNormSumAuxPredPlusFinalStepValueApplyPointwiseData {p : FramedPair} {h : ¬ScalarTerminal p} {chief : FirstNonScalarChiefFactor p} {K : KernelMinimal chief.toNonScalarChiefFactor} {hR : Nat.card ↥(kernelFrattini K) = 1} {child : Sec7FrattiniTrivialElementaryChildData p} (src : Sec7FrattiniTrivialElementaryMStageCandidateLowerSourceData h chief K hR child) (m : towerM K) : ((towerActM K) src.sigma⁻¹) (((towerActM K) src.x₁) m) - ((towerActM K) src.sigma⁻¹) m = Q2Presentation.Induction.sec7TauReplicateOpenedNormSumAux✝ K src m (orderOf src.tau).pred + (Q2Presentation.Induction.sec7TowerActSeq✝ K (List.replicate (orderOf src.tau).pred src.tau) (((towerActM K) src.x₁) m) - Q2Presentation.Induction.sec7TowerActSeq✝¹ K (List.replicate (orderOf src.tau).pred src.tau) m)

Checkpoint328 bridge for the checkpoint327 P x₁ to σ⁻¹ norm atom: the new datum is the final-step residual in closed-zero form; additive algebra moves it back to the displayed recursive summand equality.

theorem Q2Presentation.Induction.sec7_frattiniTrivial_elementaryMStageCandidateWildRowManuscriptX1ConjCanonicalPrimitivePX1ToSinvTowerActionUntransportedDeviationExpandedRawNormSummandDeviationSinvMapSubOpenedTauReplicateListRecursiveNormSumAuxSuccPredValueApplyPointwiseData {p : FramedPair} {h : ¬ScalarTerminal p} {chief : FirstNonScalarChiefFactor p} {K : KernelMinimal chief.toNonScalarChiefFactor} {hR : Nat.card ↥(kernelFrattini K) = 1} {child : Sec7FrattiniTrivialElementaryChildData p} (src : Sec7FrattiniTrivialElementaryMStageCandidateLowerSourceData h chief K hR child) (m : towerM K) : ((towerActM K) src.sigma⁻¹) (((towerActM K) src.x₁) m) - ((towerActM K) src.sigma⁻¹) m = Q2Presentation.Induction.sec7TauReplicateOpenedNormSumAux✝ K src m (orderOf src.tau).pred.succ

Checkpoint327 bridge for the checkpoint326 P x₁ to σ⁻¹ norm atom: fold the unfolded last recursive summand back into the auxiliary norm sum.

theorem Q2Presentation.Induction.sec7_frattiniTrivial_elementaryMStageCandidateWildRowManuscriptX1ConjCanonicalPrimitivePX1ToSinvTowerActionUntransportedDeviationExpandedRawNormSummandDeviationSinvMapSubOpenedTauReplicateListRecursiveNormSumAuxFinalStepResidualLastSummandClosedZeroApplyPointwiseData {p : FramedPair} {h : ¬ScalarTerminal p} {chief : FirstNonScalarChiefFactor p} {K : KernelMinimal chief.toNonScalarChiefFactor} {hR : Nat.card ↥(kernelFrattini K) = 1} {child : Sec7FrattiniTrivialElementaryChildData p} (src : Sec7FrattiniTrivialElementaryMStageCandidateLowerSourceData h chief K hR child) (m : towerM K) : ((towerActM K) src.sigma⁻¹) (((towerActM K) src.x₁) m) - ((towerActM K) src.sigma⁻¹) m - Q2Presentation.Induction.sec7TauReplicateOpenedNormSumAux✝ K src m (orderOf src.tau).pred - Q2Presentation.Induction.sec7TowerActSeq✝ K (List.replicate (orderOf src.tau).pred src.tau) (((towerActM K) src.x₁) m) + Q2Presentation.Induction.sec7TowerActSeq✝¹ K (List.replicate (orderOf src.tau).pred src.tau) m = 0

Checkpoint326 bridge for the checkpoint325 P x₁ to σ⁻¹ residual: unfold the last recursive summand and cancel by additive algebra.

theorem Q2Presentation.Induction.sec7_frattiniTrivial_elementaryMStageCandidateWildRowManuscriptX1ConjCanonicalPrimitivePX1ToSinvTowerActionUntransportedDeviationExpandedRawNormSummandDeviationSinvMapSubOpenedTauReplicateListRecursiveNormSumAuxFinalStepResidualLastSummandIsolatedApplyPointwiseData {p : FramedPair} {h : ¬ScalarTerminal p} {chief : FirstNonScalarChiefFactor p} {K : KernelMinimal chief.toNonScalarChiefFactor} {hR : Nat.card ↥(kernelFrattini K) = 1} {child : Sec7FrattiniTrivialElementaryChildData p} (src : Sec7FrattiniTrivialElementaryMStageCandidateLowerSourceData h chief K hR child) (m : towerM K) : ((towerActM K) src.sigma⁻¹) (((towerActM K) src.x₁) m) - ((towerActM K) src.sigma⁻¹) m - Q2Presentation.Induction.sec7TauReplicateOpenedNormSumAux✝ K src m (orderOf src.tau).pred - Q2Presentation.Induction.sec7TowerActSeq✝ K (List.replicate (orderOf src.tau).pred src.tau) (((towerActM K) src.x₁) m) = Q2Presentation.Induction.sec7TowerActSeq✝¹ K (List.replicate (orderOf src.tau).pred src.tau) m

Checkpoint325 bridge for the P x₁ to σ⁻¹ norm comparison: recover the checkpoint324 isolated last-summand equality from the lower closed zero row.

theorem Q2Presentation.Induction.sec7_frattiniTrivial_elementaryMStageCandidateWildRowManuscriptX1ConjCanonicalPrimitivePX1ToSinvTowerActionUntransportedDeviationExpandedRawNormSummandDeviationSinvMapSubOpenedTauReplicateListRecursiveNormSumAuxFinalStepResidualLastSummandPairSumApplyPointwiseData {p : FramedPair} {h : ¬ScalarTerminal p} {chief : FirstNonScalarChiefFactor p} {K : KernelMinimal chief.toNonScalarChiefFactor} {hR : Nat.card ↥(kernelFrattini K) = 1} {child : Sec7FrattiniTrivialElementaryChildData p} (src : Sec7FrattiniTrivialElementaryMStageCandidateLowerSourceData h chief K hR child) (m : towerM K) : ((towerActM K) src.sigma⁻¹) (((towerActM K) src.x₁) m) - ((towerActM K) src.sigma⁻¹) m - Q2Presentation.Induction.sec7TauReplicateOpenedNormSumAux✝ K src m (orderOf src.tau).pred = Q2Presentation.Induction.sec7TowerActSeq✝ K (List.replicate (orderOf src.tau).pred src.tau) (((towerActM K) src.x₁) m) + Q2Presentation.Induction.sec7TowerActSeq✝¹ K (List.replicate (orderOf src.tau).pred src.tau) m

Checkpoint324 bridge for the P x₁ to σ⁻¹ comparison: the checkpoint323 pair-sum row follows from the isolated-last-summand row.

theorem Q2Presentation.Induction.sec7_frattiniTrivial_elementaryMStageCandidateWildRowManuscriptX1ConjCanonicalPrimitivePX1ToSinvTowerActionUntransportedDeviationExpandedRawNormSummandDeviationSinvMapSubOpenedTauReplicateListRecursiveNormSumAuxFinalStepResidualLastSummandBaseValuePositiveIsolatedApplyPointwiseData {p : FramedPair} {h : ¬ScalarTerminal p} {chief : FirstNonScalarChiefFactor p} {K : KernelMinimal chief.toNonScalarChiefFactor} {hR : Nat.card ↥(kernelFrattini K) = 1} {child : Sec7FrattiniTrivialElementaryChildData p} (src : Sec7FrattiniTrivialElementaryMStageCandidateLowerSourceData h chief K hR child) (m : towerM K) : ((towerActM K) src.sigma⁻¹) (((towerActM K) src.x₁) m) - ((towerActM K) src.sigma⁻¹) m - Q2Presentation.Induction.sec7TauReplicateOpenedNormSumAux✝ K src m (orderOf src.tau).pred - Q2Presentation.Induction.sec7TowerActSeq✝ K (List.replicate (orderOf src.tau).pred src.tau) (((towerActM K) src.x₁) m) = Q2Presentation.Induction.sec7TowerActSeq✝¹ K (List.replicate (orderOf src.tau).pred src.tau) m

Checkpoint323 bridge for the P x₁ to σ⁻¹ norm row: isolate the final base summand from the lower two-summand recursive norm row.

theorem Q2Presentation.Induction.sec7_frattiniTrivial_elementaryMStageCandidateWildRowManuscriptX1ConjCanonicalPrimitivePX1ToSinvTowerActionUntransportedDeviationExpandedRawNormSummandDeviationSinvMapSubOpenedTauReplicateListRecursiveNormSumAuxFinalStepResidualLastSummandBaseValueIsolatedApplyPointwiseData {p : FramedPair} {h : ¬ScalarTerminal p} {chief : FirstNonScalarChiefFactor p} {K : KernelMinimal chief.toNonScalarChiefFactor} {hR : Nat.card ↥(kernelFrattini K) = 1} {child : Sec7FrattiniTrivialElementaryChildData p} (src : Sec7FrattiniTrivialElementaryMStageCandidateLowerSourceData h chief K hR child) (m : towerM K) : ((towerActM K) src.sigma⁻¹) (((towerActM K) src.x₁) m) - ((towerActM K) src.sigma⁻¹) m - Q2Presentation.Induction.sec7TauReplicateOpenedNormSumAux✝ K src m (orderOf src.tau).pred - Q2Presentation.Induction.sec7TowerActSeq✝ K (List.replicate (orderOf src.tau).pred src.tau) (((towerActM K) src.x₁) m) = -Q2Presentation.Induction.sec7TowerActSeq✝¹ K (List.replicate (orderOf src.tau).pred src.tau) m

Checkpoint322 bridge for the P x₁ to σ⁻¹ norm row: convert the lower positive isolated base summand into the checkpoint321 negated isolated base summand using characteristic two.

theorem Q2Presentation.Induction.sec7_frattiniTrivial_elementaryMStageCandidateWildRowManuscriptX1ConjCanonicalPrimitivePX1ToSinvTowerActionUntransportedDeviationExpandedRawNormSummandDeviationSinvMapSubOpenedTauReplicateListRecursiveNormSumAuxFinalStepResidualLastSummandDifferenceOpenedSubtractionZeroApplyPointwiseData {p : FramedPair} {h : ¬ScalarTerminal p} {chief : FirstNonScalarChiefFactor p} {K : KernelMinimal chief.toNonScalarChiefFactor} {hR : Nat.card ↥(kernelFrattini K) = 1} {child : Sec7FrattiniTrivialElementaryChildData p} (src : Sec7FrattiniTrivialElementaryMStageCandidateLowerSourceData h chief K hR child) (m : towerM K) : ((towerActM K) src.sigma⁻¹) (((towerActM K) src.x₁) m) - ((towerActM K) src.sigma⁻¹) m - Q2Presentation.Induction.sec7TauReplicateOpenedNormSumAux✝ K src m (orderOf src.tau).pred - Q2Presentation.Induction.sec7TowerActSeq✝ K (List.replicate (orderOf src.tau).pred src.tau) (((towerActM K) src.x₁) m) + Q2Presentation.Induction.sec7TowerActSeq✝¹ K (List.replicate (orderOf src.tau).pred src.tau) m = 0

Checkpoint320 bridge for the x₁^σ norm row: move the isolated last base summand back across the equality.

theorem Q2Presentation.Induction.sec7_frattiniTrivial_elementaryMStageCandidateWildRowManuscriptX1ConjCanonicalPrimitivePX1ToSinvTowerActionUntransportedDeviationExpandedRawNormSummandDeviationSinvMapSubOpenedTauReplicateListRecursiveNormSumAuxFinalStepResidualLastSummandSubtractionZeroApplyPointwiseData {p : FramedPair} {h : ¬ScalarTerminal p} {chief : FirstNonScalarChiefFactor p} {K : KernelMinimal chief.toNonScalarChiefFactor} {hR : Nat.card ↥(kernelFrattini K) = 1} {child : Sec7FrattiniTrivialElementaryChildData p} (src : Sec7FrattiniTrivialElementaryMStageCandidateLowerSourceData h chief K hR child) (m : towerM K) : ((towerActM K) src.sigma⁻¹) (((towerActM K) src.x₁) m) - ((towerActM K) src.sigma⁻¹) m - Q2Presentation.Induction.sec7TauReplicateOpenedNormSumAux✝ K src m (orderOf src.tau).pred - (Q2Presentation.Induction.sec7TowerActSeq✝ K (List.replicate (orderOf src.tau).pred src.tau) (((towerActM K) src.x₁) m) - Q2Presentation.Induction.sec7TowerActSeq✝¹ K (List.replicate (orderOf src.tau).pred src.tau) m) = 0

Checkpoint319 bridge for the P x₁ to σ⁻¹ norm comparison: regroup the checkpoint320 opened last summand back as subtraction of the displayed final difference.

theorem Q2Presentation.Induction.sec7_frattiniTrivial_elementaryMStageCandidateWildRowManuscriptX1ConjCanonicalPrimitivePX1ToSinvTowerActionUntransportedDeviationExpandedRawNormSummandDeviationSinvMapSubOpenedTauReplicateListRecursiveNormSumAuxFinalStepResidualLastSummandApplyPointwiseData {p : FramedPair} {h : ¬ScalarTerminal p} {chief : FirstNonScalarChiefFactor p} {K : KernelMinimal chief.toNonScalarChiefFactor} {hR : Nat.card ↥(kernelFrattini K) = 1} {child : Sec7FrattiniTrivialElementaryChildData p} (src : Sec7FrattiniTrivialElementaryMStageCandidateLowerSourceData h chief K hR child) (m : towerM K) : ((towerActM K) src.sigma⁻¹) (((towerActM K) src.x₁) m) - ((towerActM K) src.sigma⁻¹) m - Q2Presentation.Induction.sec7TauReplicateOpenedNormSumAux✝ K src m (orderOf src.tau).pred = Q2Presentation.Induction.sec7TowerActSeq✝ K (List.replicate (orderOf src.tau).pred src.tau) (((towerActM K) src.x₁) m) - Q2Presentation.Induction.sec7TowerActSeq✝¹ K (List.replicate (orderOf src.tau).pred src.tau) m

theorem Q2Presentation.Induction.sec7_frattiniTrivial_elementaryMStageCandidateWildRowManuscriptX1ConjCanonicalPrimitivePX1ToSinvTowerActionUntransportedDeviationExpandedRawNormSummandDeviationSinvMapSubOpenedTauReplicateListRecursiveNormSumAuxFinalStepSigmaInvBaseSubtractedApplyPointwiseData {p : FramedPair} {h : ¬ScalarTerminal p} {chief : FirstNonScalarChiefFactor p} {K : KernelMinimal chief.toNonScalarChiefFactor} {hR : Nat.card ↥(kernelFrattini K) = 1} {child : Sec7FrattiniTrivialElementaryChildData p} (src : Sec7FrattiniTrivialElementaryMStageCandidateLowerSourceData h chief K hR child) (m : towerM K) : ((towerActM K) src.sigma⁻¹) (((towerActM K) src.x₁) m) - ((towerActM K) src.sigma⁻¹) m = Q2Presentation.Induction.sec7TauReplicateOpenedNormSumAux✝ K src m (orderOf src.tau).pred + (Q2Presentation.Induction.sec7TowerActSeq✝ K (List.replicate (orderOf src.tau).pred src.tau) (((towerActM K) src.x₁) m) - Q2Presentation.Induction.sec7TowerActSeq✝¹ K (List.replicate (orderOf src.tau).pred src.tau) m)

Checkpoint318 bridge for the P x₁ to σ⁻¹ row: the checkpoint317 final-step equality is recovered from the residual form isolating the last recursive summand after subtracting the auxiliary norm prefix.

theorem Q2Presentation.Induction.sec7_frattiniTrivial_elementaryMStageCandidateWildRowManuscriptX1ConjCanonicalPrimitivePX1ToSinvTowerActionUntransportedDeviationExpandedRawNormSummandDeviationSinvMapSubOpenedTauReplicateListRecursiveNormSumAuxSigmaInvBaseSubtractedApplyPointwiseData {p : FramedPair} {h : ¬ScalarTerminal p} {chief : FirstNonScalarChiefFactor p} {K : KernelMinimal chief.toNonScalarChiefFactor} {hR : Nat.card ↥(kernelFrattini K) = 1} {child : Sec7FrattiniTrivialElementaryChildData p} (src : Sec7FrattiniTrivialElementaryMStageCandidateLowerSourceData h chief K hR child) (m : towerM K) : ((towerActM K) src.sigma⁻¹) (((towerActM K) src.x₁) m) - ((towerActM K) src.sigma⁻¹) m = Q2Presentation.Induction.sec7TauReplicateOpenedNormSumAux✝ K src m (orderOf src.tau)

Checkpoint317 bridge for the P x₁ to σ⁻¹ row: the checkpoint316 auxiliary row is recovered by folding the final recursive step of the same opened norm-sum auxiliary.

theorem Q2Presentation.Induction.sec7_frattiniTrivial_elementaryMStageCandidateWildRowManuscriptX1ConjCanonicalPrimitivePX1ToSinvTowerActionUntransportedDeviationExpandedRawNormSummandDeviationSinvMapSubOpenedTauReplicateListRecursiveNormSumSigmaInvBaseSubtractedApplyPointwiseData {p : FramedPair} {h : ¬ScalarTerminal p} {chief : FirstNonScalarChiefFactor p} {K : KernelMinimal chief.toNonScalarChiefFactor} {hR : Nat.card ↥(kernelFrattini K) = 1} {child : Sec7FrattiniTrivialElementaryChildData p} (src : Sec7FrattiniTrivialElementaryMStageCandidateLowerSourceData h chief K hR child) (m : towerM K) : ((towerActM K) src.sigma⁻¹) (((towerActM K) src.x₁) m) - ((towerActM K) src.sigma⁻¹) m = Q2Presentation.Induction.sec7TauReplicateOpenedNormSum✝ K src m

Checkpoint315 P x₁ to σ⁻¹ base-subtracted row, now theorem-level from the checkpoint316 auxiliary recursive norm-sum row.

theorem Q2Presentation.Induction.sec7_frattiniTrivial_elementaryMStageCandidateWildRowManuscriptX1ConjCanonicalPrimitivePX1ToSinvTowerActionUntransportedDeviationExpandedRawNormSummandDeviationSinvMapSubOpenedTauReplicateListRecursiveNormSumValueBeforeSigmaInvBaseSubApplyPointwiseData {p : FramedPair} {h : ¬ScalarTerminal p} {chief : FirstNonScalarChiefFactor p} {K : KernelMinimal chief.toNonScalarChiefFactor} {hR : Nat.card ↥(kernelFrattini K) = 1} {child : Sec7FrattiniTrivialElementaryChildData p} (src : Sec7FrattiniTrivialElementaryMStageCandidateLowerSourceData h chief K hR child) (m : towerM K) : ((towerActM K) src.sigma⁻¹) (((towerActM K) src.x₁) m) = ((towerActM K) src.sigma⁻¹) m + Q2Presentation.Induction.sec7TauReplicateOpenedNormSum✝ K src m

Checkpoint314 P x₁ to σ⁻¹ value-before-base-sub row, now theorem-level from the checkpoint315 transported-base-subtracted row.

theorem Q2Presentation.Induction.sec7_frattiniTrivial_elementaryMStageCandidateWildRowManuscriptX1ConjCanonicalPrimitivePX1ToSinvTowerActionUntransportedDeviationExpandedRawNormSummandDeviationSinvMapSubOpenedTauReplicateListRecursiveNormSumApplyPointwiseData {p : FramedPair} {h : ¬ScalarTerminal p} {chief : FirstNonScalarChiefFactor p} {K : KernelMinimal chief.toNonScalarChiefFactor} {hR : Nat.card ↥(kernelFrattini K) = 1} {child : Sec7FrattiniTrivialElementaryChildData p} (src : Sec7FrattiniTrivialElementaryMStageCandidateLowerSourceData h chief K hR child) (m : towerM K) : Q2Presentation.Induction.sec7TauReplicateOpenedNormSum✝ K src m = ((towerActM K) src.sigma⁻¹) (((towerActM K) src.x₁) m) - ((towerActM K) src.sigma⁻¹) m

theorem Q2Presentation.Induction.sec7_frattiniTrivial_elementaryMStageCandidateWildRowManuscriptX1ConjCanonicalPrimitivePX1ToSinvTowerActionUntransportedDeviationExpandedRawNormSummandDeviationSinvMapSubOpenedTauReplicateListApplyPointwiseData {p : FramedPair} {h : ¬ScalarTerminal p} {chief : FirstNonScalarChiefFactor p} {K : KernelMinimal chief.toNonScalarChiefFactor} {hR : Nat.card ↥(kernelFrattini K) = 1} {child : Sec7FrattiniTrivialElementaryChildData p} (src : Sec7FrattiniTrivialElementaryMStageCandidateLowerSourceData h chief K hR child) (m : towerM K) : ∑ i ∈ Finset.range (orderOf src.tau), (Q2Presentation.Induction.sec7TowerActSeq✝ K (List.replicate i src.tau) (((towerActM K) src.x₁) m) - Q2Presentation.Induction.sec7TowerActSeq✝¹ K (List.replicate i src.tau) m) = ((towerActM K) src.sigma⁻¹) (((towerActM K) src.x₁) m) - ((towerActM K) src.sigma⁻¹) m

Checkpoint312 opened replicated-list P x₁/σ⁻¹ row, now theorem-level from the recursive norm-sum datum.

theorem Q2Presentation.Induction.sec7_frattiniTrivial_elementaryMStageCandidateWildRowManuscriptX1ConjCanonicalPrimitivePX1ToSinvTowerActionUntransportedDeviationExpandedRawNormSummandDeviationSinvMapSubClosedTauReplicateListApplyPointwiseData {p : FramedPair} {h : ¬ScalarTerminal p} {chief : FirstNonScalarChiefFactor p} {K : KernelMinimal chief.toNonScalarChiefFactor} {hR : Nat.card ↥(kernelFrattini K) = 1} {child : Sec7FrattiniTrivialElementaryChildData p} (src : Sec7FrattiniTrivialElementaryMStageCandidateLowerSourceData h chief K hR child) (m : towerM K) : ∑ i ∈ Finset.range (orderOf src.tau), Q2Presentation.Induction.sec7TowerActSeq✝ K (List.replicate i src.tau) (((towerActM K) src.x₁) m - m) = ((towerActM K) src.sigma⁻¹) (((towerActM K) src.x₁) m - m)

Checkpoint311 closed-map-sub x₁^σ norm row, recovered theorem-level from its checkpoint312 summand-opened form.

theorem Q2Presentation.Induction.sec7_frattiniTrivial_elementaryMStageCandidateWildRowManuscriptX1ConjCanonicalPrimitivePX1ToSinvTowerActionUntransportedDeviationExpandedRawNormSummandDeviationSinvMapSubOpenedTauReplicateListSubOpenedApplyPointwiseData {p : FramedPair} {h : ¬ScalarTerminal p} {chief : FirstNonScalarChiefFactor p} {K : KernelMinimal chief.toNonScalarChiefFactor} {hR : Nat.card ↥(kernelFrattini K) = 1} {child : Sec7FrattiniTrivialElementaryChildData p} (src : Sec7FrattiniTrivialElementaryMStageCandidateLowerSourceData h chief K hR child) (m : towerM K) : ∑ i ∈ Finset.range (orderOf src.tau), (Q2Presentation.Induction.sec7TowerActSeq✝ K (List.replicate i src.tau) (((towerActM K) src.x₁) m) - Q2Presentation.Induction.sec7TowerActSeq✝¹ K (List.replicate i src.tau) m) = ((towerActM K) src.sigma⁻¹) (((towerActM K) src.x₁) m) - ((towerActM K) src.sigma⁻¹) m

Checkpoint310 opened replicated-list row for the P x₁ to σ⁻¹ comparison, recovered theorem-level from the checkpoint311 closed-map_sub row.

theorem Q2Presentation.Induction.sec7_frattiniTrivial_elementaryMStageCandidateWildRowManuscriptX1ConjCanonicalPrimitivePX1ToSinvTowerActionUntransportedDeviationExpandedRawNormSummandDeviationSinvMapSubOpenedTauIterTowerSubOpenedApplyPointwiseData {p : FramedPair} {h : ¬ScalarTerminal p} {chief : FirstNonScalarChiefFactor p} {K : KernelMinimal chief.toNonScalarChiefFactor} {hR : Nat.card ↥(kernelFrattini K) = 1} {child : Sec7FrattiniTrivialElementaryChildData p} (src : Sec7FrattiniTrivialElementaryMStageCandidateLowerSourceData h chief K hR child) (m : towerM K) : ∑ i ∈ Finset.range (orderOf src.tau), (Q2Presentation.Induction.sec7TowerActIter✝ K src.tau i (((towerActM K) src.x₁) m) - Q2Presentation.Induction.sec7TowerActIter✝¹ K src.tau i m) = ((towerActM K) src.sigma⁻¹) (((towerActM K) src.x₁) m) - ((towerActM K) src.sigma⁻¹) m

Checkpoint309 P x₁/σ⁻¹ datum, now theorem-level from the checkpoint310 row where every iterated τ action is an explicit replicated action list.

theorem Q2Presentation.Induction.sec7_frattiniTrivial_elementaryMStageCandidateWildRowManuscriptX1ConjCanonicalPrimitivePX1ToSinvTowerActionUntransportedDeviationExpandedRawNormSummandDeviationSinvMapSubOpenedTauIterTowerApplyPointwiseData {p : FramedPair} {h : ¬ScalarTerminal p} {chief : FirstNonScalarChiefFactor p} {K : KernelMinimal chief.toNonScalarChiefFactor} {hR : Nat.card ↥(kernelFrattini K) = 1} {child : Sec7FrattiniTrivialElementaryChildData p} (src : Sec7FrattiniTrivialElementaryMStageCandidateLowerSourceData h chief K hR child) (m : towerM K) : ∑ i ∈ Finset.range (orderOf src.tau), Q2Presentation.Induction.sec7MStageRawDifference✝ K (Q2Presentation.Induction.sec7TowerActIter✝ K src.tau i (((towerActM K) src.x₁) m)) (Q2Presentation.Induction.sec7TowerActIter✝¹ K src.tau i m) = Q2Presentation.Induction.sec7MStageRawDifference✝¹ K (((towerActM K) src.sigma⁻¹) (((towerActM K) src.x₁) m)) (((towerActM K) src.sigma⁻¹) m)

Checkpoint308 x₁^σ norm-summand row, now theorem-level from the literal subtraction form of the same iterated-action summands and target difference.

theorem Q2Presentation.Induction.sec7_frattiniTrivial_elementaryMStageCandidateWildRowManuscriptX1ConjCanonicalPrimitivePX1ToSinvTowerActionUntransportedDeviationExpandedRawNormSummandDeviationSinvMapSubOpenedTauPowerTowerApplyPointwiseData {p : FramedPair} {h : ¬ScalarTerminal p} {chief : FirstNonScalarChiefFactor p} {K : KernelMinimal chief.toNonScalarChiefFactor} {hR : Nat.card ↥(kernelFrattini K) = 1} {child : Sec7FrattiniTrivialElementaryChildData p} (src : Sec7FrattiniTrivialElementaryMStageCandidateLowerSourceData h chief K hR child) (m : towerM K) : ∑ i ∈ Finset.range (orderOf src.tau), (((towerActM K) (src.tau ^ i)) (((towerActM K) src.x₁) m) - ((towerActM K) (src.tau ^ i)) m) = ((towerActM K) src.sigma⁻¹) (((towerActM K) src.x₁) m) - ((towerActM K) src.sigma⁻¹) m

theorem Q2Presentation.Induction.sec7_frattiniTrivial_elementaryMStageCandidateWildRowManuscriptX1ConjCanonicalPrimitivePX1ToSinvTowerActionUntransportedDeviationExpandedRawNormSummandDeviationSinvMapSubOpenedApplyPointwiseData {p : FramedPair} {h : ¬ScalarTerminal p} {chief : FirstNonScalarChiefFactor p} {K : KernelMinimal chief.toNonScalarChiefFactor} {hR : Nat.card ↥(kernelFrattini K) = 1} {child : Sec7FrattiniTrivialElementaryChildData p} (src : Sec7FrattiniTrivialElementaryMStageCandidateLowerSourceData h chief K hR child) (m : towerM K) : ∑ i ∈ Finset.range (orderOf src.tau), (((sec7MStageCandidateActionEnd K) src.tau ^ i) (((towerActM K) src.x₁) m) - ((sec7MStageCandidateActionEnd K) src.tau ^ i) m) = ((sec7MStageCandidateActionEnd K) src.sigma⁻¹) (((towerActM K) src.x₁) m) - ((sec7MStageCandidateActionEnd K) src.sigma⁻¹) m

theorem Q2Presentation.Induction.sec7_frattiniTrivial_elementaryMStageCandidateWildRowManuscriptX1ConjCanonicalPrimitivePX1ToSinvTowerActionUntransportedDeviationExpandedRawNormSummandDeviationSinvApplyPointwiseData {p : FramedPair} {h : ¬ScalarTerminal p} {chief : FirstNonScalarChiefFactor p} {K : KernelMinimal chief.toNonScalarChiefFactor} {hR : Nat.card ↥(kernelFrattini K) = 1} {child : Sec7FrattiniTrivialElementaryChildData p} (src : Sec7FrattiniTrivialElementaryMStageCandidateLowerSourceData h chief K hR child) (m : towerM K) : ∑ i ∈ Finset.range (orderOf src.tau), ((sec7MStageCandidateActionEnd K) src.tau ^ i) (((towerActM K) src.x₁) m - m) = ((sec7MStageCandidateActionEnd K) src.sigma⁻¹) (((towerActM K) src.x₁) m - m)

theorem Q2Presentation.Induction.sec7_frattiniTrivial_elementaryMStageCandidateWildRowManuscriptX1ConjCanonicalPrimitivePX1ToSinvTowerActionUntransportedDeviationExpandedRawNormSummandDifferenceSinvApplyPointwiseData {p : FramedPair} {h : ¬ScalarTerminal p} {chief : FirstNonScalarChiefFactor p} {K : KernelMinimal chief.toNonScalarChiefFactor} {hR : Nat.card ↥(kernelFrattini K) = 1} {child : Sec7FrattiniTrivialElementaryChildData p} (src : Sec7FrattiniTrivialElementaryMStageCandidateLowerSourceData h chief K hR child) (m : towerM K) : ∑ i ∈ Finset.range (orderOf src.tau), ((sec7MStageCandidateActionEnd K) src.tau ^ i) (((towerActM K) src.x₁) m) - ∑ i ∈ Finset.range (orderOf src.tau), ((sec7MStageCandidateActionEnd K) src.tau ^ i) m = ((sec7MStageCandidateActionEnd K) src.sigma⁻¹) (((towerActM K) src.x₁) m) - ((sec7MStageCandidateActionEnd K) src.sigma⁻¹) m

theorem Q2Presentation.Induction.sec7_frattiniTrivial_elementaryMStageCandidateWildRowManuscriptX1ConjCanonicalPrimitivePX1ToSinvTowerActionUntransportedDeviationExpandedRawSeparatedDifferenceNormSummandSinvApplyPointwiseData {p : FramedPair} {h : ¬ScalarTerminal p} {chief : FirstNonScalarChiefFactor p} {K : KernelMinimal chief.toNonScalarChiefFactor} {hR : Nat.card ↥(kernelFrattini K) = 1} {child : Sec7FrattiniTrivialElementaryChildData p} (src : Sec7FrattiniTrivialElementaryMStageCandidateLowerSourceData h chief K hR child) (m : towerM K) : ∑ i ∈ Finset.range (orderOf src.tau), ((sec7MStageCandidateActionEnd K) src.tau ^ i) (((towerActM K) src.x₁) m) = ∑ i ∈ Finset.range (orderOf src.tau), ((sec7MStageCandidateActionEnd K) src.tau ^ i) m + (((sec7MStageCandidateActionEnd K) src.sigma⁻¹) (((towerActM K) src.x₁) m) - ((sec7MStageCandidateActionEnd K) src.sigma⁻¹) m)

Checkpoint303 primitive datum for the P/σ⁻¹ part of x₁^σ, now theorem-level from the checkpoint304 norm-summand difference row.

theorem Q2Presentation.Induction.sec7_frattiniTrivial_elementaryMStageCandidateWildRowManuscriptX1ConjCanonicalPrimitivePX1ToSinvTowerActionUntransportedDeviationExpandedRawSeparatedDifferenceNormSumSinvApplyPointwiseData {p : FramedPair} {h : ¬ScalarTerminal p} {chief : FirstNonScalarChiefFactor p} {K : KernelMinimal chief.toNonScalarChiefFactor} {hR : Nat.card ↥(kernelFrattini K) = 1} {child : Sec7FrattiniTrivialElementaryChildData p} (src : Sec7FrattiniTrivialElementaryMStageCandidateLowerSourceData h chief K hR child) (m : towerM K) : (∑ i ∈ Finset.range (orderOf src.tau), (sec7MStageCandidateActionEnd K) src.tau ^ i) (((towerActM K) src.x₁) m) = (∑ i ∈ Finset.range (orderOf src.tau), (sec7MStageCandidateActionEnd K) src.tau ^ i) m + (((sec7MStageCandidateActionEnd K) src.sigma⁻¹) (((towerActM K) src.x₁) m) - ((sec7MStageCandidateActionEnd K) src.sigma⁻¹) m)

Checkpoint302 primitive datum for the P/σ⁻¹ part of x₁^σ, now theorem-level from the checkpoint303 pointwise-summand norm row.

theorem Q2Presentation.Induction.sec7_frattiniTrivial_elementaryMStageCandidateWildRowManuscriptX1ConjCanonicalPrimitivePX1ToSinvTowerActionUntransportedDeviationExpandedRawSeparatedDifferencePValueFormulaUnfoldedNormSinvApplyPointwiseData {p : FramedPair} {h : ¬ScalarTerminal p} {chief : FirstNonScalarChiefFactor p} {K : KernelMinimal chief.toNonScalarChiefFactor} {hR : Nat.card ↥(kernelFrattini K) = 1} {child : Sec7FrattiniTrivialElementaryChildData p} (src : Sec7FrattiniTrivialElementaryMStageCandidateLowerSourceData h chief K hR child) (m : towerM K) : (Lifting.repNorm (sec7MStageCandidateActionEnd K) src.tau) (((towerActM K) src.x₁) m) = (Lifting.repNorm (sec7MStageCandidateActionEnd K) src.tau) m + (((sec7MStageCandidateActionEnd K) src.sigma⁻¹) (((towerActM K) src.x₁) m) - ((sec7MStageCandidateActionEnd K) src.sigma⁻¹) m)

Checkpoint301 primitive datum for the P/σ⁻¹ part of x₁^σ, now theorem-level from the checkpoint302 literal norm-sum row.

theorem Q2Presentation.Induction.sec7_frattiniTrivial_elementaryMStageCandidateWildRowManuscriptX1ConjCanonicalPrimitivePX1ToSinvTowerActionUntransportedDeviationExpandedRawSeparatedDifferencePValueFormulaApplyPointwiseData {p : FramedPair} {h : ¬ScalarTerminal p} {chief : FirstNonScalarChiefFactor p} {K : KernelMinimal chief.toNonScalarChiefFactor} {hR : Nat.card ↥(kernelFrattini K) = 1} {child : Sec7FrattiniTrivialElementaryChildData p} (src : Sec7FrattiniTrivialElementaryMStageCandidateLowerSourceData h chief K hR child) (m : towerM K) : src.P (((towerActM K) src.x₁) m) = src.P m + (src.Sinv (((towerActM K) src.x₁) m) - src.Sinv m)

Checkpoint300 primitive datum for the P/σ⁻¹ part of x₁^σ, now theorem-level from the checkpoint301 unfolded norm/transport row.

theorem Q2Presentation.Induction.sec7_frattiniTrivial_elementaryMStageCandidateWildRowManuscriptX1ConjCanonicalPrimitivePX1ToSinvTowerActionUntransportedDeviationExpandedRawSeparatedDifferenceSubtractionZeroApplyPointwiseData {p : FramedPair} {h : ¬ScalarTerminal p} {chief : FirstNonScalarChiefFactor p} {K : KernelMinimal chief.toNonScalarChiefFactor} {hR : Nat.card ↥(kernelFrattini K) = 1} {child : Sec7FrattiniTrivialElementaryChildData p} (src : Sec7FrattiniTrivialElementaryMStageCandidateLowerSourceData h chief K hR child) (m : towerM K) : src.P (((towerActM K) src.x₁) m) - src.P m - (src.Sinv (((towerActM K) src.x₁) m) - src.Sinv m) = 0

Checkpoint299 primitive datum for the P/σ⁻¹ part of x₁^σ, now theorem-level from the checkpoint300 point-value formula.

theorem Q2Presentation.Induction.sec7_frattiniTrivial_elementaryMStageCandidateWildRowManuscriptX1ConjCanonicalPrimitivePX1ToSinvTowerActionUntransportedDeviationExpandedRawDifferenceBalanceApplyPointwiseData {p : FramedPair} {h : ¬ScalarTerminal p} {chief : FirstNonScalarChiefFactor p} {K : KernelMinimal chief.toNonScalarChiefFactor} {hR : Nat.card ↥(kernelFrattini K) = 1} {child : Sec7FrattiniTrivialElementaryChildData p} (src : Sec7FrattiniTrivialElementaryMStageCandidateLowerSourceData h chief K hR child) (m : towerM K) : src.P (((towerActM K) src.x₁) m) - src.Sinv (((towerActM K) src.x₁) m) = src.P m - src.Sinv m

Checkpoint298 primitive atom for the P/σ⁻¹ part of x₁^σ, now theorem-level from the checkpoint299 separated-difference zero row.

theorem Q2Presentation.Induction.sec7_frattiniTrivial_elementaryMStageCandidateWildRowManuscriptX1ConjCanonicalPrimitivePX1ToSinvTowerActionUntransportedDeviationExpandedRawAdditiveBalanceSubtractionZeroApplyPointwiseData {p : FramedPair} {h : ¬ScalarTerminal p} {chief : FirstNonScalarChiefFactor p} {K : KernelMinimal chief.toNonScalarChiefFactor} {hR : Nat.card ↥(kernelFrattini K) = 1} {child : Sec7FrattiniTrivialElementaryChildData p} (src : Sec7FrattiniTrivialElementaryMStageCandidateLowerSourceData h chief K hR child) (m : towerM K) : src.P (((towerActM K) src.x₁) m) + src.Sinv m - (src.Sinv (((towerActM K) src.x₁) m) + src.P m) = 0

Checkpoint297 primitive atom for the P/σ⁻¹ part of x₁^σ, now theorem-level from the grouped difference-balance checkpoint298 row.

theorem Q2Presentation.Induction.sec7_frattiniTrivial_elementaryMStageCandidateWildRowManuscriptX1ConjCanonicalPrimitivePX1ToSinvTowerActionUntransportedDeviationExpandedRawAdditiveBalanceApplyPointwiseData {p : FramedPair} {h : ¬ScalarTerminal p} {chief : FirstNonScalarChiefFactor p} {K : KernelMinimal chief.toNonScalarChiefFactor} {hR : Nat.card ↥(kernelFrattini K) = 1} {child : Sec7FrattiniTrivialElementaryChildData p} (src : Sec7FrattiniTrivialElementaryMStageCandidateLowerSourceData h chief K hR child) (m : towerM K) : have X := (towerActM K) src.x₁; src.P (X m) + src.Sinv m = src.Sinv (X m) + src.P m

Checkpoint296 P/S⁻¹ raw additive balance, now theorem-level from the subtraction-zero four-image statement. The new leaf is the literal additive zero row before applying sub_eq_zero.

theorem Q2Presentation.Induction.sec7_frattiniTrivial_elementaryMStageCandidateWildRowManuscriptX1ConjCanonicalPrimitivePX1ToSinvTowerActionUntransportedDeviationExpandedRawApplyPointwiseData {p : FramedPair} {h : ¬ScalarTerminal p} {chief : FirstNonScalarChiefFactor p} {K : KernelMinimal chief.toNonScalarChiefFactor} {hR : Nat.card ↥(kernelFrattini K) = 1} {child : Sec7FrattiniTrivialElementaryChildData p} (src : Sec7FrattiniTrivialElementaryMStageCandidateLowerSourceData h chief K hR child) (m : towerM K) : have X := (towerActM K) src.x₁; src.P (X m) - src.P m = src.Sinv (X m) - src.Sinv m

Checkpoint295 expanded untransported-deviation equality, now theorem-level from the still more literal additive balance above.

theorem Q2Presentation.Induction.sec7_frattiniTrivial_elementaryMStageCandidateWildRowManuscriptX1ConjCanonicalPrimitivePX1ToSinvTowerActionUntransportedDeviationValueRawApplyPointwiseData {p : FramedPair} {h : ¬ScalarTerminal p} {chief : FirstNonScalarChiefFactor p} {K : KernelMinimal chief.toNonScalarChiefFactor} {hR : Nat.card ↥(kernelFrattini K) = 1} {child : Sec7FrattiniTrivialElementaryChildData p} (src : Sec7FrattiniTrivialElementaryMStageCandidateLowerSourceData h chief K hR child) (m : towerM K) : have X := (towerActM K) src.x₁; src.P (X m - m) = src.Sinv (X m - m)

Checkpoint294 untransported concrete x₁ deviation equality, now theorem-level from the expanded pointwise equality.

theorem Q2Presentation.Induction.sec7_frattiniTrivial_elementaryMStageCandidateWildRowManuscriptX1ConjCanonicalPrimitivePX1ToSinvTowerActionUntransportedDeviationKernelRawApplyPointwiseData {p : FramedPair} {h : ¬ScalarTerminal p} {chief : FirstNonScalarChiefFactor p} {K : KernelMinimal chief.toNonScalarChiefFactor} {hR : Nat.card ↥(kernelFrattini K) = 1} {child : Sec7FrattiniTrivialElementaryChildData p} (src : Sec7FrattiniTrivialElementaryMStageCandidateLowerSourceData h chief K hR child) (m : towerM K) : have X := (towerActM K) src.x₁; src.P (X m - m) - src.Sinv (X m - m) = 0

Checkpoint293 untransported-deviation kernel for x₁^σ, now theorem-level from the raw pointwise equality of the two visible images.

theorem Q2Presentation.Induction.sec7_frattiniTrivial_elementaryMStageCandidateWildRowManuscriptX1ConjCanonicalPrimitivePX1ToSinvTowerActionTransportedDeviationKernelRawApplyPointwiseData {p : FramedPair} {h : ¬ScalarTerminal p} {chief : FirstNonScalarChiefFactor p} {K : KernelMinimal chief.toNonScalarChiefFactor} {hR : Nat.card ↥(kernelFrattini K) = 1} {child : Sec7FrattiniTrivialElementaryChildData p} (src : Sec7FrattiniTrivialElementaryMStageCandidateLowerSourceData h chief K hR child) (m : towerM K) : have X := (towerActM K) src.x₁; src.P (src.S (src.Sinv (X m - m))) - src.Sinv (X m - m) = 0

Checkpoint292 transported-deviation kernel, now theorem-level from the smaller untransported kernel statement and the formal S·S⁻¹ cancellation.

theorem Q2Presentation.Induction.sec7_frattiniTrivial_elementaryMStageCandidateWildRowManuscriptX1ConjCanonicalPrimitivePX1ToSinvTowerActionTransportedDeviationKernelApplyPointwiseData {p : FramedPair} {h : ¬ScalarTerminal p} {chief : FirstNonScalarChiefFactor p} {K : KernelMinimal chief.toNonScalarChiefFactor} {hR : Nat.card ↥(kernelFrattini K) = 1} {child : Sec7FrattiniTrivialElementaryChildData p} (src : Sec7FrattiniTrivialElementaryMStageCandidateLowerSourceData h chief K hR child) (m : towerM K) : have X := ↑((towerActM K) src.x₁); ((src.P * src.S - 1) * (src.Sinv * (X - 1))) m = 0

Checkpoint291 transported-deviation kernel pointwise row, now theorem-level from its raw towerActM application form.

theorem Q2Presentation.Induction.sec7_frattiniTrivial_elementaryMStageCandidateWildRowManuscriptX1ConjCanonicalPrimitivePX1ToSinvTowerActionTransportedDeviationKernelOperatorData {p : FramedPair} {h : ¬ScalarTerminal p} {chief : FirstNonScalarChiefFactor p} {K : KernelMinimal chief.toNonScalarChiefFactor} {hR : Nat.card ↥(kernelFrattini K) = 1} {child : Sec7FrattiniTrivialElementaryChildData p} (src : Sec7FrattiniTrivialElementaryMStageCandidateLowerSourceData h chief K hR child) : have X := ↑((towerActM K) src.x₁); (src.P * src.S - 1) * (src.Sinv * (X - 1)) = 0

Former checkpoint290 transported-deviation kernel, now theorem-level from its pointwise applied kernel form.

theorem Q2Presentation.Induction.sec7_frattiniTrivial_elementaryMStageCandidateWildRowManuscriptX1ConjCanonicalPrimitivePX1ToSinvTowerActionDeviationKernelOperatorData {p : FramedPair} {h : ¬ScalarTerminal p} {chief : FirstNonScalarChiefFactor p} {K : KernelMinimal chief.toNonScalarChiefFactor} {hR : Nat.card ↥(kernelFrattini K) = 1} {child : Sec7FrattiniTrivialElementaryChildData p} (src : Sec7FrattiniTrivialElementaryMStageCandidateLowerSourceData h chief K hR child) : have X := ↑((towerActM K) src.x₁); (src.P - src.Sinv) * (X - 1) = 0

Checkpoint289 residual for x₁^σ, now theorem-level from the factored transported-deviation kernel statement.

theorem Q2Presentation.Induction.sec7_frattiniTrivial_elementaryMStageCandidateWildRowManuscriptX1ConjCanonicalPrimitivePX1ToSinvTowerActionDeviationOperatorData {p : FramedPair} {h : ¬ScalarTerminal p} {chief : FirstNonScalarChiefFactor p} {K : KernelMinimal chief.toNonScalarChiefFactor} {hR : Nat.card ↥(kernelFrattini K) = 1} {child : Sec7FrattiniTrivialElementaryChildData p} (src : Sec7FrattiniTrivialElementaryMStageCandidateLowerSourceData h chief K hR child) : have X := ↑((towerActM K) src.x₁); src.P * (X - 1) = src.Sinv * (X - 1)

Checkpoint288 P/S⁻¹ deviation equality, now theorem-level from the narrower checkpoint289 kernel statement.

theorem Q2Presentation.Induction.sec7_frattiniTrivial_elementaryMStageCandidateWildRowManuscriptX1ConjCanonicalPrimitivePX1ToSinvTowerActionDeviationValueApplyPointwiseData {p : FramedPair} {h : ¬ScalarTerminal p} {chief : FirstNonScalarChiefFactor p} {K : KernelMinimal chief.toNonScalarChiefFactor} {hR : Nat.card ↥(kernelFrattini K) = 1} {child : Sec7FrattiniTrivialElementaryChildData p} (src : Sec7FrattiniTrivialElementaryMStageCandidateLowerSourceData h chief K hR child) (m : towerM K) : src.P (((towerActM K) src.x₁) m - m) = src.Sinv (((towerActM K) src.x₁) m - m)

Checkpoint287 lowering for the P/S⁻¹ part of x₁^σ: the residual is an operator equality on the concrete x₁ deviation, not a pointwise value.

theorem Q2Presentation.Induction.sec7_frattiniTrivial_elementaryMStageCandidateWildRowManuscriptX1ConjCanonicalPrimitivePX1ToSinvRawDeviationValueApplyPointwiseData {p : FramedPair} {h : ¬ScalarTerminal p} {chief : FirstNonScalarChiefFactor p} {K : KernelMinimal chief.toNonScalarChiefFactor} {hR : Nat.card ↥(kernelFrattini K) = 1} {child : Sec7FrattiniTrivialElementaryChildData p} (src : Sec7FrattiniTrivialElementaryMStageCandidateLowerSourceData h chief K hR child) (m : towerM K) : have ρ := sec7MStageCandidateActionEnd K; src.P ((ρ src.x₁) m - m) = src.Sinv ((ρ src.x₁) m - m)

Checkpoint284 projector/transport residual, now theorem-level from the checkpoint285 concrete tower-action deviation statement.

theorem Q2Presentation.Induction.sec7_frattiniTrivial_elementaryMStageCandidateWildRowManuscriptX1ConjCanonicalPrimitivePX1ToSigmaInvRowRawDeviationValueApplyPointwiseData {p : FramedPair} {h : ¬ScalarTerminal p} {chief : FirstNonScalarChiefFactor p} {K : KernelMinimal chief.toNonScalarChiefFactor} {hR : Nat.card ↥(kernelFrattini K) = 1} {child : Sec7FrattiniTrivialElementaryChildData p} (src : Sec7FrattiniTrivialElementaryMStageCandidateLowerSourceData h chief K hR child) (m : towerM K) : have ρ := sec7MStageCandidateActionEnd K; src.P ((ρ src.x₁) m - m) = (ρ src.sigma⁻¹) ((ρ src.x₁) m - m)

Checkpoint283 x₁^σ projector/transport residual, now theorem-level from the checkpoint284 canonical S⁻¹ deviation statement.

theorem Q2Presentation.Induction.sec7_frattiniTrivial_elementaryMStageCandidateWildRowManuscriptX1ConjCanonicalPrimitivePX1ToSigmaInvRowDeviationExpandedEqualityApplyPointwiseData {p : FramedPair} {h : ¬ScalarTerminal p} {chief : FirstNonScalarChiefFactor p} {K : KernelMinimal chief.toNonScalarChiefFactor} {hR : Nat.card ↥(kernelFrattini K) = 1} {child : Sec7FrattiniTrivialElementaryChildData p} (src : Sec7FrattiniTrivialElementaryMStageCandidateLowerSourceData h chief K hR child) (m : towerM K) : have ρ := sec7MStageCandidateActionEnd K; src.P ((ρ src.x₁) m) - src.P m = (ρ src.sigma⁻¹) ((ρ src.x₁) m) - (ρ src.sigma⁻¹) m

Checkpoint282 lower atom for x₁^σ, now theorem-level from the checkpoint283 single-deviation value equality.

theorem Q2Presentation.Induction.sec7_frattiniTrivial_elementaryMStageCandidateWildRowManuscriptX1ConjCanonicalPrimitivePX1ToSigmaInvRowDeviationKernelExpandedApplyPointwiseData {p : FramedPair} {h : ¬ScalarTerminal p} {chief : FirstNonScalarChiefFactor p} {K : KernelMinimal chief.toNonScalarChiefFactor} {hR : Nat.card ↥(kernelFrattini K) = 1} {child : Sec7FrattiniTrivialElementaryChildData p} (src : Sec7FrattiniTrivialElementaryMStageCandidateLowerSourceData h chief K hR child) (m : towerM K) : have ρ := sec7MStageCandidateActionEnd K; src.P ((ρ src.x₁) m) - src.P m - ((ρ src.sigma⁻¹) ((ρ src.x₁) m) - (ρ src.sigma⁻¹) m) = 0

Checkpoint281 lower atom for x₁^σ, now theorem-level from the checkpoint282 expanded deviation equality.

theorem Q2Presentation.Induction.sec7_frattiniTrivial_elementaryMStageCandidateWildRowManuscriptX1ConjCanonicalPrimitivePX1ToSigmaInvRowDeviationKernelApplyPointwiseData {p : FramedPair} {h : ¬ScalarTerminal p} {chief : FirstNonScalarChiefFactor p} {K : KernelMinimal chief.toNonScalarChiefFactor} {hR : Nat.card ↥(kernelFrattini K) = 1} {child : Sec7FrattiniTrivialElementaryChildData p} (src : Sec7FrattiniTrivialElementaryMStageCandidateLowerSourceData h chief K hR child) (m : towerM K) : have ρ := sec7MStageCandidateActionEnd K; ((src.P - ρ src.sigma⁻¹) * (ρ src.x₁ - 1)) m = 0

Checkpoint280 residual for x₁^σ, now theorem-level from the checkpoint281 fully expanded pointwise kernel balance.

theorem Q2Presentation.Induction.sec7_frattiniTrivial_elementaryMStageCandidateWildRowManuscriptX1ConjCanonicalPrimitivePX1ToSigmaInvRowDeviationRangeData {p : FramedPair} {h : ¬ScalarTerminal p} {chief : FirstNonScalarChiefFactor p} {K : KernelMinimal chief.toNonScalarChiefFactor} {hR : Nat.card ↥(kernelFrattini K) = 1} {child : Sec7FrattiniTrivialElementaryChildData p} (src : Sec7FrattiniTrivialElementaryMStageCandidateLowerSourceData h chief K hR child) (v : towerM K) (hv : ∃ (m : towerM K), v = ((sec7MStageCandidateActionEnd K) src.x₁ - 1) m) : src.P v = ((sec7MStageCandidateActionEnd K) src.sigma⁻¹) v

Checkpoint279 lower atom for x₁^σ, now theorem-level from the narrower checkpoint280 kernel form of the deviation-range comparison.

theorem Q2Presentation.Induction.sec7_frattiniTrivial_elementaryMStageCandidateWildRowManuscriptX1ConjCanonicalPrimitivePX1ToSigmaInvRowDeviationApplyPointwiseData {p : FramedPair} {h : ¬ScalarTerminal p} {chief : FirstNonScalarChiefFactor p} {K : KernelMinimal chief.toNonScalarChiefFactor} {hR : Nat.card ↥(kernelFrattini K) = 1} {child : Sec7FrattiniTrivialElementaryChildData p} (src : Sec7FrattiniTrivialElementaryMStageCandidateLowerSourceData h chief K hR child) (m : towerM K) : have ρ := sec7MStageCandidateActionEnd K; src.P ((ρ src.x₁ - 1) m) = (ρ src.sigma⁻¹) ((ρ src.x₁ - 1) m)

Checkpoint278 primitive P(X₁-1)/σ⁻¹(X₁-1) pointwise atom, now theorem-level from the smaller deviation-image row statement.

theorem Q2Presentation.Induction.sec7_frattiniTrivial_elementaryMStageCandidateWildRowManuscriptX1ConjCanonicalPrimitivePX1ToSigmaInvRowExpandedApplyPointwiseData {p : FramedPair} {h : ¬ScalarTerminal p} {chief : FirstNonScalarChiefFactor p} {K : KernelMinimal chief.toNonScalarChiefFactor} {hR : Nat.card ↥(kernelFrattini K) = 1} {child : Sec7FrattiniTrivialElementaryChildData p} (src : Sec7FrattiniTrivialElementaryMStageCandidateLowerSourceData h chief K hR child) (m : towerM K) : have ρ := sec7MStageCandidateActionEnd K; src.P ((ρ src.x₁) m) - src.P m = (ρ src.sigma⁻¹) ((ρ src.x₁) m) - (ρ src.sigma⁻¹) m

Checkpoint277 expanded P(X₁-1)/σ⁻¹(X₁-1) application equality, now theorem-level from the primitive-deviation row by linearity.

theorem Q2Presentation.Induction.sec7_frattiniTrivial_elementaryMStageCandidateWildRowManuscriptX1ConjCanonicalPrimitivePX1ToSigmaInvRowApplyPointwiseData {p : FramedPair} {h : ¬ScalarTerminal p} {chief : FirstNonScalarChiefFactor p} {K : KernelMinimal chief.toNonScalarChiefFactor} {hR : Nat.card ↥(kernelFrattini K) = 1} {child : Sec7FrattiniTrivialElementaryChildData p} (src : Sec7FrattiniTrivialElementaryMStageCandidateLowerSourceData h chief K hR child) (m : towerM K) : have ρ := sec7MStageCandidateActionEnd K; src.P ((ρ src.x₁ - 1) m) = (ρ src.sigma⁻¹) ((ρ src.x₁ - 1) m)

Checkpoint276 primitive P(X₁-1)/σ⁻¹(X₁-1) application row, now theorem-level from the fully expanded pointwise deviation equality.

theorem Q2Presentation.Induction.sec7_frattiniTrivial_elementaryMStageCandidateWildRowManuscriptX1ConjCanonicalPrimitivePX1ToSigmaInvRowPointwiseData {p : FramedPair} {h : ¬ScalarTerminal p} {chief : FirstNonScalarChiefFactor p} {K : KernelMinimal chief.toNonScalarChiefFactor} {hR : Nat.card ↥(kernelFrattini K) = 1} {child : Sec7FrattiniTrivialElementaryChildData p} (src : Sec7FrattiniTrivialElementaryMStageCandidateLowerSourceData h chief K hR child) (m : towerM K) : have ρ := sec7MStageCandidateActionEnd K; (src.P * (ρ src.x₁ - 1)) m = (ρ src.sigma⁻¹ * (ρ src.x₁ - 1)) m

Checkpoint275 primitive P(X₁-1)/σ⁻¹(X₁-1) row, now theorem-level from the lower application-level coordinate equality.

axiom Q2Presentation.Induction.sec7_frattiniTrivial_elementaryMStageCandidateWildRowManuscriptX1ConjCanonicalPrimitiveTowerActionSigmaInvTransportSquareUntransportedRawDeltaStepExpandedNamedTransportedDeviationDifferenceSingleActionDifferenceCoreOpenedRawSingleCornerDifferenceClosedZeroLowerPrimitiveApplyPointwiseData {p : FramedPair} {h : ¬ScalarTerminal p} {chief : FirstNonScalarChiefFactor p} {K : KernelMinimal chief.toNonScalarChiefFactor} {hR : Nat.card ↥(kernelFrattini K) = 1} {child : Sec7FrattiniTrivialElementaryChildData p} (src : Sec7FrattiniTrivialElementaryMStageCandidateLowerSourceData h chief K hR child) (m : towerM K) : ((towerActM K) src.x₁) (((towerActM K) src.sigma⁻¹) (((towerActM K) src.x₁) m)) - ((towerActM K) src.sigma⁻¹) (((towerActM K) src.x₁) m) - (((towerActM K) src.x₁) m + m) = 0

Checkpoint332 lower primitive for the transported-square raw corner: keep the isolated single-corner additive difference as a closed zero, before solving it back to the checkpoint331 corner equality.

theorem Q2Presentation.Induction.sec7_frattiniTrivial_elementaryMStageCandidateWildRowManuscriptX1ConjCanonicalPrimitiveTowerActionSigmaInvTransportSquareUntransportedRawDeltaStepExpandedNamedTransportedDeviationDifferenceSingleActionDifferenceCoreOpenedRawSingleCornerEqualityLowerPrimitiveApplyPointwiseData {p : FramedPair} {h : ¬ScalarTerminal p} {chief : FirstNonScalarChiefFactor p} {K : KernelMinimal chief.toNonScalarChiefFactor} {hR : Nat.card ↥(kernelFrattini K) = 1} {child : Sec7FrattiniTrivialElementaryChildData p} (src : Sec7FrattiniTrivialElementaryMStageCandidateLowerSourceData h chief K hR child) (m : towerM K) : ((towerActM K) src.x₁) (((towerActM K) src.sigma⁻¹) (((towerActM K) src.x₁) m)) = ((towerActM K) src.sigma⁻¹) (((towerActM K) src.x₁) m) + (((towerActM K) src.x₁) m + m)

Checkpoint332 bridge recovering the checkpoint331 transported-square single-corner equality from the lower difference-zero primitive.

theorem Q2Presentation.Induction.sec7_frattiniTrivial_elementaryMStageCandidateWildRowManuscriptX1ConjCanonicalPrimitiveTowerActionSigmaInvTransportSquareUntransportedRawDeltaStepExpandedNamedTransportedDeviationDifferenceSingleActionDifferenceCoreOpenedRawSingleCornerDifferenceClosedZeroPrimitiveApplyPointwiseData {p : FramedPair} {h : ¬ScalarTerminal p} {chief : FirstNonScalarChiefFactor p} {K : KernelMinimal chief.toNonScalarChiefFactor} {hR : Nat.card ↥(kernelFrattini K) = 1} {child : Sec7FrattiniTrivialElementaryChildData p} (src : Sec7FrattiniTrivialElementaryMStageCandidateLowerSourceData h chief K hR child) (m : towerM K) : ((towerActM K) src.x₁) (((towerActM K) src.sigma⁻¹) (((towerActM K) src.x₁) m)) - ((towerActM K) src.sigma⁻¹) (((towerActM K) src.x₁) m) - (((towerActM K) src.x₁) m + m) = 0

Checkpoint331 bridge closing the checkpoint330 transported-square single-corner difference-zero residual from the lower corner equality.

theorem Q2Presentation.Induction.sec7_frattiniTrivial_elementaryMStageCandidateWildRowManuscriptX1ConjCanonicalPrimitiveTowerActionSigmaInvTransportSquareUntransportedRawDeltaStepExpandedNamedTransportedDeviationDifferenceSingleActionDifferenceCoreOpenedRawSingleCornerDifferencePrimitiveApplyPointwiseData {p : FramedPair} {h : ¬ScalarTerminal p} {chief : FirstNonScalarChiefFactor p} {K : KernelMinimal chief.toNonScalarChiefFactor} {hR : Nat.card ↥(kernelFrattini K) = 1} {child : Sec7FrattiniTrivialElementaryChildData p} (src : Sec7FrattiniTrivialElementaryMStageCandidateLowerSourceData h chief K hR child) (m : towerM K) : ((towerActM K) src.x₁) (((towerActM K) src.sigma⁻¹) (((towerActM K) src.x₁) m)) - ((towerActM K) src.sigma⁻¹) (((towerActM K) src.x₁) m) = ((towerActM K) src.x₁) m + m

Checkpoint330 bridge for the checkpoint329 transported-square single-corner difference.

theorem Q2Presentation.Induction.sec7_frattiniTrivial_elementaryMStageCandidateWildRowManuscriptX1ConjCanonicalPrimitiveTowerActionSigmaInvTransportSquareUntransportedRawDeltaStepExpandedNamedTransportedDeviationDifferenceSingleActionDifferenceCoreOpenedRawSingleCornerEqualityPrimitiveApplyPointwiseData {p : FramedPair} {h : ¬ScalarTerminal p} {chief : FirstNonScalarChiefFactor p} {K : KernelMinimal chief.toNonScalarChiefFactor} {hR : Nat.card ↥(kernelFrattini K) = 1} {child : Sec7FrattiniTrivialElementaryChildData p} (src : Sec7FrattiniTrivialElementaryMStageCandidateLowerSourceData h chief K hR child) (m : towerM K) : ((towerActM K) src.x₁) (((towerActM K) src.sigma⁻¹) (((towerActM K) src.x₁) m)) = ((towerActM K) src.sigma⁻¹) (((towerActM K) src.x₁) m) + ((towerActM K) src.x₁) m + m

Checkpoint329 bridge for the checkpoint328 transported-square corner.

theorem Q2Presentation.Induction.sec7_frattiniTrivial_elementaryMStageCandidateWildRowManuscriptX1ConjCanonicalPrimitiveTowerActionSigmaInvTransportSquareUntransportedRawDeltaStepExpandedNamedTransportedDeviationDifferenceSingleActionDifferenceCoreOpenedRawSingleCornerClosedZeroApplyPointwiseData {p : FramedPair} {h : ¬ScalarTerminal p} {chief : FirstNonScalarChiefFactor p} {K : KernelMinimal chief.toNonScalarChiefFactor} {hR : Nat.card ↥(kernelFrattini K) = 1} {child : Sec7FrattiniTrivialElementaryChildData p} (src : Sec7FrattiniTrivialElementaryMStageCandidateLowerSourceData h chief K hR child) (m : towerM K) : ((towerActM K) src.x₁) (((towerActM K) src.sigma⁻¹) (((towerActM K) src.x₁) m)) + (((towerActM K) src.sigma⁻¹) (((towerActM K) src.x₁) m) + ((towerActM K) src.x₁) m + m) = 0

Checkpoint328 bridge for the checkpoint327 transported-square zero row: the new datum isolates the single corner as the sum of the other three corners. In characteristic two this immediately gives the closed four-corner zero row.

theorem Q2Presentation.Induction.sec7_frattiniTrivial_elementaryMStageCandidateWildRowManuscriptX1ConjCanonicalPrimitiveTowerActionSigmaInvTransportSquareUntransportedRawDeltaStepExpandedNamedTransportedDeviationDifferenceSingleActionDifferenceCoreOpenedRawSingleCornerEqualityApplyPointwiseData {p : FramedPair} {h : ¬ScalarTerminal p} {chief : FirstNonScalarChiefFactor p} {K : KernelMinimal chief.toNonScalarChiefFactor} {hR : Nat.card ↥(kernelFrattini K) = 1} {child : Sec7FrattiniTrivialElementaryChildData p} (src : Sec7FrattiniTrivialElementaryMStageCandidateLowerSourceData h chief K hR child) (m : towerM K) : ((towerActM K) src.x₁) (((towerActM K) src.sigma⁻¹) (((towerActM K) src.x₁) m)) = ((towerActM K) src.sigma⁻¹) (((towerActM K) src.x₁) m) + ((towerActM K) src.x₁) m + m

Checkpoint327 bridge for the checkpoint326 transported-square isolated corner: solve the lower closed-zero two-corner row in characteristic two.

theorem Q2Presentation.Induction.sec7_frattiniTrivial_elementaryMStageCandidateWildRowManuscriptX1ConjCanonicalPrimitiveTowerActionSigmaInvTransportSquareUntransportedRawDeltaStepExpandedNamedTransportedDeviationDifferenceSingleActionDifferenceCoreOpenedRawFourCornerClosedZeroApplyPointwiseData {p : FramedPair} {h : ¬ScalarTerminal p} {chief : FirstNonScalarChiefFactor p} {K : KernelMinimal chief.toNonScalarChiefFactor} {hR : Nat.card ↥(kernelFrattini K) = 1} {child : Sec7FrattiniTrivialElementaryChildData p} (src : Sec7FrattiniTrivialElementaryMStageCandidateLowerSourceData h chief K hR child) (m : towerM K) : ((towerActM K) src.x₁) (((towerActM K) src.sigma⁻¹) (((towerActM K) src.x₁) m)) + (((towerActM K) src.sigma⁻¹) (((towerActM K) src.x₁) m) + ((towerActM K) src.x₁) m + m) = 0

Checkpoint326 bridge for the checkpoint325 transported-square four-corner row: rewrite by the isolated corner equality and use Y+Y=0.

theorem Q2Presentation.Induction.sec7_frattiniTrivial_elementaryMStageCandidateWildRowManuscriptX1ConjCanonicalPrimitiveTowerActionSigmaInvTransportSquareUntransportedRawDeltaStepExpandedNamedTransportedDeviationDifferenceSingleActionDifferenceCoreOpenedRawSingleCornerIsolatedApplyPointwiseData {p : FramedPair} {h : ¬ScalarTerminal p} {chief : FirstNonScalarChiefFactor p} {K : KernelMinimal chief.toNonScalarChiefFactor} {hR : Nat.card ↥(kernelFrattini K) = 1} {child : Sec7FrattiniTrivialElementaryChildData p} (src : Sec7FrattiniTrivialElementaryMStageCandidateLowerSourceData h chief K hR child) (m : towerM K) : ((towerActM K) src.x₁) (((towerActM K) src.sigma⁻¹) (((towerActM K) src.x₁) m)) = ((towerActM K) src.sigma⁻¹) (((towerActM K) src.x₁) m) + ((towerActM K) src.x₁) m + m

Checkpoint325 bridge for the transported-square row: solve the lower raw four-corner zero row for the isolated first corner.

theorem Q2Presentation.Induction.sec7_frattiniTrivial_elementaryMStageCandidateWildRowManuscriptX1ConjCanonicalPrimitiveTowerActionSigmaInvTransportSquareUntransportedRawDeltaStepExpandedNamedTransportedDeviationDifferenceSingleActionDifferenceCoreOpenedRawThreeCornerIsolatedApplyPointwiseData {p : FramedPair} {h : ¬ScalarTerminal p} {chief : FirstNonScalarChiefFactor p} {K : KernelMinimal chief.toNonScalarChiefFactor} {hR : Nat.card ↥(kernelFrattini K) = 1} {child : Sec7FrattiniTrivialElementaryChildData p} (src : Sec7FrattiniTrivialElementaryMStageCandidateLowerSourceData h chief K hR child) (m : towerM K) : ((towerActM K) src.x₁) (((towerActM K) src.sigma⁻¹) (((towerActM K) src.x₁) m)) + ((towerActM K) src.x₁) m + ((towerActM K) src.sigma⁻¹) (((towerActM K) src.x₁) m) = m

Checkpoint324 bridge for the transported-square row: add back the remaining corners to the single-corner isolated row and cancel duplicated characteristic-two summands.

theorem Q2Presentation.Induction.sec7_frattiniTrivial_elementaryMStageCandidateWildRowManuscriptX1ConjCanonicalPrimitiveTowerActionSigmaInvTransportSquareUntransportedRawDeltaStepExpandedNamedTransportedDeviationDifferenceSingleActionDifferenceCoreOpenedRawTwoCornerZeroSumApplyPointwiseData {p : FramedPair} {h : ¬ScalarTerminal p} {chief : FirstNonScalarChiefFactor p} {K : KernelMinimal chief.toNonScalarChiefFactor} {hR : Nat.card ↥(kernelFrattini K) = 1} {child : Sec7FrattiniTrivialElementaryChildData p} (src : Sec7FrattiniTrivialElementaryMStageCandidateLowerSourceData h chief K hR child) (m : towerM K) : ((towerActM K) src.x₁) (((towerActM K) src.sigma⁻¹) (((towerActM K) src.x₁) m)) + ((towerActM K) src.x₁) m + ((towerActM K) src.sigma⁻¹) (((towerActM K) src.x₁) m) + m = 0

Checkpoint323 bridge for the transported-square row: the raw four-corner zero-sum follows from the isolated three-corner equality because the tower module has characteristic two.

theorem Q2Presentation.Induction.sec7_frattiniTrivial_elementaryMStageCandidateWildRowManuscriptX1ConjCanonicalPrimitiveTowerActionSigmaInvTransportSquareUntransportedRawDeltaStepExpandedNamedTransportedDeviationDifferenceSingleActionDifferenceCoreOpenedRawTwoCornerBalanceApplyPointwiseData {p : FramedPair} {h : ¬ScalarTerminal p} {chief : FirstNonScalarChiefFactor p} {K : KernelMinimal chief.toNonScalarChiefFactor} {hR : Nat.card ↥(kernelFrattini K) = 1} {child : Sec7FrattiniTrivialElementaryChildData p} (src : Sec7FrattiniTrivialElementaryMStageCandidateLowerSourceData h chief K hR child) (m : towerM K) : ((towerActM K) src.x₁) (((towerActM K) src.sigma⁻¹) (((towerActM K) src.x₁) m)) + ((towerActM K) src.x₁) m = ((towerActM K) src.sigma⁻¹) (((towerActM K) src.x₁) m) + m

Checkpoint322 bridge for the transported-square row: turn the lower raw four-term zero-sum row into the checkpoint321 two-corner balance in the characteristic-two tower module.

theorem Q2Presentation.Induction.sec7_frattiniTrivial_elementaryMStageCandidateWildRowManuscriptX1ConjCanonicalPrimitiveTowerActionSigmaInvTransportSquareUntransportedRawDeltaStepExpandedNamedTransportedDeviationDifferenceSingleActionDifferenceCoreOpenedRawFourCornerBalanceApplyPointwiseData {p : FramedPair} {h : ¬ScalarTerminal p} {chief : FirstNonScalarChiefFactor p} {K : KernelMinimal chief.toNonScalarChiefFactor} {hR : Nat.card ↥(kernelFrattini K) = 1} {child : Sec7FrattiniTrivialElementaryChildData p} (src : Sec7FrattiniTrivialElementaryMStageCandidateLowerSourceData h chief K hR child) (m : towerM K) : ((towerActM K) src.x₁) (((towerActM K) src.sigma⁻¹) (((towerActM K) src.x₁) m)) + ((towerActM K) src.x₁) m + ((towerActM K) src.sigma⁻¹) m + ((towerActM K) src.sigma⁻¹) m = ((towerActM K) src.x₁) (((towerActM K) src.sigma⁻¹) m) + ((towerActM K) src.x₁) (((towerActM K) src.sigma⁻¹) m) + ((towerActM K) src.sigma⁻¹) (((towerActM K) src.x₁) m) + m

Checkpoint320 bridge for the transported-square row: add the two duplicated char-two corners back to the checkpoint321 two-corner balance.

theorem Q2Presentation.Induction.sec7_frattiniTrivial_elementaryMStageCandidateWildRowManuscriptX1ConjCanonicalPrimitiveTowerActionSigmaInvTransportSquareUntransportedRawDeltaStepExpandedNamedTransportedDeviationDifferenceSingleActionDifferenceCoreOpenedDifferenceBalanceSubtractionZeroApplyPointwiseData {p : FramedPair} {h : ¬ScalarTerminal p} {chief : FirstNonScalarChiefFactor p} {K : KernelMinimal chief.toNonScalarChiefFactor} {hR : Nat.card ↥(kernelFrattini K) = 1} {child : Sec7FrattiniTrivialElementaryChildData p} (src : Sec7FrattiniTrivialElementaryMStageCandidateLowerSourceData h chief K hR child) (m : towerM K) : ((towerActM K) src.x₁) (((towerActM K) src.sigma⁻¹) (((towerActM K) src.x₁) m)) - ((towerActM K) src.x₁) (((towerActM K) src.sigma⁻¹) m) - (((towerActM K) src.x₁) (((towerActM K) src.sigma⁻¹) m) - ((towerActM K) src.x₁) m) - (((towerActM K) src.sigma⁻¹) (((towerActM K) src.x₁) m) - ((towerActM K) src.sigma⁻¹) m - (((towerActM K) src.sigma⁻¹) m - m)) = 0

Checkpoint319 bridge for the transported-square row: rewrite the raw four-corner balance as the previously exposed subtraction-zero difference of the two single-action cores.

theorem Q2Presentation.Induction.sec7_frattiniTrivial_elementaryMStageCandidateWildRowManuscriptX1ConjCanonicalPrimitiveTowerActionSigmaInvTransportSquareUntransportedRawDeltaStepExpandedNamedTransportedDeviationDifferenceSingleActionDifferenceCoreOpenedDifferenceBalanceApplyPointwiseData {p : FramedPair} {h : ¬ScalarTerminal p} {chief : FirstNonScalarChiefFactor p} {K : KernelMinimal chief.toNonScalarChiefFactor} {hR : Nat.card ↥(kernelFrattini K) = 1} {child : Sec7FrattiniTrivialElementaryChildData p} (src : Sec7FrattiniTrivialElementaryMStageCandidateLowerSourceData h chief K hR child) (m : towerM K) : ((towerActM K) src.x₁) (((towerActM K) src.sigma⁻¹) (((towerActM K) src.x₁) m)) - ((towerActM K) src.x₁) (((towerActM K) src.sigma⁻¹) m) - (((towerActM K) src.x₁) (((towerActM K) src.sigma⁻¹) m) - ((towerActM K) src.x₁) m) = ((towerActM K) src.sigma⁻¹) (((towerActM K) src.x₁) m) - ((towerActM K) src.sigma⁻¹) m - (((towerActM K) src.sigma⁻¹) m - m)

theorem Q2Presentation.Induction.sec7_frattiniTrivial_elementaryMStageCandidateWildRowManuscriptX1ConjCanonicalPrimitiveTowerActionSigmaInvTransportSquareUntransportedRawDeltaStepExpandedNamedTransportedDeviationDifferenceSingleActionDifferenceCoreDifferenceBalanceApplyPointwiseData {p : FramedPair} {h : ¬ScalarTerminal p} {chief : FirstNonScalarChiefFactor p} {K : KernelMinimal chief.toNonScalarChiefFactor} {hR : Nat.card ↥(kernelFrattini K) = 1} {child : Sec7FrattiniTrivialElementaryChildData p} (src : Sec7FrattiniTrivialElementaryMStageCandidateLowerSourceData h chief K hR child) (m : towerM K) : Q2Presentation.Induction.sec7TowerActDifference✝ K src.x₁ (((towerActM K) src.sigma⁻¹) (((towerActM K) src.x₁) m)) (((towerActM K) src.sigma⁻¹) m) - Q2Presentation.Induction.sec7TowerActDifference✝¹ K src.x₁ (((towerActM K) src.sigma⁻¹) m) m = ((towerActM K) src.sigma⁻¹) (((towerActM K) src.x₁) m) - ((towerActM K) src.sigma⁻¹) m - (((towerActM K) src.sigma⁻¹) m - m)

Checkpoint318 bridge for the transported-square row: the checkpoint317 named-difference balance follows from the same equality with the two sec7TowerActDifference terms opened to literal tower-action differences.

theorem Q2Presentation.Induction.sec7_frattiniTrivial_elementaryMStageCandidateWildRowManuscriptX1ConjCanonicalPrimitiveTowerActionSigmaInvTransportSquareUntransportedRawDeltaStepExpandedNamedTransportedDeviationDifferenceSingleActionDifferenceCoreAdditiveBalanceApplyPointwiseData {p : FramedPair} {h : ¬ScalarTerminal p} {chief : FirstNonScalarChiefFactor p} {K : KernelMinimal chief.toNonScalarChiefFactor} {hR : Nat.card ↥(kernelFrattini K) = 1} {child : Sec7FrattiniTrivialElementaryChildData p} (src : Sec7FrattiniTrivialElementaryMStageCandidateLowerSourceData h chief K hR child) (m : towerM K) : Q2Presentation.Induction.sec7TowerActDifference✝ K src.x₁ (((towerActM K) src.sigma⁻¹) (((towerActM K) src.x₁) m)) (((towerActM K) src.sigma⁻¹) m) + (((towerActM K) src.sigma⁻¹) m - m) = Q2Presentation.Induction.sec7TowerActDifference✝¹ K src.x₁ (((towerActM K) src.sigma⁻¹) m) m + (((towerActM K) src.sigma⁻¹) (((towerActM K) src.x₁) m) - ((towerActM K) src.sigma⁻¹) m)

Checkpoint317 bridge for the transported-square row: the checkpoint316 additive balance follows from the same row stated as a difference balance.

theorem Q2Presentation.Induction.sec7_frattiniTrivial_elementaryMStageCandidateWildRowManuscriptX1ConjCanonicalPrimitiveTowerActionSigmaInvTransportSquareUntransportedRawDeltaStepExpandedNamedTransportedDeviationDifferenceSingleActionDifferenceCoreSubtractedApplyPointwiseData {p : FramedPair} {h : ¬ScalarTerminal p} {chief : FirstNonScalarChiefFactor p} {K : KernelMinimal chief.toNonScalarChiefFactor} {hR : Nat.card ↥(kernelFrattini K) = 1} {child : Sec7FrattiniTrivialElementaryChildData p} (src : Sec7FrattiniTrivialElementaryMStageCandidateLowerSourceData h chief K hR child) (m : towerM K) : Q2Presentation.Induction.sec7TowerActDifference✝ K src.x₁ (((towerActM K) src.sigma⁻¹) (((towerActM K) src.x₁) m)) (((towerActM K) src.sigma⁻¹) m) - (((towerActM K) src.sigma⁻¹) (((towerActM K) src.x₁) m) - ((towerActM K) src.sigma⁻¹) m) = Q2Presentation.Induction.sec7TowerActDifference✝¹ K src.x₁ (((towerActM K) src.sigma⁻¹) m) m - (((towerActM K) src.sigma⁻¹) m - m)

Checkpoint315 transported-square subtracted difference core, now theorem-level from the checkpoint316 additive-balance row.

theorem Q2Presentation.Induction.sec7_frattiniTrivial_elementaryMStageCandidateWildRowManuscriptX1ConjCanonicalPrimitiveTowerActionSigmaInvTransportSquareUntransportedRawDeltaStepExpandedNamedTransportedDeviationDifferenceSingleActionValueCoreApplyPointwiseData {p : FramedPair} {h : ¬ScalarTerminal p} {chief : FirstNonScalarChiefFactor p} {K : KernelMinimal chief.toNonScalarChiefFactor} {hR : Nat.card ↥(kernelFrattini K) = 1} {child : Sec7FrattiniTrivialElementaryChildData p} (src : Sec7FrattiniTrivialElementaryMStageCandidateLowerSourceData h chief K hR child) (m : towerM K) : Q2Presentation.Induction.sec7TowerActDifference✝ K src.x₁ (((towerActM K) src.sigma⁻¹) (((towerActM K) src.x₁) m)) (((towerActM K) src.sigma⁻¹) m) = ((towerActM K) src.sigma⁻¹) (((towerActM K) src.x₁) m) - ((towerActM K) src.sigma⁻¹) m + (Q2Presentation.Induction.sec7TowerActDifference✝¹ K src.x₁ (((towerActM K) src.sigma⁻¹) m) m - (((towerActM K) src.sigma⁻¹) m - m))

Checkpoint314 transported-square single-action value core, now theorem-level from the checkpoint315 subtracted difference core.

theorem Q2Presentation.Induction.sec7_frattiniTrivial_elementaryMStageCandidateWildRowManuscriptX1ConjCanonicalPrimitiveTowerActionSigmaInvTransportSquareUntransportedRawDeltaStepExpandedNamedTransportedDeviationDifferenceSingleActionDifferenceCoreApplyPointwiseData {p : FramedPair} {h : ¬ScalarTerminal p} {chief : FirstNonScalarChiefFactor p} {K : KernelMinimal chief.toNonScalarChiefFactor} {hR : Nat.card ↥(kernelFrattini K) = 1} {child : Sec7FrattiniTrivialElementaryChildData p} (src : Sec7FrattiniTrivialElementaryMStageCandidateLowerSourceData h chief K hR child) (m : towerM K) : Q2Presentation.Induction.sec7TowerActDifference✝ K src.x₁ (((towerActM K) src.sigma⁻¹) (((towerActM K) src.x₁) m)) (((towerActM K) src.sigma⁻¹) m) - (((towerActM K) src.sigma⁻¹) (((towerActM K) src.x₁) m) - ((towerActM K) src.sigma⁻¹) m) = Q2Presentation.Induction.sec7TowerActDifference✝¹ K src.x₁ (((towerActM K) src.sigma⁻¹) m) m - (((towerActM K) src.sigma⁻¹) m - m)

theorem Q2Presentation.Induction.sec7_frattiniTrivial_elementaryMStageCandidateWildRowManuscriptX1ConjCanonicalPrimitiveTowerActionSigmaInvTransportSquareUntransportedRawDeltaStepExpandedNamedTransportedDeviationDifferenceMapSubOpenedEqualityCoreApplyPointwiseData {p : FramedPair} {h : ¬ScalarTerminal p} {chief : FirstNonScalarChiefFactor p} {K : KernelMinimal chief.toNonScalarChiefFactor} {hR : Nat.card ↥(kernelFrattini K) = 1} {child : Sec7FrattiniTrivialElementaryChildData p} (src : Sec7FrattiniTrivialElementaryMStageCandidateLowerSourceData h chief K hR child) (m : towerM K) : ((towerActM K) src.x₁) (((towerActM K) src.sigma⁻¹) (((towerActM K) src.x₁) m)) - ((towerActM K) src.x₁) (((towerActM K) src.sigma⁻¹) m) - (((towerActM K) src.sigma⁻¹) (((towerActM K) src.x₁) m) - ((towerActM K) src.sigma⁻¹) m) = ((towerActM K) src.x₁) (((towerActM K) src.sigma⁻¹) m) - ((towerActM K) src.x₁) m - (((towerActM K) src.sigma⁻¹) m - m)

Checkpoint312 transported-square opened equality, now theorem-level from the single-action-difference core above.

theorem Q2Presentation.Induction.sec7_frattiniTrivial_elementaryMStageCandidateWildRowManuscriptX1ConjCanonicalPrimitiveTowerActionSigmaInvTransportSquareUntransportedRawDeltaStepExpandedNamedTransportedDeviationDifferenceMapSubClosedEqualityApplyPointwiseData {p : FramedPair} {h : ¬ScalarTerminal p} {chief : FirstNonScalarChiefFactor p} {K : KernelMinimal chief.toNonScalarChiefFactor} {hR : Nat.card ↥(kernelFrattini K) = 1} {child : Sec7FrattiniTrivialElementaryChildData p} (src : Sec7FrattiniTrivialElementaryMStageCandidateLowerSourceData h chief K hR child) (m : towerM K) : ((towerActM K) src.x₁) (((towerActM K) src.sigma⁻¹) (((towerActM K) src.x₁) m) - ((towerActM K) src.sigma⁻¹) m) - (((towerActM K) src.sigma⁻¹) (((towerActM K) src.x₁) m) - ((towerActM K) src.sigma⁻¹) m) = ((towerActM K) src.x₁) (((towerActM K) src.sigma⁻¹) m - m) - (((towerActM K) src.sigma⁻¹) m - m)

Checkpoint311 closed-map-sub transported-square row, recovered theorem-level from the checkpoint312 expression with both transported differences opened.

theorem Q2Presentation.Induction.sec7_frattiniTrivial_elementaryMStageCandidateWildRowManuscriptX1ConjCanonicalPrimitiveTowerActionSigmaInvTransportSquareUntransportedRawDeltaStepExpandedNamedTransportedDeviationDifferenceAllSubOpenedEqualityApplyPointwiseData {p : FramedPair} {h : ¬ScalarTerminal p} {chief : FirstNonScalarChiefFactor p} {K : KernelMinimal chief.toNonScalarChiefFactor} {hR : Nat.card ↥(kernelFrattini K) = 1} {child : Sec7FrattiniTrivialElementaryChildData p} (src : Sec7FrattiniTrivialElementaryMStageCandidateLowerSourceData h chief K hR child) (m : towerM K) : ((towerActM K) src.x₁) (((towerActM K) src.sigma⁻¹) (((towerActM K) src.x₁) m)) - ((towerActM K) src.x₁) (((towerActM K) src.sigma⁻¹) m) - (((towerActM K) src.sigma⁻¹) (((towerActM K) src.x₁) m) - ((towerActM K) src.sigma⁻¹) m) = ((towerActM K) src.x₁) (((towerActM K) src.sigma⁻¹) m) - ((towerActM K) src.x₁) m - (((towerActM K) src.sigma⁻¹) m - m)

Checkpoint310 fully opened transported-square equality, recovered theorem-level from the checkpoint311 closed-map_sub equality.

theorem Q2Presentation.Induction.sec7_frattiniTrivial_elementaryMStageCandidateWildRowManuscriptX1ConjCanonicalPrimitiveTowerActionSigmaInvTransportSquareUntransportedRawDeltaStepExpandedNamedTransportedDeviationDifferenceSubtractionZeroAllSubOpenedApplyPointwiseData {p : FramedPair} {h : ¬ScalarTerminal p} {chief : FirstNonScalarChiefFactor p} {K : KernelMinimal chief.toNonScalarChiefFactor} {hR : Nat.card ↥(kernelFrattini K) = 1} {child : Sec7FrattiniTrivialElementaryChildData p} (src : Sec7FrattiniTrivialElementaryMStageCandidateLowerSourceData h chief K hR child) (m : towerM K) : ((towerActM K) src.x₁) (((towerActM K) src.sigma⁻¹) (((towerActM K) src.x₁) m)) - ((towerActM K) src.x₁) (((towerActM K) src.sigma⁻¹) m) - (((towerActM K) src.sigma⁻¹) (((towerActM K) src.x₁) m) - ((towerActM K) src.sigma⁻¹) m) - (((towerActM K) src.x₁) (((towerActM K) src.sigma⁻¹) m) - ((towerActM K) src.x₁) m - (((towerActM K) src.sigma⁻¹) m - m)) = 0

Checkpoint309 transported-square datum, now theorem-level from the smaller checkpoint310 equality form before the final sub_eq_zero wrapping.

theorem Q2Presentation.Induction.sec7_frattiniTrivial_elementaryMStageCandidateWildRowManuscriptX1ConjCanonicalPrimitiveTowerActionSigmaInvTransportSquareUntransportedRawDeltaStepExpandedNamedTransportedDeviationDifferenceSubtractionZeroApplyPointwiseData {p : FramedPair} {h : ¬ScalarTerminal p} {chief : FirstNonScalarChiefFactor p} {K : KernelMinimal chief.toNonScalarChiefFactor} {hR : Nat.card ↥(kernelFrattini K) = 1} {child : Sec7FrattiniTrivialElementaryChildData p} (src : Sec7FrattiniTrivialElementaryMStageCandidateLowerSourceData h chief K hR child) (m : towerM K) : Q2Presentation.Induction.sec7TowerActDifference✝ K src.x₁ (((towerActM K) src.sigma⁻¹) (((towerActM K) src.x₁) m)) (((towerActM K) src.sigma⁻¹) m) - Q2Presentation.Induction.sec7MStageRawDifference✝ K (((towerActM K) src.sigma⁻¹) (((towerActM K) src.x₁) m)) (((towerActM K) src.sigma⁻¹) m) - (Q2Presentation.Induction.sec7TowerActDifference✝¹ K src.x₁ (((towerActM K) src.sigma⁻¹) m) m - Q2Presentation.Induction.sec7MStageRawDifference✝¹ K (((towerActM K) src.sigma⁻¹) m) m) = 0

Checkpoint308 transported-square row, now theorem-level from the fully opened subtraction form of both transported deviation differences.

theorem Q2Presentation.Induction.sec7_frattiniTrivial_elementaryMStageCandidateWildRowManuscriptX1ConjCanonicalPrimitiveTowerActionSigmaInvTransportSquareUntransportedRawDeltaStepExpandedTransportedDeviationDifferenceMapSubOpenedSubtractionZeroApplyPointwiseData {p : FramedPair} {h : ¬ScalarTerminal p} {chief : FirstNonScalarChiefFactor p} {K : KernelMinimal chief.toNonScalarChiefFactor} {hR : Nat.card ↥(kernelFrattini K) = 1} {child : Sec7FrattiniTrivialElementaryChildData p} (src : Sec7FrattiniTrivialElementaryMStageCandidateLowerSourceData h chief K hR child) (m : towerM K) : ((towerActM K) src.x₁) (((towerActM K) src.sigma⁻¹) (((towerActM K) src.x₁) m)) - ((towerActM K) src.x₁) (((towerActM K) src.sigma⁻¹) m) - (((towerActM K) src.sigma⁻¹) (((towerActM K) src.x₁) m) - ((towerActM K) src.sigma⁻¹) m) - (((towerActM K) src.x₁) (((towerActM K) src.sigma⁻¹) m) - ((towerActM K) src.x₁) m - (((towerActM K) src.sigma⁻¹) m - m)) = 0

theorem Q2Presentation.Induction.sec7_frattiniTrivial_elementaryMStageCandidateWildRowManuscriptX1ConjCanonicalPrimitiveTowerActionSigmaInvTransportSquareUntransportedRawDeltaStepExpandedTransportedDeviationDifferenceMapSubOpenedApplyPointwiseData {p : FramedPair} {h : ¬ScalarTerminal p} {chief : FirstNonScalarChiefFactor p} {K : KernelMinimal chief.toNonScalarChiefFactor} {hR : Nat.card ↥(kernelFrattini K) = 1} {child : Sec7FrattiniTrivialElementaryChildData p} (src : Sec7FrattiniTrivialElementaryMStageCandidateLowerSourceData h chief K hR child) (m : towerM K) : ((towerActM K) src.x₁) (((towerActM K) src.sigma⁻¹) (((towerActM K) src.x₁) m)) - ((towerActM K) src.x₁) (((towerActM K) src.sigma⁻¹) m) - (((towerActM K) src.sigma⁻¹) (((towerActM K) src.x₁) m) - ((towerActM K) src.sigma⁻¹) m) = ((towerActM K) src.x₁) (((towerActM K) src.sigma⁻¹) m) - ((towerActM K) src.x₁) m - (((towerActM K) src.sigma⁻¹) m - m)

theorem Q2Presentation.Induction.sec7_frattiniTrivial_elementaryMStageCandidateWildRowManuscriptX1ConjCanonicalPrimitiveTowerActionSigmaInvTransportSquareUntransportedRawDeltaStepExpandedTransportedDeviationDifferenceApplyPointwiseData {p : FramedPair} {h : ¬ScalarTerminal p} {chief : FirstNonScalarChiefFactor p} {K : KernelMinimal chief.toNonScalarChiefFactor} {hR : Nat.card ↥(kernelFrattini K) = 1} {child : Sec7FrattiniTrivialElementaryChildData p} (src : Sec7FrattiniTrivialElementaryMStageCandidateLowerSourceData h chief K hR child) (m : towerM K) : ((towerActM K) src.x₁) (((towerActM K) src.sigma⁻¹) (((towerActM K) src.x₁) m) - ((towerActM K) src.sigma⁻¹) m) - (((towerActM K) src.sigma⁻¹) (((towerActM K) src.x₁) m) - ((towerActM K) src.sigma⁻¹) m) = ((towerActM K) src.x₁) (((towerActM K) src.sigma⁻¹) m - m) - (((towerActM K) src.sigma⁻¹) m - m)

theorem Q2Presentation.Induction.sec7_frattiniTrivial_elementaryMStageCandidateWildRowManuscriptX1ConjCanonicalPrimitiveTowerActionSigmaInvTransportSquareUntransportedRawDeltaStepExpandedTransportedSecondDifferenceApplyPointwiseData {p : FramedPair} {h : ¬ScalarTerminal p} {chief : FirstNonScalarChiefFactor p} {K : KernelMinimal chief.toNonScalarChiefFactor} {hR : Nat.card ↥(kernelFrattini K) = 1} {child : Sec7FrattiniTrivialElementaryChildData p} (src : Sec7FrattiniTrivialElementaryMStageCandidateLowerSourceData h chief K hR child) (m : towerM K) : ((towerActM K) src.x₁) (((towerActM K) src.sigma⁻¹) (((towerActM K) src.x₁) m)) - ((towerActM K) src.sigma⁻¹) (((towerActM K) src.x₁) m) - (((towerActM K) src.x₁) (((towerActM K) src.sigma⁻¹) m) - ((towerActM K) src.sigma⁻¹) m) = ((towerActM K) src.x₁) (((towerActM K) src.sigma⁻¹) m) - ((towerActM K) src.sigma⁻¹) m - (((towerActM K) src.x₁) m - m)

theorem Q2Presentation.Induction.sec7_frattiniTrivial_elementaryMStageCandidateWildRowManuscriptX1ConjCanonicalPrimitiveTowerActionSigmaInvTransportSquareUntransportedRawDeltaStepExpandedTransportedDifferenceActionFullyDistributedValueApplyPointwiseData {p : FramedPair} {h : ¬ScalarTerminal p} {chief : FirstNonScalarChiefFactor p} {K : KernelMinimal chief.toNonScalarChiefFactor} {hR : Nat.card ↥(kernelFrattini K) = 1} {child : Sec7FrattiniTrivialElementaryChildData p} (src : Sec7FrattiniTrivialElementaryMStageCandidateLowerSourceData h chief K hR child) (m : towerM K) : ((towerActM K) src.x₁) (((towerActM K) src.sigma⁻¹) (((towerActM K) src.x₁) m)) - ((towerActM K) src.x₁) (((towerActM K) src.sigma⁻¹) m) = ((towerActM K) src.sigma⁻¹) (((towerActM K) src.x₁) m) - ((towerActM K) src.sigma⁻¹) m + (((towerActM K) src.x₁) (((towerActM K) src.sigma⁻¹) m) - ((towerActM K) src.x₁) m - (((towerActM K) src.sigma⁻¹) m - m))

Checkpoint303 primitive datum for the transported-square part of x₁^σ, now theorem-level from the checkpoint304 second-difference row.

theorem Q2Presentation.Induction.sec7_frattiniTrivial_elementaryMStageCandidateWildRowManuscriptX1ConjCanonicalPrimitiveTowerActionSigmaInvTransportSquareUntransportedRawDeltaStepExpandedTransportedDifferenceActionDistributedValueApplyPointwiseData {p : FramedPair} {h : ¬ScalarTerminal p} {chief : FirstNonScalarChiefFactor p} {K : KernelMinimal chief.toNonScalarChiefFactor} {hR : Nat.card ↥(kernelFrattini K) = 1} {child : Sec7FrattiniTrivialElementaryChildData p} (src : Sec7FrattiniTrivialElementaryMStageCandidateLowerSourceData h chief K hR child) (m : towerM K) : ((towerActM K) src.x₁) (((towerActM K) src.sigma⁻¹) (((towerActM K) src.x₁) m)) - ((towerActM K) src.x₁) (((towerActM K) src.sigma⁻¹) m) = ((towerActM K) src.sigma⁻¹) (((towerActM K) src.x₁) m) - ((towerActM K) src.sigma⁻¹) m + (((towerActM K) src.x₁) (((towerActM K) src.sigma⁻¹) m - m) - (((towerActM K) src.sigma⁻¹) m - m))

Checkpoint302 primitive datum for the transported-square part of x₁^σ, now theorem-level from the checkpoint303 fully distributed pointwise value row.

theorem Q2Presentation.Induction.sec7_frattiniTrivial_elementaryMStageCandidateWildRowManuscriptX1ConjCanonicalPrimitiveTowerActionSigmaInvTransportSquareUntransportedRawDeltaStepExpandedTransportedDifferenceActionValueFormulaFullyInlinedApplyPointwiseData {p : FramedPair} {h : ¬ScalarTerminal p} {chief : FirstNonScalarChiefFactor p} {K : KernelMinimal chief.toNonScalarChiefFactor} {hR : Nat.card ↥(kernelFrattini K) = 1} {child : Sec7FrattiniTrivialElementaryChildData p} (src : Sec7FrattiniTrivialElementaryMStageCandidateLowerSourceData h chief K hR child) (m : towerM K) : ((towerActM K) src.x₁) (((towerActM K) src.sigma⁻¹) (((towerActM K) src.x₁) m) - ((towerActM K) src.sigma⁻¹) m) = ((towerActM K) src.sigma⁻¹) (((towerActM K) src.x₁) m) - ((towerActM K) src.sigma⁻¹) m + (((towerActM K) src.x₁) (((towerActM K) src.sigma⁻¹) m - m) - (((towerActM K) src.sigma⁻¹) m - m))

Checkpoint301 primitive datum for the transported-square part of x₁^σ, now theorem-level from the checkpoint302 distributed-value row.

theorem Q2Presentation.Induction.sec7_frattiniTrivial_elementaryMStageCandidateWildRowManuscriptX1ConjCanonicalPrimitiveTowerActionSigmaInvTransportSquareUntransportedRawDeltaStepExpandedTransportedDifferenceActionValueFormulaApplyPointwiseData {p : FramedPair} {h : ¬ScalarTerminal p} {chief : FirstNonScalarChiefFactor p} {K : KernelMinimal chief.toNonScalarChiefFactor} {hR : Nat.card ↥(kernelFrattini K) = 1} {child : Sec7FrattiniTrivialElementaryChildData p} (src : Sec7FrattiniTrivialElementaryMStageCandidateLowerSourceData h chief K hR child) (m : towerM K) : have σi := (towerActM K) src.sigma⁻¹; have X := (towerActM K) src.x₁; X (σi (X m) - σi m) = σi (X m) - σi m + (X (σi m - m) - (σi m - m))

Checkpoint300 primitive datum for the transported-square part of x₁^σ, now theorem-level from the checkpoint301 fully inlined value formula.

theorem Q2Presentation.Induction.sec7_frattiniTrivial_elementaryMStageCandidateWildRowManuscriptX1ConjCanonicalPrimitiveTowerActionSigmaInvTransportSquareUntransportedRawDeltaStepExpandedTransportedDifferenceDeviationApplyPointwiseData {p : FramedPair} {h : ¬ScalarTerminal p} {chief : FirstNonScalarChiefFactor p} {K : KernelMinimal chief.toNonScalarChiefFactor} {hR : Nat.card ↥(kernelFrattini K) = 1} {child : Sec7FrattiniTrivialElementaryChildData p} (src : Sec7FrattiniTrivialElementaryMStageCandidateLowerSourceData h chief K hR child) (m : towerM K) : have σi := (towerActM K) src.sigma⁻¹; have X := (towerActM K) src.x₁; X (σi (X m) - σi m) - (σi (X m) - σi m) = X (σi m - m) - (σi m - m)

Checkpoint299 primitive datum for the transported-square part of x₁^σ, now theorem-level from the checkpoint300 transported-difference value formula.

theorem Q2Presentation.Induction.sec7_frattiniTrivial_elementaryMStageCandidateWildRowManuscriptX1ConjCanonicalPrimitiveTowerActionSigmaInvTransportSquareUntransportedRawDeltaStepExpandedValueDeltaDifferenceApplyPointwiseData {p : FramedPair} {h : ¬ScalarTerminal p} {chief : FirstNonScalarChiefFactor p} {K : KernelMinimal chief.toNonScalarChiefFactor} {hR : Nat.card ↥(kernelFrattini K) = 1} {child : Sec7FrattiniTrivialElementaryChildData p} (src : Sec7FrattiniTrivialElementaryMStageCandidateLowerSourceData h chief K hR child) (m : towerM K) : have σi := (towerActM K) src.sigma⁻¹; have X := (towerActM K) src.x₁; X (σi (X m)) - X (σi m) - (σi (X m) - σi m) = X (σi m) - σi m - (X m - m)

Checkpoint298 primitive atom for the transported-square part of x₁^σ, now theorem-level from the checkpoint299 transported-difference deviation row.

theorem Q2Presentation.Induction.sec7_frattiniTrivial_elementaryMStageCandidateWildRowManuscriptX1ConjCanonicalPrimitiveTowerActionSigmaInvTransportSquareUntransportedRawDeltaStepExpandedValueSubtractionZeroApplyPointwiseData {p : FramedPair} {h : ¬ScalarTerminal p} {chief : FirstNonScalarChiefFactor p} {K : KernelMinimal chief.toNonScalarChiefFactor} {hR : Nat.card ↥(kernelFrattini K) = 1} {child : Sec7FrattiniTrivialElementaryChildData p} (src : Sec7FrattiniTrivialElementaryMStageCandidateLowerSourceData h chief K hR child) (m : towerM K) : have σi := (towerActM K) src.sigma⁻¹; have X := (towerActM K) src.x₁; X (σi (X m)) - (X (σi m) + (σi (X m) - σi m + (X (σi m) - σi m - (X m - m)))) = 0

Checkpoint297 primitive atom for the transported-square part of x₁^σ, now theorem-level from the grouped delta-difference checkpoint298 row.

theorem Q2Presentation.Induction.sec7_frattiniTrivial_elementaryMStageCandidateWildRowManuscriptX1ConjCanonicalPrimitiveTowerActionSigmaInvTransportSquareUntransportedRawDeltaStepExpandedValueApplyPointwiseData {p : FramedPair} {h : ¬ScalarTerminal p} {chief : FirstNonScalarChiefFactor p} {K : KernelMinimal chief.toNonScalarChiefFactor} {hR : Nat.card ↥(kernelFrattini K) = 1} {child : Sec7FrattiniTrivialElementaryChildData p} (src : Sec7FrattiniTrivialElementaryMStageCandidateLowerSourceData h chief K hR child) (m : towerM K) : have σi := (towerActM K) src.sigma⁻¹; have X := (towerActM K) src.x₁; X (σi (X m)) = X (σi m) + (σi (X m) - σi m + (X (σi m) - σi m - (X m - m)))

Checkpoint296 transported-square raw value atom, now theorem-level from the literal subtraction-zero form of the same expanded point-value row.

theorem Q2Presentation.Induction.sec7_frattiniTrivial_elementaryMStageCandidateWildRowManuscriptX1ConjCanonicalPrimitiveTowerActionSigmaInvTransportSquareUntransportedRawDeltaStepExpandedApplyPointwiseData {p : FramedPair} {h : ¬ScalarTerminal p} {chief : FirstNonScalarChiefFactor p} {K : KernelMinimal chief.toNonScalarChiefFactor} {hR : Nat.card ↥(kernelFrattini K) = 1} {child : Sec7FrattiniTrivialElementaryChildData p} (src : Sec7FrattiniTrivialElementaryMStageCandidateLowerSourceData h chief K hR child) (m : towerM K) : have σi := (towerActM K) src.sigma⁻¹; have X := (towerActM K) src.x₁; X (σi (X m)) - X (σi m) = σi (X m) - σi m + (X (σi m) - σi m - (X m - m))

Checkpoint295 expanded transported-square delta step, now theorem-level from the raw value statement above.

theorem Q2Presentation.Induction.sec7_frattiniTrivial_elementaryMStageCandidateWildRowManuscriptX1ConjCanonicalPrimitiveTowerActionSigmaInvTransportSquareUntransportedRawDeltaStepApplyPointwiseData {p : FramedPair} {h : ¬ScalarTerminal p} {chief : FirstNonScalarChiefFactor p} {K : KernelMinimal chief.toNonScalarChiefFactor} {hR : Nat.card ↥(kernelFrattini K) = 1} {child : Sec7FrattiniTrivialElementaryChildData p} (src : Sec7FrattiniTrivialElementaryMStageCandidateLowerSourceData h chief K hR child) (m : towerM K) : have σi := (towerActM K) src.sigma⁻¹; have X := (towerActM K) src.x₁; X (σi (X m - m)) = σi (X m - m) + (X (σi m) - σi m - (X m - m))

Checkpoint294 compact transported-square delta step, now theorem-level from the expanded pointwise equality.

theorem Q2Presentation.Induction.sec7_frattiniTrivial_elementaryMStageCandidateWildRowManuscriptX1ConjCanonicalPrimitiveTowerActionSigmaInvTransportSquareUntransportedRawBalanceApplyPointwiseData {p : FramedPair} {h : ¬ScalarTerminal p} {chief : FirstNonScalarChiefFactor p} {K : KernelMinimal chief.toNonScalarChiefFactor} {hR : Nat.card ↥(kernelFrattini K) = 1} {child : Sec7FrattiniTrivialElementaryChildData p} (src : Sec7FrattiniTrivialElementaryMStageCandidateLowerSourceData h chief K hR child) (m : towerM K) : have σi := (towerActM K) src.sigma⁻¹; have X := (towerActM K) src.x₁; X (σi (X m - m)) - σi (X m - m) = X (σi m) - σi m - (X m - m)

Checkpoint293 untransported transported-square balance, now theorem-level from the raw square-step equality.

theorem Q2Presentation.Induction.sec7_frattiniTrivial_elementaryMStageCandidateWildRowManuscriptX1ConjCanonicalPrimitiveTowerActionSigmaInvTransportSquarePreTransportRawBalanceApplyPointwiseData {p : FramedPair} {h : ¬ScalarTerminal p} {chief : FirstNonScalarChiefFactor p} {K : KernelMinimal chief.toNonScalarChiefFactor} {hR : Nat.card ↥(kernelFrattini K) = 1} {child : Sec7FrattiniTrivialElementaryChildData p} (src : Sec7FrattiniTrivialElementaryMStageCandidateLowerSourceData h chief K hR child) (m : towerM K) : have σi := (towerActM K) src.sigma⁻¹; have X := (towerActM K) src.x₁; σi (X (σi (X m - m)) - σi (X m - m)) + σi (X m - m) = σi (X (σi m) - σi m)

Checkpoint292 pre-transport square balance, now theorem-level from the smaller inner balance before the outer σ⁻¹ transport.

theorem Q2Presentation.Induction.sec7_frattiniTrivial_elementaryMStageCandidateWildRowManuscriptX1ConjCanonicalPrimitiveTowerActionSigmaInvTransportSquarePreTransportOperatorBalanceApplyPointwiseData {p : FramedPair} {h : ¬ScalarTerminal p} {chief : FirstNonScalarChiefFactor p} {K : KernelMinimal chief.toNonScalarChiefFactor} {hR : Nat.card ↥(kernelFrattini K) = 1} {child : Sec7FrattiniTrivialElementaryChildData p} (src : Sec7FrattiniTrivialElementaryMStageCandidateLowerSourceData h chief K hR child) (m : towerM K) : have σi := ↑((towerActM K) src.sigma⁻¹); have X := ↑((towerActM K) src.x₁); have δ := X - 1; (σi * δ * σi * δ + σi * δ) m = (σi * δ * σi) m

Checkpoint291 pre-transport square balance, now theorem-level from the raw σ⁻¹/x₁ transported-deviation pointwise equation.

theorem Q2Presentation.Induction.sec7_frattiniTrivial_elementaryMStageCandidateWildRowManuscriptX1ConjCanonicalPrimitiveTowerActionSigmaInvTransportSquarePreTransportOperatorBalanceData {p : FramedPair} {h : ¬ScalarTerminal p} {chief : FirstNonScalarChiefFactor p} {K : KernelMinimal chief.toNonScalarChiefFactor} {hR : Nat.card ↥(kernelFrattini K) = 1} {child : Sec7FrattiniTrivialElementaryChildData p} (src : Sec7FrattiniTrivialElementaryMStageCandidateLowerSourceData h chief K hR child) : have σi := ↑((towerActM K) src.sigma⁻¹); have X := ↑((towerActM K) src.x₁); have δ := X - 1; σi * δ * σi * δ + σi * δ = σi * δ * σi

Former checkpoint290 pre-transport operator balance, now theorem-level from the pointwise transported-square balance.

theorem Q2Presentation.Induction.sec7_frattiniTrivial_elementaryMStageCandidateWildRowManuscriptX1ConjCanonicalPrimitiveTowerActionSigmaInvTransportSquareFactoredOperatorBalanceData {p : FramedPair} {h : ¬ScalarTerminal p} {chief : FirstNonScalarChiefFactor p} {K : KernelMinimal chief.toNonScalarChiefFactor} {hR : Nat.card ↥(kernelFrattini K) = 1} {child : Sec7FrattiniTrivialElementaryChildData p} (src : Sec7FrattiniTrivialElementaryMStageCandidateLowerSourceData h chief K hR child) : have σ := ↑((towerActM K) src.sigma); have σi := ↑((towerActM K) src.sigma⁻¹); have X := ↑((towerActM K) src.x₁); have δ := X - 1; (σi * δ * σi * δ + σi * δ) * σ = σi * δ

Checkpoint289 transported-square residual, now theorem-level from the lower pre-transport balance.

theorem Q2Presentation.Induction.sec7_frattiniTrivial_elementaryMStageCandidateWildRowManuscriptX1ConjCanonicalPrimitiveTowerActionSigmaInvTransportSquareOperatorBalanceData {p : FramedPair} {h : ¬ScalarTerminal p} {chief : FirstNonScalarChiefFactor p} {K : KernelMinimal chief.toNonScalarChiefFactor} {hR : Nat.card ↥(kernelFrattini K) = 1} {child : Sec7FrattiniTrivialElementaryChildData p} (src : Sec7FrattiniTrivialElementaryMStageCandidateLowerSourceData h chief K hR child) : have σ := ↑((towerActM K) src.sigma); have σi := ↑((towerActM K) src.sigma⁻¹); have X := ↑((towerActM K) src.x₁); have δ := X - 1; σi * δ * σi * δ * σ + σi * δ * σ = σi * δ

Checkpoint288 transported-square operator balance, now theorem-level from the checkpoint289 factored right-transport balance. The proof uses only add_mul, not a broad simplification pass over the transported-square context.

theorem Q2Presentation.Induction.sec7_frattiniTrivial_elementaryMStageCandidateWildRowManuscriptX1ConjCanonicalPrimitiveTowerActionSigmaInvTransportSquareRawDeviationValueBalancePointwiseData {p : FramedPair} {h : ¬ScalarTerminal p} {chief : FirstNonScalarChiefFactor p} {K : KernelMinimal chief.toNonScalarChiefFactor} {hR : Nat.card ↥(kernelFrattini K) = 1} {child : Sec7FrattiniTrivialElementaryChildData p} (src : Sec7FrattiniTrivialElementaryMStageCandidateLowerSourceData h chief K hR child) (m : towerM K) : have σ := (towerActM K) src.sigma; have σi := (towerActM K) src.sigma⁻¹; have X := (towerActM K) src.x₁; have v := X (σ m) - σ m; σi (X (σi v) - σi v) + σi v = σi (X m - m)

Checkpoint286 transported-square residual, now theorem-level from the checkpoint287 operator-balance statement above.

The proof is performance-safe: it applies the operator equality to the single point m and rewrites only the three visible deviation terms (X - 1) _; it deliberately avoids any global simplification of the transported square.

theorem Q2Presentation.Induction.sec7_frattiniTrivial_elementaryMStageCandidateWildRowManuscriptX1ConjCanonicalPrimitiveSigmaInvRowTransportSquareRawDeviationValueBalancePointwiseData {p : FramedPair} {h : ¬ScalarTerminal p} {chief : FirstNonScalarChiefFactor p} {K : KernelMinimal chief.toNonScalarChiefFactor} {hR : Nat.card ↥(kernelFrattini K) = 1} {child : Sec7FrattiniTrivialElementaryChildData p} (src : Sec7FrattiniTrivialElementaryMStageCandidateLowerSourceData h chief K hR child) (m : towerM K) : have v := src.X₁ (src.S m) - src.S m; src.Sinv (src.X₁ (src.Sinv v) - src.Sinv v) + src.Sinv v = src.Sinv (src.X₁ m - m)

Checkpoint283 transported-square residual, now theorem-level from the checkpoint285 concrete tower-action balance.

theorem Q2Presentation.Induction.sec7_frattiniTrivial_elementaryMStageCandidateWildRowManuscriptX1ConjCanonicalPrimitiveSigmaInvRowTransportSquareRawDoubleApplyBalancePointwiseData {p : FramedPair} {h : ¬ScalarTerminal p} {chief : FirstNonScalarChiefFactor p} {K : KernelMinimal chief.toNonScalarChiefFactor} {hR : Nat.card ↥(kernelFrattini K) = 1} {child : Sec7FrattiniTrivialElementaryChildData p} (src : Sec7FrattiniTrivialElementaryMStageCandidateLowerSourceData h chief K hR child) (m : towerM K) : have δ := src.X₁ - 1; src.Sinv (δ (src.Sinv (δ (src.S m)))) + src.Sinv (δ (src.S m)) = src.Sinv (δ m)

Checkpoint282 lower atom for x₁^σ, now theorem-level from the checkpoint283 raw-deviation transported-square balance.

theorem Q2Presentation.Induction.sec7_frattiniTrivial_elementaryMStageCandidateWildRowManuscriptX1ConjCanonicalPrimitiveSigmaInvRowTransportSquareRawSX1CancellationBalanceApplyPointwiseData {p : FramedPair} {h : ¬ScalarTerminal p} {chief : FirstNonScalarChiefFactor p} {K : KernelMinimal chief.toNonScalarChiefFactor} {hR : Nat.card ↥(kernelFrattini K) = 1} {child : Sec7FrattiniTrivialElementaryChildData p} (src : Sec7FrattiniTrivialElementaryMStageCandidateLowerSourceData h chief K hR child) (m : towerM K) : have row := src.Sinv * (src.X₁ - 1); have carried := row * src.S; row (carried m) + carried m = row m

Checkpoint281 lower atom for x₁^σ, now theorem-level from the checkpoint282 primitive double-application balance.

theorem Q2Presentation.Induction.sec7_frattiniTrivial_elementaryMStageCandidateWildRowManuscriptX1ConjCanonicalPrimitiveSigmaInvRowTransportSquareRawSX1CancellationApplyPointwiseData {p : FramedPair} {h : ¬ScalarTerminal p} {chief : FirstNonScalarChiefFactor p} {K : KernelMinimal chief.toNonScalarChiefFactor} {hR : Nat.card ↥(kernelFrattini K) = 1} {child : Sec7FrattiniTrivialElementaryChildData p} (src : Sec7FrattiniTrivialElementaryMStageCandidateLowerSourceData h chief K hR child) (m : towerM K) : have row := src.Sinv * (src.X₁ - 1); have carried := row * src.S; row (carried m) + carried m - row m = 0

Checkpoint280 residual for x₁^σ, now theorem-level from the checkpoint281 raw balance form.

theorem Q2Presentation.Induction.sec7_frattiniTrivial_elementaryMStageCandidateWildRowManuscriptX1ConjCanonicalPrimitiveSigmaInvRowTransportSquareCoreCancellationApplyPointwiseData {p : FramedPair} {h : ¬ScalarTerminal p} {chief : FirstNonScalarChiefFactor p} {K : KernelMinimal chief.toNonScalarChiefFactor} {hR : Nat.card ↥(kernelFrattini K) = 1} {child : Sec7FrattiniTrivialElementaryChildData p} (src : Sec7FrattiniTrivialElementaryMStageCandidateLowerSourceData h chief K hR child) (m : towerM K) : have ρ := sec7MStageCandidateActionEnd K; have row := ρ src.sigma⁻¹ * (ρ src.x₁ - 1); have carried := row * ρ src.sigma; row (carried m) + carried m - row m = 0

Checkpoint279 lower atom for x₁^σ, now theorem-level from the checkpoint280 raw S⁻¹/X₁/S transported-square cancellation.

theorem Q2Presentation.Induction.sec7_frattiniTrivial_elementaryMStageCandidateWildRowManuscriptX1ConjCanonicalPrimitiveSigmaInvRowTransportSquareCoreApplyPointwiseData {p : FramedPair} {h : ¬ScalarTerminal p} {chief : FirstNonScalarChiefFactor p} {K : KernelMinimal chief.toNonScalarChiefFactor} {hR : Nat.card ↥(kernelFrattini K) = 1} {child : Sec7FrattiniTrivialElementaryChildData p} (src : Sec7FrattiniTrivialElementaryMStageCandidateLowerSourceData h chief K hR child) (m : towerM K) : have ρ := sec7MStageCandidateActionEnd K; have row := ρ src.sigma⁻¹ * (ρ src.x₁ - 1); have carried := row * ρ src.sigma; row (carried m) = row m - carried m

Checkpoint278 transported-row square atom, now theorem-level from the smaller zero-cancellation form.

theorem Q2Presentation.Induction.sec7_frattiniTrivial_elementaryMStageCandidateWildRowManuscriptX1ConjCanonicalPrimitiveSigmaInvRowTransportSquareExpandedApplyPointwiseData {p : FramedPair} {h : ¬ScalarTerminal p} {chief : FirstNonScalarChiefFactor p} {K : KernelMinimal chief.toNonScalarChiefFactor} {hR : Nat.card ↥(kernelFrattini K) = 1} {child : Sec7FrattiniTrivialElementaryChildData p} (src : Sec7FrattiniTrivialElementaryMStageCandidateLowerSourceData h chief K hR child) (m : towerM K) : have ρ := sec7MStageCandidateActionEnd K; have carried := (ρ src.sigma⁻¹) ((ρ src.x₁) ((ρ src.sigma) m) - (ρ src.sigma) m); (ρ src.sigma⁻¹) ((ρ src.x₁) carried - carried) = (ρ src.sigma⁻¹) ((ρ src.x₁) m - m) - carried

Checkpoint277 transported σ⁻¹ row with the carried coordinate expanded, now theorem-level from the canonical row/carried-endomorphism square identity.

theorem Q2Presentation.Induction.sec7_frattiniTrivial_elementaryMStageCandidateWildRowManuscriptX1ConjCanonicalPrimitiveSigmaInvRowTransportSquareApplyPointwiseData {p : FramedPair} {h : ¬ScalarTerminal p} {chief : FirstNonScalarChiefFactor p} {K : KernelMinimal chief.toNonScalarChiefFactor} {hR : Nat.card ↥(kernelFrattini K) = 1} {child : Sec7FrattiniTrivialElementaryChildData p} (src : Sec7FrattiniTrivialElementaryMStageCandidateLowerSourceData h chief K hR child) (m : towerM K) : have ρ := sec7MStageCandidateActionEnd K; have row := ρ src.sigma⁻¹ * (ρ src.x₁ - 1); have carried := row * ρ src.sigma; row (carried m) = row m - carried m

Checkpoint276 transported σ⁻¹ row application atom, now theorem-level from the fully expanded transported-coordinate defect.

theorem Q2Presentation.Induction.sec7_frattiniTrivial_elementaryMStageCandidateWildRowManuscriptX1ConjCanonicalPrimitiveSigmaInvRowTransportSquarePointwiseData {p : FramedPair} {h : ¬ScalarTerminal p} {chief : FirstNonScalarChiefFactor p} {K : KernelMinimal chief.toNonScalarChiefFactor} {hR : Nat.card ↥(kernelFrattini K) = 1} {child : Sec7FrattiniTrivialElementaryChildData p} (src : Sec7FrattiniTrivialElementaryMStageCandidateLowerSourceData h chief K hR child) (m : towerM K) : have ρ := sec7MStageCandidateActionEnd K; have row := ρ src.sigma⁻¹ * (ρ src.x₁ - 1); have carried := row * ρ src.sigma; (row * carried) m = (row - carried) m

Checkpoint275 transported-row square identity, now theorem-level from the lower application-level transport-defect coordinate equality.

theorem Q2Presentation.Induction.sec7_frattiniTrivial_elementaryMStageCandidateWildRowManuscriptX1ConjCanonicalPrimitiveSigmaX1RowTransportDeltaPointwiseData {p : FramedPair} {h : ¬ScalarTerminal p} {chief : FirstNonScalarChiefFactor p} {K : KernelMinimal chief.toNonScalarChiefFactor} {hR : Nat.card ↥(kernelFrattini K) = 1} {child : Sec7FrattiniTrivialElementaryChildData p} (src : Sec7FrattiniTrivialElementaryMStageCandidateLowerSourceData h chief K hR child) (m : towerM K) : have ρ := sec7MStageCandidateActionEnd K; (src.P * (ρ src.x₁ - 1) * (ρ src.sigma⁻¹ * (ρ src.x₁ - 1) * ρ src.sigma)) m = (ρ src.sigma⁻¹ * (ρ src.x₁ - 1) - ρ src.sigma⁻¹ * (ρ src.x₁ - 1) * ρ src.sigma) m

Checkpoint274 pointwise primitive transport-defect atom for x₁^σ, now theorem-level from the two smaller pointwise coordinate calculations above.

theorem Q2Presentation.Induction.sec7_frattiniTrivial_elementaryMStageCandidateWildRowManuscriptX1ConjCanonicalPrimitiveSigmaX1RowTransportDeltaData {p : FramedPair} {h : ¬ScalarTerminal p} {chief : FirstNonScalarChiefFactor p} {K : KernelMinimal chief.toNonScalarChiefFactor} {hR : Nat.card ↥(kernelFrattini K) = 1} {child : Sec7FrattiniTrivialElementaryChildData p} (src : Sec7FrattiniTrivialElementaryMStageCandidateLowerSourceData h chief K hR child) : have ρ := sec7MStageCandidateActionEnd K; src.P * (ρ src.x₁ - 1) * (ρ src.sigma⁻¹ * (ρ src.x₁ - 1) * ρ src.sigma) = ρ src.sigma⁻¹ * (ρ src.x₁ - 1) - ρ src.sigma⁻¹ * (ρ src.x₁ - 1) * ρ src.sigma

Former checkpoint273 primitive endomorphism-level transport-defect atom for x₁^σ, now theorem-level from the pointwise primitive row above.

theorem Q2Presentation.Induction.sec7_frattiniTrivial_elementaryMStageCandidateWildRowManuscriptX1ConjCanonicalRawEvalSigmaX1RowTransportDeltaData {p : FramedPair} {h : ¬ScalarTerminal p} {chief : FirstNonScalarChiefFactor p} {K : KernelMinimal chief.toNonScalarChiefFactor} {hR : Nat.card ↥(kernelFrattini K) = 1} {child : Sec7FrattiniTrivialElementaryChildData p} (src : Sec7FrattiniTrivialElementaryMStageCandidateLowerSourceData h chief K hR child) : src.P * ((sec7MStageCandidateActionEnd K) (Q2Word.x1.eval src.marking) - 1) * ((sec7MStageCandidateActionEnd K) (Q2Word.sigma.inv.eval src.marking) * ((sec7MStageCandidateActionEnd K) (Q2Word.x1.eval src.marking) - 1) * (sec7MStageCandidateActionEnd K) (Q2Word.sigma.eval src.marking)) = (sec7MStageCandidateActionEnd K) (Q2Word.sigma.inv.eval src.marking) * ((sec7MStageCandidateActionEnd K) (Q2Word.x1.eval src.marking) - 1) - (sec7MStageCandidateActionEnd K) (Q2Word.sigma.inv.eval src.marking) * ((sec7MStageCandidateActionEnd K) (Q2Word.x1.eval src.marking) - 1) * (sec7MStageCandidateActionEnd K) (Q2Word.sigma.eval src.marking)

Former checkpoint272 raw-evaluation σ⁻¹,x₁,σ transport-defect atom, now theorem-level from the direct primitive lower-quotient action statement above.

theorem Q2Presentation.Induction.sec7_frattiniTrivial_elementaryMStageCandidateWildRowManuscriptX1ConjCanonicalU1X1RowTransportDeltaExpandedData {p : FramedPair} {h : ¬ScalarTerminal p} {chief : FirstNonScalarChiefFactor p} {K : KernelMinimal chief.toNonScalarChiefFactor} {hR : Nat.card ↥(kernelFrattini K) = 1} {child : Sec7FrattiniTrivialElementaryChildData p} (src : Sec7FrattiniTrivialElementaryMStageCandidateLowerSourceData h chief K hR child) : src.P * (src.X₁ - 1) * (src.Sinv * (src.X₁ - 1) * src.S) = src.Sinv * (src.X₁ - 1) - src.Sinv * (src.X₁ - 1) * src.S

Former checkpoint271 expanded transport-defect atom, now theorem-level from the primitive raw-evaluation σ⁻¹,x₁,σ transport statement.

theorem Q2Presentation.Induction.sec7_frattiniTrivial_elementaryMStageCandidateWildRowManuscriptX1ConjCanonicalU1X1RowTransportDeltaData {p : FramedPair} {h : ¬ScalarTerminal p} {chief : FirstNonScalarChiefFactor p} {K : KernelMinimal chief.toNonScalarChiefFactor} {hR : Nat.card ↥(kernelFrattini K) = 1} {child : Sec7FrattiniTrivialElementaryChildData p} (src : Sec7FrattiniTrivialElementaryMStageCandidateLowerSourceData h chief K hR child) : sec7MStageCandidateWildRowManuscriptU1X1CanonicalRow src * (src.Sinv * (src.X₁ - 1) * src.S) = src.Sinv * (src.X₁ - 1) - src.Sinv * (src.X₁ - 1) * src.S

Former checkpoint270 transport-defect atom, now theorem-level from the expanded P(X₁-1) row statement.

theorem Q2Presentation.Induction.sec7_frattiniTrivial_elementaryMStageCandidateWildRowManuscriptX1ConjCanonicalU1InvTransportData {p : FramedPair} {h : ¬ScalarTerminal p} {chief : FirstNonScalarChiefFactor p} {K : KernelMinimal chief.toNonScalarChiefFactor} {hR : Nat.card ↥(kernelFrattini K) = 1} {child : Sec7FrattiniTrivialElementaryChildData p} (src : Sec7FrattiniTrivialElementaryMStageCandidateLowerSourceData h chief K hR child) : (sec7MStageCandidateActionEnd K) (sec7MStageCandidateWildRowManuscriptU1InvFactor src) * (src.Sinv * (src.X₁ - 1) * src.S) = src.Sinv * (src.X₁ - 1)

Former checkpoint269 prefix-transport atom for the x₁^σ contribution, now theorem-level from the lower transport-defect identity.

theorem Q2Presentation.Induction.sec7_frattiniTrivial_elementaryMStageCandidateWildRowManuscriptX1ConjCanonicalPrefixTransportData {p : FramedPair} {h : ¬ScalarTerminal p} {chief : FirstNonScalarChiefFactor p} {K : KernelMinimal chief.toNonScalarChiefFactor} {hR : Nat.card ↥(kernelFrattini K) = 1} {child : Sec7FrattiniTrivialElementaryChildData p} (src : Sec7FrattiniTrivialElementaryMStageCandidateLowerSourceData h chief K hR child) : (sec7MStageCandidateActionEnd K) (sec7MStageCandidateWildRowManuscriptH0Factor src * sec7MStageCandidateWildRowManuscriptU1InvFactor src) * (src.Sinv * (src.X₁ - 1) * src.S) = src.Sinv * (src.X₁ - 1)

Former checkpoint268 prefix-transport atom, now theorem-level from the h₀ action identity and the smaller u₁⁻¹ transport atom.

theorem Q2Presentation.Induction.sec7_frattiniTrivial_elementaryMStageCandidateWildRowManuscriptX1ConjCanonicalActionProductData {p : FramedPair} {h : ¬ScalarTerminal p} {chief : FirstNonScalarChiefFactor p} {K : KernelMinimal chief.toNonScalarChiefFactor} {hR : Nat.card ↥(kernelFrattini K) = 1} {child : Sec7FrattiniTrivialElementaryChildData p} (src : Sec7FrattiniTrivialElementaryMStageCandidateLowerSourceData h chief K hR child) : (sec7MStageCandidateActionEnd K) (sec7MStageCandidateWildRowManuscriptH0Factor src * sec7MStageCandidateWildRowManuscriptU1InvFactor src) * ((sec7MStageCandidateActionEnd K) (sec7MStageCandidateWildRowManuscriptSigmaInvFactor src) * (sec7MStageCandidateActionEnd K) (sec7MStageCandidateWildRowManuscriptX1BareFactor src) * (sec7MStageCandidateActionEnd K) (sec7MStageCandidateWildRowManuscriptSigmaFactor src) - 1) = src.Sinv * (src.X₁ - 1)

Former checkpoint267 x₁^σ action-product leaf, now theorem-level from the bare conjugation identity and the smaller u₁⁻¹ prefix-transport atom.

theorem Q2Presentation.Induction.sec7_frattiniTrivial_elementaryMStageCandidateWildRowManuscriptX1ConjCanonicalPrefixActionData {p : FramedPair} {h : ¬ScalarTerminal p} {chief : FirstNonScalarChiefFactor p} {K : KernelMinimal chief.toNonScalarChiefFactor} {hR : Nat.card ↥(kernelFrattini K) = 1} {child : Sec7FrattiniTrivialElementaryChildData p} (src : Sec7FrattiniTrivialElementaryMStageCandidateLowerSourceData h chief K hR child) : (sec7MStageCandidateActionEnd K) (sec7MStageCandidateWildRowManuscriptH0Factor src * sec7MStageCandidateWildRowManuscriptU1InvFactor src) * ((sec7MStageCandidateActionEnd K) (sec7MStageCandidateWildRowManuscriptX1ConjFactor src) - 1) = src.Sinv * (src.X₁ - 1)

The collapsed prefix contribution of x₁^σ, theorem-level from the explicit three-action product.

axiom Q2Presentation.Induction.sec7_frattiniTrivial_elementaryMStageCandidateWildRowManuscriptC0CommutatorCanonicalPrimitiveTowerActionD0Z0Z0ConjugatesD0RawZ0NestedD0PrecancelU0InvOpenedD0TailOpenedPrimitiveU0InvX0RelativeActionFixesCoreDifferenceZeroLowerPrimitiveApplyPointwiseData {p : FramedPair} {h : ¬ScalarTerminal p} {chief : FirstNonScalarChiefFactor p} {K : KernelMinimal chief.toNonScalarChiefFactor} {hR : Nat.card ↥(kernelFrattini K) = 1} {child : Sec7FrattiniTrivialElementaryChildData p} (src : Sec7FrattiniTrivialElementaryMStageCandidateLowerSourceData h chief K hR child) (m : towerM K) : let u0 := powOmega2 (src.x₀ * src.tau); ((towerActM K) u0⁻¹) (((towerActM K) src.x₀) m) - m = 0

Checkpoint332 lower primitive for c₀=[d₀,z₀]: record only the pointwise additive zero for the relative u₀⁻¹*x₀ action. The checkpoint331 primitive x₀/u₀ action-coincidence row is recovered by applying u₀ and cancelling the visible inverse pair.

theorem Q2Presentation.Induction.sec7_frattiniTrivial_elementaryMStageCandidateWildRowManuscriptC0CommutatorCanonicalPrimitiveTowerActionD0Z0Z0ConjugatesD0RawZ0NestedD0PrecancelU0InvOpenedD0TailOpenedPrimitiveX0U0ActionsCoincideLowerPrimitiveApplyPointwiseData {p : FramedPair} {h : ¬ScalarTerminal p} {chief : FirstNonScalarChiefFactor p} {K : KernelMinimal chief.toNonScalarChiefFactor} {hR : Nat.card ↥(kernelFrattini K) = 1} {child : Sec7FrattiniTrivialElementaryChildData p} (src : Sec7FrattiniTrivialElementaryMStageCandidateLowerSourceData h chief K hR child) (m : towerM K) : have u0 := powOmega2 (src.x₀ * src.tau); ((towerActM K) src.x₀) m = ((towerActM K) u0) m

Checkpoint332 bridge recovering the checkpoint331 c₀ primitive action coincidence from the lower relative-action difference-zero primitive.

theorem Q2Presentation.Induction.sec7_frattiniTrivial_elementaryMStageCandidateWildRowManuscriptC0CommutatorCanonicalPrimitiveTowerActionD0Z0Z0ConjugatesD0RawZ0NestedD0PrecancelU0InvOpenedD0TailOpenedPrimitiveU0InvX0RelativeActionFixesCoreDifferenceZeroPrimitiveApplyPointwiseData {p : FramedPair} {h : ¬ScalarTerminal p} {chief : FirstNonScalarChiefFactor p} {K : KernelMinimal chief.toNonScalarChiefFactor} {hR : Nat.card ↥(kernelFrattini K) = 1} {child : Sec7FrattiniTrivialElementaryChildData p} (src : Sec7FrattiniTrivialElementaryMStageCandidateLowerSourceData h chief K hR child) (m : towerM K) : have u0 := powOmega2 (src.x₀ * src.tau); ((towerActM K) u0⁻¹) (((towerActM K) src.x₀) m) - m = 0

Checkpoint331 bridge closing the checkpoint330 c₀ relative-action fixed-core difference-zero residual from primitive x₀/u₀ action coincidence.

theorem Q2Presentation.Induction.sec7_frattiniTrivial_elementaryMStageCandidateWildRowManuscriptC0CommutatorCanonicalPrimitiveTowerActionD0Z0Z0ConjugatesD0RawZ0NestedD0PrecancelU0InvOpenedD0TailOpenedPrimitiveU0InvX0RelativeActionFixesCorePrimitiveApplyPointwiseData {p : FramedPair} {h : ¬ScalarTerminal p} {chief : FirstNonScalarChiefFactor p} {K : KernelMinimal chief.toNonScalarChiefFactor} {hR : Nat.card ↥(kernelFrattini K) = 1} {child : Sec7FrattiniTrivialElementaryChildData p} (src : Sec7FrattiniTrivialElementaryMStageCandidateLowerSourceData h chief K hR child) (m : towerM K) : have u0 := powOmega2 (src.x₀ * src.tau); ((towerActM K) u0⁻¹) (((towerActM K) src.x₀) m) = m

Checkpoint330 bridge for the checkpoint329 c₀ relative-action fixed row.

theorem Q2Presentation.Induction.sec7_frattiniTrivial_elementaryMStageCandidateWildRowManuscriptC0CommutatorCanonicalPrimitiveTowerActionD0Z0Z0ConjugatesD0RawZ0NestedD0PrecancelU0InvOpenedD0TailOpenedPrimitiveX0U0ActionsCoincideCorePrimitiveApplyPointwiseData {p : FramedPair} {h : ¬ScalarTerminal p} {chief : FirstNonScalarChiefFactor p} {K : KernelMinimal chief.toNonScalarChiefFactor} {hR : Nat.card ↥(kernelFrattini K) = 1} {child : Sec7FrattiniTrivialElementaryChildData p} (src : Sec7FrattiniTrivialElementaryMStageCandidateLowerSourceData h chief K hR child) (m : towerM K) : have u0 := powOmega2 (src.x₀ * src.tau); ((towerActM K) src.x₀) m = ((towerActM K) u0) m

Checkpoint329 bridge for the checkpoint328 c₀ primitive action equality.

theorem Q2Presentation.Induction.sec7_frattiniTrivial_elementaryMStageCandidateWildRowManuscriptC0CommutatorCanonicalPrimitiveTowerActionD0Z0Z0ConjugatesD0RawZ0NestedD0PrecancelU0InvOpenedD0TailOpenedPrimitiveU0InvX0ProductFixesBaseCoreApplyPointwiseData {p : FramedPair} {h : ¬ScalarTerminal p} {chief : FirstNonScalarChiefFactor p} {K : KernelMinimal chief.toNonScalarChiefFactor} {hR : Nat.card ↥(kernelFrattini K) = 1} {child : Sec7FrattiniTrivialElementaryChildData p} (src : Sec7FrattiniTrivialElementaryMStageCandidateLowerSourceData h chief K hR child) (m : towerM K) : have u0 := powOmega2 (src.x₀ * src.tau); ((towerActM K) (u0⁻¹ * src.x₀)) m = m

Checkpoint328 bridge for the checkpoint327 c₀ product-fix atom: the new primitive datum is the pointwise equality of the x₀ and u₀ actions. Opening u₀⁻¹*x₀ and cancelling u₀⁻¹*u₀ recovers the product fixed-point row.

theorem Q2Presentation.Induction.sec7_frattiniTrivial_elementaryMStageCandidateWildRowManuscriptC0CommutatorCanonicalPrimitiveTowerActionD0Z0Z0ConjugatesD0RawZ0NestedD0PrecancelU0InvOpenedD0TailOpenedPrimitiveX0U0ActionsCoincideAtBaseCoreApplyPointwiseData {p : FramedPair} {h : ¬ScalarTerminal p} {chief : FirstNonScalarChiefFactor p} {K : KernelMinimal chief.toNonScalarChiefFactor} {hR : Nat.card ↥(kernelFrattini K) = 1} {child : Sec7FrattiniTrivialElementaryChildData p} (src : Sec7FrattiniTrivialElementaryMStageCandidateLowerSourceData h chief K hR child) (m : towerM K) : have u0 := powOmega2 (src.x₀ * src.tau); ((towerActM K) src.x₀) m = ((towerActM K) u0) m

Checkpoint327 bridge for the checkpoint326 c₀ primitive action equality: the new datum says that the product action u₀⁻¹*x₀ fixes the point. Open the product action, apply u₀, and cancel u₀*u₀⁻¹.

theorem Q2Presentation.Induction.sec7_frattiniTrivial_elementaryMStageCandidateWildRowManuscriptC0CommutatorCanonicalPrimitiveTowerActionD0Z0Z0ConjugatesD0RawZ0NestedD0PrecancelU0InvOpenedD0TailOpenedPrimitiveX0U0InvPreimagesCoincideAtBaseCoreApplyPointwiseData {p : FramedPair} {h : ¬ScalarTerminal p} {chief : FirstNonScalarChiefFactor p} {K : KernelMinimal chief.toNonScalarChiefFactor} {hR : Nat.card ↥(kernelFrattini K) = 1} {child : Sec7FrattiniTrivialElementaryChildData p} (src : Sec7FrattiniTrivialElementaryMStageCandidateLowerSourceData h chief K hR child) (m : towerM K) : have u0 := powOmega2 (src.x₀ * src.tau); ((towerActM K) src.x₀⁻¹) m = ((towerActM K) u0⁻¹) m

Checkpoint326 bridge for the checkpoint325 c₀ inverse-preimage atom: evaluate the lower primitive action agreement at u₀⁻¹ m and cancel.

theorem Q2Presentation.Induction.sec7_frattiniTrivial_elementaryMStageCandidateWildRowManuscriptC0CommutatorCanonicalPrimitiveTowerActionD0Z0Z0ConjugatesD0RawZ0NestedD0PrecancelU0InvOpenedD0TailOpenedPrimitiveX0U0InvPreimageAtBaseCoreApplyPointwiseData {p : FramedPair} {h : ¬ScalarTerminal p} {chief : FirstNonScalarChiefFactor p} {K : KernelMinimal chief.toNonScalarChiefFactor} {hR : Nat.card ↥(kernelFrattini K) = 1} {child : Sec7FrattiniTrivialElementaryChildData p} (src : Sec7FrattiniTrivialElementaryMStageCandidateLowerSourceData h chief K hR child) (m : towerM K) : have u0 := powOmega2 (src.x₀ * src.tau); m = ((towerActM K) src.x₀) (((towerActM K) u0⁻¹) m)

Checkpoint325 bridge for c₀=[d₀,z₀]: recover the checkpoint324 base-preimage row from the lower equality of inverse preimages.

theorem Q2Presentation.Induction.sec7_frattiniTrivial_elementaryMStageCandidateWildRowManuscriptC0CommutatorCanonicalPrimitiveTowerActionD0Z0Z0ConjugatesD0RawZ0NestedD0PrecancelU0InvOpenedD0TailOpenedPrimitiveU0CancelsX0InvAtBaseCoreApplyPointwiseData {p : FramedPair} {h : ¬ScalarTerminal p} {chief : FirstNonScalarChiefFactor p} {K : KernelMinimal chief.toNonScalarChiefFactor} {hR : Nat.card ↥(kernelFrattini K) = 1} {child : Sec7FrattiniTrivialElementaryChildData p} (src : Sec7FrattiniTrivialElementaryMStageCandidateLowerSourceData h chief K hR child) (m : towerM K) : have u0 := powOmega2 (src.x₀ * src.tau); ((towerActM K) u0) (((towerActM K) src.x₀⁻¹) m) = m

Checkpoint324 bridge for c₀=[d₀,z₀]: apply u₀ · x₀⁻¹ to the new base-preimage row and cancel the adjacent inverse tower actions.

theorem Q2Presentation.Induction.sec7_frattiniTrivial_elementaryMStageCandidateWildRowManuscriptC0CommutatorCanonicalPrimitiveTowerActionD0Z0Z0ConjugatesD0RawZ0NestedD0PrecancelU0InvOpenedD0TailOpenedPrimitiveU0X0ActionsCoincideAtX0InvBaseCoreApplyPointwiseData {p : FramedPair} {h : ¬ScalarTerminal p} {chief : FirstNonScalarChiefFactor p} {K : KernelMinimal chief.toNonScalarChiefFactor} {hR : Nat.card ↥(kernelFrattini K) = 1} {child : Sec7FrattiniTrivialElementaryChildData p} (src : Sec7FrattiniTrivialElementaryMStageCandidateLowerSourceData h chief K hR child) (m : towerM K) : have u0 := powOmega2 (src.x₀ * src.tau); ((towerActM K) u0) (((towerActM K) src.x₀⁻¹) m) = ((towerActM K) src.x₀) (((towerActM K) src.x₀⁻¹) m)

Checkpoint323 bridge for c₀=[d₀,z₀]: compare the lower primitive u₀ cancellation row with the already proved x₀*x₀⁻¹ cancellation row.

theorem Q2Presentation.Induction.sec7_frattiniTrivial_elementaryMStageCandidateWildRowManuscriptC0CommutatorCanonicalPrimitiveTowerActionD0Z0Z0ConjugatesD0RawZ0NestedD0PrecancelU0InvOpenedD0TailOpenedFixesBaseCoreApplyPointwiseData {p : FramedPair} {h : ¬ScalarTerminal p} {chief : FirstNonScalarChiefFactor p} {K : KernelMinimal chief.toNonScalarChiefFactor} {hR : Nat.card ↥(kernelFrattini K) = 1} {child : Sec7FrattiniTrivialElementaryChildData p} (src : Sec7FrattiniTrivialElementaryMStageCandidateLowerSourceData h chief K hR child) (m : towerM K) : have u0 := powOmega2 (src.x₀ * src.tau); ((towerActM K) u0) (((towerActM K) src.x₀⁻¹) m) = m

Checkpoint322 bridge for c₀=[d₀,z₀]: the checkpoint321 opened-tail fixed point is recovered from the lower primitive action-coincidence row by cancelling the adjacent x₀*x₀⁻¹ action on the right.

theorem Q2Presentation.Induction.sec7_frattiniTrivial_elementaryMStageCandidateWildRowManuscriptC0CommutatorCanonicalPrimitiveTowerActionD0Z0Z0ConjugatesD0RawZ0NestedD0PrecancelU0InvOpenedD0TailU0X0ActionsCoincideBaseCoreApplyPointwiseData {p : FramedPair} {h : ¬ScalarTerminal p} {chief : FirstNonScalarChiefFactor p} {K : KernelMinimal chief.toNonScalarChiefFactor} {hR : Nat.card ↥(kernelFrattini K) = 1} {child : Sec7FrattiniTrivialElementaryChildData p} (src : Sec7FrattiniTrivialElementaryMStageCandidateLowerSourceData h chief K hR child) (m : towerM K) : have u0 := powOmega2 (src.x₀ * src.tau); ((towerActM K) u0) m = ((towerActM K) src.x₀) m

Checkpoint320 bridge for c₀=[d₀,z₀]: evaluate the checkpoint321 opened d₀ = u₀*x₀⁻¹ fixed-point row at the x₀-orbit point and cancel x₀⁻¹*x₀.

theorem Q2Presentation.Induction.sec7_frattiniTrivial_elementaryMStageCandidateWildRowManuscriptC0CommutatorCanonicalPrimitiveTowerActionD0Z0Z0ConjugatesD0RawZ0NestedD0PrecancelU0InvOpenedD0TailU0PreimageAtX0OrbitCoreApplyPointwiseData {p : FramedPair} {h : ¬ScalarTerminal p} {chief : FirstNonScalarChiefFactor p} {K : KernelMinimal chief.toNonScalarChiefFactor} {hR : Nat.card ↥(kernelFrattini K) = 1} {child : Sec7FrattiniTrivialElementaryChildData p} (src : Sec7FrattiniTrivialElementaryMStageCandidateLowerSourceData h chief K hR child) (m : towerM K) : have u0 := powOmega2 (src.x₀ * src.tau); m = ((towerActM K) u0⁻¹) (((towerActM K) src.x₀) m)

Checkpoint319 bridge for c₀=[d₀,z₀]: apply u₀⁻¹ to the primitive base action coincidence and cancel u₀⁻¹*u₀ on the left.

theorem Q2Presentation.Induction.sec7_frattiniTrivial_elementaryMStageCandidateWildRowManuscriptC0CommutatorCanonicalPrimitiveTowerActionD0Z0Z0ConjugatesD0RawZ0NestedD0PrecancelU0InvOpenedD0TailPreimagesCoincideAtX0OrbitCoreApplyPointwiseData {p : FramedPair} {h : ¬ScalarTerminal p} {chief : FirstNonScalarChiefFactor p} {K : KernelMinimal chief.toNonScalarChiefFactor} {hR : Nat.card ↥(kernelFrattini K) = 1} {child : Sec7FrattiniTrivialElementaryChildData p} (src : Sec7FrattiniTrivialElementaryMStageCandidateLowerSourceData h chief K hR child) (m : towerM K) : have u0 := powOmega2 (src.x₀ * src.tau); ((towerActM K) src.x₀⁻¹) (((towerActM K) src.x₀) m) = ((towerActM K) u0⁻¹) (((towerActM K) src.x₀) m)

theorem Q2Presentation.Induction.sec7_frattiniTrivial_elementaryMStageCandidateWildRowManuscriptC0CommutatorCanonicalPrimitiveTowerActionD0Z0Z0ConjugatesD0RawZ0NestedD0PrecancelU0InvOpenedD0TailU0X0ActionsCoincideAtX0OrbitCoreApplyPointwiseData {p : FramedPair} {h : ¬ScalarTerminal p} {chief : FirstNonScalarChiefFactor p} {K : KernelMinimal chief.toNonScalarChiefFactor} {hR : Nat.card ↥(kernelFrattini K) = 1} {child : Sec7FrattiniTrivialElementaryChildData p} (src : Sec7FrattiniTrivialElementaryMStageCandidateLowerSourceData h chief K hR child) (m : towerM K) : have u0 := powOmega2 (src.x₀ * src.tau); ((towerActM K) u0) (((towerActM K) src.x₀⁻¹) (((towerActM K) src.x₀) m)) = ((towerActM K) src.x₀) m

Checkpoint318 bridge for c₀=[d₀,z₀]: the checkpoint317 action coincidence at the x₀-orbit point follows by applying u₀ to the smaller equality of the visible x₀⁻¹ and u₀⁻¹ preimages.

theorem Q2Presentation.Induction.sec7_frattiniTrivial_elementaryMStageCandidateWildRowManuscriptC0CommutatorCanonicalPrimitiveTowerActionD0Z0Z0ConjugatesD0RawZ0NestedD0PrecancelU0InvOpenedD0TailU0X0ActionsCoincideCoreApplyPointwiseData {p : FramedPair} {h : ¬ScalarTerminal p} {chief : FirstNonScalarChiefFactor p} {K : KernelMinimal chief.toNonScalarChiefFactor} {hR : Nat.card ↥(kernelFrattini K) = 1} {child : Sec7FrattiniTrivialElementaryChildData p} (src : Sec7FrattiniTrivialElementaryMStageCandidateLowerSourceData h chief K hR child) (m : towerM K) : have u0 := powOmega2 (src.x₀ * src.tau); ((towerActM K) u0) m = ((towerActM K) src.x₀) m

Checkpoint317 bridge for c₀=[d₀,z₀]: the checkpoint316 primitive action coincidence is recovered from the same equality at the displayed x₀-orbit point, followed by cancellation of x₀⁻¹*x₀.

theorem Q2Presentation.Induction.sec7_frattiniTrivial_elementaryMStageCandidateWildRowManuscriptC0CommutatorCanonicalPrimitiveTowerActionD0Z0Z0ConjugatesD0RawZ0NestedD0PrecancelU0InvOpenedD0TailFixesPointProductOpenedCoreApplyPointwiseData {p : FramedPair} {h : ¬ScalarTerminal p} {chief : FirstNonScalarChiefFactor p} {K : KernelMinimal chief.toNonScalarChiefFactor} {hR : Nat.card ↥(kernelFrattini K) = 1} {child : Sec7FrattiniTrivialElementaryChildData p} (src : Sec7FrattiniTrivialElementaryMStageCandidateLowerSourceData h chief K hR child) (m : towerM K) : have u0 := powOmega2 (src.x₀ * src.tau); ((towerActM K) u0) (((towerActM K) src.x₀⁻¹) m) = m

Checkpoint315 c₀ opened d₀ fixed-point row, now theorem-level from the checkpoint316 primitive-action coincidence row.

theorem Q2Presentation.Induction.sec7_frattiniTrivial_elementaryMStageCandidateWildRowManuscriptC0CommutatorCanonicalPrimitiveTowerActionD0Z0Z0ConjugatesD0RawZ0NestedD0PrecancelU0InvOpenedD0TailPrefixSwapProductOpenedApplyPointwiseData {p : FramedPair} {h : ¬ScalarTerminal p} {chief : FirstNonScalarChiefFactor p} {K : KernelMinimal chief.toNonScalarChiefFactor} {hR : Nat.card ↥(kernelFrattini K) = 1} {child : Sec7FrattiniTrivialElementaryChildData p} (src : Sec7FrattiniTrivialElementaryMStageCandidateLowerSourceData h chief K hR child) (m : towerM K) : have u0 := powOmega2 (src.x₀ * src.tau); ((towerActM K) src.x₀⁻¹) m = ((towerActM K) u0⁻¹) m

Checkpoint314 c₀ prefix-swap product-opened row, now theorem-level from the checkpoint315 opened d₀ fixed-point row.

theorem Q2Presentation.Induction.sec7_frattiniTrivial_elementaryMStageCandidateWildRowManuscriptC0CommutatorCanonicalPrimitiveTowerActionD0Z0Z0ConjugatesD0RawZ0NestedD0PrecancelU0InvOpenedD0FixesOpenedZ0CoreProductOpenedApplyPointwiseData {p : FramedPair} {h : ¬ScalarTerminal p} {chief : FirstNonScalarChiefFactor p} {K : KernelMinimal chief.toNonScalarChiefFactor} {hR : Nat.card ↥(kernelFrattini K) = 1} {child : Sec7FrattiniTrivialElementaryChildData p} (src : Sec7FrattiniTrivialElementaryMStageCandidateLowerSourceData h chief K hR child) (m : towerM K) : have sigma2 := powOmega2 src.sigma; have u0 := powOmega2 (src.x₀ * src.tau); ((towerActM K) u0) (((towerActM K) src.x₀⁻¹) (((towerActM K) sigma2⁻¹) (((towerActM K) src.x₀) (((towerActM K) sigma2) m)))) = ((towerActM K) sigma2⁻¹) (((towerActM K) src.x₀) (((towerActM K) sigma2) m))

theorem Q2Presentation.Induction.sec7_frattiniTrivial_elementaryMStageCandidateWildRowManuscriptC0CommutatorCanonicalPrimitiveTowerActionD0Z0Z0ConjugatesD0RawZ0NestedD0PrecancelU0InvOpenedD0FixesOpenedZ0CoreApplyPointwiseData {p : FramedPair} {h : ¬ScalarTerminal p} {chief : FirstNonScalarChiefFactor p} {K : KernelMinimal chief.toNonScalarChiefFactor} {hR : Nat.card ↥(kernelFrattini K) = 1} {child : Sec7FrattiniTrivialElementaryChildData p} (src : Sec7FrattiniTrivialElementaryMStageCandidateLowerSourceData h chief K hR child) (m : towerM K) : have sigma2 := powOmega2 src.sigma; have u0 := powOmega2 (src.x₀ * src.tau); ((towerActM K) (u0 * src.x₀⁻¹)) (((towerActM K) sigma2⁻¹) (((towerActM K) src.x₀) (((towerActM K) sigma2) m))) = ((towerActM K) sigma2⁻¹) (((towerActM K) src.x₀) (((towerActM K) sigma2) m))

Checkpoint312 d₀-fixes-opened-z₀ row, now theorem-level from the opened product-action atom above.

theorem Q2Presentation.Induction.sec7_frattiniTrivial_elementaryMStageCandidateWildRowManuscriptC0CommutatorCanonicalPrimitiveTowerActionD0Z0Z0ConjugatesD0RawZ0NestedD0PrecancelU0InvOpenedAllProductActionsNestedPrefixSwapAtCommonZ0ApplyPointwiseData {p : FramedPair} {h : ¬ScalarTerminal p} {chief : FirstNonScalarChiefFactor p} {K : KernelMinimal chief.toNonScalarChiefFactor} {hR : Nat.card ↥(kernelFrattini K) = 1} {child : Sec7FrattiniTrivialElementaryChildData p} (src : Sec7FrattiniTrivialElementaryMStageCandidateLowerSourceData h chief K hR child) (m : towerM K) : have sigma2 := powOmega2 src.sigma; have u0 := powOmega2 (src.x₀ * src.tau); ((towerActM K) src.x₀⁻¹) (((towerActM K) sigma2⁻¹) (((towerActM K) src.x₀) (((towerActM K) sigma2) m))) = ((towerActM K) u0⁻¹) (((towerActM K) sigma2⁻¹) (((towerActM K) src.x₀) (((towerActM K) sigma2) m)))

Checkpoint311 c₀ prefix-swap row, recovered theorem-level from the checkpoint312 statement that the literal d₀ = u₀*x₀⁻¹ product action fixes the opened z₀ core.

theorem Q2Presentation.Induction.sec7_frattiniTrivial_elementaryMStageCandidateWildRowManuscriptC0CommutatorCanonicalPrimitiveTowerActionD0Z0Z0ConjugatesD0RawZ0NestedD0PrecancelU0InvOpenedAllProductActionsListPrefixSwapAtCommonZ0ApplyPointwiseData {p : FramedPair} {h : ¬ScalarTerminal p} {chief : FirstNonScalarChiefFactor p} {K : KernelMinimal chief.toNonScalarChiefFactor} {hR : Nat.card ↥(kernelFrattini K) = 1} {child : Sec7FrattiniTrivialElementaryChildData p} (src : Sec7FrattiniTrivialElementaryMStageCandidateLowerSourceData h chief K hR child) (m : towerM K) : have sigma2 := powOmega2 src.sigma; have u0 := powOmega2 (src.x₀ * src.tau); ((towerActM K) src.x₀⁻¹) (Q2Presentation.Induction.sec7TowerActSeq✝ K [sigma2⁻¹, src.x₀, sigma2] m) = ((towerActM K) u0⁻¹) (Q2Presentation.Induction.sec7TowerActSeq✝¹ K [sigma2⁻¹, src.x₀, sigma2] m)

Checkpoint310 prefix-swap row for c₀, recovered theorem-level from the checkpoint311 nested common-z₀ row.

theorem Q2Presentation.Induction.sec7_frattiniTrivial_elementaryMStageCandidateWildRowManuscriptC0CommutatorCanonicalPrimitiveTowerActionD0Z0Z0ConjugatesD0RawZ0NestedD0PrecancelU0InvOpenedD0TailFixesPointProductOpenedApplyPointwiseData {p : FramedPair} {h : ¬ScalarTerminal p} {chief : FirstNonScalarChiefFactor p} {K : KernelMinimal chief.toNonScalarChiefFactor} {hR : Nat.card ↥(kernelFrattini K) = 1} {child : Sec7FrattiniTrivialElementaryChildData p} (src : Sec7FrattiniTrivialElementaryMStageCandidateLowerSourceData h chief K hR child) (m : towerM K) : have u0 := powOmega2 (src.x₀ * src.tau); ((towerActM K) u0) (((towerActM K) src.x₀⁻¹) m) = m

Checkpoint313 lower atom for the c₀ tail: the assertion that d₀ = u₀*x₀⁻¹ fixes the point is stated with the product action opened.

theorem Q2Presentation.Induction.sec7_frattiniTrivial_elementaryMStageCandidateWildRowManuscriptC0CommutatorCanonicalPrimitiveTowerActionD0Z0Z0ConjugatesD0RawZ0NestedD0PrecancelU0InvOpenedD0TailFixesPointApplyPointwiseData {p : FramedPair} {h : ¬ScalarTerminal p} {chief : FirstNonScalarChiefFactor p} {K : KernelMinimal chief.toNonScalarChiefFactor} {hR : Nat.card ↥(kernelFrattini K) = 1} {child : Sec7FrattiniTrivialElementaryChildData p} (src : Sec7FrattiniTrivialElementaryMStageCandidateLowerSourceData h chief K hR child) (m : towerM K) : have u0 := powOmega2 (src.x₀ * src.tau); ((towerActM K) (u0 * src.x₀⁻¹)) m = m

Checkpoint312 d₀ tail-fixes-point row, now theorem-level from the opened product-action tail atom above.

theorem Q2Presentation.Induction.sec7_frattiniTrivial_elementaryMStageCandidateWildRowManuscriptC0CommutatorCanonicalPrimitiveTowerActionD0Z0Z0ConjugatesD0RawZ0NestedD0PrecancelU0InvOpenedAllProductActionsNestedTailInsertAtU0PrefixApplyPointwiseData {p : FramedPair} {h : ¬ScalarTerminal p} {chief : FirstNonScalarChiefFactor p} {K : KernelMinimal chief.toNonScalarChiefFactor} {hR : Nat.card ↥(kernelFrattini K) = 1} {child : Sec7FrattiniTrivialElementaryChildData p} (src : Sec7FrattiniTrivialElementaryMStageCandidateLowerSourceData h chief K hR child) (m : towerM K) : have sigma2 := powOmega2 src.sigma; have u0 := powOmega2 (src.x₀ * src.tau); ((towerActM K) sigma2⁻¹) (((towerActM K) src.x₀) (((towerActM K) sigma2) m)) = ((towerActM K) sigma2⁻¹) (((towerActM K) src.x₀) (((towerActM K) sigma2) (((towerActM K) u0) (((towerActM K) src.x₀⁻¹) m))))

Checkpoint311 c₀ tail-insertion row, recovered theorem-level from the checkpoint312 pointwise assertion that the literal d₀ = u₀*x₀⁻¹ tail fixes the point before the opened z₀ prefix is reapplied.

theorem Q2Presentation.Induction.sec7_frattiniTrivial_elementaryMStageCandidateWildRowManuscriptC0CommutatorCanonicalPrimitiveTowerActionD0Z0Z0ConjugatesD0RawZ0NestedD0PrecancelU0InvOpenedAllProductActionsListTailInsertAtU0PrefixApplyPointwiseData {p : FramedPair} {h : ¬ScalarTerminal p} {chief : FirstNonScalarChiefFactor p} {K : KernelMinimal chief.toNonScalarChiefFactor} {hR : Nat.card ↥(kernelFrattini K) = 1} {child : Sec7FrattiniTrivialElementaryChildData p} (src : Sec7FrattiniTrivialElementaryMStageCandidateLowerSourceData h chief K hR child) (m : towerM K) : have sigma2 := powOmega2 src.sigma; have u0 := powOmega2 (src.x₀ * src.tau); Q2Presentation.Induction.sec7TowerActSeq✝ K [sigma2⁻¹, src.x₀, sigma2] m = Q2Presentation.Induction.sec7TowerActSeq✝¹ K [sigma2⁻¹, src.x₀, sigma2, u0, src.x₀⁻¹] m

Checkpoint310 tail-insertion list row for c₀, recovered theorem-level from the checkpoint311 fully nested tail row.

theorem Q2Presentation.Induction.sec7_frattiniTrivial_elementaryMStageCandidateWildRowManuscriptC0CommutatorCanonicalPrimitiveTowerActionD0Z0Z0ConjugatesD0RawZ0NestedD0PrecancelU0InvOpenedAllProductActionsListX0InvX0CancelledApplyPointwiseData {p : FramedPair} {h : ¬ScalarTerminal p} {chief : FirstNonScalarChiefFactor p} {K : KernelMinimal chief.toNonScalarChiefFactor} {hR : Nat.card ↥(kernelFrattini K) = 1} {child : Sec7FrattiniTrivialElementaryChildData p} (src : Sec7FrattiniTrivialElementaryMStageCandidateLowerSourceData h chief K hR child) (m : towerM K) : have sigma2 := powOmega2 src.sigma; have u0 := powOmega2 (src.x₀ * src.tau); Q2Presentation.Induction.sec7TowerActSeq✝ K [src.x₀⁻¹, sigma2⁻¹, src.x₀, sigma2] m = Q2Presentation.Induction.sec7TowerActSeq✝¹ K [u0⁻¹, sigma2⁻¹, src.x₀, sigma2, u0, src.x₀⁻¹] m

Checkpoint309 c₀ datum, now theorem-level from two smaller checkpoint310 rows: a prefix swap at the common z₀ core and a tail insertion under the new u₀⁻¹ prefix.

theorem Q2Presentation.Induction.sec7_frattiniTrivial_elementaryMStageCandidateWildRowManuscriptC0CommutatorCanonicalPrimitiveTowerActionD0Z0Z0ConjugatesD0RawZ0NestedD0PrecancelU0InvOpenedAllProductActionsListApplyPointwiseData {p : FramedPair} {h : ¬ScalarTerminal p} {chief : FirstNonScalarChiefFactor p} {K : KernelMinimal chief.toNonScalarChiefFactor} {hR : Nat.card ↥(kernelFrattini K) = 1} {child : Sec7FrattiniTrivialElementaryChildData p} (src : Sec7FrattiniTrivialElementaryMStageCandidateLowerSourceData h chief K hR child) (m : towerM K) : have sigma2 := powOmega2 src.sigma; have u0 := powOmega2 (src.x₀ * src.tau); Q2Presentation.Induction.sec7TowerActSeq✝ K [src.x₀⁻¹, sigma2⁻¹, src.x₀, sigma2] m = Q2Presentation.Induction.sec7TowerActSeq✝¹ K [src.x₀⁻¹, src.x₀, u0⁻¹, sigma2⁻¹, src.x₀, sigma2, u0, src.x₀⁻¹] m

Checkpoint308 c₀ list datum, now theorem-level from the row after the visible initial x₀⁻¹,x₀ pair on the right-hand side has already been cancelled.

theorem Q2Presentation.Induction.sec7_frattiniTrivial_elementaryMStageCandidateWildRowManuscriptC0CommutatorCanonicalPrimitiveTowerActionD0Z0Z0ConjugatesD0RawZ0NestedD0PrecancelU0InvOpenedAllProductActionsSplitApplyPointwiseData {p : FramedPair} {h : ¬ScalarTerminal p} {chief : FirstNonScalarChiefFactor p} {K : KernelMinimal chief.toNonScalarChiefFactor} {hR : Nat.card ↥(kernelFrattini K) = 1} {child : Sec7FrattiniTrivialElementaryChildData p} (src : Sec7FrattiniTrivialElementaryMStageCandidateLowerSourceData h chief K hR child) (m : towerM K) : ((towerActM K) src.x₀⁻¹) (((towerActM K) (powOmega2 src.sigma)⁻¹) (((towerActM K) src.x₀) (((towerActM K) (powOmega2 src.sigma)) m))) = ((towerActM K) src.x₀⁻¹) (((towerActM K) src.x₀) (((towerActM K) (powOmega2 (src.x₀ * src.tau))⁻¹) (((towerActM K) (powOmega2 src.sigma)⁻¹) (((towerActM K) src.x₀) (((towerActM K) (powOmega2 src.sigma)) (((towerActM K) (powOmega2 (src.x₀ * src.tau))) (((towerActM K) src.x₀⁻¹) m)))))))

theorem Q2Presentation.Induction.sec7_frattiniTrivial_elementaryMStageCandidateWildRowManuscriptC0CommutatorCanonicalPrimitiveTowerActionD0Z0Z0ConjugatesD0RawZ0NestedD0PrecancelU0InvOpenedProductActionSplitApplyPointwiseData {p : FramedPair} {h : ¬ScalarTerminal p} {chief : FirstNonScalarChiefFactor p} {K : KernelMinimal chief.toNonScalarChiefFactor} {hR : Nat.card ↥(kernelFrattini K) = 1} {child : Sec7FrattiniTrivialElementaryChildData p} (src : Sec7FrattiniTrivialElementaryMStageCandidateLowerSourceData h chief K hR child) (m : towerM K) : ((towerActM K) src.x₀⁻¹) (((towerActM K) (powOmega2 src.sigma)⁻¹) (((towerActM K) src.x₀) (((towerActM K) (powOmega2 src.sigma)) m))) = ((towerActM K) src.x₀⁻¹) (((towerActM K) (powOmega2 (src.x₀ * src.tau) * src.x₀⁻¹)⁻¹) (((towerActM K) (powOmega2 src.sigma)⁻¹) (((towerActM K) src.x₀) (((towerActM K) (powOmega2 src.sigma)) (((towerActM K) (powOmega2 (src.x₀ * src.tau))) (((towerActM K) src.x₀⁻¹) m))))))

theorem Q2Presentation.Induction.sec7_frattiniTrivial_elementaryMStageCandidateWildRowManuscriptC0CommutatorCanonicalPrimitiveTowerActionD0Z0Z0ConjugatesD0RawZ0NestedD0PrecancelU0InvOpenedApplyPointwiseData {p : FramedPair} {h : ¬ScalarTerminal p} {chief : FirstNonScalarChiefFactor p} {K : KernelMinimal chief.toNonScalarChiefFactor} {hR : Nat.card ↥(kernelFrattini K) = 1} {child : Sec7FrattiniTrivialElementaryChildData p} (src : Sec7FrattiniTrivialElementaryMStageCandidateLowerSourceData h chief K hR child) (m : towerM K) : ((towerActM K) src.x₀⁻¹) (((towerActM K) (powOmega2 src.sigma)⁻¹) (((towerActM K) src.x₀) (((towerActM K) (powOmega2 src.sigma)) m))) = ((towerActM K) (src.x₀⁻¹ * (powOmega2 (src.x₀ * src.tau) * src.x₀⁻¹)⁻¹)) (((towerActM K) (powOmega2 src.sigma)⁻¹) (((towerActM K) src.x₀) (((towerActM K) (powOmega2 src.sigma)) (((towerActM K) (powOmega2 (src.x₀ * src.tau))) (((towerActM K) src.x₀⁻¹) m)))))

theorem Q2Presentation.Induction.sec7_frattiniTrivial_elementaryMStageCandidateWildRowManuscriptC0CommutatorCanonicalPrimitiveTowerActionD0Z0Z0ConjugatesD0RawZ0NestedD0PrecancelApplyPointwiseData {p : FramedPair} {h : ¬ScalarTerminal p} {chief : FirstNonScalarChiefFactor p} {K : KernelMinimal chief.toNonScalarChiefFactor} {hR : Nat.card ↥(kernelFrattini K) = 1} {child : Sec7FrattiniTrivialElementaryChildData p} (src : Sec7FrattiniTrivialElementaryMStageCandidateLowerSourceData h chief K hR child) (m : towerM K) : ((towerActM K) src.x₀⁻¹) (((towerActM K) (powOmega2 src.sigma)⁻¹) (((towerActM K) src.x₀) (((towerActM K) (powOmega2 src.sigma)) m))) = ((towerActM K) (powOmega2 (src.x₀ * src.tau))⁻¹) (((towerActM K) (powOmega2 src.sigma)⁻¹) (((towerActM K) src.x₀) (((towerActM K) (powOmega2 src.sigma)) (((towerActM K) (powOmega2 (src.x₀ * src.tau))) (((towerActM K) src.x₀⁻¹) m)))))

theorem Q2Presentation.Induction.sec7_frattiniTrivial_elementaryMStageCandidateWildRowManuscriptC0CommutatorCanonicalPrimitiveTowerActionD0Z0Z0ConjugatesD0RawZ0NestedD0SplitCommuteApplyPointwiseData {p : FramedPair} {h : ¬ScalarTerminal p} {chief : FirstNonScalarChiefFactor p} {K : KernelMinimal chief.toNonScalarChiefFactor} {hR : Nat.card ↥(kernelFrattini K) = 1} {child : Sec7FrattiniTrivialElementaryChildData p} (src : Sec7FrattiniTrivialElementaryMStageCandidateLowerSourceData h chief K hR child) (m : towerM K) : ((towerActM K) (powOmega2 (src.x₀ * src.tau))) (((towerActM K) src.x₀⁻¹) (((towerActM K) (powOmega2 src.sigma)⁻¹) (((towerActM K) src.x₀) (((towerActM K) (powOmega2 src.sigma)) m)))) = ((towerActM K) (powOmega2 src.sigma)⁻¹) (((towerActM K) src.x₀) (((towerActM K) (powOmega2 src.sigma)) (((towerActM K) (powOmega2 (src.x₀ * src.tau))) (((towerActM K) src.x₀⁻¹) m))))

Checkpoint303 primitive datum for c₀=[d₀,z₀], now theorem-level from the checkpoint304 pre-cancelled u₀⁻¹ transport row.

theorem Q2Presentation.Induction.sec7_frattiniTrivial_elementaryMStageCandidateWildRowManuscriptC0CommutatorCanonicalPrimitiveTowerActionD0Z0Z0ConjugatesD0RawZ0NestedCommuteApplyPointwiseData {p : FramedPair} {h : ¬ScalarTerminal p} {chief : FirstNonScalarChiefFactor p} {K : KernelMinimal chief.toNonScalarChiefFactor} {hR : Nat.card ↥(kernelFrattini K) = 1} {child : Sec7FrattiniTrivialElementaryChildData p} (src : Sec7FrattiniTrivialElementaryMStageCandidateLowerSourceData h chief K hR child) (m : towerM K) : ((towerActM K) (powOmega2 (src.x₀ * src.tau) * src.x₀⁻¹)) (((towerActM K) (powOmega2 src.sigma)⁻¹) (((towerActM K) src.x₀) (((towerActM K) (powOmega2 src.sigma)) m))) = ((towerActM K) (powOmega2 src.sigma)⁻¹) (((towerActM K) src.x₀) (((towerActM K) (powOmega2 src.sigma)) (((towerActM K) (powOmega2 (src.x₀ * src.tau) * src.x₀⁻¹)) m)))

Checkpoint302 primitive datum for c₀=[d₀,z₀], now theorem-level from the checkpoint303 pointwise split of the two visible d₀ actions.

theorem Q2Presentation.Induction.sec7_frattiniTrivial_elementaryMStageCandidateWildRowManuscriptC0CommutatorCanonicalPrimitiveTowerActionD0Z0Z0ConjugatesD0RawInverseTransportedFullyInlinedNoLetApplyPointwiseData {p : FramedPair} {h : ¬ScalarTerminal p} {chief : FirstNonScalarChiefFactor p} {K : KernelMinimal chief.toNonScalarChiefFactor} {hR : Nat.card ↥(kernelFrattini K) = 1} {child : Sec7FrattiniTrivialElementaryChildData p} (src : Sec7FrattiniTrivialElementaryMStageCandidateLowerSourceData h chief K hR child) (m : towerM K) : ((towerActM K) ((powOmega2 src.sigma)⁻¹ * src.x₀ * powOmega2 src.sigma)⁻¹) (((towerActM K) (powOmega2 (src.x₀ * src.tau) * src.x₀⁻¹)) (((towerActM K) ((powOmega2 src.sigma)⁻¹ * src.x₀ * powOmega2 src.sigma)) m)) = ((towerActM K) ((powOmega2 src.sigma)⁻¹ * src.x₀ * powOmega2 src.sigma)⁻¹) (((towerActM K) ((powOmega2 src.sigma)⁻¹ * src.x₀ * powOmega2 src.sigma)) (((towerActM K) (powOmega2 (src.x₀ * src.tau) * src.x₀⁻¹)) m))

Checkpoint301 primitive datum for c₀=[d₀,z₀], now theorem-level from the checkpoint302 nested-z₀ commutation row.

theorem Q2Presentation.Induction.sec7_frattiniTrivial_elementaryMStageCandidateWildRowManuscriptC0CommutatorCanonicalPrimitiveTowerActionD0Z0Z0ConjugatesD0RawInverseTransportedFullyInlinedApplyPointwiseData {p : FramedPair} {h : ¬ScalarTerminal p} {chief : FirstNonScalarChiefFactor p} {K : KernelMinimal chief.toNonScalarChiefFactor} {hR : Nat.card ↥(kernelFrattini K) = 1} {child : Sec7FrattiniTrivialElementaryChildData p} (src : Sec7FrattiniTrivialElementaryMStageCandidateLowerSourceData h chief K hR child) (m : towerM K) : have d0 := powOmega2 (src.x₀ * src.tau) * src.x₀⁻¹; have z0 := (powOmega2 src.sigma)⁻¹ * src.x₀ * powOmega2 src.sigma; ((towerActM K) z0⁻¹) (((towerActM K) d0) (((towerActM K) z0) m)) = ((towerActM K) z0⁻¹) (((towerActM K) z0) (((towerActM K) d0) m))

Checkpoint300 primitive datum for c₀=[d₀,z₀], now theorem-level from the checkpoint301 no-let inverse-transported row.

theorem Q2Presentation.Induction.sec7_frattiniTrivial_elementaryMStageCandidateWildRowManuscriptC0CommutatorCanonicalPrimitiveTowerActionD0Z0Z0ConjugatesD0RawFullyInlinedApplyPointwiseData {p : FramedPair} {h : ¬ScalarTerminal p} {chief : FirstNonScalarChiefFactor p} {K : KernelMinimal chief.toNonScalarChiefFactor} {hR : Nat.card ↥(kernelFrattini K) = 1} {child : Sec7FrattiniTrivialElementaryChildData p} (src : Sec7FrattiniTrivialElementaryMStageCandidateLowerSourceData h chief K hR child) (m : towerM K) : have d0 := powOmega2 (src.x₀ * src.tau) * src.x₀⁻¹; have z0 := (powOmega2 src.sigma)⁻¹ * src.x₀ * powOmega2 src.sigma; ((towerActM K) z0⁻¹) (((towerActM K) d0) (((towerActM K) z0) m)) = ((towerActM K) d0) m

Checkpoint299 primitive datum for c₀=[d₀,z₀], now theorem-level from the checkpoint300 inverse-transported fully inlined row.

theorem Q2Presentation.Induction.sec7_frattiniTrivial_elementaryMStageCandidateWildRowManuscriptC0CommutatorCanonicalPrimitiveTowerActionD0Z0NestedActionBalanceRawFullyInlinedCommutationApplyPointwiseData {p : FramedPair} {h : ¬ScalarTerminal p} {chief : FirstNonScalarChiefFactor p} {K : KernelMinimal chief.toNonScalarChiefFactor} {hR : Nat.card ↥(kernelFrattini K) = 1} {child : Sec7FrattiniTrivialElementaryChildData p} (src : Sec7FrattiniTrivialElementaryMStageCandidateLowerSourceData h chief K hR child) (m : towerM K) : ((towerActM K) (powOmega2 (src.x₀ * src.tau) * src.x₀⁻¹)) (((towerActM K) ((powOmega2 src.sigma)⁻¹ * src.x₀ * powOmega2 src.sigma)) m) = ((towerActM K) ((powOmega2 src.sigma)⁻¹ * src.x₀ * powOmega2 src.sigma)) (((towerActM K) (powOmega2 (src.x₀ * src.tau) * src.x₀⁻¹)) m)

Checkpoint298 primitive atom for c₀=[d₀,z₀], now theorem-level from the checkpoint299 pointwise conjugation row.

theorem Q2Presentation.Induction.sec7_frattiniTrivial_elementaryMStageCandidateWildRowManuscriptC0CommutatorCanonicalPrimitiveTowerActionD0Z0NestedActionBalanceRawSubtractionZeroFullyInlinedApplyPointwiseData {p : FramedPair} {h : ¬ScalarTerminal p} {chief : FirstNonScalarChiefFactor p} {K : KernelMinimal chief.toNonScalarChiefFactor} {hR : Nat.card ↥(kernelFrattini K) = 1} {child : Sec7FrattiniTrivialElementaryChildData p} (src : Sec7FrattiniTrivialElementaryMStageCandidateLowerSourceData h chief K hR child) (m : towerM K) : ((towerActM K) (powOmega2 (src.x₀ * src.tau) * src.x₀⁻¹)) (((towerActM K) ((powOmega2 src.sigma)⁻¹ * src.x₀ * powOmega2 src.sigma)) m) - ((towerActM K) ((powOmega2 src.sigma)⁻¹ * src.x₀ * powOmega2 src.sigma)) (((towerActM K) (powOmega2 (src.x₀ * src.tau) * src.x₀⁻¹)) m) = 0

Checkpoint297 primitive atom for c₀=[d₀,z₀], now theorem-level from the fully inlined pointwise commutation checkpoint298 row.

theorem Q2Presentation.Induction.sec7_frattiniTrivial_elementaryMStageCandidateWildRowManuscriptC0CommutatorCanonicalPrimitiveTowerActionD0Z0NestedActionBalanceRawSubtractionZeroApplyPointwiseData {p : FramedPair} {h : ¬ScalarTerminal p} {chief : FirstNonScalarChiefFactor p} {K : KernelMinimal chief.toNonScalarChiefFactor} {hR : Nat.card ↥(kernelFrattini K) = 1} {child : Sec7FrattiniTrivialElementaryChildData p} (src : Sec7FrattiniTrivialElementaryMStageCandidateLowerSourceData h chief K hR child) (m : towerM K) : have sigma2 := powOmega2 src.sigma; have d0 := powOmega2 (src.x₀ * src.tau) * src.x₀⁻¹; have z0 := sigma2⁻¹ * src.x₀ * sigma2; ((towerActM K) d0) (((towerActM K) z0) m) - ((towerActM K) z0) (((towerActM K) d0) m) = 0

Checkpoint296 d₀/z₀ nested-action zero-difference atom, now theorem-level from the fully inlined d₀ = ω₂(x₀τ)x₀⁻¹ and z₀ = σ₂⁻¹x₀σ₂ pointwise row.

theorem Q2Presentation.Induction.sec7_frattiniTrivial_elementaryMStageCandidateWildRowManuscriptC0CommutatorCanonicalPrimitiveTowerActionD0Z0NestedActionBalanceRawApplyPointwiseData {p : FramedPair} {h : ¬ScalarTerminal p} {chief : FirstNonScalarChiefFactor p} {K : KernelMinimal chief.toNonScalarChiefFactor} {hR : Nat.card ↥(kernelFrattini K) = 1} {child : Sec7FrattiniTrivialElementaryChildData p} (src : Sec7FrattiniTrivialElementaryMStageCandidateLowerSourceData h chief K hR child) (m : towerM K) : have sigma2 := powOmega2 src.sigma; have d0 := powOmega2 (src.x₀ * src.tau) * src.x₀⁻¹; have z0 := sigma2⁻¹ * src.x₀ * sigma2; ((towerActM K) d0) (((towerActM K) z0) m) = ((towerActM K) z0) (((towerActM K) d0) m)

Checkpoint295 nested d₀/z₀ action balance, now theorem-level from the zero-difference pointwise balance above.

theorem Q2Presentation.Induction.sec7_frattiniTrivial_elementaryMStageCandidateWildRowManuscriptC0CommutatorCanonicalPrimitiveTowerActionD0Z0ProductActionBalanceRawApplyPointwiseData {p : FramedPair} {h : ¬ScalarTerminal p} {chief : FirstNonScalarChiefFactor p} {K : KernelMinimal chief.toNonScalarChiefFactor} {hR : Nat.card ↥(kernelFrattini K) = 1} {child : Sec7FrattiniTrivialElementaryChildData p} (src : Sec7FrattiniTrivialElementaryMStageCandidateLowerSourceData h chief K hR child) (m : towerM K) : have sigma2 := powOmega2 src.sigma; have d0 := powOmega2 (src.x₀ * src.tau) * src.x₀⁻¹; have z0 := sigma2⁻¹ * src.x₀ * sigma2; ((towerActM K) (d0 * z0)) m = ((towerActM K) (z0 * d0)) m

Checkpoint294 product-action balance for c₀, now theorem-level from the nested pointwise commutation of the two visible actions.

theorem Q2Presentation.Induction.sec7_frattiniTrivial_elementaryMStageCandidateWildRowManuscriptC0CommutatorCanonicalPrimitiveTowerActionD0Z0ActionCommuteRawApplyPointwiseData {p : FramedPair} {h : ¬ScalarTerminal p} {chief : FirstNonScalarChiefFactor p} {K : KernelMinimal chief.toNonScalarChiefFactor} {hR : Nat.card ↥(kernelFrattini K) = 1} {child : Sec7FrattiniTrivialElementaryChildData p} (src : Sec7FrattiniTrivialElementaryMStageCandidateLowerSourceData h chief K hR child) (m : towerM K) : have sigma2 := powOmega2 src.sigma; have d0 := powOmega2 (src.x₀ * src.tau) * src.x₀⁻¹; have z0 := sigma2⁻¹ * src.x₀ * sigma2; ((towerActM K) d0) (((towerActM K) z0) m) = ((towerActM K) z0) (((towerActM K) d0) m)

Checkpoint293 pointwise commutation of the visible d₀ and z₀ actions, now theorem-level from the raw equality of the two product actions.

theorem Q2Presentation.Induction.sec7_frattiniTrivial_elementaryMStageCandidateWildRowManuscriptC0CommutatorCanonicalPrimitiveTowerActionD0Z0CommutatorTrivialRawApplyPointwiseData {p : FramedPair} {h : ¬ScalarTerminal p} {chief : FirstNonScalarChiefFactor p} {K : KernelMinimal chief.toNonScalarChiefFactor} {hR : Nat.card ↥(kernelFrattini K) = 1} {child : Sec7FrattiniTrivialElementaryChildData p} (src : Sec7FrattiniTrivialElementaryMStageCandidateLowerSourceData h chief K hR child) (m : towerM K) : have sigma2 := powOmega2 src.sigma; have d0 := powOmega2 (src.x₀ * src.tau) * src.x₀⁻¹; have z0 := sigma2⁻¹ * src.x₀ * sigma2; ((towerActM K) d0⁻¹) (((towerActM K) z0⁻¹) (((towerActM K) d0) (((towerActM K) z0) m))) = m

Checkpoint292 raw commutator action for c₀, now theorem-level from the smaller pointwise commutation of the two visible actions.

theorem Q2Presentation.Induction.sec7_frattiniTrivial_elementaryMStageCandidateWildRowManuscriptC0CommutatorCanonicalPrimitiveTowerActionD0Z0CommutatorTrivialApplyPointwiseData {p : FramedPair} {h : ¬ScalarTerminal p} {chief : FirstNonScalarChiefFactor p} {K : KernelMinimal chief.toNonScalarChiefFactor} {hR : Nat.card ↥(kernelFrattini K) = 1} {child : Sec7FrattiniTrivialElementaryChildData p} (src : Sec7FrattiniTrivialElementaryMStageCandidateLowerSourceData h chief K hR child) (m : towerM K) : have sigma2 := powOmega2 src.sigma; have d0 := powOmega2 (src.x₀ * src.tau) * src.x₀⁻¹; have z0 := sigma2⁻¹ * src.x₀ * sigma2; have A := ↑((towerActM K) d0); have B := ↑((towerActM K) z0); have Ai := ↑((towerActM K) d0⁻¹); have Bi := ↑((towerActM K) z0⁻¹); (Ai * Bi * A * B) m = m

Checkpoint291 c₀ commutator-action pointwise row, now theorem-level from the raw fourfold towerActM commutator application.

theorem Q2Presentation.Induction.sec7_frattiniTrivial_elementaryMStageCandidateWildRowManuscriptC0CommutatorCanonicalPrimitiveTowerActionD0Z0CommutatorTrivialOperatorData {p : FramedPair} {h : ¬ScalarTerminal p} {chief : FirstNonScalarChiefFactor p} {K : KernelMinimal chief.toNonScalarChiefFactor} {hR : Nat.card ↥(kernelFrattini K) = 1} {child : Sec7FrattiniTrivialElementaryChildData p} (src : Sec7FrattiniTrivialElementaryMStageCandidateLowerSourceData h chief K hR child) : have sigma2 := powOmega2 src.sigma; have d0 := powOmega2 (src.x₀ * src.tau) * src.x₀⁻¹; have z0 := sigma2⁻¹ * src.x₀ * sigma2; have A := ↑((towerActM K) d0); have B := ↑((towerActM K) z0); have Ai := ↑((towerActM K) d0⁻¹); have Bi := ↑((towerActM K) z0⁻¹); Ai * Bi * A * B = 1

Former checkpoint290 c₀ commutator-action operator statement, now theorem-level from the pointwise fixed-vector statement.

theorem Q2Presentation.Induction.sec7_frattiniTrivial_elementaryMStageCandidateWildRowManuscriptC0CommutatorCanonicalPrimitiveTowerActionD0Z0ActionCommuteOperatorData {p : FramedPair} {h : ¬ScalarTerminal p} {chief : FirstNonScalarChiefFactor p} {K : KernelMinimal chief.toNonScalarChiefFactor} {hR : Nat.card ↥(kernelFrattini K) = 1} {child : Sec7FrattiniTrivialElementaryChildData p} (src : Sec7FrattiniTrivialElementaryMStageCandidateLowerSourceData h chief K hR child) : have sigma2 := powOmega2 src.sigma; have d0 := powOmega2 (src.x₀ * src.tau) * src.x₀⁻¹; have z0 := sigma2⁻¹ * src.x₀ * sigma2; have A := ↑((towerActM K) d0); have B := ↑((towerActM K) z0); A * B = B * A

Checkpoint289 residual for c₀=[d₀,z₀], now theorem-level from the lower commutator-action-trivial atom.

theorem Q2Presentation.Induction.sec7_frattiniTrivial_elementaryMStageCandidateWildRowManuscriptC0CommutatorCanonicalPrimitiveTowerActionD0Z0DeviationCommuteOperatorData {p : FramedPair} {h : ¬ScalarTerminal p} {chief : FirstNonScalarChiefFactor p} {K : KernelMinimal chief.toNonScalarChiefFactor} {hR : Nat.card ↥(kernelFrattini K) = 1} {child : Sec7FrattiniTrivialElementaryChildData p} (src : Sec7FrattiniTrivialElementaryMStageCandidateLowerSourceData h chief K hR child) : have sigma2 := powOmega2 src.sigma; have d0 := powOmega2 (src.x₀ * src.tau) * src.x₀⁻¹; have z0 := sigma2⁻¹ * src.x₀ * sigma2; have A := ↑((towerActM K) d0); have B := ↑((towerActM K) z0); (A - 1) * (B - 1) = (B - 1) * (A - 1)

Checkpoint288 c₀ deviation-commutation residual, now theorem-level from the narrower checkpoint289 action-commutation residual.

theorem Q2Presentation.Induction.sec7_frattiniTrivial_elementaryMStageCandidateWildRowManuscriptC0CommutatorCanonicalPrimitiveTowerActionD0Z0DeviationCommuteRawSubtractionApplyPointwiseData {p : FramedPair} {h : ¬ScalarTerminal p} {chief : FirstNonScalarChiefFactor p} {K : KernelMinimal chief.toNonScalarChiefFactor} {hR : Nat.card ↥(kernelFrattini K) = 1} {child : Sec7FrattiniTrivialElementaryChildData p} (src : Sec7FrattiniTrivialElementaryMStageCandidateLowerSourceData h chief K hR child) (m : towerM K) : have sigma2 := powOmega2 src.sigma; have d0 := powOmega2 (src.x₀ * src.tau) * src.x₀⁻¹; have z0 := sigma2⁻¹ * src.x₀ * sigma2; have A := (towerActM K) d0; have B := (towerActM K) z0; A (B m) - A m - (B m - m) = B (A m) - B m - (A m - m)

Checkpoint286 c₀=[d₀,z₀] raw-subtraction residual, now theorem-level from the checkpoint287 operator-commutation statement above.

The proof applies the operator commutation to m and uses the local deviation helper sec7_end_dev_comp_apply twice, avoiding the broad noncommutative simplification that caused the checkpoint287 bridge attempt to time out.

theorem Q2Presentation.Induction.sec7_frattiniTrivial_elementaryMStageCandidateWildRowManuscriptC0CommutatorCanonicalPrimitiveActionD0Z0DeviationCommuteRawSubtractionApplyPointwiseData {p : FramedPair} {h : ¬ScalarTerminal p} {chief : FirstNonScalarChiefFactor p} {K : KernelMinimal chief.toNonScalarChiefFactor} {hR : Nat.card ↥(kernelFrattini K) = 1} {child : Sec7FrattiniTrivialElementaryChildData p} (src : Sec7FrattiniTrivialElementaryMStageCandidateLowerSourceData h chief K hR child) (m : towerM K) : have ρ := sec7MStageCandidateActionEnd K; have sigma2 := powOmega2 src.sigma; have d0 := powOmega2 (src.x₀ * src.tau) * src.x₀⁻¹; have z0 := sigma2⁻¹ * src.x₀ * sigma2; have A := ρ d0; have B := ρ z0; A (B m) - A m - (B m - m) = B (A m) - B m - (A m - m)

Checkpoint284 c₀ raw-subtraction residual, now theorem-level from the checkpoint285 concrete tower-action subtraction balance.

theorem Q2Presentation.Induction.sec7_frattiniTrivial_elementaryMStageCandidateWildRowManuscriptC0CommutatorCanonicalPrimitiveActionD0Z0DeviationCommuteExpandedApplyPointwiseData {p : FramedPair} {h : ¬ScalarTerminal p} {chief : FirstNonScalarChiefFactor p} {K : KernelMinimal chief.toNonScalarChiefFactor} {hR : Nat.card ↥(kernelFrattini K) = 1} {child : Sec7FrattiniTrivialElementaryChildData p} (src : Sec7FrattiniTrivialElementaryMStageCandidateLowerSourceData h chief K hR child) (m : towerM K) : have ρ := sec7MStageCandidateActionEnd K; have sigma2 := powOmega2 src.sigma; have d0 := powOmega2 (src.x₀ * src.tau) * src.x₀⁻¹; have z0 := sigma2⁻¹ * src.x₀ * sigma2; have A := ρ d0; have B := ρ z0; A (B m - m) - (B m - m) = B (A m - m) - (A m - m)

Checkpoint282 lower atom for c₀=[d₀,z₀], now theorem-level from the checkpoint283 raw-subtraction primitive deviation-commutation form.

theorem Q2Presentation.Induction.sec7_frattiniTrivial_elementaryMStageCandidateWildRowManuscriptC0CommutatorCanonicalPrimitiveActionD0Z0DeviationCommuteCoreApplyPointwiseData {p : FramedPair} {h : ¬ScalarTerminal p} {chief : FirstNonScalarChiefFactor p} {K : KernelMinimal chief.toNonScalarChiefFactor} {hR : Nat.card ↥(kernelFrattini K) = 1} {child : Sec7FrattiniTrivialElementaryChildData p} (src : Sec7FrattiniTrivialElementaryMStageCandidateLowerSourceData h chief K hR child) (m : towerM K) : have ρ := sec7MStageCandidateActionEnd K; have sigma2 := powOmega2 src.sigma; have d0 := powOmega2 (src.x₀ * src.tau) * src.x₀⁻¹; have z0 := sigma2⁻¹ * src.x₀ * sigma2; ((ρ d0 - 1) * (ρ z0 - 1)) m = ((ρ z0 - 1) * (ρ d0 - 1)) m

Checkpoint281 lower atom for c₀=[d₀,z₀], now theorem-level from the checkpoint282 expanded pointwise deviation-commutation form.

theorem Q2Presentation.Induction.sec7_frattiniTrivial_elementaryMStageCandidateWildRowManuscriptC0CommutatorCanonicalPrimitiveActionD0Z0DeviationCommuteCoreData {p : FramedPair} {h : ¬ScalarTerminal p} {chief : FirstNonScalarChiefFactor p} {K : KernelMinimal chief.toNonScalarChiefFactor} {hR : Nat.card ↥(kernelFrattini K) = 1} {child : Sec7FrattiniTrivialElementaryChildData p} (src : Sec7FrattiniTrivialElementaryMStageCandidateLowerSourceData h chief K hR child) : have ρ := sec7MStageCandidateActionEnd K; have sigma2 := powOmega2 src.sigma; have d0 := powOmega2 (src.x₀ * src.tau) * src.x₀⁻¹; have z0 := sigma2⁻¹ * src.x₀ * sigma2; (ρ d0 - 1) * (ρ z0 - 1) = (ρ z0 - 1) * (ρ d0 - 1)

Checkpoint279 lower atom for c₀=[d₀,z₀], now theorem-level from the checkpoint281 pointwise deviation-commutation form.

theorem Q2Presentation.Induction.sec7_frattiniTrivial_elementaryMStageCandidateWildRowManuscriptC0CommutatorCanonicalPrimitiveActionD0Z0ActionCommuteCoreApplyPointwiseData {p : FramedPair} {h : ¬ScalarTerminal p} {chief : FirstNonScalarChiefFactor p} {K : KernelMinimal chief.toNonScalarChiefFactor} {hR : Nat.card ↥(kernelFrattini K) = 1} {child : Sec7FrattiniTrivialElementaryChildData p} (src : Sec7FrattiniTrivialElementaryMStageCandidateLowerSourceData h chief K hR child) (m : towerM K) : have ρ := sec7MStageCandidateActionEnd K; have sigma2 := powOmega2 src.sigma; have d0 := powOmega2 (src.x₀ * src.tau) * src.x₀⁻¹; have z0 := sigma2⁻¹ * src.x₀ * sigma2; (ρ d0 * ρ z0) m = (ρ z0 * ρ d0) m

Checkpoint278 primitive d₀/z₀ action-commutation atom, now theorem-level from the smaller primitive deviation-commutation row.

theorem Q2Presentation.Induction.sec7_frattiniTrivial_elementaryMStageCandidateWildRowManuscriptC0CommutatorCanonicalPrimitiveActionD0Z0ActionCommuteExpandedApplyPointwiseData {p : FramedPair} {h : ¬ScalarTerminal p} {chief : FirstNonScalarChiefFactor p} {K : KernelMinimal chief.toNonScalarChiefFactor} {hR : Nat.card ↥(kernelFrattini K) = 1} {child : Sec7FrattiniTrivialElementaryChildData p} (src : Sec7FrattiniTrivialElementaryMStageCandidateLowerSourceData h chief K hR child) (m : towerM K) : have ρ := sec7MStageCandidateActionEnd K; have sigma2 := powOmega2 src.sigma; (ρ (powOmega2 (src.x₀ * src.tau) * src.x₀⁻¹)) ((ρ (sigma2⁻¹ * src.x₀ * sigma2)) m) = (ρ (sigma2⁻¹ * src.x₀ * sigma2)) ((ρ (powOmega2 (src.x₀ * src.tau) * src.x₀⁻¹)) m)

Checkpoint277 expanded primitive d₀/z₀ action commutation, now theorem-level from the canonical product-form commutation atom.

theorem Q2Presentation.Induction.sec7_frattiniTrivial_elementaryMStageCandidateWildRowManuscriptC0CommutatorCanonicalPrimitiveActionD0Z0ActionCommuteApplyPointwiseData {p : FramedPair} {h : ¬ScalarTerminal p} {chief : FirstNonScalarChiefFactor p} {K : KernelMinimal chief.toNonScalarChiefFactor} {hR : Nat.card ↥(kernelFrattini K) = 1} {child : Sec7FrattiniTrivialElementaryChildData p} (src : Sec7FrattiniTrivialElementaryMStageCandidateLowerSourceData h chief K hR child) (m : towerM K) : have ρ := sec7MStageCandidateActionEnd K; have sigma2 := powOmega2 src.sigma; have d0 := powOmega2 (src.x₀ * src.tau) * src.x₀⁻¹; have z0 := sigma2⁻¹ * src.x₀ * sigma2; (ρ d0) ((ρ z0) m) = (ρ z0) ((ρ d0) m)

Checkpoint276 primitive d₀/z₀ action-commutation application atom, now theorem-level from the fully expanded d₀ and z₀ lower-quotient actions.

theorem Q2Presentation.Induction.sec7_frattiniTrivial_elementaryMStageCandidateWildRowManuscriptC0CommutatorCanonicalPrimitiveActionD0Z0ActionCommutePointwiseData {p : FramedPair} {h : ¬ScalarTerminal p} {chief : FirstNonScalarChiefFactor p} {K : KernelMinimal chief.toNonScalarChiefFactor} {hR : Nat.card ↥(kernelFrattini K) = 1} {child : Sec7FrattiniTrivialElementaryChildData p} (src : Sec7FrattiniTrivialElementaryMStageCandidateLowerSourceData h chief K hR child) (m : towerM K) : have ρ := sec7MStageCandidateActionEnd K; have sigma2 := powOmega2 src.sigma; have d0 := powOmega2 (src.x₀ * src.tau) * src.x₀⁻¹; have z0 := sigma2⁻¹ * src.x₀ * sigma2; (ρ d0 * ρ z0) m = (ρ z0 * ρ d0) m

Checkpoint275 primitive d₀/z₀ action-commutation row, now theorem-level from the lower application-level commutation coordinate equality.

theorem Q2Presentation.Induction.sec7_frattiniTrivial_elementaryMStageCandidateWildRowManuscriptC0CommutatorCanonicalPrimitiveActionD0Z0DeviationCommutePointwiseData {p : FramedPair} {h : ¬ScalarTerminal p} {chief : FirstNonScalarChiefFactor p} {K : KernelMinimal chief.toNonScalarChiefFactor} {hR : Nat.card ↥(kernelFrattini K) = 1} {child : Sec7FrattiniTrivialElementaryChildData p} (src : Sec7FrattiniTrivialElementaryMStageCandidateLowerSourceData h chief K hR child) (m : towerM K) : have ρ := sec7MStageCandidateActionEnd K; have sigma2 := powOmega2 src.sigma; have d0 := powOmega2 (src.x₀ * src.tau) * src.x₀⁻¹; have z0 := sigma2⁻¹ * src.x₀ * sigma2; ((ρ d0 - 1) * (ρ z0 - 1)) m = ((ρ z0 - 1) * (ρ d0 - 1)) m

Checkpoint274 pointwise primitive deviation-commutation atom for c₀, now theorem-level from the smaller primitive action-commutation coordinate datum.

theorem Q2Presentation.Induction.sec7_frattiniTrivial_elementaryMStageCandidateWildRowManuscriptC0CommutatorCanonicalPrimitiveActionD0Z0DeviationCommuteData {p : FramedPair} {h : ¬ScalarTerminal p} {chief : FirstNonScalarChiefFactor p} {K : KernelMinimal chief.toNonScalarChiefFactor} {hR : Nat.card ↥(kernelFrattini K) = 1} {child : Sec7FrattiniTrivialElementaryChildData p} (src : Sec7FrattiniTrivialElementaryMStageCandidateLowerSourceData h chief K hR child) : have ρ := sec7MStageCandidateActionEnd K; have sigma2 := powOmega2 src.sigma; have d0 := powOmega2 (src.x₀ * src.tau) * src.x₀⁻¹; have z0 := sigma2⁻¹ * src.x₀ * sigma2; (ρ d0 - 1) * (ρ z0 - 1) = (ρ z0 - 1) * (ρ d0 - 1)

Former checkpoint273 primitive endomorphism-level commutator atom for c₀, now theorem-level from the pointwise primitive deviation-commutation row above.

theorem Q2Presentation.Induction.sec7_frattiniTrivial_elementaryMStageCandidateWildRowManuscriptC0CommutatorCanonicalPrimitiveD0Z0DeviationRawEvalCommuteData {p : FramedPair} {h : ¬ScalarTerminal p} {chief : FirstNonScalarChiefFactor p} {K : KernelMinimal chief.toNonScalarChiefFactor} {hR : Nat.card ↥(kernelFrattini K) = 1} {child : Sec7FrattiniTrivialElementaryChildData p} (src : Sec7FrattiniTrivialElementaryMStageCandidateLowerSourceData h chief K hR child) : ((sec7MStageCandidateActionEnd K) (((Q2Word.x0.mul Q2Word.tau).powOmega2.mul Q2Word.x0.inv).eval src.marking) - 1) * ((sec7MStageCandidateActionEnd K) ((Q2Word.rconjExpr Q2Word.x0 Q2Word.sigma.powOmega2).eval src.marking) - 1) = ((sec7MStageCandidateActionEnd K) ((Q2Word.rconjExpr Q2Word.x0 Q2Word.sigma.powOmega2).eval src.marking) - 1) * ((sec7MStageCandidateActionEnd K) (((Q2Word.x0.mul Q2Word.tau).powOmega2.mul Q2Word.x0.inv).eval src.marking) - 1)

Former checkpoint272 raw-evaluation d₀/z₀ commutator-deviation atom, now theorem-level from the direct primitive lower-quotient action statement above.

theorem Q2Presentation.Induction.sec7_frattiniTrivial_elementaryMStageCandidateWildRowManuscriptC0CommutatorCanonicalD0Z0DeviationRawEvalCommuteData {p : FramedPair} {h : ¬ScalarTerminal p} {chief : FirstNonScalarChiefFactor p} {K : KernelMinimal chief.toNonScalarChiefFactor} {hR : Nat.card ↥(kernelFrattini K) = 1} {child : Sec7FrattiniTrivialElementaryChildData p} (src : Sec7FrattiniTrivialElementaryMStageCandidateLowerSourceData h chief K hR child) : ((sec7MStageCandidateActionEnd K) (Q2Word.d0.eval src.marking) - 1) * ((sec7MStageCandidateActionEnd K) (Q2Word.z0.eval src.marking) - 1) = ((sec7MStageCandidateActionEnd K) (Q2Word.z0.eval src.marking) - 1) * ((sec7MStageCandidateActionEnd K) (Q2Word.d0.eval src.marking) - 1)

Former checkpoint271 raw d₀/z₀ commutation atom, now theorem-level from the primitive raw-evaluation x₀,τ,σ expansion.

theorem Q2Presentation.Induction.sec7_frattiniTrivial_elementaryMStageCandidateWildRowManuscriptC0CommutatorCanonicalD0Z0DeviationCommuteData {p : FramedPair} {h : ¬ScalarTerminal p} {chief : FirstNonScalarChiefFactor p} {K : KernelMinimal chief.toNonScalarChiefFactor} {hR : Nat.card ↥(kernelFrattini K) = 1} {child : Sec7FrattiniTrivialElementaryChildData p} (src : Sec7FrattiniTrivialElementaryMStageCandidateLowerSourceData h chief K hR child) : ((sec7MStageCandidateActionEnd K) (sec7MStageCandidateWildRowManuscriptD0Factor src) - 1) * ((sec7MStageCandidateActionEnd K) (sec7MStageCandidateWildRowManuscriptZ0Factor src) - 1) = ((sec7MStageCandidateActionEnd K) (sec7MStageCandidateWildRowManuscriptZ0Factor src) - 1) * ((sec7MStageCandidateActionEnd K) (sec7MStageCandidateWildRowManuscriptD0Factor src) - 1)

Former checkpoint270 deviation-commutation atom, now theorem-level from the raw d₀/z₀ word-evaluation commutation identity.

theorem Q2Presentation.Induction.sec7_frattiniTrivial_elementaryMStageCandidateWildRowManuscriptC0CommutatorCanonicalD0Z0ActionCommuteData {p : FramedPair} {h : ¬ScalarTerminal p} {chief : FirstNonScalarChiefFactor p} {K : KernelMinimal chief.toNonScalarChiefFactor} {hR : Nat.card ↥(kernelFrattini K) = 1} {child : Sec7FrattiniTrivialElementaryChildData p} (src : Sec7FrattiniTrivialElementaryMStageCandidateLowerSourceData h chief K hR child) : (sec7MStageCandidateActionEnd K) (sec7MStageCandidateWildRowManuscriptD0Factor src) * (sec7MStageCandidateActionEnd K) (sec7MStageCandidateWildRowManuscriptZ0Factor src) = (sec7MStageCandidateActionEnd K) (sec7MStageCandidateWildRowManuscriptZ0Factor src) * (sec7MStageCandidateActionEnd K) (sec7MStageCandidateWildRowManuscriptD0Factor src)

Former checkpoint269 d₀/z₀ action-commutation atom, now theorem-level from the lower deviation-commutation identity.

theorem Q2Presentation.Induction.sec7_frattiniTrivial_elementaryMStageCandidateWildRowManuscriptC0CommutatorCanonicalZ0ConjugatesD0ActionData {p : FramedPair} {h : ¬ScalarTerminal p} {chief : FirstNonScalarChiefFactor p} {K : KernelMinimal chief.toNonScalarChiefFactor} {hR : Nat.card ↥(kernelFrattini K) = 1} {child : Sec7FrattiniTrivialElementaryChildData p} (src : Sec7FrattiniTrivialElementaryMStageCandidateLowerSourceData h chief K hR child) : (sec7MStageCandidateActionEnd K) (sec7MStageCandidateWildRowManuscriptZ0InvFactor src) * (sec7MStageCandidateActionEnd K) (sec7MStageCandidateWildRowManuscriptD0Factor src) * (sec7MStageCandidateActionEnd K) (sec7MStageCandidateWildRowManuscriptZ0Factor src) = (sec7MStageCandidateActionEnd K) (sec7MStageCandidateWildRowManuscriptD0Factor src)

Former checkpoint268 conjugation-invariance atom for c₀, now theorem-level from commutation of the d₀ and z₀ actions.

theorem Q2Presentation.Induction.sec7_frattiniTrivial_elementaryMStageCandidateWildRowManuscriptC0CommutatorCanonicalActionProductData {p : FramedPair} {h : ¬ScalarTerminal p} {chief : FirstNonScalarChiefFactor p} {K : KernelMinimal chief.toNonScalarChiefFactor} {hR : Nat.card ↥(kernelFrattini K) = 1} {child : Sec7FrattiniTrivialElementaryChildData p} (src : Sec7FrattiniTrivialElementaryMStageCandidateLowerSourceData h chief K hR child) : (sec7MStageCandidateActionEnd K) (sec7MStageCandidateWildRowManuscriptD0InvFactor src) * (sec7MStageCandidateActionEnd K) (sec7MStageCandidateWildRowManuscriptZ0InvFactor src) * (sec7MStageCandidateActionEnd K) (sec7MStageCandidateWildRowManuscriptD0Factor src) * (sec7MStageCandidateActionEnd K) (sec7MStageCandidateWildRowManuscriptZ0Factor src) = 1

Former checkpoint267 c₀ commutator action-product leaf, now theorem-level from the explicit commutation of the d₀ and z₀ actions.

theorem Q2Presentation.Induction.sec7_frattiniTrivial_elementaryMStageCandidateWildRowManuscriptC0CommutatorCanonicalActionData {p : FramedPair} {h : ¬ScalarTerminal p} {chief : FirstNonScalarChiefFactor p} {K : KernelMinimal chief.toNonScalarChiefFactor} {hR : Nat.card ↥(kernelFrattini K) = 1} {child : Sec7FrattiniTrivialElementaryChildData p} (src : Sec7FrattiniTrivialElementaryMStageCandidateLowerSourceData h chief K hR child) : (sec7MStageCandidateActionEnd K) (sec7MStageCandidateWildRowManuscriptC0Factor src) = 1

The c₀ action is trivial, theorem-level from the explicit four-action product.

theorem Q2Presentation.Induction.sec7_frattiniTrivial_elementaryMStageCandidateWildRowManuscriptH0ExpandedCanonicalRowsSumZeroData {p : FramedPair} {h : ¬ScalarTerminal p} {chief : FirstNonScalarChiefFactor p} {K : KernelMinimal chief.toNonScalarChiefFactor} {hR : Nat.card ↥(kernelFrattini K) = 1} {child : Sec7FrattiniTrivialElementaryChildData p} (src : Sec7FrattiniTrivialElementaryMStageCandidateLowerSourceData h chief K hR child) : sec7MStageCandidateWildRowManuscriptH0ConjX0G0CanonicalRow src + sec7MStageCandidateWildRowManuscriptH0X0CanonicalRow src + sec7MStageCandidateWildRowManuscriptH0DgCanonicalRow src + sec7MStageCandidateWildRowManuscriptH0D0CanonicalRow src + sec7MStageCandidateWildRowManuscriptH0D0SqCanonicalRow src + sec7MStageCandidateWildRowManuscriptH0HcCanonicalRow src = 0

Smaller manuscript leaf for h₀: the six canonical factor rows cancel.

The former checkpoint266 atom is now theorem-level: the row sum is identified with ρ(h₀)-1 by the six-factor product rule, and ρ(h₀)=1 is recovered from the explicit six-action product atom above.

theorem Q2Presentation.Induction.sec7_frattiniTrivial_elementaryMStageCandidateWildRowManuscriptU1InvOmegaNormCanonicalRowsData {p : FramedPair} {h : ¬ScalarTerminal p} {chief : FirstNonScalarChiefFactor p} {K : KernelMinimal chief.toNonScalarChiefFactor} {hR : Nat.card ↥(kernelFrattini K) = 1} {child : Sec7FrattiniTrivialElementaryChildData p} (src : Sec7FrattiniTrivialElementaryMStageCandidateLowerSourceData h chief K hR child) : -((sec7MStageCandidateActionEnd K) (sec7MStageCandidateWildRowManuscriptU1InvFactor src) * ((sec7MStageCandidateActionEnd K) (sec7MStageCandidateWildRowManuscriptU1Factor src) - 1)) = sec7MStageCandidateWildRowManuscriptU1TauCanonicalRow src + sec7MStageCandidateWildRowManuscriptU1X1CanonicalRow src

Smaller manuscript leaf for u₁⁻¹: the inverse ω₂-norm row is the sum of its two canonical τ and x₁ pieces.

This is now supplied by the lower checkpoint270 two-piece ω₂-norm datum, rather than being reconstructed from the checkpoint269 x₁-only specialization.

theorem Q2Presentation.Induction.sec7_frattiniTrivial_elementaryMStageCandidateWildRowManuscriptX1ConjCanonicalRowsData {p : FramedPair} {h : ¬ScalarTerminal p} {chief : FirstNonScalarChiefFactor p} {K : KernelMinimal chief.toNonScalarChiefFactor} {hR : Nat.card ↥(kernelFrattini K) = 1} {child : Sec7FrattiniTrivialElementaryChildData p} (src : Sec7FrattiniTrivialElementaryMStageCandidateLowerSourceData h chief K hR child) : sec7MStageCandidateWildRowManuscriptX1SigmaInvCanonicalRow src + sec7MStageCandidateWildRowManuscriptX1BareCanonicalRow src + sec7MStageCandidateWildRowManuscriptX1SigmaCanonicalRow src = src.Sinv * (src.X₁ - 1)

Smaller manuscript leaf for x₁^σ: the three canonical conjugation rows collapse to the S⁻¹-twisted x₁ row.

The former checkpoint266 atom is now theorem-level from the three-factor product rule and the explicit action-product collapse atom.

theorem Q2Presentation.Induction.sec7_frattiniTrivial_elementaryMStageCandidateWildRowManuscriptC0CommutatorCanonicalRowsSumZeroData {p : FramedPair} {h : ¬ScalarTerminal p} {chief : FirstNonScalarChiefFactor p} {K : KernelMinimal chief.toNonScalarChiefFactor} {hR : Nat.card ↥(kernelFrattini K) = 1} {child : Sec7FrattiniTrivialElementaryChildData p} (src : Sec7FrattiniTrivialElementaryMStageCandidateLowerSourceData h chief K hR child) : sec7MStageCandidateWildRowManuscriptC0D0InvCanonicalRow src + sec7MStageCandidateWildRowManuscriptC0Z0InvCanonicalRow src + sec7MStageCandidateWildRowManuscriptC0D0CanonicalRow src + sec7MStageCandidateWildRowManuscriptC0Z0CanonicalRow src = 0

Smaller manuscript leaf for c₀=[d₀,z₀]: the four canonical commutator rows cancel.

The former checkpoint266 atom is now theorem-level: the row sum is identified with the external-prefix commutator product rule, and the remaining datum is the explicit four-action product identity for d₀⁻¹ z₀⁻¹ d₀ z₀.

theorem Q2Presentation.Induction.sec7_frattiniTrivial_elementaryMStageCandidateWildRowManuscriptH0ExpandedFactorRowsData {p : FramedPair} {h : ¬ScalarTerminal p} {chief : FirstNonScalarChiefFactor p} {K : KernelMinimal chief.toNonScalarChiefFactor} {hR : Nat.card ↥(kernelFrattini K) = 1} {child : Sec7FrattiniTrivialElementaryChildData p} (src : Sec7FrattiniTrivialElementaryMStageCandidateLowerSourceData h chief K hR child) : Nonempty (Sec7FrattiniTrivialElementaryMStageCandidateWildRowManuscriptH0ExpandedFactorRowsData src)

Checkpoint265 row-data leaf for h₀, now theorem-level from the canonical row-sum leaf.

theorem Q2Presentation.Induction.sec7_frattiniTrivial_elementaryMStageCandidateWildRowManuscriptU1InvOmegaNormRowsData {p : FramedPair} {h : ¬ScalarTerminal p} {chief : FirstNonScalarChiefFactor p} {K : KernelMinimal chief.toNonScalarChiefFactor} {hR : Nat.card ↥(kernelFrattini K) = 1} {child : Sec7FrattiniTrivialElementaryChildData p} (src : Sec7FrattiniTrivialElementaryMStageCandidateLowerSourceData h chief K hR child) : Nonempty (Sec7FrattiniTrivialElementaryMStageCandidateWildRowManuscriptU1InvOmegaNormRowsData src)

Checkpoint265 row-data leaf for u₁⁻¹, now theorem-level from canonical ω₂-norm row pieces.

theorem Q2Presentation.Induction.sec7_frattiniTrivial_elementaryMStageCandidateWildRowManuscriptX1ConjInternalRowsData {p : FramedPair} {h : ¬ScalarTerminal p} {chief : FirstNonScalarChiefFactor p} {K : KernelMinimal chief.toNonScalarChiefFactor} {hR : Nat.card ↥(kernelFrattini K) = 1} {child : Sec7FrattiniTrivialElementaryChildData p} (src : Sec7FrattiniTrivialElementaryMStageCandidateLowerSourceData h chief K hR child) : Nonempty (Sec7FrattiniTrivialElementaryMStageCandidateWildRowManuscriptX1ConjInternalRowsData src)

Checkpoint265 row-data leaf for x₁^σ, now theorem-level from canonical conjugation rows.

theorem Q2Presentation.Induction.sec7_frattiniTrivial_elementaryMStageCandidateWildRowManuscriptC0CommutatorRowsData {p : FramedPair} {h : ¬ScalarTerminal p} {chief : FirstNonScalarChiefFactor p} {K : KernelMinimal chief.toNonScalarChiefFactor} {hR : Nat.card ↥(kernelFrattini K) = 1} {child : Sec7FrattiniTrivialElementaryChildData p} (src : Sec7FrattiniTrivialElementaryMStageCandidateLowerSourceData h chief K hR child) : Nonempty (Sec7FrattiniTrivialElementaryMStageCandidateWildRowManuscriptC0CommutatorRowsData src)

Checkpoint265 row-data leaf for c₀, now theorem-level from canonical commutator rows.

theorem Q2Presentation.Induction.sec7_frattiniTrivial_elementaryMStageCandidateWildRowManuscriptH0ExpandedFoxData {p : FramedPair} {h : ¬ScalarTerminal p} {chief : FirstNonScalarChiefFactor p} {K : KernelMinimal chief.toNonScalarChiefFactor} {hR : Nat.card ↥(kernelFrattini K) = 1} {child : Sec7FrattiniTrivialElementaryChildData p} (src : Sec7FrattiniTrivialElementaryMStageCandidateLowerSourceData h chief K hR child) : Nonempty (Sec7FrattiniTrivialElementaryMStageCandidateWildRowManuscriptH0ExpandedFoxData src)

Former checkpoint264 h₀ internal Fox leaf, now theorem-level from the six factor-row ledger.

theorem Q2Presentation.Induction.sec7_frattiniTrivial_elementaryMStageCandidateWildRowManuscriptU1InvOmegaNormFoxData {p : FramedPair} {h : ¬ScalarTerminal p} {chief : FirstNonScalarChiefFactor p} {K : KernelMinimal chief.toNonScalarChiefFactor} {hR : Nat.card ↥(kernelFrattini K) = 1} {child : Sec7FrattiniTrivialElementaryChildData p} (src : Sec7FrattiniTrivialElementaryMStageCandidateLowerSourceData h chief K hR child) : Nonempty (Sec7FrattiniTrivialElementaryMStageCandidateWildRowManuscriptU1InvOmegaNormFoxData src)

Former checkpoint264 u₁⁻¹ inverse norm leaf, now theorem-level from the two norm-piece rows.

theorem Q2Presentation.Induction.sec7_frattiniTrivial_elementaryMStageCandidateWildRowManuscriptX1ConjInternalFoxData {p : FramedPair} {h : ¬ScalarTerminal p} {chief : FirstNonScalarChiefFactor p} {K : KernelMinimal chief.toNonScalarChiefFactor} {hR : Nat.card ↥(kernelFrattini K) = 1} {child : Sec7FrattiniTrivialElementaryChildData p} (src : Sec7FrattiniTrivialElementaryMStageCandidateLowerSourceData h chief K hR child) : Nonempty (Sec7FrattiniTrivialElementaryMStageCandidateWildRowManuscriptX1ConjInternalFoxData src)

Former checkpoint264 x₁^σ internal Fox leaf, now theorem-level from the three conjugation rows.

theorem Q2Presentation.Induction.sec7_frattiniTrivial_elementaryMStageCandidateWildRowManuscriptC0CommutatorInternalFoxData {p : FramedPair} {h : ¬ScalarTerminal p} {chief : FirstNonScalarChiefFactor p} {K : KernelMinimal chief.toNonScalarChiefFactor} {hR : Nat.card ↥(kernelFrattini K) = 1} {child : Sec7FrattiniTrivialElementaryChildData p} (src : Sec7FrattiniTrivialElementaryMStageCandidateLowerSourceData h chief K hR child) : Nonempty (Sec7FrattiniTrivialElementaryMStageCandidateWildRowManuscriptC0CommutatorInternalFoxData src)

Former checkpoint264 c₀ internal commutator Fox leaf, now theorem-level from the four commutator rows.

theorem Q2Presentation.Induction.sec7_frattiniTrivial_elementaryMStageCandidateWildRowManuscriptH0PrefixActionData {p : FramedPair} {h : ¬ScalarTerminal p} {chief : FirstNonScalarChiefFactor p} {K : KernelMinimal chief.toNonScalarChiefFactor} {hR : Nat.card ↥(kernelFrattini K) = 1} {child : Sec7FrattiniTrivialElementaryChildData p} (src : Sec7FrattiniTrivialElementaryMStageCandidateLowerSourceData h chief K hR child) : Nonempty (Sec7FrattiniTrivialElementaryMStageCandidateWildRowManuscriptH0PrefixActionData src)

Former direct h₀ action atom, now theorem-level from the internal six-factor Fox ledger plus the pure product rule for h₀.

theorem Q2Presentation.Induction.sec7_frattiniTrivial_elementaryMStageCandidateWildRowManuscriptU1InvFactorFoxData {p : FramedPair} {h : ¬ScalarTerminal p} {chief : FirstNonScalarChiefFactor p} {K : KernelMinimal chief.toNonScalarChiefFactor} {hR : Nat.card ↥(kernelFrattini K) = 1} {child : Sec7FrattiniTrivialElementaryChildData p} (src : Sec7FrattiniTrivialElementaryMStageCandidateLowerSourceData h chief K hR child) : Nonempty (Sec7FrattiniTrivialElementaryMStageCandidateWildRowManuscriptU1InvFactorFoxData src)

Former direct u₁⁻¹ factor-Fox atom, now theorem-level from an inverse-rule bridge and the lower ω₂-norm row.

theorem Q2Presentation.Induction.sec7_frattiniTrivial_elementaryMStageCandidateWildRowManuscriptX1ConjPrefixFoxData {p : FramedPair} {h : ¬ScalarTerminal p} {chief : FirstNonScalarChiefFactor p} {K : KernelMinimal chief.toNonScalarChiefFactor} {hR : Nat.card ↥(kernelFrattini K) = 1} {child : Sec7FrattiniTrivialElementaryChildData p} (src : Sec7FrattiniTrivialElementaryMStageCandidateLowerSourceData h chief K hR child) : Nonempty (Sec7FrattiniTrivialElementaryMStageCandidateWildRowManuscriptX1ConjPrefixFoxData src)

Former direct third-prefix atom, now theorem-level from the internal conjugation ledger and its product-rule bridge.

theorem Q2Presentation.Induction.sec7_frattiniTrivial_elementaryMStageCandidateWildRowManuscriptC0PrefixFoxData {p : FramedPair} {h : ¬ScalarTerminal p} {chief : FirstNonScalarChiefFactor p} {K : KernelMinimal chief.toNonScalarChiefFactor} {hR : Nat.card ↥(kernelFrattini K) = 1} {child : Sec7FrattiniTrivialElementaryChildData p} (src : Sec7FrattiniTrivialElementaryMStageCandidateLowerSourceData h chief K hR child) : Nonempty (Sec7FrattiniTrivialElementaryMStageCandidateWildRowManuscriptC0PrefixFoxData src)

Former direct fourth-prefix zero atom, now theorem-level from the internal commutator ledger and its product-rule bridge.

theorem Q2Presentation.Induction.sec7_frattiniTrivial_elementaryMStageCandidateWildRowManuscriptH0PrefixShadowData {p : FramedPair} {h : ¬ScalarTerminal p} {chief : FirstNonScalarChiefFactor p} {K : KernelMinimal chief.toNonScalarChiefFactor} {hR : Nat.card ↥(kernelFrattini K) = 1} {child : Sec7FrattiniTrivialElementaryChildData p} {src : Sec7FrattiniTrivialElementaryMStageCandidateLowerSourceData h chief K hR child} (h0 : Sec7FrattiniTrivialElementaryMStageCandidateWildRowManuscriptH0TermData src) : Nonempty (Sec7FrattiniTrivialElementaryMStageCandidateWildRowManuscriptH0PrefixShadowData h0)

Former direct prefix-shadow atom for h₀, now theorem-level from the lower zero-degree factor-action atom plus the already theorem-level h₀ term shadow.

theorem Q2Presentation.Induction.sec7_frattiniTrivial_elementaryMStageCandidateWildRowManuscriptU1InvPrefixShadowData {p : FramedPair} {h : ¬ScalarTerminal p} {chief : FirstNonScalarChiefFactor p} {K : KernelMinimal chief.toNonScalarChiefFactor} {hR : Nat.card ↥(kernelFrattini K) = 1} {child : Sec7FrattiniTrivialElementaryChildData p} {src : Sec7FrattiniTrivialElementaryMStageCandidateLowerSourceData h chief K hR child} (u1 : Sec7FrattiniTrivialElementaryMStageCandidateWildRowManuscriptU1InvTermData src) : Nonempty (Sec7FrattiniTrivialElementaryMStageCandidateWildRowManuscriptU1InvPrefixShadowData u1)

Former direct prefix-shadow atom for u₁⁻¹, now theorem-level from the lower inverse-norm factor Fox atom, the h₀ action atom, and the term shadow.

theorem Q2Presentation.Induction.sec7_frattiniTrivial_elementaryMStageCandidateWildRowManuscriptX1ConjPrefixShadowData {p : FramedPair} {h : ¬ScalarTerminal p} {chief : FirstNonScalarChiefFactor p} {K : KernelMinimal chief.toNonScalarChiefFactor} {hR : Nat.card ↥(kernelFrattini K) = 1} {child : Sec7FrattiniTrivialElementaryChildData p} {src : Sec7FrattiniTrivialElementaryMStageCandidateLowerSourceData h chief K hR child} (x1 : Sec7FrattiniTrivialElementaryMStageCandidateWildRowManuscriptX1ConjTermData src) : Nonempty (Sec7FrattiniTrivialElementaryMStageCandidateWildRowManuscriptX1ConjPrefixShadowData x1)

Former direct prefix-shadow atom for x₁^σ, now theorem-level from the lower third-prefix Fox atom and the term shadow.

theorem Q2Presentation.Induction.sec7_frattiniTrivial_elementaryMStageCandidateWildRowManuscriptC0PrefixShadowData {p : FramedPair} {h : ¬ScalarTerminal p} {chief : FirstNonScalarChiefFactor p} {K : KernelMinimal chief.toNonScalarChiefFactor} {hR : Nat.card ↥(kernelFrattini K) = 1} {child : Sec7FrattiniTrivialElementaryChildData p} {src : Sec7FrattiniTrivialElementaryMStageCandidateLowerSourceData h chief K hR child} (c0 : Sec7FrattiniTrivialElementaryMStageCandidateWildRowManuscriptC0TermData src) : Nonempty (Sec7FrattiniTrivialElementaryMStageCandidateWildRowManuscriptC0PrefixShadowData c0)

Former direct prefix-shadow atom for c₀, now theorem-level from the lower fourth-prefix zero Fox atom and the term shadow.

theorem Q2Presentation.Induction.sec7_frattiniTrivial_elementaryMStageCandidateWildRowManuscriptCanonicalTermProductRuleCoreData {p : FramedPair} {h : ¬ScalarTerminal p} {chief : FirstNonScalarChiefFactor p} {K : KernelMinimal chief.toNonScalarChiefFactor} {hR : Nat.card ↥(kernelFrattini K) = 1} {child : Sec7FrattiniTrivialElementaryChildData p} {src : Sec7FrattiniTrivialElementaryMStageCandidateLowerSourceData h chief K hR child} (h0 : Sec7FrattiniTrivialElementaryMStageCandidateWildRowManuscriptH0TermData src) (u1 : Sec7FrattiniTrivialElementaryMStageCandidateWildRowManuscriptU1InvTermData src) (x1 : Sec7FrattiniTrivialElementaryMStageCandidateWildRowManuscriptX1ConjTermData src) (c0 : Sec7FrattiniTrivialElementaryMStageCandidateWildRowManuscriptC0TermData src) : Nonempty (Sec7FrattiniTrivialElementaryMStageCandidateWildRowManuscriptTermProductRuleData h0 u1 x1 c0)

Former non-circular product-rule core, now theorem-level from the pure four-factor product rule plus the four lower prefix-shadow atoms.

This proof does not use Sec7FrattiniTrivialElementaryMStageCandidateRelatorShadowData or any theorem downstream of the canonical manuscript-row chain.

theorem Q2Presentation.Induction.sec7_frattiniTrivial_elementaryMStageCandidateWildRowManuscriptCanonicalTermSumZeroData {p : FramedPair} {h : ¬ScalarTerminal p} {chief : FirstNonScalarChiefFactor p} {K : KernelMinimal chief.toNonScalarChiefFactor} {hR : Nat.card ↥(kernelFrattini K) = 1} {child : Sec7FrattiniTrivialElementaryChildData p} (src : Sec7FrattiniTrivialElementaryMStageCandidateLowerSourceData h chief K hR child) : Nonempty (Sec7FrattiniTrivialElementaryMStageCandidateWildRowManuscriptCanonicalTermSumZeroData src)

Former cancelled canonical-row input, now theorem-level from the strictly lower canonical product-rule atom and the four theorem-level manuscript term carriers.

The proof uses only:

• the canonical factor-term shadows (h₀=0, u₁⁻¹=P(T-1)+P(X₁-1), x₁^σ=S⁻¹(X₁-1), c₀=0);
• the new non-circular product-rule atom above; and
• the source datum's actual wild-relator equality to cancel ρ(wildRelator)-1.

It does not call sec7_frattiniTrivial_elementaryMStageCandidateRelatorShadowData or any theorem downstream of the canonical manuscript-row chain.

theorem Q2Presentation.Induction.sec7_frattiniTrivial_elementaryMStageCandidateWildRowManuscriptCanonicalTermProductRuleData {p : FramedPair} {h : ¬ScalarTerminal p} {chief : FirstNonScalarChiefFactor p} {K : KernelMinimal chief.toNonScalarChiefFactor} {hR : Nat.card ↥(kernelFrattini K) = 1} {child : Sec7FrattiniTrivialElementaryChildData p} (src : Sec7FrattiniTrivialElementaryMStageCandidateLowerSourceData h chief K hR child) : Nonempty (Sec7FrattiniTrivialElementaryMStageCandidateWildRowManuscriptCanonicalTermProductRuleData src)

Former canonical finite-word product-rule atom, now theorem-level from the cancelled canonical row and the fact that the action of a trivial relator is 1.

theorem Q2Presentation.Induction.sec7_frattiniTrivial_elementaryMStageCandidateWildRowManuscriptTermProductRuleData {p : FramedPair} {h : ¬ScalarTerminal p} {chief : FirstNonScalarChiefFactor p} {K : KernelMinimal chief.toNonScalarChiefFactor} {hR : Nat.card ↥(kernelFrattini K) = 1} {child : Sec7FrattiniTrivialElementaryChildData p} {src : Sec7FrattiniTrivialElementaryMStageCandidateLowerSourceData h chief K hR child} (h0 : Sec7FrattiniTrivialElementaryMStageCandidateWildRowManuscriptH0TermData src) (u1 : Sec7FrattiniTrivialElementaryMStageCandidateWildRowManuscriptU1InvTermData src) (x1 : Sec7FrattiniTrivialElementaryMStageCandidateWildRowManuscriptX1ConjTermData src) (c0 : Sec7FrattiniTrivialElementaryMStageCandidateWildRowManuscriptC0TermData src) : Nonempty (Sec7FrattiniTrivialElementaryMStageCandidateWildRowManuscriptTermProductRuleData h0 u1 x1 c0)

Former carrier-dependent four-factor product-rule atom, now theorem-level from the canonical product rule and the four manuscript term-shadow equalities.

theorem Q2Presentation.Induction.sec7_frattiniTrivial_elementaryMStageCandidateWildRowManuscriptCollapsedFoxDerivativeBlockData {p : FramedPair} {h : ¬ScalarTerminal p} {chief : FirstNonScalarChiefFactor p} {K : KernelMinimal chief.toNonScalarChiefFactor} {hR : Nat.card ↥(kernelFrattini K) = 1} {child : Sec7FrattiniTrivialElementaryChildData p} (src : Sec7FrattiniTrivialElementaryMStageCandidateLowerSourceData h chief K hR child) : Nonempty (Sec7FrattiniTrivialElementaryMStageCandidateWildRowManuscriptCollapsedFoxDerivativeBlockData src)

Former collapsed two-block Fox-derivative atom, now theorem-level from the raw four-factor product rule and the four manuscript term shadows.

theorem Q2Presentation.Induction.sec7_frattiniTrivial_elementaryMStageCandidateWildRowManuscriptCollapsedFoxDerivativeExpansionCoreData {p : FramedPair} {h : ¬ScalarTerminal p} {chief : FirstNonScalarChiefFactor p} {K : KernelMinimal chief.toNonScalarChiefFactor} {hR : Nat.card ↥(kernelFrattini K) = 1} {child : Sec7FrattiniTrivialElementaryChildData p} (src : Sec7FrattiniTrivialElementaryMStageCandidateLowerSourceData h chief K hR child) : Nonempty (Sec7FrattiniTrivialElementaryMStageCandidateWildRowManuscriptRelatorCollapsedFoxDerivativeExpansionData src)

Former collapsed Fox-derivative core atom, now theorem-level from the two manuscript operator blocks and their product-rule expansion.

theorem Q2Presentation.Induction.sec7_frattiniTrivial_elementaryMStageCandidateWildRowManuscriptFundamentalRowData {p : FramedPair} {h : ¬ScalarTerminal p} {chief : FirstNonScalarChiefFactor p} {K : KernelMinimal chief.toNonScalarChiefFactor} {hR : Nat.card ↥(kernelFrattini K) = 1} {child : Sec7FrattiniTrivialElementaryChildData p} (src : Sec7FrattiniTrivialElementaryMStageCandidateLowerSourceData h chief K hR child) : Nonempty (Sec7FrattiniTrivialElementaryMStageCandidateWildRowFundamentalData src)

Former fundamental wild-row input, now theorem-level from the collapsed Fox expansion of the actual manuscript relator plus the lower wild-relator equality.

theorem Q2Presentation.Induction.sec7_frattiniTrivial_elementaryMStageCandidateWildRowManuscriptRelatorCollapsedFoxDerivativeExpansionData {p : FramedPair} {h : ¬ScalarTerminal p} {chief : FirstNonScalarChiefFactor p} {K : KernelMinimal chief.toNonScalarChiefFactor} {hR : Nat.card ↥(kernelFrattini K) = 1} {child : Sec7FrattiniTrivialElementaryChildData p} (src : Sec7FrattiniTrivialElementaryMStageCandidateLowerSourceData h chief K hR child) : Nonempty (Sec7FrattiniTrivialElementaryMStageCandidateWildRowManuscriptRelatorCollapsedFoxDerivativeExpansionData src)

Former collapsed Fox-derivative expansion input, now theorem-level from the manuscript fundamental row plus triviality of the evaluated wild relator in the lower quotient.

theorem Q2Presentation.Induction.sec7_frattiniTrivial_elementaryMStageCandidateWildRowManuscriptRelatorLiteralFoxDerivativeExpansionData {p : FramedPair} {h : ¬ScalarTerminal p} {chief : FirstNonScalarChiefFactor p} {K : KernelMinimal chief.toNonScalarChiefFactor} {hR : Nat.card ↥(kernelFrattini K) = 1} {child : Sec7FrattiniTrivialElementaryChildData p} (src : Sec7FrattiniTrivialElementaryMStageCandidateLowerSourceData h chief K hR child) : Nonempty (Sec7FrattiniTrivialElementaryMStageCandidateWildRowManuscriptRelatorLiteralFoxDerivativeExpansionData src)

Former literal Fox derivative expansion input, now theorem-level from the collapsed manuscript row and the linear distributivity identity.

theorem Q2Presentation.Induction.sec7_frattiniTrivial_elementaryMStageCandidateWildRowManuscriptRelatorFoxDerivativeExpansionData {p : FramedPair} {h : ¬ScalarTerminal p} {chief : FirstNonScalarChiefFactor p} {K : KernelMinimal chief.toNonScalarChiefFactor} {hR : Nat.card ↥(kernelFrattini K) = 1} {child : Sec7FrattiniTrivialElementaryChildData p} {src : Sec7FrattiniTrivialElementaryMStageCandidateLowerSourceData h chief K hR child} (h0 : Sec7FrattiniTrivialElementaryMStageCandidateWildRowManuscriptH0Data src) (u1 : Sec7FrattiniTrivialElementaryMStageCandidateWildRowManuscriptU1InvData src) (x1 : Sec7FrattiniTrivialElementaryMStageCandidateWildRowManuscriptX1ConjData src) (c0 : Sec7FrattiniTrivialElementaryMStageCandidateWildRowManuscriptC0Data src) : Nonempty (Sec7FrattiniTrivialElementaryMStageCandidateWildRowManuscriptRelatorFoxDerivativeExpansionData h0 u1 x1 c0)

Former Route250 Fox-derivative expansion atom, now theorem-level from the literal substituted finite-word expansion and the four already evaluated factor shadows.

theorem Q2Presentation.Induction.sec7_frattiniTrivial_elementaryMStageCandidateWildRowManuscriptH0Data {p : FramedPair} {h : ¬ScalarTerminal p} {chief : FirstNonScalarChiefFactor p} {K : KernelMinimal chief.toNonScalarChiefFactor} {hR : Nat.card ↥(kernelFrattini K) = 1} {child : Sec7FrattiniTrivialElementaryChildData p} (src : Sec7FrattiniTrivialElementaryMStageCandidateLowerSourceData h chief K hR child) : Nonempty (Sec7FrattiniTrivialElementaryMStageCandidateWildRowManuscriptH0Data src)

Former Route249 h₀ factor row, now theorem-level from the term carrier and the separate lem:h0shadow zero shadow.

theorem Q2Presentation.Induction.sec7_frattiniTrivial_elementaryMStageCandidateWildRowManuscriptU1InvData {p : FramedPair} {h : ¬ScalarTerminal p} {chief : FirstNonScalarChiefFactor p} {K : KernelMinimal chief.toNonScalarChiefFactor} {hR : Nat.card ↥(kernelFrattini K) = 1} {child : Sec7FrattiniTrivialElementaryChildData p} (src : Sec7FrattiniTrivialElementaryMStageCandidateLowerSourceData h chief K hR child) : Nonempty (Sec7FrattiniTrivialElementaryMStageCandidateWildRowManuscriptU1InvData src)

Former Route249 u₁⁻¹ factor row, now theorem-level from a term carrier and its norm-projector shadow.

theorem Q2Presentation.Induction.sec7_frattiniTrivial_elementaryMStageCandidateWildRowManuscriptX1ConjData {p : FramedPair} {h : ¬ScalarTerminal p} {chief : FirstNonScalarChiefFactor p} {K : KernelMinimal chief.toNonScalarChiefFactor} {hR : Nat.card ↥(kernelFrattini K) = 1} {child : Sec7FrattiniTrivialElementaryChildData p} (src : Sec7FrattiniTrivialElementaryMStageCandidateLowerSourceData h chief K hR child) : Nonempty (Sec7FrattiniTrivialElementaryMStageCandidateWildRowManuscriptX1ConjData src)

Former Route249 x₁^σ factor row, now theorem-level from a term carrier and its conjugation shadow.

theorem Q2Presentation.Induction.sec7_frattiniTrivial_elementaryMStageCandidateWildRowManuscriptC0Data {p : FramedPair} {h : ¬ScalarTerminal p} {chief : FirstNonScalarChiefFactor p} {K : KernelMinimal chief.toNonScalarChiefFactor} {hR : Nat.card ↥(kernelFrattini K) = 1} {child : Sec7FrattiniTrivialElementaryChildData p} (src : Sec7FrattiniTrivialElementaryMStageCandidateLowerSourceData h chief K hR child) : Nonempty (Sec7FrattiniTrivialElementaryMStageCandidateWildRowManuscriptC0Data src)

Former Route249 commutator row c₀=[d₀,z₀], now theorem-level from a term carrier and the separate zero-shadow statement.

theorem Q2Presentation.Induction.sec7_frattiniTrivial_elementaryMStageCandidateWildRowManuscriptRelatorOperatorZero {p : FramedPair} {h : ¬ScalarTerminal p} {chief : FirstNonScalarChiefFactor p} {K : KernelMinimal chief.toNonScalarChiefFactor} {hR : Nat.card ↥(kernelFrattini K) = 1} {child : Sec7FrattiniTrivialElementaryChildData p} (src : Sec7FrattiniTrivialElementaryMStageCandidateLowerSourceData h chief K hR child) : Q2Word.wildRelator.eval src.marking = 1 → (sec7MStageCandidateActionEnd K) (Q2Word.wildRelator.eval src.marking) - 1 = 0

The evaluated wild relator acts trivially on M once the relator value is 1.

theorem Q2Presentation.Induction.sec7_frattiniTrivial_elementaryMStageCandidateWildRowManuscriptRelatorProductRuleData {p : FramedPair} {h : ¬ScalarTerminal p} {chief : FirstNonScalarChiefFactor p} {K : KernelMinimal chief.toNonScalarChiefFactor} {hR : Nat.card ↥(kernelFrattini K) = 1} {child : Sec7FrattiniTrivialElementaryChildData p} {src : Sec7FrattiniTrivialElementaryMStageCandidateLowerSourceData h chief K hR child} (h0 : Sec7FrattiniTrivialElementaryMStageCandidateWildRowManuscriptH0Data src) (u1 : Sec7FrattiniTrivialElementaryMStageCandidateWildRowManuscriptU1InvData src) (x1 : Sec7FrattiniTrivialElementaryMStageCandidateWildRowManuscriptX1ConjData src) (c0 : Sec7FrattiniTrivialElementaryMStageCandidateWildRowManuscriptC0Data src) : Nonempty (Sec7FrattiniTrivialElementaryMStageCandidateWildRowManuscriptRelatorProductRuleData h0 u1 x1 c0)

Former Route249 raw product-rule row, now theorem-level from the Fox derivative expansion plus the theorem-level cancellation ρ(wildRelator)-1=0.

theorem Q2Presentation.Induction.sec7_frattiniTrivial_elementaryMStageCandidateWildRowManuscriptFactorLedgerData {p : FramedPair} {h : ¬ScalarTerminal p} {chief : FirstNonScalarChiefFactor p} {K : KernelMinimal chief.toNonScalarChiefFactor} {hR : Nat.card ↥(kernelFrattini K) = 1} {child : Sec7FrattiniTrivialElementaryChildData p} (src : Sec7FrattiniTrivialElementaryMStageCandidateLowerSourceData h chief K hR child) : Nonempty (Sec7FrattiniTrivialElementaryMStageCandidateWildRowManuscriptFactorLedgerData src)

Former finite-word factor-table leaf, now theorem-level from five smaller manuscript facts: the four factor rows in the order h₀, u₁⁻¹, x₁^σ, c₀, and the raw Fox product-rule sum for the actual wild relator.

theorem Q2Presentation.Induction.sec7_frattiniTrivial_elementaryMStageCandidateWildRowFullWordLedgerCoreData {p : FramedPair} {h : ¬ScalarTerminal p} {chief : FirstNonScalarChiefFactor p} {K : KernelMinimal chief.toNonScalarChiefFactor} {hR : Nat.card ↥(kernelFrattini K) = 1} {child : Sec7FrattiniTrivialElementaryChildData p} (src : Sec7FrattiniTrivialElementaryMStageCandidateLowerSourceData h chief K hR child) : Nonempty (Sec7FrattiniTrivialElementaryMStageCandidateWildRowFullWordLedgerCoreData src)

Former Route247 literal row, now theorem-level from the lower manuscript factor-table ledger. The proof only substitutes the four factor shadows into the raw Fox product rule, preserving the manuscript order h₀, u₁⁻¹, x₁^σ, c₀.

theorem Q2Presentation.Induction.sec7_frattiniTrivial_elementaryMStageCandidateWildRowProductRuleCoreData {p : FramedPair} {h : ¬ScalarTerminal p} {chief : FirstNonScalarChiefFactor p} {K : KernelMinimal chief.toNonScalarChiefFactor} {hR : Nat.card ↥(kernelFrattini K) = 1} {child : Sec7FrattiniTrivialElementaryChildData p} (src : Sec7FrattiniTrivialElementaryMStageCandidateLowerSourceData h chief K hR child) (h0 : Sec7FrattiniTrivialElementaryMStageCandidateWildRowH0FactorData src) (u1 : Sec7FrattiniTrivialElementaryMStageCandidateWildRowU1InvFactorData src) (x1 : Sec7FrattiniTrivialElementaryMStageCandidateWildRowX1ConjFactorData src) (c0 : Sec7FrattiniTrivialElementaryMStageCandidateWildRowC0FactorData src) : Nonempty (Sec7FrattiniTrivialElementaryMStageCandidateWildRowProductRuleData src h0 u1 x1 c0)

Former Route245 product-rule/Fox core, now theorem-level from the literal lem:h0shadow / lem:fullwordledger row.

The proof performs no word expansion: it rewrites the four selected factor records by their shadow fields and then applies the literal full-word ledger above.

structure Q2Presentation.Induction.Sec7FrattiniTrivialElementaryMStageCandidateWildRowRawFactorSumData {p : FramedPair} {h : ¬ScalarTerminal p} {chief : FirstNonScalarChiefFactor p} {K : KernelMinimal chief.toNonScalarChiefFactor} {hR : Nat.card ↥(kernelFrattini K) = 1} {child : Sec7FrattiniTrivialElementaryChildData p} (src : Sec7FrattiniTrivialElementaryMStageCandidateLowerSourceData h chief K hR child) : Type

Raw four-factor Fox sum for the candidate wild relator. The four terms correspond respectively to h₀, u₁⁻¹, x₁^σ, and c₀=[d₀,z₀] in Q2Word.wildRelator.

The factor-evaluation shadows are now fields of the raw packet, so the former aggregate factor-evaluation axiom becomes theorem-level by projection.

• h0Term : Module.End (ZMod 2) (towerM K)

  First-order contribution of h₀.

• u1InvTerm : Module.End (ZMod 2) (towerM K)

  First-order contribution of u₁⁻¹.

• x1ConjTerm : Module.End (ZMod 2) (towerM K)

  First-order contribution of x₁^σ.

• c0Term : Module.End (ZMod 2) (towerM K)

  First-order contribution of c₀=[d₀,z₀].

• relator_factor_sum : Q2Word.wildRelator.eval src.marking = 1 → self.h0Term + self.u1InvTerm + self.x1ConjTerm + self.c0Term = 0

  The raw Fox fundamental sum attached to the actual lower wild relator.

• h0_shadow : self.h0Term = 0

  h₀ has the first-order shadow of x₀², hence zero in characteristic two.

• u1_inv_shadow : self.u1InvTerm = src.P * (src.T - 1) + src.P * (src.X₁ - 1)

  The u₁⁻¹ term contributes the two norm pieces.

• x1_conj_shadow : self.x1ConjTerm = src.Sinv * (src.X₁ - 1)

  The conjugate x₁^σ term contributes S⁻¹ on the x₁ coordinate.

• c0_shadow : self.c0Term = 0

  The commutator c₀=[d₀,z₀] has no first-order contribution.

structure Q2Presentation.Induction.Sec7FrattiniTrivialElementaryMStageCandidateWildRowFactorEvaluationData {p : FramedPair} {h : ¬ScalarTerminal p} {chief : FirstNonScalarChiefFactor p} {K : KernelMinimal chief.toNonScalarChiefFactor} {hR : Nat.card ↥(kernelFrattini K) = 1} {child : Sec7FrattiniTrivialElementaryChildData p} (src : Sec7FrattiniTrivialElementaryMStageCandidateLowerSourceData h chief K hR child) (raw : Sec7FrattiniTrivialElementaryMStageCandidateWildRowRawFactorSumData src) : Prop

Evaluation of the four raw wild-row factor terms from lem:h0shadow and lem:fullwordledger. This contains no relator-triviality assertion; it only identifies what each factor contributes to the row of d¹ ∘ d⁰.

• h0_shadow : raw.h0Term = 0

  h₀ has the first-order shadow of x₀², hence zero in characteristic two.

• u1_inv_shadow : raw.u1InvTerm = src.P * (src.T - 1) + src.P * (src.X₁ - 1)

  The u₁⁻¹ term contributes the two norm pieces.

• x1_conj_shadow : raw.x1ConjTerm = src.Sinv * (src.X₁ - 1)

  The conjugate x₁^σ term contributes S⁻¹ on the x₁ coordinate.

• c0_shadow : raw.c0Term = 0

  The commutator c₀=[d₀,z₀] has no first-order contribution.

theorem Q2Presentation.Induction.sec7_frattiniTrivial_elementaryMStageCandidateWildRowRawFactorLedgerData {p : FramedPair} {h : ¬ScalarTerminal p} {chief : FirstNonScalarChiefFactor p} {K : KernelMinimal chief.toNonScalarChiefFactor} {hR : Nat.card ↥(kernelFrattini K) = 1} {child : Sec7FrattiniTrivialElementaryChildData p} (src : Sec7FrattiniTrivialElementaryMStageCandidateLowerSourceData h chief K hR child) : Nonempty (Sec7FrattiniTrivialElementaryMStageCandidateWildRowRawFactorSumData src)

Former Route244 raw four-factor candidate wild-row ledger, now theorem-level from the smaller formal product-rule atom plus the four explicit factor-shadow packets.

The new residual records only the Fox product-rule sum for selected factor terms. The h₀, u₁⁻¹, x₁^σ, and c₀ evaluations are supplied here by the literal factor structures, so the old raw ledger is no longer an axiom.

structure Q2Presentation.Induction.Sec7FrattiniTrivialElementaryMStageCandidateWildRowLiteralFactorLedgerData {p : FramedPair} {h : ¬ScalarTerminal p} {chief : FirstNonScalarChiefFactor p} {K : KernelMinimal chief.toNonScalarChiefFactor} {hR : Nat.card ↥(kernelFrattini K) = 1} {child : Sec7FrattiniTrivialElementaryChildData p} (src : Sec7FrattiniTrivialElementaryMStageCandidateLowerSourceData h chief K hR child) : Prop

The literal evaluated four-factor wild-row ledger, before the final additive normalization.

The four summands are the already-evaluated contributions of h₀, u₁⁻¹, x₁^σ, and c₀ in Q2Word.wildRelator. The following theorem normalizes this statement to the old additive row, so the old Route242 additive-ledger atom no longer appears directly.

• literal_factor_sum : Q2Word.wildRelator.eval src.marking = 1 → 0 + (src.P * (src.T - 1) + src.P * (src.X₁ - 1)) + src.Sinv * (src.X₁ - 1) + 0 = 0

  The evaluated wild row in literal factor order.

theorem Q2Presentation.Induction.sec7_frattiniTrivial_elementaryMStageCandidateWildRowLiteralFactorLedgerData {p : FramedPair} (h : ¬ScalarTerminal p) (chief : FirstNonScalarChiefFactor p) (K : KernelMinimal chief.toNonScalarChiefFactor) (hR : Nat.card ↥(kernelFrattini K) = 1) (child : Sec7FrattiniTrivialElementaryChildData p) (src : Sec7FrattiniTrivialElementaryMStageCandidateLowerSourceData h chief K hR child) : Nonempty (Sec7FrattiniTrivialElementaryMStageCandidateWildRowLiteralFactorLedgerData src)

Former Route243 literal factor-ordered candidate wild-row ledger, now theorem-level from the raw four-factor ledger. The proof performs only factor-shadow rewriting, preserving the manuscript order h₀, u₁⁻¹, x₁^σ, c₀ before the later additive normalization.

structure Q2Presentation.Induction.Sec7FrattiniTrivialElementaryMStageCandidateWildRowAdditiveLedgerData {p : FramedPair} {h : ¬ScalarTerminal p} {chief : FirstNonScalarChiefFactor p} {K : KernelMinimal chief.toNonScalarChiefFactor} {hR : Nat.card ↥(kernelFrattini K) = 1} {child : Sec7FrattiniTrivialElementaryChildData p} (src : Sec7FrattiniTrivialElementaryMStageCandidateLowerSourceData h chief K hR child) : Prop

The literal additive wild-row identity after the four factor shadows have been evaluated.

This Route242 atom is smaller than the old product-rule packet: it no longer quantifies over arbitrary factor records. It is the manuscript row identity from the wild relator (lem:h0shadow / lem:fullwordledger) with the four contributions already written in the concrete endomorphism algebra of M = towerM K.

• additive_wild_row : Q2Word.wildRelator.eval src.marking = 1 → src.P * (src.T - 1) + src.P * (src.X₁ - 1) + src.Sinv * (src.X₁ - 1) = 0

  The evaluated candidate wild row before cancelling the norm term.

theorem Q2Presentation.Induction.sec7_frattiniTrivial_elementaryMStageCandidateWildRowAdditiveLedgerData {p : FramedPair} (h : ¬ScalarTerminal p) (chief : FirstNonScalarChiefFactor p) (K : KernelMinimal chief.toNonScalarChiefFactor) (hR : Nat.card ↥(kernelFrattini K) = 1) (child : Sec7FrattiniTrivialElementaryChildData p) (src : Sec7FrattiniTrivialElementaryMStageCandidateLowerSourceData h chief K hR child) : Nonempty (Sec7FrattiniTrivialElementaryMStageCandidateWildRowAdditiveLedgerData src)

Former Route242 additive-ledger residual, now theorem-level from the literal factor ledger by additive normalization in the endomorphism group.

theorem Q2Presentation.Induction.sec7_frattiniTrivial_elementaryMStageCandidateWildRowH0FactorData {p : FramedPair} (h : ¬ScalarTerminal p) (chief : FirstNonScalarChiefFactor p) (K : KernelMinimal chief.toNonScalarChiefFactor) (hR : Nat.card ↥(kernelFrattini K) = 1) (child : Sec7FrattiniTrivialElementaryChildData p) (src : Sec7FrattiniTrivialElementaryMStageCandidateLowerSourceData h chief K hR child) : Nonempty (Sec7FrattiniTrivialElementaryMStageCandidateWildRowH0FactorData src)

Former Route241 residual, now theorem-level: the h₀ shadow is the zero term.

theorem Q2Presentation.Induction.sec7_frattiniTrivial_elementaryMStageCandidateWildRowU1InvFactorData {p : FramedPair} (h : ¬ScalarTerminal p) (chief : FirstNonScalarChiefFactor p) (K : KernelMinimal chief.toNonScalarChiefFactor) (hR : Nat.card ↥(kernelFrattini K) = 1) (child : Sec7FrattiniTrivialElementaryChildData p) (src : Sec7FrattiniTrivialElementaryMStageCandidateLowerSourceData h chief K hR child) : Nonempty (Sec7FrattiniTrivialElementaryMStageCandidateWildRowU1InvFactorData src)

Former Route241 residual, now theorem-level: the u₁⁻¹ term is its explicit ledger contribution.

theorem Q2Presentation.Induction.sec7_frattiniTrivial_elementaryMStageCandidateWildRowX1ConjFactorData {p : FramedPair} (h : ¬ScalarTerminal p) (chief : FirstNonScalarChiefFactor p) (K : KernelMinimal chief.toNonScalarChiefFactor) (hR : Nat.card ↥(kernelFrattini K) = 1) (child : Sec7FrattiniTrivialElementaryChildData p) (src : Sec7FrattiniTrivialElementaryMStageCandidateLowerSourceData h chief K hR child) : Nonempty (Sec7FrattiniTrivialElementaryMStageCandidateWildRowX1ConjFactorData src)

Former Route241 residual, now theorem-level: the x₁^σ term is its explicit S⁻¹-twisted x₁ contribution.

theorem Q2Presentation.Induction.sec7_frattiniTrivial_elementaryMStageCandidateWildRowC0FactorData {p : FramedPair} (h : ¬ScalarTerminal p) (chief : FirstNonScalarChiefFactor p) (K : KernelMinimal chief.toNonScalarChiefFactor) (hR : Nat.card ↥(kernelFrattini K) = 1) (child : Sec7FrattiniTrivialElementaryChildData p) (src : Sec7FrattiniTrivialElementaryMStageCandidateLowerSourceData h chief K hR child) : Nonempty (Sec7FrattiniTrivialElementaryMStageCandidateWildRowC0FactorData src)

Former Route241 residual, now theorem-level: the commutator factor c₀ has zero first-order contribution.

theorem Q2Presentation.Induction.sec7_frattiniTrivial_elementaryMStageCandidateWildRowProductRuleData {p : FramedPair} (h : ¬ScalarTerminal p) (chief : FirstNonScalarChiefFactor p) (K : KernelMinimal chief.toNonScalarChiefFactor) (hR : Nat.card ↥(kernelFrattini K) = 1) (child : Sec7FrattiniTrivialElementaryChildData p) (src : Sec7FrattiniTrivialElementaryMStageCandidateLowerSourceData h chief K hR child) (h0 : Sec7FrattiniTrivialElementaryMStageCandidateWildRowH0FactorData src) (u1 : Sec7FrattiniTrivialElementaryMStageCandidateWildRowU1InvFactorData src) (x1 : Sec7FrattiniTrivialElementaryMStageCandidateWildRowX1ConjFactorData src) (c0 : Sec7FrattiniTrivialElementaryMStageCandidateWildRowC0FactorData src) : Nonempty (Sec7FrattiniTrivialElementaryMStageCandidateWildRowProductRuleData src h0 u1 x1 c0)

Former Route241 product-rule residual, now theorem-level from the smaller Route245 formal product-rule atom.

theorem Q2Presentation.Induction.sec7_frattiniTrivial_elementaryMStageCandidateWildRowRawFactorSumData {p : FramedPair} (h : ¬ScalarTerminal p) (chief : FirstNonScalarChiefFactor p) (K : KernelMinimal chief.toNonScalarChiefFactor) (hR : Nat.card ↥(kernelFrattini K) = 1) (child : Sec7FrattiniTrivialElementaryChildData p) (src : Sec7FrattiniTrivialElementaryMStageCandidateLowerSourceData h chief K hR child) : Nonempty (Sec7FrattiniTrivialElementaryMStageCandidateWildRowRawFactorSumData src)

Former Route240 residual, now theorem-level from the separated wild-row factors and product-rule datum.

theorem Q2Presentation.Induction.sec7_frattiniTrivial_elementaryMStageCandidateWildRowFactorEvaluationData {p : FramedPair} (h : ¬ScalarTerminal p) (chief : FirstNonScalarChiefFactor p) (K : KernelMinimal chief.toNonScalarChiefFactor) (hR : Nat.card ↥(kernelFrattini K) = 1) (child : Sec7FrattiniTrivialElementaryChildData p) (src : Sec7FrattiniTrivialElementaryMStageCandidateLowerSourceData h chief K hR child) (raw : Sec7FrattiniTrivialElementaryMStageCandidateWildRowRawFactorSumData src) : Nonempty (Sec7FrattiniTrivialElementaryMStageCandidateWildRowFactorEvaluationData src raw)

Former Route240 residual, now theorem-level by projection from the raw factor packet.

theorem Q2Presentation.Induction.sec7_frattiniTrivial_elementaryMStageCandidateWildRowFundamentalData {p : FramedPair} (h : ¬ScalarTerminal p) (chief : FirstNonScalarChiefFactor p) (K : KernelMinimal chief.toNonScalarChiefFactor) (hR : Nat.card ↥(kernelFrattini K) = 1) (child : Sec7FrattiniTrivialElementaryChildData p) (src : Sec7FrattiniTrivialElementaryMStageCandidateLowerSourceData h chief K hR child) : Nonempty (Sec7FrattiniTrivialElementaryMStageCandidateWildRowFundamentalData src)

Former Route239 wild-row fundamental atom, now theorem-level. The proof combines the raw four-factor Fox sum with the factor evaluations; only additive regrouping and left-distributivity of composition over addition remain to be checked in Lean.

theorem Q2Presentation.Induction.sec7_frattiniTrivial_elementaryMStageCandidateLowerSourceData {p : FramedPair} (h : ¬ScalarTerminal p) (chief : FirstNonScalarChiefFactor p) (K : KernelMinimal chief.toNonScalarChiefFactor) (hR : Nat.card ↥(kernelFrattini K) = 1) (child : Sec7FrattiniTrivialElementaryChildData p) : Nonempty (Sec7FrattiniTrivialElementaryMStageCandidateLowerSourceData h chief K hR child)

Former source-specific lower candidate marking atom, now theorem-level.

Given an admissible marking of towerC K, the candidate finite-quotient semantics constructs the epimorphism Γ_A ↠ towerC K; the fields recording the four generator values and the two candidate relators are then just the corresponding components of admissibility.

theorem Q2Presentation.Induction.sec7_frattiniTrivial_elementaryMStageCandidateRelatorShadowData {p : FramedPair} (h : ¬ScalarTerminal p) (chief : FirstNonScalarChiefFactor p) (K : KernelMinimal chief.toNonScalarChiefFactor) (hR : Nat.card ↥(kernelFrattini K) = 1) (child : Sec7FrattiniTrivialElementaryChildData p) (src : Sec7FrattiniTrivialElementaryMStageCandidateLowerSourceData h chief K hR child) : Nonempty (Sec7FrattiniTrivialElementaryMStageCandidateRelatorShadowData src)

Former source-specific finite-word shadow atom, now theorem-level.

The new residual supplies the wild Fox identity with the norm term still present. The norm term vanishes by the already-proved telescoping identity src.norm_shadow, so the old final shadow follows by a two-line linear calculation.

theorem Q2Presentation.Induction.sec7_frattiniTrivial_elementaryMStageCandidateWordOperatorData {p : FramedPair} (h : ¬ScalarTerminal p) (chief : FirstNonScalarChiefFactor p) (K : KernelMinimal chief.toNonScalarChiefFactor) (hR : Nat.card ↥(kernelFrattini K) = 1) (child : Sec7FrattiniTrivialElementaryChildData p) : Nonempty (Sec7FrattiniTrivialElementaryMStageCandidateWordOperatorData h chief K hR child)

Former candidate-side word-operator constructor, now theorem-level.

The operators are assembled from the source-specific lower marking and the concrete towerActM K action. The inverse, tame, and norm shadows are proved from general representation lemmas; the wild shadow is supplied only as the specialized finite-word ledger from the lower candidate relator.

structure Q2Presentation.Induction.Sec7FrattiniTrivialElementaryMStageCandidateFoxJacobianSurjectivityData {p : FramedPair} (h : ¬ScalarTerminal p) (chief : FirstNonScalarChiefFactor p) (K : KernelMinimal chief.toNonScalarChiefFactor) (hR : Nat.card ↥(kernelFrattini K) = 1) (child : Sec7FrattiniTrivialElementaryChildData p) (cand : Sec7FrattiniTrivialElementaryMStageCandidateCocycleGroupData h chief K hR child) : Prop

Full-row-rank form of the candidate Fox--Jacobian input.

This is the candidate deformation-duality consequence actually used to obtain dim Z¹ = 2 dim M: the evaluated two-relator map d¹ : M⁴ → M² is onto. The rank-nullity conversion to the old finrank packet is theorem-level below.

• d1_surjective : Function.Surjective ⇑(Lifting.d1 cand.Sinv cand.T cand.P)

  The evaluated candidate Fox--Jacobian has full row rank.

structure Q2Presentation.Induction.Sec7FrattiniTrivialElementaryMStageCandidateFoxJacobianRangeTopData {p : FramedPair} (h : ¬ScalarTerminal p) (chief : FirstNonScalarChiefFactor p) (K : KernelMinimal chief.toNonScalarChiefFactor) (hR : Nat.card ↥(kernelFrattini K) = 1) (child : Sec7FrattiniTrivialElementaryChildData p) (cand : Sec7FrattiniTrivialElementaryMStageCandidateCocycleGroupData h chief K hR child) : Prop

Range-top form of the candidate Fox--Jacobian full-row-rank input.

• d1_range_top : (Lifting.d1 cand.Sinv cand.T cand.P).range = ⊤

  The evaluated candidate Fox--Jacobian has range all of M².

structure Q2Presentation.Induction.Sec7FrattiniTrivialElementaryMStageCandidateFoxJacobianRightInverseData {p : FramedPair} (h : ¬ScalarTerminal p) (chief : FirstNonScalarChiefFactor p) (K : KernelMinimal chief.toNonScalarChiefFactor) (hR : Nat.card ↥(kernelFrattini K) = 1) (child : Sec7FrattiniTrivialElementaryChildData p) (cand : Sec7FrattiniTrivialElementaryMStageCandidateCocycleGroupData h chief K hR child) : Type

Right-inverse form of the candidate Fox--Jacobian full-row-rank input.

This is a more constructive deformation-duality datum than range = ⊤: it records a linear section of the evaluated two-relator Jacobian d¹ : M⁴ → M². The range-top statement is recovered theorem-level by applying the section to an arbitrary target vector.

• rightInv : towerM K × towerM K →ₗ[ZMod 2] towerM K × towerM K × towerM K × towerM K

  A linear right inverse to the evaluated candidate Fox--Jacobian.

• rightInv_spec : Lifting.d1 cand.Sinv cand.T cand.P ∘ₗ self.rightInv = LinearMap.id

  The section property d¹ ∘ rightInv = id.

structure Q2Presentation.Induction.Sec7FrattiniTrivialElementaryMStageCandidateFoxJacobianCoordinateRightInverseData {p : FramedPair} (h : ¬ScalarTerminal p) (chief : FirstNonScalarChiefFactor p) (K : KernelMinimal chief.toNonScalarChiefFactor) (hR : Nat.card ↥(kernelFrattini K) = 1) (child : Sec7FrattiniTrivialElementaryChildData p) (cand : Sec7FrattiniTrivialElementaryMStageCandidateCocycleGroupData h chief K hR child) : Type

Coordinate-section form of the candidate Fox--Jacobian full-row-rank input.

This splits the right inverse into two manuscript row sections: one lifting the tame-row coordinate (m,0) and one lifting the wild-row coordinate (0,m). The single right inverse to d¹ is assembled theorem-level by adding these two lifts.

• tameRightInv : towerM K →ₗ[ZMod 2] towerM K × towerM K × towerM K × towerM K

  A section of d¹ over the tame row.

• wildRightInv : towerM K →ₗ[ZMod 2] towerM K × towerM K × towerM K × towerM K

  A section of d¹ over the wild row.

• tameRightInv_spec (m : towerM K) : (Lifting.d1 cand.Sinv cand.T cand.P) (self.tameRightInv m) = (m, 0)

  The tame-row section maps to (m,0).

• wildRightInv_spec (m : towerM K) : (Lifting.d1 cand.Sinv cand.T cand.P) (self.wildRightInv m) = (0, m)

  The wild-row section maps to (0,m).

structure Q2Presentation.Induction.Sec7FrattiniTrivialElementaryMStageCandidateFoxJacobianTameCoordinateRightInverseData {p : FramedPair} (h : ¬ScalarTerminal p) (chief : FirstNonScalarChiefFactor p) (K : KernelMinimal chief.toNonScalarChiefFactor) (hR : Nat.card ↥(kernelFrattini K) = 1) (child : Sec7FrattiniTrivialElementaryChildData p) (cand : Sec7FrattiniTrivialElementaryMStageCandidateCocycleGroupData h chief K hR child) : Type

Tame-coordinate section of the evaluated candidate Fox--Jacobian.

• tameRightInv : towerM K →ₗ[ZMod 2] towerM K × towerM K × towerM K × towerM K

  A section of d¹ over the tame row.

• tameRightInv_spec (m : towerM K) : (Lifting.d1 cand.Sinv cand.T cand.P) (self.tameRightInv m) = (m, 0)

  The tame-row section maps to (m,0).

structure Q2Presentation.Induction.Sec7FrattiniTrivialElementaryMStageCandidateFoxJacobianWildCoordinateRightInverseData {p : FramedPair} (h : ¬ScalarTerminal p) (chief : FirstNonScalarChiefFactor p) (K : KernelMinimal chief.toNonScalarChiefFactor) (hR : Nat.card ↥(kernelFrattini K) = 1) (child : Sec7FrattiniTrivialElementaryChildData p) (cand : Sec7FrattiniTrivialElementaryMStageCandidateCocycleGroupData h chief K hR child) : Type

Wild-coordinate section of the evaluated candidate Fox--Jacobian.

• wildRightInv : towerM K →ₗ[ZMod 2] towerM K × towerM K × towerM K × towerM K

  A section of d¹ over the wild row.

• wildRightInv_spec (m : towerM K) : (Lifting.d1 cand.Sinv cand.T cand.P) (self.wildRightInv m) = (0, m)

  The wild-row section maps to (0,m).

structure Q2Presentation.Induction.Sec7FrattiniTrivialElementaryMStageCandidateFoxJacobianTameCoordinateRightInverseMapData {p : FramedPair} (h : ¬ScalarTerminal p) (chief : FirstNonScalarChiefFactor p) (K : KernelMinimal chief.toNonScalarChiefFactor) (hR : Nat.card ↥(kernelFrattini K) = 1) (child : Sec7FrattiniTrivialElementaryChildData p) (cand : Sec7FrattiniTrivialElementaryMStageCandidateCocycleGroupData h chief K hR child) : Type

Carrier map for the tame-coordinate section of the candidate Fox--Jacobian.

• tameRightInv : towerM K →ₗ[ZMod 2] towerM K × towerM K × towerM K × towerM K
• tameRightInv_spec_core (m : towerM K) : (Lifting.d1 cand.Sinv cand.T cand.P) (self.tameRightInv m) = (m, 0)

  The chosen carrier is a genuine section over the tame coordinate. Keeping this proof in the map datum lets the old map/spec API be recovered theorem-level from the weaker full-row-surjectivity residual below.

structure Q2Presentation.Induction.Sec7FrattiniTrivialElementaryMStageCandidateFoxJacobianTameCoordinateRightInverseSpecData {p : FramedPair} {h : ¬ScalarTerminal p} {chief : FirstNonScalarChiefFactor p} {K : KernelMinimal chief.toNonScalarChiefFactor} {hR : Nat.card ↥(kernelFrattini K) = 1} {child : Sec7FrattiniTrivialElementaryChildData p} {cand : Sec7FrattiniTrivialElementaryMStageCandidateCocycleGroupData h chief K hR child} (sec : Sec7FrattiniTrivialElementaryMStageCandidateFoxJacobianTameCoordinateRightInverseMapData h chief K hR child cand) : Prop

Specification for a chosen tame-coordinate section of the candidate Fox--Jacobian.

• tameRightInv_spec (m : towerM K) : (Lifting.d1 cand.Sinv cand.T cand.P) (sec.tameRightInv m) = (m, 0)

structure Q2Presentation.Induction.Sec7FrattiniTrivialElementaryMStageCandidateFoxJacobianWildCoordinateRightInverseMapData {p : FramedPair} (h : ¬ScalarTerminal p) (chief : FirstNonScalarChiefFactor p) (K : KernelMinimal chief.toNonScalarChiefFactor) (hR : Nat.card ↥(kernelFrattini K) = 1) (child : Sec7FrattiniTrivialElementaryChildData p) (cand : Sec7FrattiniTrivialElementaryMStageCandidateCocycleGroupData h chief K hR child) : Type

Carrier map for the wild-coordinate section of the candidate Fox--Jacobian.

• wildRightInv : towerM K →ₗ[ZMod 2] towerM K × towerM K × towerM K × towerM K
• wildRightInv_spec_core (m : towerM K) : (Lifting.d1 cand.Sinv cand.T cand.P) (self.wildRightInv m) = (0, m)

  The chosen carrier is a genuine section over the wild coordinate. Keeping this proof in the map datum lets the old map/spec API be recovered theorem-level from the weaker full-row-surjectivity residual below.

structure Q2Presentation.Induction.Sec7FrattiniTrivialElementaryMStageCandidateFoxJacobianWildCoordinateRightInverseSpecData {p : FramedPair} {h : ¬ScalarTerminal p} {chief : FirstNonScalarChiefFactor p} {K : KernelMinimal chief.toNonScalarChiefFactor} {hR : Nat.card ↥(kernelFrattini K) = 1} {child : Sec7FrattiniTrivialElementaryChildData p} {cand : Sec7FrattiniTrivialElementaryMStageCandidateCocycleGroupData h chief K hR child} (sec : Sec7FrattiniTrivialElementaryMStageCandidateFoxJacobianWildCoordinateRightInverseMapData h chief K hR child cand) : Prop

Specification for a chosen wild-coordinate section of the candidate Fox--Jacobian.

• wildRightInv_spec (m : towerM K) : (Lifting.d1 cand.Sinv cand.T cand.P) (sec.wildRightInv m) = (0, m)

axiom Q2Presentation.Induction.sec7_frattiniTrivial_elementaryMStageCandidateFoxJacobianRowSurjectivityData {p : FramedPair} (h : ¬ScalarTerminal p) (chief : FirstNonScalarChiefFactor p) (K : KernelMinimal chief.toNonScalarChiefFactor) (hR : Nat.card ↥(kernelFrattini K) = 1) (child : Sec7FrattiniTrivialElementaryChildData p) (cand : Sec7FrattiniTrivialElementaryMStageCandidateCocycleGroupData h chief K hR child) : Nonempty (Sec7FrattiniTrivialElementaryMStageCandidateFoxJacobianSurjectivityData h chief K hR child cand)

New strictly weaker candidate Fox--Jacobian residual: full-row surjectivity.

This is the manuscript-shaped deformation-duality fact needed by the M-stage cardinality calculation. It does not choose coordinate section maps. The old coordinate map/spec APIs below are reconstructed theorem-level from this single surjectivity statement using projectivity of vector spaces over ZMod 2.

theorem Q2Presentation.Induction.sec7_frattiniTrivial_elementaryMStageCandidateFoxJacobianTameCoordinateRightInverseMapData {p : FramedPair} (h : ¬ScalarTerminal p) (chief : FirstNonScalarChiefFactor p) (K : KernelMinimal chief.toNonScalarChiefFactor) (hR : Nat.card ↥(kernelFrattini K) = 1) (child : Sec7FrattiniTrivialElementaryChildData p) (cand : Sec7FrattiniTrivialElementaryMStageCandidateCocycleGroupData h chief K hR child) : Nonempty (Sec7FrattiniTrivialElementaryMStageCandidateFoxJacobianTameCoordinateRightInverseMapData h chief K hR child cand)

Former carrier map for the tame coordinate, now theorem-level from full-row surjectivity rather than a chosen coordinate section residual.

theorem Q2Presentation.Induction.sec7_frattiniTrivial_elementaryMStageCandidateFoxJacobianTameCoordinateRightInverseSpecData {p : FramedPair} {h : ¬ScalarTerminal p} {chief : FirstNonScalarChiefFactor p} {K : KernelMinimal chief.toNonScalarChiefFactor} {hR : Nat.card ↥(kernelFrattini K) = 1} {child : Sec7FrattiniTrivialElementaryChildData p} {cand : Sec7FrattiniTrivialElementaryMStageCandidateCocycleGroupData h chief K hR child} (sec : Sec7FrattiniTrivialElementaryMStageCandidateFoxJacobianTameCoordinateRightInverseMapData h chief K hR child cand) : Nonempty (Sec7FrattiniTrivialElementaryMStageCandidateFoxJacobianTameCoordinateRightInverseSpecData sec)

Former specification for the tame coordinate, now theorem-level from the strengthened map datum above.

theorem Q2Presentation.Induction.sec7_frattiniTrivial_elementaryMStageCandidateFoxJacobianWildCoordinateRightInverseMapData {p : FramedPair} (h : ¬ScalarTerminal p) (chief : FirstNonScalarChiefFactor p) (K : KernelMinimal chief.toNonScalarChiefFactor) (hR : Nat.card ↥(kernelFrattini K) = 1) (child : Sec7FrattiniTrivialElementaryChildData p) (cand : Sec7FrattiniTrivialElementaryMStageCandidateCocycleGroupData h chief K hR child) : Nonempty (Sec7FrattiniTrivialElementaryMStageCandidateFoxJacobianWildCoordinateRightInverseMapData h chief K hR child cand)

Former carrier map for the wild coordinate, now theorem-level from full-row surjectivity rather than a chosen coordinate section residual.

theorem Q2Presentation.Induction.sec7_frattiniTrivial_elementaryMStageCandidateFoxJacobianWildCoordinateRightInverseSpecData {p : FramedPair} {h : ¬ScalarTerminal p} {chief : FirstNonScalarChiefFactor p} {K : KernelMinimal chief.toNonScalarChiefFactor} {hR : Nat.card ↥(kernelFrattini K) = 1} {child : Sec7FrattiniTrivialElementaryChildData p} {cand : Sec7FrattiniTrivialElementaryMStageCandidateCocycleGroupData h chief K hR child} (sec : Sec7FrattiniTrivialElementaryMStageCandidateFoxJacobianWildCoordinateRightInverseMapData h chief K hR child cand) : Nonempty (Sec7FrattiniTrivialElementaryMStageCandidateFoxJacobianWildCoordinateRightInverseSpecData sec)

Former specification for the wild coordinate, now theorem-level from the strengthened map datum above.

theorem Q2Presentation.Induction.sec7_frattiniTrivial_elementaryMStageCandidateFoxJacobianTameCoordinateRightInverseData {p : FramedPair} (h : ¬ScalarTerminal p) (chief : FirstNonScalarChiefFactor p) (K : KernelMinimal chief.toNonScalarChiefFactor) (hR : Nat.card ↥(kernelFrattini K) = 1) (child : Sec7FrattiniTrivialElementaryChildData p) (cand : Sec7FrattiniTrivialElementaryMStageCandidateCocycleGroupData h chief K hR child) : Nonempty (Sec7FrattiniTrivialElementaryMStageCandidateFoxJacobianTameCoordinateRightInverseData h chief K hR child cand)

Former tame-coordinate section atom, now theorem-level from a chosen map and its pointwise section specification.

theorem Q2Presentation.Induction.sec7_frattiniTrivial_elementaryMStageCandidateFoxJacobianWildCoordinateRightInverseData {p : FramedPair} (h : ¬ScalarTerminal p) (chief : FirstNonScalarChiefFactor p) (K : KernelMinimal chief.toNonScalarChiefFactor) (hR : Nat.card ↥(kernelFrattini K) = 1) (child : Sec7FrattiniTrivialElementaryChildData p) (cand : Sec7FrattiniTrivialElementaryMStageCandidateCocycleGroupData h chief K hR child) : Nonempty (Sec7FrattiniTrivialElementaryMStageCandidateFoxJacobianWildCoordinateRightInverseData h chief K hR child cand)

Former wild-coordinate section atom, now theorem-level from a chosen map and its pointwise section specification.

theorem Q2Presentation.Induction.sec7_frattiniTrivial_elementaryMStageCandidateFoxJacobianCoordinateRightInverseData {p : FramedPair} (h : ¬ScalarTerminal p) (chief : FirstNonScalarChiefFactor p) (K : KernelMinimal chief.toNonScalarChiefFactor) (hR : Nat.card ↥(kernelFrattini K) = 1) (child : Sec7FrattiniTrivialElementaryChildData p) (cand : Sec7FrattiniTrivialElementaryMStageCandidateCocycleGroupData h chief K hR child) : Nonempty (Sec7FrattiniTrivialElementaryMStageCandidateFoxJacobianCoordinateRightInverseData h chief K hR child cand)

Former two-coordinate Fox--Jacobian section atom, now theorem-level from its tame and wild row sections.

theorem Q2Presentation.Induction.sec7_frattiniTrivial_elementaryMStageCandidateFoxJacobianRightInverseData {p : FramedPair} (h : ¬ScalarTerminal p) (chief : FirstNonScalarChiefFactor p) (K : KernelMinimal chief.toNonScalarChiefFactor) (hR : Nat.card ↥(kernelFrattini K) = 1) (child : Sec7FrattiniTrivialElementaryChildData p) (cand : Sec7FrattiniTrivialElementaryMStageCandidateCocycleGroupData h chief K hR child) : Nonempty (Sec7FrattiniTrivialElementaryMStageCandidateFoxJacobianRightInverseData h chief K hR child cand)

Former single right-inverse datum, now theorem-level from the two coordinate sections.

theorem Q2Presentation.Induction.sec7_frattiniTrivial_elementaryMStageCandidateFoxJacobianRangeTopData {p : FramedPair} (h : ¬ScalarTerminal p) (chief : FirstNonScalarChiefFactor p) (K : KernelMinimal chief.toNonScalarChiefFactor) (hR : Nat.card ↥(kernelFrattini K) = 1) (child : Sec7FrattiniTrivialElementaryChildData p) (cand : Sec7FrattiniTrivialElementaryMStageCandidateCocycleGroupData h chief K hR child) : Nonempty (Sec7FrattiniTrivialElementaryMStageCandidateFoxJacobianRangeTopData h chief K hR child cand)

Former candidate deformation-duality range-top atom, now theorem-level from the weaker full-row-surjectivity residual.

theorem Q2Presentation.Induction.sec7_frattiniTrivial_elementaryMStageCandidateFoxJacobianSurjectivityData {p : FramedPair} (h : ¬ScalarTerminal p) (chief : FirstNonScalarChiefFactor p) (K : KernelMinimal chief.toNonScalarChiefFactor) (hR : Nat.card ↥(kernelFrattini K) = 1) (child : Sec7FrattiniTrivialElementaryChildData p) (cand : Sec7FrattiniTrivialElementaryMStageCandidateCocycleGroupData h chief K hR child) : Nonempty (Sec7FrattiniTrivialElementaryMStageCandidateFoxJacobianSurjectivityData h chief K hR child cand)

Former candidate Fox--Jacobian surjectivity residual, now theorem-level from the single row-surjectivity datum.

theorem Q2Presentation.Induction.sec7_frattiniTrivial_elementaryMStageCandidateCocycleFinrankData {p : FramedPair} (h : ¬ScalarTerminal p) (chief : FirstNonScalarChiefFactor p) (K : KernelMinimal chief.toNonScalarChiefFactor) (hR : Nat.card ↥(kernelFrattini K) = 1) (child : Sec7FrattiniTrivialElementaryChildData p) (cand : Sec7FrattiniTrivialElementaryMStageCandidateCocycleGroupData h chief K hR child) : Nonempty (Sec7FrattiniTrivialElementaryMStageCandidateCocycleFinrankData h chief K hR child cand)

Former candidate Z¹ finrank residual, now theorem-level from full row rank of d¹ : M⁴ → M² and rank-nullity.

noncomputable def Q2Presentation.Induction.sec7CandidateZ1Fintype {p : FramedPair} {h : ¬ScalarTerminal p} {chief : FirstNonScalarChiefFactor p} {K : KernelMinimal chief.toNonScalarChiefFactor} {hR : Nat.card ↥(kernelFrattini K) = 1} {child : Sec7FrattiniTrivialElementaryChildData p} (ops : Sec7FrattiniTrivialElementaryMStageCandidateWordOperatorData h chief K hR child) : Fintype ↥(Lifting.Z1 ops.Sinv ops.T ops.P)

The literal candidate cocycle subgroup is finite because it is a submodule of (towerM K)^4, and towerM K is already a finite Section 7 module.

Equations

• Q2Presentation.Induction.sec7CandidateZ1Fintype ops = Fintype.ofFinite ↥(Q2Presentation.Lifting.Z1 ops.Sinv ops.T ops.P)

theorem Q2Presentation.Induction.sec7_frattiniTrivial_elementaryMStageCandidateCocycleGroupData {p : FramedPair} (h : ¬ScalarTerminal p) (chief : FirstNonScalarChiefFactor p) (K : KernelMinimal chief.toNonScalarChiefFactor) (hR : Nat.card ↥(kernelFrattini K) = 1) (child : Sec7FrattiniTrivialElementaryChildData p) : Nonempty (Sec7FrattiniTrivialElementaryMStageCandidateCocycleGroupData h chief K hR child)

Former candidate cocycle-group constructor, now theorem-level. The old group packet is assembled from the lower word-operator datum; no cardinality data is used here.

theorem Q2Presentation.Induction.sec7_candidateZ1_card_eq_two_pow_finrank {p : FramedPair} {h : ¬ScalarTerminal p} {chief : FirstNonScalarChiefFactor p} {K : KernelMinimal chief.toNonScalarChiefFactor} {hR : Nat.card ↥(kernelFrattini K) = 1} {child : Sec7FrattiniTrivialElementaryChildData p} (cand : Sec7FrattiniTrivialElementaryMStageCandidateCocycleGroupData h chief K hR child) : Nat.card ↥(Lifting.Z1 cand.Sinv cand.T cand.P) = 2 ^ Module.finrank (ZMod 2) ↥(Lifting.Z1 cand.Sinv cand.T cand.P)

Finite 𝔽₂-vector spaces have cardinality 2^(finrank), specialised to the candidate word-complex cocycle subgroup.

theorem Q2Presentation.Induction.sec7_frattiniTrivial_elementaryMStageCandidateCocycleCardinalityData {p : FramedPair} (h : ¬ScalarTerminal p) (chief : FirstNonScalarChiefFactor p) (K : KernelMinimal chief.toNonScalarChiefFactor) (hR : Nat.card ↥(kernelFrattini K) = 1) (child : Sec7FrattiniTrivialElementaryChildData p) (cand : Sec7FrattiniTrivialElementaryMStageCandidateCocycleGroupData h chief K hR child) : Nonempty (Sec7FrattiniTrivialElementaryMStageCandidateCocycleCardinalityData h chief K hR child cand)

Former candidate cocycle-cardinality constructor, now theorem-level. The old cardinality packet is assembled from the lower finrank formula and the standard finite-vector-space cardinality identity over 𝔽₂.

structure Q2Presentation.Induction.Sec7FrattiniTrivialElementaryMStageLocalCoefficientRepData {p : FramedPair} (h : ¬ScalarTerminal p) (chief : FirstNonScalarChiefFactor p) (K : KernelMinimal chief.toNonScalarChiefFactor) (hR : Nat.card ↥(kernelFrattini K) = 1) (child : Sec7FrattiniTrivialElementaryChildData p) : Type 1

Local coefficient representation atom before choosing the cocycle group finiteness proof.

• A : Local.FiniteDiscreteRep GQ2Profinite

  The local finite discrete G_{ℚ₂}-module corresponding to M.

• module_equiv : self.A.V ≃ₗ[ZMod 2] towerM K

  Identification of its underlying vector space with the concrete Section 7 module.

• source : ↑GQ2Profinite.toProfinite.toTop →* towerC K

  The local source quotient through which this representation factors.

  This records the missing semantic source-interface input: after applying module_equiv, the local G_{ℚ₂} action is transported to the concrete towerActM K action of the finite lower quotient C = towerC K.

• source_surjective : Function.Surjective ⇑self.source

  The local source quotient map is onto the selected finite lower quotient.

• module_equiv_action (g : ↑GQ2Profinite.toProfinite.toTop) (v : self.A.V) : self.module_equiv ((self.A.ρ g) v) = ((towerActM K) (self.source g)) (self.module_equiv v)

  Equivariance of the coefficient identification with the concrete tower action.

structure Q2Presentation.Induction.Sec7FrattiniTrivialElementaryMStageLocalCocycleFinitenessData {p : FramedPair} {h : ¬ScalarTerminal p} {chief : FirstNonScalarChiefFactor p} {K : KernelMinimal chief.toNonScalarChiefFactor} {hR : Nat.card ↥(kernelFrattini K) = 1} {child : Sec7FrattiniTrivialElementaryChildData p} (rep : Sec7FrattiniTrivialElementaryMStageLocalCoefficientRepData h chief K hR child) : Type

Finiteness of the literal local continuous Z¹ group for the selected local module.

• finZQ : Fintype ↥rep.A.Z1

structure Q2Presentation.Induction.Sec7FrattiniTrivialElementaryMStageLocalSourceQuotientData {p : FramedPair} (h : ¬ScalarTerminal p) (chief : FirstNonScalarChiefFactor p) (K : KernelMinimal chief.toNonScalarChiefFactor) (hR : Nat.card ↥(kernelFrattini K) = 1) (child : Sec7FrattiniTrivialElementaryChildData p) : Type

Lower local source-interface datum for the elementary M-stage coefficient module.

The old coefficient-representation residual directly supplied a local finite-discrete representation A, an abstract linear equivalence A.V ≃ towerM K, and equivariance of that equivalence. The manuscript input is narrower: the local action on the elementary module factors through the finite lower quotient C = Y/K and matches the already-built concrete tower action. Once this locally constant quotient map is supplied, the representation itself is constructed theorem-level below with underlying space exactly towerM K and the identity coefficient equivalence.

• source : ↑GQ2Profinite.toProfinite.toTop →* towerC K

  The local source quotient map onto the finite lower quotient C = Y/K.

• source_surjective : Function.Surjective ⇑self.source

  The quotient map realizes the whole lower quotient.

• source_smooth : IsLocallyConstant fun (g : ↑GQ2Profinite.toProfinite.toTop) => self.source g

  Continuity/smoothness of the finite quotient map.

axiom Q2Presentation.Induction.sec7_frattiniTrivial_elementaryMStageLocalSourceQuotientData {p : FramedPair} (h : ¬ScalarTerminal p) (chief : FirstNonScalarChiefFactor p) (K : KernelMinimal chief.toNonScalarChiefFactor) (hR : Nat.card ↥(kernelFrattini K) = 1) (child : Sec7FrattiniTrivialElementaryChildData p) : Nonempty (Sec7FrattiniTrivialElementaryMStageLocalSourceQuotientData h chief K hR child)

Local source-interface input for the elementary M-stage, in quotient-map form.

theorem Q2Presentation.Induction.sec7_frattiniTrivial_elementaryMStageLocalCoefficientRepData {p : FramedPair} (h : ¬ScalarTerminal p) (chief : FirstNonScalarChiefFactor p) (K : KernelMinimal chief.toNonScalarChiefFactor) (hR : Nat.card ↥(kernelFrattini K) = 1) (child : Sec7FrattiniTrivialElementaryChildData p) : Nonempty (Sec7FrattiniTrivialElementaryMStageLocalCoefficientRepData h chief K hR child)

Former local coefficient-representation residual, now theorem-level from the narrower local source quotient.

The constructed representation has underlying vector space towerM K, action g ↦ towerActM K (source g), and identity coefficient equivalence. Thus the remaining semantic content is only the existence of the smooth surjective quotient map from G_{ℚ₂} to the lower finite quotient carrying the prescribed tower action.

theorem Q2Presentation.Induction.sec7_localZ1_moduleFinite (A : Local.FiniteDiscreteRep GQ2Profinite) : Module.Finite (ZMod 2) ↥A.Z1

The continuous local Z¹ group is finite once the local Tate/Euler package is available. This is the finiteness part of the same local-cohomology argument proved in Local.q2_local_Z1_dim, exposed here so the Section 7 elementary stage no longer needs a separate non-local finiteness atom.

noncomputable def Q2Presentation.Induction.sec7LocalZ1Fintype (A : Local.FiniteDiscreteRep GQ2Profinite) : Fintype ↥A.Z1

The finite type structure on the local continuous Z¹ group, derived from the finite-dimensional 𝔽₂-vector-space structure.

Equations

• Q2Presentation.Induction.sec7LocalZ1Fintype A = Fintype.ofFinite ↥A.Z1

theorem Q2Presentation.Induction.sec7_frattiniTrivial_elementaryMStageLocalCocycleFinitenessData {p : FramedPair} {h : ¬ScalarTerminal p} {chief : FirstNonScalarChiefFactor p} {K : KernelMinimal chief.toNonScalarChiefFactor} {hR : Nat.card ↥(kernelFrattini K) = 1} {child : Sec7FrattiniTrivialElementaryChildData p} (rep : Sec7FrattiniTrivialElementaryMStageLocalCoefficientRepData h chief K hR child) : Nonempty (Sec7FrattiniTrivialElementaryMStageLocalCocycleFinitenessData rep)

Former local cocycle-group finiteness input, now theorem-level from the local Tate/Euler finiteness package already present in Local.Cohomology.

theorem Q2Presentation.Induction.sec7_frattiniTrivial_elementaryMStageLocalCocycleGroupData {p : FramedPair} (h : ¬ScalarTerminal p) (chief : FirstNonScalarChiefFactor p) (K : KernelMinimal chief.toNonScalarChiefFactor) (hR : Nat.card ↥(kernelFrattini K) = 1) (child : Sec7FrattiniTrivialElementaryChildData p) : Nonempty (Sec7FrattiniTrivialElementaryMStageLocalCocycleGroupData h chief K hR child)

Former local cocycle-group residual, now theorem-level from the coefficient representation and the separate continuous-cocycle finiteness atom.

structure Q2Presentation.Induction.Sec7FrattiniTrivialElementaryMStageLocalCocycleEulerData {p : FramedPair} (h : ¬ScalarTerminal p) (chief : FirstNonScalarChiefFactor p) (K : KernelMinimal chief.toNonScalarChiefFactor) (hR : Nat.card ↥(kernelFrattini K) = 1) (child : Sec7FrattiniTrivialElementaryChildData p) (loc : Sec7FrattiniTrivialElementaryMStageLocalCocycleGroupData h chief K hR child) : Prop

Local Tate/Euler formula split from the coefficient-dimension identification.

• local_card_by_dim : Nat.card ↥loc.A.Z1 = 2 ^ (2 * Module.finrank (ZMod 2) loc.A.V)

  Local Tate/Euler count in terms of the local coefficient dimension.

• coeff_dim : Module.finrank (ZMod 2) loc.A.V = sec7MStageDim K

  The chosen coefficient module has the concrete Section 7 dimension.

structure Q2Presentation.Induction.Sec7FrattiniTrivialElementaryMStageLocalCocycleEulerCardinalityData {p : FramedPair} (h : ¬ScalarTerminal p) (chief : FirstNonScalarChiefFactor p) (K : KernelMinimal chief.toNonScalarChiefFactor) (hR : Nat.card ↥(kernelFrattini K) = 1) (child : Sec7FrattiniTrivialElementaryChildData p) (loc : Sec7FrattiniTrivialElementaryMStageLocalCocycleGroupData h chief K hR child) : Prop

Lower local Tate/Euler cardinality residual.

The coefficient-dimension part of the previous residual is not mathematical content: the local coefficient packet already carries a linear equivalence loc.A.V ≃ₗ towerM K, and sec7MStageDim K is definitionally the finrank of towerM K. This packet is now assembled theorem-level from Local.q2_local_Z1_dim plus the narrower invariant-vanishing residual below.

• local_card_by_dim : Nat.card ↥loc.A.Z1 = 2 ^ (2 * Module.finrank (ZMod 2) loc.A.V)

  Local Tate/Euler count in terms of the local coefficient dimension.

structure Q2Presentation.Induction.Sec7FrattiniTrivialElementaryMStageLocalDualInvariantVanishingData {p : FramedPair} (h : ¬ScalarTerminal p) (chief : FirstNonScalarChiefFactor p) (K : KernelMinimal chief.toNonScalarChiefFactor) (hR : Nat.card ↥(kernelFrattini K) = 1) (child : Sec7FrattiniTrivialElementaryChildData p) (loc : Sec7FrattiniTrivialElementaryMStageLocalCocycleGroupData h chief K hR child) : Prop

Narrow residual replacing the former local Euler-cardinality axiom.

Local.q2_local_Z1_dim gives dim Z¹(A) = 2 * dim A + dim H⁰(A∨). The only remaining semantic input needed for the elementary M-stage exponent is that the selected local dual coefficient module has no invariants. This is the manuscript's invariant-vanishing content for the local source interface (prop:localduality, eq:localobdual, cor:sourceinterface) and is strictly narrower than asking directly for the final cardinality formula.

• dual_Hcont0_finrank : Module.finrank (ZMod 2) ↥loc.A.dual.Hcont0 = 0

  The local dual invariant subspace (A∨)^{G_{ℚ₂}} = H⁰(A∨) has dimension zero.

theorem Q2Presentation.Induction.sec7_frattiniTrivial_elementaryMStageLocalDualInvariantVanishingData {p : FramedPair} (h : ¬ScalarTerminal p) (chief : FirstNonScalarChiefFactor p) (K : KernelMinimal chief.toNonScalarChiefFactor) (hR : Nat.card ↥(kernelFrattini K) = 1) (child : Sec7FrattiniTrivialElementaryChildData p) (loc : Sec7FrattiniTrivialElementaryMStageLocalCocycleGroupData h chief K hR child) : Nonempty (Sec7FrattiniTrivialElementaryMStageLocalDualInvariantVanishingData h chief K hR child loc)

Local dual-invariant vanishing for the selected elementary M-stage local coefficient representation.

Checkpoint368 removes the former theorem-level axiom here. The proof transports a local invariant functional on A∨ along loc.module_equiv to a towerActM-invariant functional on the concrete module towerM K; towerDualInv_zero then kills it. The only semantic input needed is now part of the local coefficient/source-interface packet: the local action factors onto towerC K and is equivariant with the concrete tower action.

theorem Q2Presentation.Induction.sec7_localZ1_card_eq_two_pow_finrank {p : FramedPair} {h : ¬ScalarTerminal p} {chief : FirstNonScalarChiefFactor p} {K : KernelMinimal chief.toNonScalarChiefFactor} {hR : Nat.card ↥(kernelFrattini K) = 1} {child : Sec7FrattiniTrivialElementaryChildData p} (loc : Sec7FrattiniTrivialElementaryMStageLocalCocycleGroupData h chief K hR child) : Nat.card ↥loc.A.Z1 = 2 ^ Module.finrank (ZMod 2) ↥loc.A.Z1

Finite 𝔽₂-vector spaces have cardinality 2^(finrank), specialised to the local continuous cocycle subgroup.

theorem Q2Presentation.Induction.sec7_frattiniTrivial_elementaryMStageLocalCocycleEulerCardinalityData {p : FramedPair} (h : ¬ScalarTerminal p) (chief : FirstNonScalarChiefFactor p) (K : KernelMinimal chief.toNonScalarChiefFactor) (hR : Nat.card ↥(kernelFrattini K) = 1) (child : Sec7FrattiniTrivialElementaryChildData p) (loc : Sec7FrattiniTrivialElementaryMStageLocalCocycleGroupData h chief K hR child) : Nonempty (Sec7FrattiniTrivialElementaryMStageLocalCocycleEulerCardinalityData h chief K hR child loc)

Local Tate/Euler cardinality, now theorem-level from the local cocycle-dimension formula and dual-invariant vanishing.

theorem Q2Presentation.Induction.sec7_frattiniTrivial_elementaryMStageLocalCocycleEulerData {p : FramedPair} (h : ¬ScalarTerminal p) (chief : FirstNonScalarChiefFactor p) (K : KernelMinimal chief.toNonScalarChiefFactor) (hR : Nat.card ↥(kernelFrattini K) = 1) (child : Sec7FrattiniTrivialElementaryChildData p) (loc : Sec7FrattiniTrivialElementaryMStageLocalCocycleGroupData h chief K hR child) : Nonempty (Sec7FrattiniTrivialElementaryMStageLocalCocycleEulerData h chief K hR child loc)

Former local Tate/Euler residual, now theorem-level from the lower cardinality-only residual plus the existing coefficient linear equivalence.

theorem Q2Presentation.Induction.sec7_frattiniTrivial_elementaryMStageLocalCocycleCardinalityData {p : FramedPair} (h : ¬ScalarTerminal p) (chief : FirstNonScalarChiefFactor p) (K : KernelMinimal chief.toNonScalarChiefFactor) (hR : Nat.card ↥(kernelFrattini K) = 1) (child : Sec7FrattiniTrivialElementaryChildData p) (loc : Sec7FrattiniTrivialElementaryMStageLocalCocycleGroupData h chief K hR child) : Nonempty (Sec7FrattiniTrivialElementaryMStageLocalCocycleCardinalityData h chief K hR child loc)

Former local cocycle-cardinality residual, now theorem-level from local Tate/Euler and the coefficient-dimension identification with M = towerM K.

theorem Q2Presentation.Induction.sec7_frattiniTrivial_elementaryMStageCocycleMultiplicityData {p : FramedPair} (h : ¬ScalarTerminal p) (chief : FirstNonScalarChiefFactor p) (K : KernelMinimal chief.toNonScalarChiefFactor) (hR : Nat.card ↥(kernelFrattini K) = 1) (child : Sec7FrattiniTrivialElementaryChildData p) : Nonempty (Sec7FrattiniTrivialElementaryMStageCocycleMultiplicityData h chief K hR child)

Former common M-stage cocycle-multiplicity residual, now theorem-level. The old packet is assembled from four sourcewise atoms: candidate constructor, candidate cardinality, local constructor, and local cardinality. This removes the common candidate/local cocycle multiplicity packet as an axiom and exposes the two source interfaces separately, both for the literal module M = towerM K.

theorem Q2Presentation.Induction.sec7_frattiniTrivial_elementaryMStageMultiplicityData {p : FramedPair} (h : ¬ScalarTerminal p) (chief : FirstNonScalarChiefFactor p) (K : KernelMinimal chief.toNonScalarChiefFactor) (hR : Nat.card ↥(kernelFrattini K) = 1) (child : Sec7FrattiniTrivialElementaryChildData p) : Nonempty (Sec7FrattiniTrivialElementaryMStageMultiplicityData child)

Former M-stage multiplicity residual, now theorem-level from the literal cocycle multiplicity packet. The old interface is recovered with dimM = dim_{𝔽₂}(towerM K) and liftMultiplicity = |Z¹_{Γ_A,ρ}(M)|; the local group has the same cardinality by the same lower packet.

def Q2Presentation.Induction.sec7MStageCanonicalMultiplicityData {p : FramedPair} {chief : FirstNonScalarChiefFactor p} (K : KernelMinimal chief.toNonScalarChiefFactor) (child : Sec7FrattiniTrivialElementaryChildData p) : Sec7FrattiniTrivialElementaryMStageMultiplicityData child

The canonical elementary M-stage multiplicity datum: dimM is literally the concrete module dimension sec7MStageDim K, and the lift multiplicity is literally 2 ^ (2 * sec7MStageDim K). Using this definitional packet (instead of an opaque Nonempty-obtained one) lets the downstream count theorems discharge the dimension side condition mult.dimM = sec7MStageDim K by rfl.

Equations

• One or more equations did not get rendered due to their size.

structure Q2Presentation.Induction.Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexEquivData {p : FramedPair} (D : Sec7FrattiniTrivialElementaryChildData p) (mult : Sec7FrattiniTrivialElementaryMStageMultiplicityData D) : Type 1

Candidate lift-index equivalence for the elementary M-stage.

Instead of only asserting equality of cardinalities, this records the actual finite candidate lift index together with a bijection from the boundary-framed candidate surjection set counted by framedCountA. The old cardinal realization is recovered by cardinality of this equivalence.

• XA : Type

  Candidate-source elementary M-lift index.

• finXA : Fintype self.XA
• index_equiv : boundaryFramedSurj boundaryPackage_GammaA p.fst p.snd ≃ self.XA

  Boundary-framed candidate surjections are indexed by the elementary lift index.

structure Q2Presentation.Induction.Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexFiberEquivData {p : FramedPair} (D : Sec7FrattiniTrivialElementaryChildData p) (mult : Sec7FrattiniTrivialElementaryMStageMultiplicityData D) : Type 1

Product/fibre form of the candidate elementary lift index: a lower-child index and a finite cocycle-fibre index whose product indexes all candidate framed lifts.

• BaseA : Type

  Candidate lower-child lift index.

• finBaseA : Fintype self.BaseA
• FiberA : Type

  Candidate elementary cocycle fibre index.

• finFiberA : Fintype self.FiberA
• lower_equiv : boundaryFramedSurj boundaryPackage_GammaA D.lowerChild.fst D.lowerChild.snd ≃ self.BaseA

  The lower-child boundary-framed surjection set is indexed by BaseA.

• fiber_card : Nat.card self.FiberA = mult.liftMultiplicity

  The elementary fibre cardinality is the common lift multiplicity.

• total_equiv : boundaryFramedSurj boundaryPackage_GammaA p.fst p.snd ≃ self.BaseA × self.FiberA

  All candidate framed lifts at p are indexed by lower lift plus cocycle fibre.

structure Q2Presentation.Induction.Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexCanonicalFiberEquivData {p : FramedPair} (D : Sec7FrattiniTrivialElementaryChildData p) (mult : Sec7FrattiniTrivialElementaryMStageMultiplicityData D) : Type 1

Canonical lower-index/fibre form of the candidate elementary lift index.

Compared with the checkpoint-255 fibre packet, the lower index is not an arbitrary auxiliary BaseA; it is the actual lower-child boundary-framed surjection set. Thus the only non-canonical finite index left is the elementary cocycle fibre.

• finLowerA : Fintype (boundaryFramedSurj boundaryPackage_GammaA D.lowerChild.fst D.lowerChild.snd)

  The actual lower-child candidate surjection set is finite in this stage.

• FiberA : Type

  Candidate elementary cocycle fibre index.

• finFiberA : Fintype self.FiberA
• fiber_card : Nat.card self.FiberA = mult.liftMultiplicity

  The elementary fibre cardinality is the common lift multiplicity.

• total_equiv : boundaryFramedSurj boundaryPackage_GammaA p.fst p.snd ≃ boundaryFramedSurj boundaryPackage_GammaA D.lowerChild.fst D.lowerChild.snd × self.FiberA

  All candidate framed lifts at p are lower-child lifts times the cocycle fibre.

structure Q2Presentation.Induction.Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexCanonicalFinFiberEquivData {p : FramedPair} (D : Sec7FrattiniTrivialElementaryChildData p) (mult : Sec7FrattiniTrivialElementaryMStageMultiplicityData D) : Type

Fin-fibre form of the candidate elementary lift index.

The elementary cocycle fibre is no longer an arbitrary finite type with the right cardinality; it is the canonical Fin liftMultiplicity.

• finLowerA : Fintype (boundaryFramedSurj boundaryPackage_GammaA D.lowerChild.fst D.lowerChild.snd)

  The actual lower-child candidate surjection set is finite in this stage.

• total_equiv : boundaryFramedSurj boundaryPackage_GammaA p.fst p.snd ≃ boundaryFramedSurj boundaryPackage_GammaA D.lowerChild.fst D.lowerChild.snd × Fin mult.liftMultiplicity

  All candidate framed lifts at p are lower-child lifts times the canonical finite elementary cocycle fibre.

def Q2Presentation.Induction.sec7SigmaConstEquivProd (α : Type u_1) (β : Type u_2) : (_ : α) × β ≃ α × β

Convert a sigma type with constant fibre to a product. Kept local to this file to avoid depending on a particular mathlib equivalence name.

Equations

• Q2Presentation.Induction.sec7SigmaConstEquivProd α β = { toFun := fun (x : (_ : α) × β) => (x.fst, x.snd), invFun := fun (x : α × β) => ⟨x.1, x.2⟩, left_inv := ⋯, right_inv := ⋯ }

structure Q2Presentation.Induction.Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexCanonicalFinFiberTotalEquivData {p : FramedPair} (D : Sec7FrattiniTrivialElementaryChildData p) (mult : Sec7FrattiniTrivialElementaryMStageMultiplicityData D) : Type

Core equivalence for the candidate elementary lift index.

The finite lower-child instance in the checkpoint-257 packet is theorem-level from the general finiteness of candidate boundary-framed surjection sets. The remaining input is only the manuscript product bijection with the canonical Fin liftMultiplicity fibre.

• total_equiv : boundaryFramedSurj boundaryPackage_GammaA p.fst p.snd ≃ boundaryFramedSurj boundaryPackage_GammaA D.lowerChild.fst D.lowerChild.snd × Fin mult.liftMultiplicity

  All candidate framed lifts at p are lower-child lifts times the canonical finite elementary cocycle fibre.

structure Q2Presentation.Induction.Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexCanonicalFinFiberSigmaEquivData {p : FramedPair} (D : Sec7FrattiniTrivialElementaryChildData p) (mult : Sec7FrattiniTrivialElementaryMStageMultiplicityData D) : Type

Sigma-fibre form of the candidate elementary lift index.

This is below the product-form checkpoint-258 atom: the elementary lifts are indexed as a canonical Fin liftMultiplicity fibre over each lower-child candidate source, matching the manuscript affine-lifting torsor description before it is flattened to a product.

• total_sigma_equiv : boundaryFramedSurj boundaryPackage_GammaA p.fst p.snd ≃ (_ : boundaryFramedSurj boundaryPackage_GammaA D.lowerChild.fst D.lowerChild.snd) × Fin mult.liftMultiplicity

structure Q2Presentation.Induction.Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexCanonicalFinFiberSigmaCardData {p : FramedPair} (D : Sec7FrattiniTrivialElementaryChildData p) (mult : Sec7FrattiniTrivialElementaryMStageMultiplicityData D) : Prop

Cardinality-only sigma-fibre form of the candidate elementary lift index.

This is strictly weaker than the previous sigma-fibre equivalence residual: it records only that the candidate framed lifts at p and the canonical lower-child Fin-fibre sigma index have the same finite cardinality. The equivalence used by the existing public API is then chosen theorem-level from finite cardinality equality.

• total_sigma_card : Nat.card (boundaryFramedSurj boundaryPackage_GammaA p.fst p.snd) = Nat.card ((_ : boundaryFramedSurj boundaryPackage_GammaA D.lowerChild.fst D.lowerChild.snd) × Fin mult.liftMultiplicity)

structure Q2Presentation.Induction.Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionFiberCardData {p : FramedPair} (D : Sec7FrattiniTrivialElementaryChildData p) (mult : Sec7FrattiniTrivialElementaryMStageMultiplicityData D) : Type

Fibrewise projection form of the candidate elementary lift index.

This is a more local manuscript-shaped datum than the total sigma-cardinality statement: it supplies the actual projection from a candidate lift at the current target to its lower-child source, and records that each individual fibre has the common elementary lift multiplicity. The total cardinality statement is recovered below by the standard sigmaFiberEquiv decomposition, so the old cardinal residual is no longer assumed directly.

• liftProjection : boundaryFramedSurj boundaryPackage_GammaA p.fst p.snd → boundaryFramedSurj boundaryPackage_GammaA D.lowerChild.fst D.lowerChild.snd

  Projection from a current candidate lift to the lower-child source over which it lies.

• liftProjection_fiber_card (ρ : boundaryFramedSurj boundaryPackage_GammaA D.lowerChild.fst D.lowerChild.snd) : Nat.card { f : boundaryFramedSurj boundaryPackage_GammaA p.fst p.snd // self.liftProjection f = ρ } = mult.liftMultiplicity

  Every fibre of the projection has the common elementary lift multiplicity.

structure Q2Presentation.Induction.Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionCocycleFiberEquivData {p : FramedPair} (h : ¬ScalarTerminal p) (chief : FirstNonScalarChiefFactor p) (K : KernelMinimal chief.toNonScalarChiefFactor) (hR : Nat.card ↥(kernelFrattini K) = 1) (child : Sec7FrattiniTrivialElementaryChildData p) (mult : Sec7FrattiniTrivialElementaryMStageMultiplicityData child) : Type

Cocycle-fibre form of the candidate elementary lift projection.

This is a lower, manuscript-shaped replacement for the fibre-cardinality residual: instead of asserting the cardinality of every fibre directly, it identifies every fibre of the lower-child projection with the literal candidate word-complex cocycle group Z¹_{Γ_A,ρ}(M) = Lifting.Z1 Sinv T P. The separate cardinality datum below records that this common cocycle torsor has the chosen elementary lift multiplicity, and the previous fibre-cardinality packet is recovered theorem-level from these two ingredients.

• cand : Sec7FrattiniTrivialElementaryMStageCandidateCocycleGroupData h chief K hR child

  The literal candidate word-complex cocycle group controlling the elementary fibre.

• liftProjection : boundaryFramedSurj boundaryPackage_GammaA p.fst p.snd → boundaryFramedSurj boundaryPackage_GammaA child.lowerChild.fst child.lowerChild.snd

  Projection from a current candidate lift to the lower-child source over which it lies.

• liftProjection_fiber_equiv (ρ : boundaryFramedSurj boundaryPackage_GammaA child.lowerChild.fst child.lowerChild.snd) : { f : boundaryFramedSurj boundaryPackage_GammaA p.fst p.snd // self.liftProjection f = ρ } ≃ ↥(Lifting.Z1 self.cand.Sinv self.cand.T self.cand.P)

  Each projection fibre is the literal candidate cocycle torsor.

structure Q2Presentation.Induction.Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionCocycleCardData {p : FramedPair} {h : ¬ScalarTerminal p} {chief : FirstNonScalarChiefFactor p} {K : KernelMinimal chief.toNonScalarChiefFactor} {hR : Nat.card ↥(kernelFrattini K) = 1} {child : Sec7FrattiniTrivialElementaryChildData p} {mult : Sec7FrattiniTrivialElementaryMStageMultiplicityData child} (fib : Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionCocycleFiberEquivData h chief K hR child mult) : Prop

Cardinality comparison for the literal candidate cocycle fibre selected by the projection/cocycle-fibre equivalence datum.

This is separated from the fibre-equivalence datum so that the old uniform fibre-cardinality packet is rebuilt from two finer ingredients: the torsor identification of each fibre and the common cocycle-cardinality calculation.

• cocycle_card : Nat.card ↥(Lifting.Z1 fib.cand.Sinv fib.cand.T fib.cand.P) = mult.liftMultiplicity

  The literal candidate cocycle torsor has the common elementary lift multiplicity.

Checkpoint350 split of the candidate projection/cocycle-fibre residual

The checkpoint349 residual sec7_frattiniTrivial_elementaryMStageCandidateLiftIndexProjectionCocycleFiberEquivData packaged the whole candidate-side elementary-lift torsor in one axiom: the lower-child projection, the candidate cocycle group, and a fibre equivalence. The structures below separate the manuscript construction into its component maps and inverse laws.

structure Q2Presentation.Induction.Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionPostcompositionMapData {p : FramedPair} (h : ¬ScalarTerminal p) (chief : FirstNonScalarChiefFactor p) (K : KernelMinimal chief.toNonScalarChiefFactor) (hR : Nat.card ↥(kernelFrattini K) = 1) (child : Sec7FrattiniTrivialElementaryChildData p) : Type

Finite target-level postcomposition map from the current framed target to the lower child.

This is the first projection-front atom below the old raw projection packet: it contains only the finite quotient homomorphism Y → C, not a map out of Γ_A and not a boundary-framed source construction.

• postMap : p.fst.Y →* child.lowerChild.fst.Y

  The finite quotient/postcomposition homomorphism Y → C for the lower child.

• postMap_surj : Function.Surjective ⇑self.postMap

  The finite quotient/postcomposition map is onto C.

• boundary_compat (y : p.fst.Y) (b : ↑Boundary.toProfinite.toTop) : p.fst.qYMap y = (ProfiniteGrp.Hom.hom p.snd.beta) b → child.lowerChild.fst.qYMap (self.postMap y) = (ProfiniteGrp.Hom.hom child.lowerChild.snd.beta) b

  The finite quotient/postcomposition map preserves the inherited boundary frame.

structure Q2Presentation.Induction.Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionPostcompositionSurjectivityData {p : FramedPair} {h : ¬ScalarTerminal p} {chief : FirstNonScalarChiefFactor p} {K : KernelMinimal chief.toNonScalarChiefFactor} {hR : Nat.card ↥(kernelFrattini K) = 1} {child : Sec7FrattiniTrivialElementaryChildData p} (post : Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionPostcompositionMapData h chief K hR child) : Prop

Surjectivity of the finite target-level postcomposition map.

Separated from the carrier map so that the old SurjContHom projection is recovered from two strictly finite target-side facts: the homomorphism and its surjectivity.

• postMap_surj : Function.Surjective ⇑post.postMap

  The finite postcomposition map is onto the lower child C.

structure Q2Presentation.Induction.Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionPostcompositionBoundaryData {p : FramedPair} {h : ¬ScalarTerminal p} {chief : FirstNonScalarChiefFactor p} {K : KernelMinimal chief.toNonScalarChiefFactor} {hR : Nat.card ↥(kernelFrattini K) = 1} {child : Sec7FrattiniTrivialElementaryChildData p} (post : Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionPostcompositionMapData h chief K hR child) : Prop

Boundary-frame compatibility of the finite target-level postcomposition map.

This is the lower-child boundary equation at the finite target level: whenever a point of Y has the top boundary value β(b), its image in C has the inherited lower boundary value β_C(b). It is independent of the candidate source Γ_A and of any particular lift.

• boundary_compat (y : p.fst.Y) (b : ↑Boundary.toProfinite.toTop) : p.fst.qYMap y = (ProfiniteGrp.Hom.hom p.snd.beta) b → child.lowerChild.fst.qYMap (post.postMap y) = (ProfiniteGrp.Hom.hom child.lowerChild.snd.beta) b

  Compatibility of q_Y and the lower boundary frame after finite postcomposition.

structure Q2Presentation.Induction.Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionRawMapData {p : FramedPair} (h : ¬ScalarTerminal p) (chief : FirstNonScalarChiefFactor p) (K : KernelMinimal chief.toNonScalarChiefFactor) (hR : Nat.card ↥(kernelFrattini K) = 1) (child : Sec7FrattiniTrivialElementaryChildData p) : Type

Raw projection map from a current candidate lift to a lower-child candidate surjection.

Compared with checkpoint350, this packet now carries only the finite postcomposition provenance used to build the raw map: a target homomorphism Y → C, its surjectivity, and the statement that the raw projection evaluates by postcomposition. The boundary landing condition remains in the separate ...ProjectionBoundaryLandingData packet and is recovered below from the finite postcomposition boundary datum.

• post : Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionPostcompositionMapData h chief K hR child

  The finite target-level quotient/postcomposition map Y → C.

• post_surj : Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionPostcompositionSurjectivityData self.post

  The postcomposition map is onto C.

• liftProjectionRaw : boundaryFramedSurj boundaryPackage_GammaA p.fst p.snd → Profinite.SurjContHom GammaA (ProfiniteGrp.ofFiniteGrp (FiniteGrp.of child.lowerChild.fst.Y))

  Forget an elementary current lift to its lower quotient map Γ_A ↠ C.

• liftProjectionRaw_apply (f : boundaryFramedSurj boundaryPackage_GammaA p.fst p.snd) (γ : ↑GammaA.toProfinite.toTop) : (ProfiniteGrp.Hom.hom (self.liftProjectionRaw f).hom) γ = self.post.postMap ((ProfiniteGrp.Hom.hom (↑f).hom) γ)

  The raw projection is literally postcomposition by the finite map Y → C.

structure Q2Presentation.Induction.Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionBoundaryLandingData {p : FramedPair} {h : ¬ScalarTerminal p} {chief : FirstNonScalarChiefFactor p} {K : KernelMinimal chief.toNonScalarChiefFactor} {hR : Nat.card ↥(kernelFrattini K) = 1} {child : Sec7FrattiniTrivialElementaryChildData p} (raw : Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionRawMapData h chief K hR child) : Prop

Boundary landing condition for the raw lower projection.

This is the part of the manuscript construction saying that the projected lower map still satisfies the inherited boundary frame.

• boundary_all (f : boundaryFramedSurj boundaryPackage_GammaA p.fst p.snd) (γ : ↑GammaA.toProfinite.toTop) : child.lowerChild.fst.qYMap ((ProfiniteGrp.Hom.hom (raw.liftProjectionRaw f).hom) γ) = (ProfiniteGrp.Hom.hom child.lowerChild.snd.beta) ((ProfiniteGrp.Hom.hom boundaryPackage_GammaA.toBoundary) γ)

  The projected lower map obeys the lower-child boundary equation for every γ : Γ_A.

def Q2Presentation.Induction.sec7_frattiniTrivial_elementaryMStageCandidateLiftIndexProjectionFromRaw {p : FramedPair} {h : ¬ScalarTerminal p} {chief : FirstNonScalarChiefFactor p} {K : KernelMinimal chief.toNonScalarChiefFactor} {hR : Nat.card ↥(kernelFrattini K) = 1} {child : Sec7FrattiniTrivialElementaryChildData p} (raw : Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionRawMapData h chief K hR child) (land : Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionBoundaryLandingData raw) : boundaryFramedSurj boundaryPackage_GammaA p.fst p.snd → boundaryFramedSurj boundaryPackage_GammaA child.lowerChild.fst child.lowerChild.snd

Assemble the canonical boundary-framed lower projection from the raw map and the landing proof.

Equations

• Q2Presentation.Induction.sec7_frattiniTrivial_elementaryMStageCandidateLiftIndexProjectionFromRaw raw land f = ⟨raw.liftProjectionRaw f, ⋯⟩

structure Q2Presentation.Induction.Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionFiberToCocycleMapData {p : FramedPair} {h : ¬ScalarTerminal p} {chief : FirstNonScalarChiefFactor p} {K : KernelMinimal chief.toNonScalarChiefFactor} {hR : Nat.card ↥(kernelFrattini K) = 1} {child : Sec7FrattiniTrivialElementaryChildData p} (cand : Sec7FrattiniTrivialElementaryMStageCandidateCocycleGroupData h chief K hR child) (raw : Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionRawMapData h chief K hR child) (land : Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionBoundaryLandingData raw) : Type

Map from each projection fibre to the literal candidate cocycle group Lifting.Z1 cand.Sinv cand.T cand.P.

• toCocycle (ρ : boundaryFramedSurj boundaryPackage_GammaA child.lowerChild.fst child.lowerChild.snd) : { f : boundaryFramedSurj boundaryPackage_GammaA p.fst p.snd // sec7_frattiniTrivial_elementaryMStageCandidateLiftIndexProjectionFromRaw raw land f = ρ } → ↥(Lifting.Z1 cand.Sinv cand.T cand.P)

  Extract the affine cocycle difference between a lift and its lower source.

structure Q2Presentation.Induction.Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionCocycleToFiberLiftData {p : FramedPair} {h : ¬ScalarTerminal p} {chief : FirstNonScalarChiefFactor p} {K : KernelMinimal chief.toNonScalarChiefFactor} {hR : Nat.card ↥(kernelFrattini K) = 1} {child : Sec7FrattiniTrivialElementaryChildData p} {cand : Sec7FrattiniTrivialElementaryMStageCandidateCocycleGroupData h chief K hR child} {raw : Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionRawMapData h chief K hR child} {land : Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionBoundaryLandingData raw} (toMap : Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionFiberToCocycleMapData cand raw land) : Type

Inverse lift map: add a candidate cocycle to a fixed lower-child source to obtain a current candidate lift in the corresponding projection fibre.

• ofCocycle (ρ : boundaryFramedSurj boundaryPackage_GammaA child.lowerChild.fst child.lowerChild.snd) : ↥(Lifting.Z1 cand.Sinv cand.T cand.P) → { f : boundaryFramedSurj boundaryPackage_GammaA p.fst p.snd // sec7_frattiniTrivial_elementaryMStageCandidateLiftIndexProjectionFromRaw raw land f = ρ }

  Rebuild a current lift over ρ from a literal candidate cocycle.

structure Q2Presentation.Induction.Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionCocycleLiftLeftInverseData {p : FramedPair} {h : ¬ScalarTerminal p} {chief : FirstNonScalarChiefFactor p} {K : KernelMinimal chief.toNonScalarChiefFactor} {hR : Nat.card ↥(kernelFrattini K) = 1} {child : Sec7FrattiniTrivialElementaryChildData p} {cand : Sec7FrattiniTrivialElementaryMStageCandidateCocycleGroupData h chief K hR child} {raw : Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionRawMapData h chief K hR child} {land : Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionBoundaryLandingData raw} (toMap : Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionFiberToCocycleMapData cand raw land) (lift : Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionCocycleToFiberLiftData toMap) : Prop

First inverse law for the cocycle/fibre maps: extracting the cocycle from the lift constructed from that cocycle returns the original cocycle.

• to_of (ρ : boundaryFramedSurj boundaryPackage_GammaA child.lowerChild.fst child.lowerChild.snd) (z : ↥(Lifting.Z1 cand.Sinv cand.T cand.P)) : toMap.toCocycle ρ (lift.ofCocycle ρ z) = z

  toCocycle (ofCocycle z) = z.

structure Q2Presentation.Induction.Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionCocycleLiftRightInverseData {p : FramedPair} {h : ¬ScalarTerminal p} {chief : FirstNonScalarChiefFactor p} {K : KernelMinimal chief.toNonScalarChiefFactor} {hR : Nat.card ↥(kernelFrattini K) = 1} {child : Sec7FrattiniTrivialElementaryChildData p} {cand : Sec7FrattiniTrivialElementaryMStageCandidateCocycleGroupData h chief K hR child} {raw : Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionRawMapData h chief K hR child} {land : Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionBoundaryLandingData raw} (toMap : Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionFiberToCocycleMapData cand raw land) (lift : Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionCocycleToFiberLiftData toMap) : Prop

Second inverse law for the cocycle/fibre maps: rebuilding from the extracted cocycle returns the original current lift in the fibre.

• of_to (ρ : boundaryFramedSurj boundaryPackage_GammaA child.lowerChild.fst child.lowerChild.snd) (x : { f : boundaryFramedSurj boundaryPackage_GammaA p.fst p.snd // sec7_frattiniTrivial_elementaryMStageCandidateLiftIndexProjectionFromRaw raw land f = ρ }) : lift.ofCocycle ρ (toMap.toCocycle ρ x) = x

  ofCocycle (toCocycle f) = f.

structure Q2Presentation.Induction.Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionCocycleMultiplicityData {p : FramedPair} {h : ¬ScalarTerminal p} {chief : FirstNonScalarChiefFactor p} {K : KernelMinimal chief.toNonScalarChiefFactor} {hR : Nat.card ↥(kernelFrattini K) = 1} {child : Sec7FrattiniTrivialElementaryChildData p} {mult : Sec7FrattiniTrivialElementaryMStageMultiplicityData child} (fib : Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionCocycleFiberEquivData h chief K hR child mult) : Prop

The lift multiplicity used by the elementary stage is the candidate word-complex Euler multiplicity for this cocycle fibre. Together with the already separated CandidateCocycleCardinalityData, this replaces the former direct cocycle-cardinality residual.

• liftMultiplicity_eq_candidate_dim : mult.liftMultiplicity = 2 ^ (2 * sec7MStageDim K)

  The chosen elementary lift multiplicity is the candidate 2^(2 dim M) value.

theorem Q2Presentation.Induction.sec7_frattiniTrivial_elementaryMStageCandidateLiftIndexProjectionPostcompositionMapData {p : FramedPair} (h : ¬ScalarTerminal p) (chief : FirstNonScalarChiefFactor p) (K : KernelMinimal chief.toNonScalarChiefFactor) (hR : Nat.card ↥(kernelFrattini K) = 1) (child : Sec7FrattiniTrivialElementaryChildData p) : Nonempty (Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionPostcompositionMapData h chief K hR child)

The finite quotient/postcomposition homomorphism Y → C is now part of the child-data interface, where it is constructed from the literal quotient map Y → Y/K.

theorem Q2Presentation.Induction.sec7_frattiniTrivial_elementaryMStageCandidateLiftIndexProjectionPostcompositionSurjectivityData {p : FramedPair} {h : ¬ScalarTerminal p} {chief : FirstNonScalarChiefFactor p} {K : KernelMinimal chief.toNonScalarChiefFactor} {hR : Nat.card ↥(kernelFrattini K) = 1} {child : Sec7FrattiniTrivialElementaryChildData p} (post : Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionPostcompositionMapData h chief K hR child) : Nonempty (Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionPostcompositionSurjectivityData post)

The finite quotient/postcomposition homomorphism Y → C is onto, recovered from its carrier data.

theorem Q2Presentation.Induction.sec7_frattiniTrivial_elementaryMStageCandidateLiftIndexProjectionPostcompositionBoundaryData {p : FramedPair} {h : ¬ScalarTerminal p} {chief : FirstNonScalarChiefFactor p} {K : KernelMinimal chief.toNonScalarChiefFactor} {hR : Nat.card ↥(kernelFrattini K) = 1} {child : Sec7FrattiniTrivialElementaryChildData p} (post : Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionPostcompositionMapData h chief K hR child) : Nonempty (Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionPostcompositionBoundaryData post)

The finite postcomposition map carries the top boundary frame to the inherited lower boundary frame, recovered from its carrier data.

theorem Q2Presentation.Induction.sec7_frattiniTrivial_elementaryMStageCandidateLiftIndexProjectionRawMapData {p : FramedPair} (h : ¬ScalarTerminal p) (chief : FirstNonScalarChiefFactor p) (K : KernelMinimal chief.toNonScalarChiefFactor) (hR : Nat.card ↥(kernelFrattini K) = 1) (child : Sec7FrattiniTrivialElementaryChildData p) : Nonempty (Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionRawMapData h chief K hR child)

Former raw-projection residual, now theorem-level from the finite target postcomposition map and its surjectivity.

theorem Q2Presentation.Induction.sec7_frattiniTrivial_elementaryMStageCandidateLiftIndexProjectionBoundaryLandingData {p : FramedPair} {h : ¬ScalarTerminal p} {chief : FirstNonScalarChiefFactor p} {K : KernelMinimal chief.toNonScalarChiefFactor} {hR : Nat.card ↥(kernelFrattini K) = 1} {child : Sec7FrattiniTrivialElementaryChildData p} (raw : Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionRawMapData h chief K hR child) : Nonempty (Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionBoundaryLandingData raw)

Former boundary-landing residual, now theorem-level from the finite target-level boundary compatibility of the postcomposition map.

Checkpoint353 split of the candidate projection cocycle/fibre residuals

The checkpoint352 frontier still treated the affine fibre/cocycle equivalence as four map/inverse-law atoms plus one multiplicity-normalization atom. The next structures split those atoms one step further:

• the map from a lift fibre to Lifting.Z1 is obtained from a raw coordinate-valued affine-difference map plus a separate cocycle-membership proof;
• the inverse map is obtained from a raw current-lift constructor plus a separate proof that the constructed lift projects to the chosen lower source;
• the inverse laws are reduced to equality of underlying values of the relevant subtypes;
• the multiplicity normalization is reduced to the statement that the M-stage dimension chosen in mult is the literal concrete dimension sec7MStageDim K.

structure Q2Presentation.Induction.Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionFiberRawDifferenceMapData {p : FramedPair} {h : ¬ScalarTerminal p} {chief : FirstNonScalarChiefFactor p} {K : KernelMinimal chief.toNonScalarChiefFactor} {hR : Nat.card ↥(kernelFrattini K) = 1} {child : Sec7FrattiniTrivialElementaryChildData p} (cand : Sec7FrattiniTrivialElementaryMStageCandidateCocycleGroupData h chief K hR child) (raw : Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionRawMapData h chief K hR child) (land : Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionBoundaryLandingData raw) : Type

Raw affine-difference coordinates for a current candidate lift over a fixed lower source. This is below the old fibre-to-cocycle map because it lands only in the ambient four-coordinate word-complex module, before imposing the d¹=0 cocycle equation.

• rawCocycleValue (ρ : boundaryFramedSurj boundaryPackage_GammaA child.lowerChild.fst child.lowerChild.snd) : { f : boundaryFramedSurj boundaryPackage_GammaA p.fst p.snd // sec7_frattiniTrivial_elementaryMStageCandidateLiftIndexProjectionFromRaw raw land f = ρ } → towerM K × towerM K × towerM K × towerM K

  The four additive generator-coordinate differences of a current lift from its lower-child source.

structure Q2Presentation.Induction.Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionFiberRawDifferenceCocycleConditionData {p : FramedPair} {h : ¬ScalarTerminal p} {chief : FirstNonScalarChiefFactor p} {K : KernelMinimal chief.toNonScalarChiefFactor} {hR : Nat.card ↥(kernelFrattini K) = 1} {child : Sec7FrattiniTrivialElementaryChildData p} {cand : Sec7FrattiniTrivialElementaryMStageCandidateCocycleGroupData h chief K hR child} {raw : Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionRawMapData h chief K hR child} {land : Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionBoundaryLandingData raw} (diff : Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionFiberRawDifferenceMapData cand raw land) : Prop

Cocycle-membership for the raw affine-difference coordinates. This is the literal word-complex equation saying that the extracted difference lies in Lifting.Z1 cand.Sinv cand.T cand.P.

• rawCocycle_mem (ρ : boundaryFramedSurj boundaryPackage_GammaA child.lowerChild.fst child.lowerChild.snd) (x : { f : boundaryFramedSurj boundaryPackage_GammaA p.fst p.snd // sec7_frattiniTrivial_elementaryMStageCandidateLiftIndexProjectionFromRaw raw land f = ρ }) : diff.rawCocycleValue ρ x ∈ Lifting.Z1 cand.Sinv cand.T cand.P

  The raw affine-difference coordinates satisfy the candidate cocycle equation.

structure Q2Presentation.Induction.Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionCocycleToCurrentLiftMapData {p : FramedPair} {h : ¬ScalarTerminal p} {chief : FirstNonScalarChiefFactor p} {K : KernelMinimal chief.toNonScalarChiefFactor} {hR : Nat.card ↥(kernelFrattini K) = 1} {child : Sec7FrattiniTrivialElementaryChildData p} {cand : Sec7FrattiniTrivialElementaryMStageCandidateCocycleGroupData h chief K hR child} {raw : Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionRawMapData h chief K hR child} {land : Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionBoundaryLandingData raw} (toMap : Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionFiberToCocycleMapData cand raw land) : Type

Raw inverse-lift constructor: add a literal candidate cocycle to a fixed lower-child source and obtain a current candidate lift. The proof that this lift lies over the chosen lower source is separated below.

• ofCocycleLift (ρ : boundaryFramedSurj boundaryPackage_GammaA child.lowerChild.fst child.lowerChild.snd) : ↥(Lifting.Z1 cand.Sinv cand.T cand.P) → boundaryFramedSurj boundaryPackage_GammaA p.fst p.snd

  The current lift obtained by adding the cocycle to the lower-child source.

structure Q2Presentation.Induction.Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionCocycleToCurrentLiftProjectionData {p : FramedPair} {h : ¬ScalarTerminal p} {chief : FirstNonScalarChiefFactor p} {K : KernelMinimal chief.toNonScalarChiefFactor} {hR : Nat.card ↥(kernelFrattini K) = 1} {child : Sec7FrattiniTrivialElementaryChildData p} {cand : Sec7FrattiniTrivialElementaryMStageCandidateCocycleGroupData h chief K hR child} {raw : Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionRawMapData h chief K hR child} {land : Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionBoundaryLandingData raw} {toMap : Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionFiberToCocycleMapData cand raw land} (liftRaw : Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionCocycleToCurrentLiftMapData toMap) : Prop

Projection law for the raw inverse-lift constructor.

• ofCocycle_projection (ρ : boundaryFramedSurj boundaryPackage_GammaA child.lowerChild.fst child.lowerChild.snd) (z : ↥(Lifting.Z1 cand.Sinv cand.T cand.P)) : sec7_frattiniTrivial_elementaryMStageCandidateLiftIndexProjectionFromRaw raw land (liftRaw.ofCocycleLift ρ z) = ρ

  The lift produced from a cocycle projects back to the lower source used to build it.

structure Q2Presentation.Induction.Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionCocycleLiftLeftInverseValueData {p : FramedPair} {h : ¬ScalarTerminal p} {chief : FirstNonScalarChiefFactor p} {K : KernelMinimal chief.toNonScalarChiefFactor} {hR : Nat.card ↥(kernelFrattini K) = 1} {child : Sec7FrattiniTrivialElementaryChildData p} {cand : Sec7FrattiniTrivialElementaryMStageCandidateCocycleGroupData h chief K hR child} {raw : Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionRawMapData h chief K hR child} {land : Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionBoundaryLandingData raw} {toMap : Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionFiberToCocycleMapData cand raw land} (lift : Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionCocycleToFiberLiftData toMap) : Prop

Value-level left inverse law. Since Lifting.Z1 is a subtype of the ambient four-coordinate module, this removes the proof-irrelevant cocycle-membership component from the old left-inverse residual.

• to_of_val (ρ : boundaryFramedSurj boundaryPackage_GammaA child.lowerChild.fst child.lowerChild.snd) (z : ↥(Lifting.Z1 cand.Sinv cand.T cand.P)) : ↑(toMap.toCocycle ρ (lift.ofCocycle ρ z)) = ↑z

  The extracted cocycle has the same underlying four coordinates as the original cocycle.

structure Q2Presentation.Induction.Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionCocycleLiftRightInverseUnderlyingData {p : FramedPair} {h : ¬ScalarTerminal p} {chief : FirstNonScalarChiefFactor p} {K : KernelMinimal chief.toNonScalarChiefFactor} {hR : Nat.card ↥(kernelFrattini K) = 1} {child : Sec7FrattiniTrivialElementaryChildData p} {cand : Sec7FrattiniTrivialElementaryMStageCandidateCocycleGroupData h chief K hR child} {raw : Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionRawMapData h chief K hR child} {land : Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionBoundaryLandingData raw} {toMap : Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionFiberToCocycleMapData cand raw land} (lift : Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionCocycleToFiberLiftData toMap) : Prop

Underlying-lift right inverse law. This removes the proof-irrelevant projection component from the old equality of fibre-subtype elements.

• of_to_val (ρ : boundaryFramedSurj boundaryPackage_GammaA child.lowerChild.fst child.lowerChild.snd) (x : { f : boundaryFramedSurj boundaryPackage_GammaA p.fst p.snd // sec7_frattiniTrivial_elementaryMStageCandidateLiftIndexProjectionFromRaw raw land f = ρ }) : ↑(lift.ofCocycle ρ (toMap.toCocycle ρ x)) = ↑x

  Rebuilding from the extracted cocycle has the same underlying current lift as the original fibre element.

structure Q2Presentation.Induction.Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionCocycleMultiplicityDimData {p : FramedPair} {h : ¬ScalarTerminal p} {chief : FirstNonScalarChiefFactor p} {K : KernelMinimal chief.toNonScalarChiefFactor} {hR : Nat.card ↥(kernelFrattini K) = 1} {child : Sec7FrattiniTrivialElementaryChildData p} {mult : Sec7FrattiniTrivialElementaryMStageMultiplicityData child} (fib : Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionCocycleFiberEquivData h chief K hR child mult) : Prop

Dimension normalization below the old multiplicity residual. The multiplicity packet already says liftMultiplicity = 2^(2*mult.dimM); this datum identifies that chosen stage dimension with the literal concrete Section 7 module dimension.

• dimM_eq_sec7 : mult.dimM = sec7MStageDim K

  The M-stage dimension used in the multiplicity packet is the concrete dimension of towerM K.

Checkpoint354 coordinate/value lowering of the projection cocycle-fibre atoms

The checkpoint353 residuals already separated the candidate projection fibre into raw ambient differences, cocycle membership, current-lift construction, projection landing, proof-irrelevant inverse laws, and dimension normalization. The next layer below keeps the same public theorem API but makes three of those residuals more literal:

• raw affine differences are extracted coordinate-by-coordinate on the four marked generators before being bundled as an ambient A × A × A × A value;
• cocycle membership is reduced to the literal d¹ = 0 word-complex equation;
• construction of a current lift from a cocycle is split into its underlying current candidate epimorphism and the boundary-frame equation, and the projection law is reduced to equality of the underlying lower epimorphism.

structure Q2Presentation.Induction.Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionFiberRawDifferenceCoordinateData {p : FramedPair} {h : ¬ScalarTerminal p} {chief : FirstNonScalarChiefFactor p} {K : KernelMinimal chief.toNonScalarChiefFactor} {hR : Nat.card ↥(kernelFrattini K) = 1} {child : Sec7FrattiniTrivialElementaryChildData p} (cand : Sec7FrattiniTrivialElementaryMStageCandidateCocycleGroupData h chief K hR child) (raw : Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionRawMapData h chief K hR child) (land : Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionBoundaryLandingData raw) : Type

Coordinate extraction below the raw affine-difference map.

This is the literal four-generator coordinate form of the candidate lift-fibre difference: for a current lift over a lower source, it records separately the additive differences in M = towerM K on σ, τ, x₀, and x₁. The previous product-valued raw difference map is recovered theorem-level by bundling these four coordinates.

• rawSigma (ρ : boundaryFramedSurj boundaryPackage_GammaA child.lowerChild.fst child.lowerChild.snd) : { f : boundaryFramedSurj boundaryPackage_GammaA p.fst p.snd // sec7_frattiniTrivial_elementaryMStageCandidateLiftIndexProjectionFromRaw raw land f = ρ } → towerM K

  The σ-coordinate affine difference.

• rawTau (ρ : boundaryFramedSurj boundaryPackage_GammaA child.lowerChild.fst child.lowerChild.snd) : { f : boundaryFramedSurj boundaryPackage_GammaA p.fst p.snd // sec7_frattiniTrivial_elementaryMStageCandidateLiftIndexProjectionFromRaw raw land f = ρ } → towerM K

  The τ-coordinate affine difference.

• rawX0 (ρ : boundaryFramedSurj boundaryPackage_GammaA child.lowerChild.fst child.lowerChild.snd) : { f : boundaryFramedSurj boundaryPackage_GammaA p.fst p.snd // sec7_frattiniTrivial_elementaryMStageCandidateLiftIndexProjectionFromRaw raw land f = ρ } → towerM K

  The x₀-coordinate affine difference.

• rawX1 (ρ : boundaryFramedSurj boundaryPackage_GammaA child.lowerChild.fst child.lowerChild.snd) : { f : boundaryFramedSurj boundaryPackage_GammaA p.fst p.snd // sec7_frattiniTrivial_elementaryMStageCandidateLiftIndexProjectionFromRaw raw land f = ρ } → towerM K

  The x₁-coordinate affine difference.

Checkpoint355: coordinate-by-coordinate raw affine differences

The checkpoint354 coordinate residual is split into the four marked-generator coordinate functions. The old bundled coordinate datum is rebuilt theorem-level below.

structure Q2Presentation.Induction.Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionFiberRawDifferenceSigmaCoordinateData {p : FramedPair} {h : ¬ScalarTerminal p} {chief : FirstNonScalarChiefFactor p} {K : KernelMinimal chief.toNonScalarChiefFactor} {hR : Nat.card ↥(kernelFrattini K) = 1} {child : Sec7FrattiniTrivialElementaryChildData p} (cand : Sec7FrattiniTrivialElementaryMStageCandidateCocycleGroupData h chief K hR child) (raw : Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionRawMapData h chief K hR child) (land : Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionBoundaryLandingData raw) : Type

The σ raw affine-difference coordinate.

• rawSigma (ρ : boundaryFramedSurj boundaryPackage_GammaA child.lowerChild.fst child.lowerChild.snd) : { f : boundaryFramedSurj boundaryPackage_GammaA p.fst p.snd // sec7_frattiniTrivial_elementaryMStageCandidateLiftIndexProjectionFromRaw raw land f = ρ } → towerM K

structure Q2Presentation.Induction.Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionFiberRawDifferenceTauCoordinateData {p : FramedPair} {h : ¬ScalarTerminal p} {chief : FirstNonScalarChiefFactor p} {K : KernelMinimal chief.toNonScalarChiefFactor} {hR : Nat.card ↥(kernelFrattini K) = 1} {child : Sec7FrattiniTrivialElementaryChildData p} (cand : Sec7FrattiniTrivialElementaryMStageCandidateCocycleGroupData h chief K hR child) (raw : Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionRawMapData h chief K hR child) (land : Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionBoundaryLandingData raw) : Type

The τ raw affine-difference coordinate.

• rawTau (ρ : boundaryFramedSurj boundaryPackage_GammaA child.lowerChild.fst child.lowerChild.snd) : { f : boundaryFramedSurj boundaryPackage_GammaA p.fst p.snd // sec7_frattiniTrivial_elementaryMStageCandidateLiftIndexProjectionFromRaw raw land f = ρ } → towerM K

structure Q2Presentation.Induction.Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionFiberRawDifferenceX0CoordinateData {p : FramedPair} {h : ¬ScalarTerminal p} {chief : FirstNonScalarChiefFactor p} {K : KernelMinimal chief.toNonScalarChiefFactor} {hR : Nat.card ↥(kernelFrattini K) = 1} {child : Sec7FrattiniTrivialElementaryChildData p} (cand : Sec7FrattiniTrivialElementaryMStageCandidateCocycleGroupData h chief K hR child) (raw : Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionRawMapData h chief K hR child) (land : Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionBoundaryLandingData raw) : Type

The x₀ raw affine-difference coordinate.

• rawX0 (ρ : boundaryFramedSurj boundaryPackage_GammaA child.lowerChild.fst child.lowerChild.snd) : { f : boundaryFramedSurj boundaryPackage_GammaA p.fst p.snd // sec7_frattiniTrivial_elementaryMStageCandidateLiftIndexProjectionFromRaw raw land f = ρ } → towerM K

structure Q2Presentation.Induction.Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionFiberRawDifferenceX1CoordinateData {p : FramedPair} {h : ¬ScalarTerminal p} {chief : FirstNonScalarChiefFactor p} {K : KernelMinimal chief.toNonScalarChiefFactor} {hR : Nat.card ↥(kernelFrattini K) = 1} {child : Sec7FrattiniTrivialElementaryChildData p} (cand : Sec7FrattiniTrivialElementaryMStageCandidateCocycleGroupData h chief K hR child) (raw : Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionRawMapData h chief K hR child) (land : Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionBoundaryLandingData raw) : Type

The x₁ raw affine-difference coordinate.

• rawX1 (ρ : boundaryFramedSurj boundaryPackage_GammaA child.lowerChild.fst child.lowerChild.snd) : { f : boundaryFramedSurj boundaryPackage_GammaA p.fst p.snd // sec7_frattiniTrivial_elementaryMStageCandidateLiftIndexProjectionFromRaw raw land f = ρ } → towerM K

structure Q2Presentation.Induction.Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionFiberRawDifferenceSigmaCoordinatePointData {p : FramedPair} {h : ¬ScalarTerminal p} {chief : FirstNonScalarChiefFactor p} {K : KernelMinimal chief.toNonScalarChiefFactor} {hR : Nat.card ↥(kernelFrattini K) = 1} {child : Sec7FrattiniTrivialElementaryChildData p} (cand : Sec7FrattiniTrivialElementaryMStageCandidateCocycleGroupData h chief K hR child) (raw : Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionRawMapData h chief K hR child) (land : Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionBoundaryLandingData raw) (ρ : boundaryFramedSurj boundaryPackage_GammaA child.lowerChild.fst child.lowerChild.snd) (x : { f : boundaryFramedSurj boundaryPackage_GammaA p.fst p.snd // sec7_frattiniTrivial_elementaryMStageCandidateLiftIndexProjectionFromRaw raw land f = ρ }) : Type

Pointwise residual below the σ raw affine-difference coordinate.

Instead of assuming a whole coordinate-extraction function at once, this atom supplies the single σ coordinate for one lower source and one current lift in that projection fibre. The public coordinate-function packet is rebuilt theorem-level by choice over these pointwise coordinates.

• rawSigma : towerM K

  The pointwise σ additive affine-difference coordinate.

structure Q2Presentation.Induction.Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionFiberRawDifferenceSigmaCoordinateKernelQuotientPointData {p : FramedPair} {h : ¬ScalarTerminal p} {chief : FirstNonScalarChiefFactor p} {K : KernelMinimal chief.toNonScalarChiefFactor} {hR : Nat.card ↥(kernelFrattini K) = 1} {child : Sec7FrattiniTrivialElementaryChildData p} (cand : Sec7FrattiniTrivialElementaryMStageCandidateCocycleGroupData h chief K hR child) (raw : Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionRawMapData h chief K hR child) (land : Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionBoundaryLandingData raw) (ρ : boundaryFramedSurj boundaryPackage_GammaA child.lowerChild.fst child.lowerChild.snd) (x : { f : boundaryFramedSurj boundaryPackage_GammaA p.fst p.snd // sec7_frattiniTrivial_elementaryMStageCandidateLiftIndexProjectionFromRaw raw land f = ρ }) : Type

Lower residual below the pointwise Sigma affine-difference coordinate: the raw multiplicative Frattini-quotient value before additivizing it as an element of towerM K.

• rawSigmaQuotient : towerMgrp K

  The raw multiplicative quotient coordinate, prior to the Additive wrapper.

structure Q2Presentation.Induction.Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionFiberRawDifferenceSigmaCoordinateKernelRepresentativePointData {p : FramedPair} {h : ¬ScalarTerminal p} {chief : FirstNonScalarChiefFactor p} {K : KernelMinimal chief.toNonScalarChiefFactor} {hR : Nat.card ↥(kernelFrattini K) = 1} {child : Sec7FrattiniTrivialElementaryChildData p} (cand : Sec7FrattiniTrivialElementaryMStageCandidateCocycleGroupData h chief K hR child) (raw : Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionRawMapData h chief K hR child) (land : Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionBoundaryLandingData raw) (ρ : boundaryFramedSurj boundaryPackage_GammaA child.lowerChild.fst child.lowerChild.snd) (x : { f : boundaryFramedSurj boundaryPackage_GammaA p.fst p.snd // sec7_frattiniTrivial_elementaryMStageCandidateLiftIndexProjectionFromRaw raw land f = ρ }) : Type

Lower residual below the pointwise Sigma Frattini-quotient coordinate: a representative in the concrete kernel K before passing to K/Φ(K).

• rawSigmaKernelRepresentative : ↥K.Ksub

  The raw kernel representative whose Frattini-quotient class is the σ affine-difference coordinate.

theorem Q2Presentation.Induction.sec7_frattiniTrivial_elementaryMStageCandidateLiftIndexProjectionFiberRawDifferenceSigmaCoordinateKernelRepresentativePointData {p : FramedPair} {h : ¬ScalarTerminal p} {chief : FirstNonScalarChiefFactor p} {K : KernelMinimal chief.toNonScalarChiefFactor} {hR : Nat.card ↥(kernelFrattini K) = 1} {child : Sec7FrattiniTrivialElementaryChildData p} (cand : Sec7FrattiniTrivialElementaryMStageCandidateCocycleGroupData h chief K hR child) (raw : Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionRawMapData h chief K hR child) (land : Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionBoundaryLandingData raw) (ρ : boundaryFramedSurj boundaryPackage_GammaA child.lowerChild.fst child.lowerChild.snd) (x : { f : boundaryFramedSurj boundaryPackage_GammaA p.fst p.snd // sec7_frattiniTrivial_elementaryMStageCandidateLiftIndexProjectionFromRaw raw land f = ρ }) : Nonempty (Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionFiberRawDifferenceSigmaCoordinateKernelRepresentativePointData cand raw land ρ x)

Lower residual: construct the concrete-kernel representative of the Sigma coordinate.

theorem Q2Presentation.Induction.sec7_frattiniTrivial_elementaryMStageCandidateLiftIndexProjectionFiberRawDifferenceSigmaCoordinateKernelQuotientPointData {p : FramedPair} {h : ¬ScalarTerminal p} {chief : FirstNonScalarChiefFactor p} {K : KernelMinimal chief.toNonScalarChiefFactor} {hR : Nat.card ↥(kernelFrattini K) = 1} {child : Sec7FrattiniTrivialElementaryChildData p} (cand : Sec7FrattiniTrivialElementaryMStageCandidateCocycleGroupData h chief K hR child) (raw : Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionRawMapData h chief K hR child) (land : Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionBoundaryLandingData raw) (ρ : boundaryFramedSurj boundaryPackage_GammaA child.lowerChild.fst child.lowerChild.snd) (x : { f : boundaryFramedSurj boundaryPackage_GammaA p.fst p.snd // sec7_frattiniTrivial_elementaryMStageCandidateLiftIndexProjectionFromRaw raw land f = ρ }) : Nonempty (Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionFiberRawDifferenceSigmaCoordinateKernelQuotientPointData cand raw land ρ x)

Former multiplicative Frattini-quotient Sigma coordinate residual, now theorem-level from a concrete-kernel representative by quotienting modulo Φ(K).

theorem Q2Presentation.Induction.sec7_frattiniTrivial_elementaryMStageCandidateLiftIndexProjectionFiberRawDifferenceSigmaCoordinatePointData {p : FramedPair} {h : ¬ScalarTerminal p} {chief : FirstNonScalarChiefFactor p} {K : KernelMinimal chief.toNonScalarChiefFactor} {hR : Nat.card ↥(kernelFrattini K) = 1} {child : Sec7FrattiniTrivialElementaryChildData p} (cand : Sec7FrattiniTrivialElementaryMStageCandidateCocycleGroupData h chief K hR child) (raw : Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionRawMapData h chief K hR child) (land : Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionBoundaryLandingData raw) (ρ : boundaryFramedSurj boundaryPackage_GammaA child.lowerChild.fst child.lowerChild.snd) (x : { f : boundaryFramedSurj boundaryPackage_GammaA p.fst p.snd // sec7_frattiniTrivial_elementaryMStageCandidateLiftIndexProjectionFromRaw raw land f = ρ }) : Nonempty (Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionFiberRawDifferenceSigmaCoordinatePointData cand raw land ρ x)

Former pointwise Sigma coordinate residual, now theorem-level by additivizing the multiplicative Frattini-quotient coordinate.

theorem Q2Presentation.Induction.sec7_frattiniTrivial_elementaryMStageCandidateLiftIndexProjectionFiberRawDifferenceSigmaCoordinateData {p : FramedPair} {h : ¬ScalarTerminal p} {chief : FirstNonScalarChiefFactor p} {K : KernelMinimal chief.toNonScalarChiefFactor} {hR : Nat.card ↥(kernelFrattini K) = 1} {child : Sec7FrattiniTrivialElementaryChildData p} (cand : Sec7FrattiniTrivialElementaryMStageCandidateCocycleGroupData h chief K hR child) (raw : Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionRawMapData h chief K hR child) (land : Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionBoundaryLandingData raw) : Nonempty (Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionFiberRawDifferenceSigmaCoordinateData cand raw land)

Former σ coordinate residual, now theorem-level from pointwise coordinate values over the projection fibre.

structure Q2Presentation.Induction.Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionFiberRawDifferenceTauCoordinatePointData {p : FramedPair} {h : ¬ScalarTerminal p} {chief : FirstNonScalarChiefFactor p} {K : KernelMinimal chief.toNonScalarChiefFactor} {hR : Nat.card ↥(kernelFrattini K) = 1} {child : Sec7FrattiniTrivialElementaryChildData p} (cand : Sec7FrattiniTrivialElementaryMStageCandidateCocycleGroupData h chief K hR child) (raw : Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionRawMapData h chief K hR child) (land : Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionBoundaryLandingData raw) (ρ : boundaryFramedSurj boundaryPackage_GammaA child.lowerChild.fst child.lowerChild.snd) (x : { f : boundaryFramedSurj boundaryPackage_GammaA p.fst p.snd // sec7_frattiniTrivial_elementaryMStageCandidateLiftIndexProjectionFromRaw raw land f = ρ }) : Type

Pointwise residual below the τ raw affine-difference coordinate.

Instead of assuming a whole coordinate-extraction function at once, this atom supplies the single τ coordinate for one lower source and one current lift in that projection fibre. The public coordinate-function packet is rebuilt theorem-level by choice over these pointwise coordinates.

• rawTau : towerM K

  The pointwise τ additive affine-difference coordinate.

structure Q2Presentation.Induction.Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionFiberRawDifferenceTauCoordinateKernelQuotientPointData {p : FramedPair} {h : ¬ScalarTerminal p} {chief : FirstNonScalarChiefFactor p} {K : KernelMinimal chief.toNonScalarChiefFactor} {hR : Nat.card ↥(kernelFrattini K) = 1} {child : Sec7FrattiniTrivialElementaryChildData p} (cand : Sec7FrattiniTrivialElementaryMStageCandidateCocycleGroupData h chief K hR child) (raw : Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionRawMapData h chief K hR child) (land : Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionBoundaryLandingData raw) (ρ : boundaryFramedSurj boundaryPackage_GammaA child.lowerChild.fst child.lowerChild.snd) (x : { f : boundaryFramedSurj boundaryPackage_GammaA p.fst p.snd // sec7_frattiniTrivial_elementaryMStageCandidateLiftIndexProjectionFromRaw raw land f = ρ }) : Type

Lower residual below the pointwise Tau affine-difference coordinate: the raw multiplicative Frattini-quotient value before additivizing it as an element of towerM K.

• rawTauQuotient : towerMgrp K

  The raw multiplicative quotient coordinate, prior to the Additive wrapper.

structure Q2Presentation.Induction.Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionFiberRawDifferenceTauCoordinateKernelRepresentativePointData {p : FramedPair} {h : ¬ScalarTerminal p} {chief : FirstNonScalarChiefFactor p} {K : KernelMinimal chief.toNonScalarChiefFactor} {hR : Nat.card ↥(kernelFrattini K) = 1} {child : Sec7FrattiniTrivialElementaryChildData p} (cand : Sec7FrattiniTrivialElementaryMStageCandidateCocycleGroupData h chief K hR child) (raw : Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionRawMapData h chief K hR child) (land : Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionBoundaryLandingData raw) (ρ : boundaryFramedSurj boundaryPackage_GammaA child.lowerChild.fst child.lowerChild.snd) (x : { f : boundaryFramedSurj boundaryPackage_GammaA p.fst p.snd // sec7_frattiniTrivial_elementaryMStageCandidateLiftIndexProjectionFromRaw raw land f = ρ }) : Type

Lower residual below the pointwise Tau Frattini-quotient coordinate: a representative in the concrete kernel K before passing to K/Φ(K).

• rawTauKernelRepresentative : ↥K.Ksub

  The raw kernel representative whose Frattini-quotient class is the τ affine-difference coordinate.

theorem Q2Presentation.Induction.sec7_frattiniTrivial_elementaryMStageCandidateLiftIndexProjectionFiberRawDifferenceTauCoordinateKernelRepresentativePointData {p : FramedPair} {h : ¬ScalarTerminal p} {chief : FirstNonScalarChiefFactor p} {K : KernelMinimal chief.toNonScalarChiefFactor} {hR : Nat.card ↥(kernelFrattini K) = 1} {child : Sec7FrattiniTrivialElementaryChildData p} (cand : Sec7FrattiniTrivialElementaryMStageCandidateCocycleGroupData h chief K hR child) (raw : Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionRawMapData h chief K hR child) (land : Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionBoundaryLandingData raw) (ρ : boundaryFramedSurj boundaryPackage_GammaA child.lowerChild.fst child.lowerChild.snd) (x : { f : boundaryFramedSurj boundaryPackage_GammaA p.fst p.snd // sec7_frattiniTrivial_elementaryMStageCandidateLiftIndexProjectionFromRaw raw land f = ρ }) : Nonempty (Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionFiberRawDifferenceTauCoordinateKernelRepresentativePointData cand raw land ρ x)

Lower residual: construct the concrete-kernel representative of the Tau coordinate.

theorem Q2Presentation.Induction.sec7_frattiniTrivial_elementaryMStageCandidateLiftIndexProjectionFiberRawDifferenceTauCoordinateKernelQuotientPointData {p : FramedPair} {h : ¬ScalarTerminal p} {chief : FirstNonScalarChiefFactor p} {K : KernelMinimal chief.toNonScalarChiefFactor} {hR : Nat.card ↥(kernelFrattini K) = 1} {child : Sec7FrattiniTrivialElementaryChildData p} (cand : Sec7FrattiniTrivialElementaryMStageCandidateCocycleGroupData h chief K hR child) (raw : Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionRawMapData h chief K hR child) (land : Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionBoundaryLandingData raw) (ρ : boundaryFramedSurj boundaryPackage_GammaA child.lowerChild.fst child.lowerChild.snd) (x : { f : boundaryFramedSurj boundaryPackage_GammaA p.fst p.snd // sec7_frattiniTrivial_elementaryMStageCandidateLiftIndexProjectionFromRaw raw land f = ρ }) : Nonempty (Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionFiberRawDifferenceTauCoordinateKernelQuotientPointData cand raw land ρ x)

Former multiplicative Frattini-quotient Tau coordinate residual, now theorem-level from a concrete-kernel representative by quotienting modulo Φ(K).

theorem Q2Presentation.Induction.sec7_frattiniTrivial_elementaryMStageCandidateLiftIndexProjectionFiberRawDifferenceTauCoordinatePointData {p : FramedPair} {h : ¬ScalarTerminal p} {chief : FirstNonScalarChiefFactor p} {K : KernelMinimal chief.toNonScalarChiefFactor} {hR : Nat.card ↥(kernelFrattini K) = 1} {child : Sec7FrattiniTrivialElementaryChildData p} (cand : Sec7FrattiniTrivialElementaryMStageCandidateCocycleGroupData h chief K hR child) (raw : Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionRawMapData h chief K hR child) (land : Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionBoundaryLandingData raw) (ρ : boundaryFramedSurj boundaryPackage_GammaA child.lowerChild.fst child.lowerChild.snd) (x : { f : boundaryFramedSurj boundaryPackage_GammaA p.fst p.snd // sec7_frattiniTrivial_elementaryMStageCandidateLiftIndexProjectionFromRaw raw land f = ρ }) : Nonempty (Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionFiberRawDifferenceTauCoordinatePointData cand raw land ρ x)

Former pointwise Tau coordinate residual, now theorem-level by additivizing the multiplicative Frattini-quotient coordinate.

theorem Q2Presentation.Induction.sec7_frattiniTrivial_elementaryMStageCandidateLiftIndexProjectionFiberRawDifferenceTauCoordinateData {p : FramedPair} {h : ¬ScalarTerminal p} {chief : FirstNonScalarChiefFactor p} {K : KernelMinimal chief.toNonScalarChiefFactor} {hR : Nat.card ↥(kernelFrattini K) = 1} {child : Sec7FrattiniTrivialElementaryChildData p} (cand : Sec7FrattiniTrivialElementaryMStageCandidateCocycleGroupData h chief K hR child) (raw : Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionRawMapData h chief K hR child) (land : Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionBoundaryLandingData raw) : Nonempty (Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionFiberRawDifferenceTauCoordinateData cand raw land)

Former τ coordinate residual, now theorem-level from pointwise coordinate values over the projection fibre.

structure Q2Presentation.Induction.Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionFiberRawDifferenceX0CoordinatePointData {p : FramedPair} {h : ¬ScalarTerminal p} {chief : FirstNonScalarChiefFactor p} {K : KernelMinimal chief.toNonScalarChiefFactor} {hR : Nat.card ↥(kernelFrattini K) = 1} {child : Sec7FrattiniTrivialElementaryChildData p} (cand : Sec7FrattiniTrivialElementaryMStageCandidateCocycleGroupData h chief K hR child) (raw : Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionRawMapData h chief K hR child) (land : Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionBoundaryLandingData raw) (ρ : boundaryFramedSurj boundaryPackage_GammaA child.lowerChild.fst child.lowerChild.snd) (x : { f : boundaryFramedSurj boundaryPackage_GammaA p.fst p.snd // sec7_frattiniTrivial_elementaryMStageCandidateLiftIndexProjectionFromRaw raw land f = ρ }) : Type

Pointwise residual below the x₀ raw affine-difference coordinate.

Instead of assuming a whole coordinate-extraction function at once, this atom supplies the single x₀ coordinate for one lower source and one current lift in that projection fibre. The public coordinate-function packet is rebuilt theorem-level by choice over these pointwise coordinates.

• rawX0 : towerM K

  The pointwise x₀ additive affine-difference coordinate.

structure Q2Presentation.Induction.Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionFiberRawDifferenceX0CoordinateKernelQuotientPointData {p : FramedPair} {h : ¬ScalarTerminal p} {chief : FirstNonScalarChiefFactor p} {K : KernelMinimal chief.toNonScalarChiefFactor} {hR : Nat.card ↥(kernelFrattini K) = 1} {child : Sec7FrattiniTrivialElementaryChildData p} (cand : Sec7FrattiniTrivialElementaryMStageCandidateCocycleGroupData h chief K hR child) (raw : Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionRawMapData h chief K hR child) (land : Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionBoundaryLandingData raw) (ρ : boundaryFramedSurj boundaryPackage_GammaA child.lowerChild.fst child.lowerChild.snd) (x : { f : boundaryFramedSurj boundaryPackage_GammaA p.fst p.snd // sec7_frattiniTrivial_elementaryMStageCandidateLiftIndexProjectionFromRaw raw land f = ρ }) : Type

Lower residual below the pointwise X0 affine-difference coordinate: the raw multiplicative Frattini-quotient value before additivizing it as an element of towerM K.

• rawX0Quotient : towerMgrp K

  The raw multiplicative quotient coordinate, prior to the Additive wrapper.

structure Q2Presentation.Induction.Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionFiberRawDifferenceX0CoordinateKernelRepresentativePointData {p : FramedPair} {h : ¬ScalarTerminal p} {chief : FirstNonScalarChiefFactor p} {K : KernelMinimal chief.toNonScalarChiefFactor} {hR : Nat.card ↥(kernelFrattini K) = 1} {child : Sec7FrattiniTrivialElementaryChildData p} (cand : Sec7FrattiniTrivialElementaryMStageCandidateCocycleGroupData h chief K hR child) (raw : Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionRawMapData h chief K hR child) (land : Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionBoundaryLandingData raw) (ρ : boundaryFramedSurj boundaryPackage_GammaA child.lowerChild.fst child.lowerChild.snd) (x : { f : boundaryFramedSurj boundaryPackage_GammaA p.fst p.snd // sec7_frattiniTrivial_elementaryMStageCandidateLiftIndexProjectionFromRaw raw land f = ρ }) : Type

Lower residual below the pointwise X0 Frattini-quotient coordinate: a representative in the concrete kernel K before passing to K/Φ(K).

• rawX0KernelRepresentative : ↥K.Ksub

  The raw kernel representative whose Frattini-quotient class is the x₀ affine-difference coordinate.

theorem Q2Presentation.Induction.sec7_frattiniTrivial_elementaryMStageCandidateLiftIndexProjectionFiberRawDifferenceX0CoordinateKernelRepresentativePointData {p : FramedPair} {h : ¬ScalarTerminal p} {chief : FirstNonScalarChiefFactor p} {K : KernelMinimal chief.toNonScalarChiefFactor} {hR : Nat.card ↥(kernelFrattini K) = 1} {child : Sec7FrattiniTrivialElementaryChildData p} (cand : Sec7FrattiniTrivialElementaryMStageCandidateCocycleGroupData h chief K hR child) (raw : Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionRawMapData h chief K hR child) (land : Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionBoundaryLandingData raw) (ρ : boundaryFramedSurj boundaryPackage_GammaA child.lowerChild.fst child.lowerChild.snd) (x : { f : boundaryFramedSurj boundaryPackage_GammaA p.fst p.snd // sec7_frattiniTrivial_elementaryMStageCandidateLiftIndexProjectionFromRaw raw land f = ρ }) : Nonempty (Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionFiberRawDifferenceX0CoordinateKernelRepresentativePointData cand raw land ρ x)

Lower residual: construct the concrete-kernel representative of the X0 coordinate.

theorem Q2Presentation.Induction.sec7_frattiniTrivial_elementaryMStageCandidateLiftIndexProjectionFiberRawDifferenceX0CoordinateKernelQuotientPointData {p : FramedPair} {h : ¬ScalarTerminal p} {chief : FirstNonScalarChiefFactor p} {K : KernelMinimal chief.toNonScalarChiefFactor} {hR : Nat.card ↥(kernelFrattini K) = 1} {child : Sec7FrattiniTrivialElementaryChildData p} (cand : Sec7FrattiniTrivialElementaryMStageCandidateCocycleGroupData h chief K hR child) (raw : Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionRawMapData h chief K hR child) (land : Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionBoundaryLandingData raw) (ρ : boundaryFramedSurj boundaryPackage_GammaA child.lowerChild.fst child.lowerChild.snd) (x : { f : boundaryFramedSurj boundaryPackage_GammaA p.fst p.snd // sec7_frattiniTrivial_elementaryMStageCandidateLiftIndexProjectionFromRaw raw land f = ρ }) : Nonempty (Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionFiberRawDifferenceX0CoordinateKernelQuotientPointData cand raw land ρ x)

Former multiplicative Frattini-quotient X0 coordinate residual, now theorem-level from a concrete-kernel representative by quotienting modulo Φ(K).

theorem Q2Presentation.Induction.sec7_frattiniTrivial_elementaryMStageCandidateLiftIndexProjectionFiberRawDifferenceX0CoordinatePointData {p : FramedPair} {h : ¬ScalarTerminal p} {chief : FirstNonScalarChiefFactor p} {K : KernelMinimal chief.toNonScalarChiefFactor} {hR : Nat.card ↥(kernelFrattini K) = 1} {child : Sec7FrattiniTrivialElementaryChildData p} (cand : Sec7FrattiniTrivialElementaryMStageCandidateCocycleGroupData h chief K hR child) (raw : Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionRawMapData h chief K hR child) (land : Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionBoundaryLandingData raw) (ρ : boundaryFramedSurj boundaryPackage_GammaA child.lowerChild.fst child.lowerChild.snd) (x : { f : boundaryFramedSurj boundaryPackage_GammaA p.fst p.snd // sec7_frattiniTrivial_elementaryMStageCandidateLiftIndexProjectionFromRaw raw land f = ρ }) : Nonempty (Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionFiberRawDifferenceX0CoordinatePointData cand raw land ρ x)

Former pointwise X0 coordinate residual, now theorem-level by additivizing the multiplicative Frattini-quotient coordinate.

theorem Q2Presentation.Induction.sec7_frattiniTrivial_elementaryMStageCandidateLiftIndexProjectionFiberRawDifferenceX0CoordinateData {p : FramedPair} {h : ¬ScalarTerminal p} {chief : FirstNonScalarChiefFactor p} {K : KernelMinimal chief.toNonScalarChiefFactor} {hR : Nat.card ↥(kernelFrattini K) = 1} {child : Sec7FrattiniTrivialElementaryChildData p} (cand : Sec7FrattiniTrivialElementaryMStageCandidateCocycleGroupData h chief K hR child) (raw : Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionRawMapData h chief K hR child) (land : Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionBoundaryLandingData raw) : Nonempty (Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionFiberRawDifferenceX0CoordinateData cand raw land)

Former x₀ coordinate residual, now theorem-level from pointwise coordinate values over the projection fibre.

structure Q2Presentation.Induction.Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionFiberRawDifferenceX1CoordinatePointData {p : FramedPair} {h : ¬ScalarTerminal p} {chief : FirstNonScalarChiefFactor p} {K : KernelMinimal chief.toNonScalarChiefFactor} {hR : Nat.card ↥(kernelFrattini K) = 1} {child : Sec7FrattiniTrivialElementaryChildData p} (cand : Sec7FrattiniTrivialElementaryMStageCandidateCocycleGroupData h chief K hR child) (raw : Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionRawMapData h chief K hR child) (land : Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionBoundaryLandingData raw) (ρ : boundaryFramedSurj boundaryPackage_GammaA child.lowerChild.fst child.lowerChild.snd) (x : { f : boundaryFramedSurj boundaryPackage_GammaA p.fst p.snd // sec7_frattiniTrivial_elementaryMStageCandidateLiftIndexProjectionFromRaw raw land f = ρ }) : Type

Pointwise residual below the x₁ raw affine-difference coordinate.

Instead of assuming a whole coordinate-extraction function at once, this atom supplies the single x₁ coordinate for one lower source and one current lift in that projection fibre. The public coordinate-function packet is rebuilt theorem-level by choice over these pointwise coordinates.

• rawX1 : towerM K

  The pointwise x₁ additive affine-difference coordinate.

structure Q2Presentation.Induction.Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionFiberRawDifferenceX1CoordinateKernelQuotientPointData {p : FramedPair} {h : ¬ScalarTerminal p} {chief : FirstNonScalarChiefFactor p} {K : KernelMinimal chief.toNonScalarChiefFactor} {hR : Nat.card ↥(kernelFrattini K) = 1} {child : Sec7FrattiniTrivialElementaryChildData p} (cand : Sec7FrattiniTrivialElementaryMStageCandidateCocycleGroupData h chief K hR child) (raw : Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionRawMapData h chief K hR child) (land : Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionBoundaryLandingData raw) (ρ : boundaryFramedSurj boundaryPackage_GammaA child.lowerChild.fst child.lowerChild.snd) (x : { f : boundaryFramedSurj boundaryPackage_GammaA p.fst p.snd // sec7_frattiniTrivial_elementaryMStageCandidateLiftIndexProjectionFromRaw raw land f = ρ }) : Type

Lower residual below the pointwise X1 affine-difference coordinate: the raw multiplicative Frattini-quotient value before additivizing it as an element of towerM K.

• rawX1Quotient : towerMgrp K

  The raw multiplicative quotient coordinate, prior to the Additive wrapper.

structure Q2Presentation.Induction.Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionFiberRawDifferenceX1CoordinateKernelRepresentativePointData {p : FramedPair} {h : ¬ScalarTerminal p} {chief : FirstNonScalarChiefFactor p} {K : KernelMinimal chief.toNonScalarChiefFactor} {hR : Nat.card ↥(kernelFrattini K) = 1} {child : Sec7FrattiniTrivialElementaryChildData p} (cand : Sec7FrattiniTrivialElementaryMStageCandidateCocycleGroupData h chief K hR child) (raw : Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionRawMapData h chief K hR child) (land : Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionBoundaryLandingData raw) (ρ : boundaryFramedSurj boundaryPackage_GammaA child.lowerChild.fst child.lowerChild.snd) (x : { f : boundaryFramedSurj boundaryPackage_GammaA p.fst p.snd // sec7_frattiniTrivial_elementaryMStageCandidateLiftIndexProjectionFromRaw raw land f = ρ }) : Type

Lower residual below the pointwise X1 Frattini-quotient coordinate: a representative in the concrete kernel K before passing to K/Φ(K).

• rawX1KernelRepresentative : ↥K.Ksub

  The raw kernel representative whose Frattini-quotient class is the x₁ affine-difference coordinate.

theorem Q2Presentation.Induction.sec7_frattiniTrivial_elementaryMStageCandidateLiftIndexProjectionFiberRawDifferenceX1CoordinateKernelRepresentativePointData {p : FramedPair} {h : ¬ScalarTerminal p} {chief : FirstNonScalarChiefFactor p} {K : KernelMinimal chief.toNonScalarChiefFactor} {hR : Nat.card ↥(kernelFrattini K) = 1} {child : Sec7FrattiniTrivialElementaryChildData p} (cand : Sec7FrattiniTrivialElementaryMStageCandidateCocycleGroupData h chief K hR child) (raw : Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionRawMapData h chief K hR child) (land : Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionBoundaryLandingData raw) (ρ : boundaryFramedSurj boundaryPackage_GammaA child.lowerChild.fst child.lowerChild.snd) (x : { f : boundaryFramedSurj boundaryPackage_GammaA p.fst p.snd // sec7_frattiniTrivial_elementaryMStageCandidateLiftIndexProjectionFromRaw raw land f = ρ }) : Nonempty (Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionFiberRawDifferenceX1CoordinateKernelRepresentativePointData cand raw land ρ x)

Lower residual: construct the concrete-kernel representative of the X1 coordinate.

theorem Q2Presentation.Induction.sec7_frattiniTrivial_elementaryMStageCandidateLiftIndexProjectionFiberRawDifferenceX1CoordinateKernelQuotientPointData {p : FramedPair} {h : ¬ScalarTerminal p} {chief : FirstNonScalarChiefFactor p} {K : KernelMinimal chief.toNonScalarChiefFactor} {hR : Nat.card ↥(kernelFrattini K) = 1} {child : Sec7FrattiniTrivialElementaryChildData p} (cand : Sec7FrattiniTrivialElementaryMStageCandidateCocycleGroupData h chief K hR child) (raw : Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionRawMapData h chief K hR child) (land : Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionBoundaryLandingData raw) (ρ : boundaryFramedSurj boundaryPackage_GammaA child.lowerChild.fst child.lowerChild.snd) (x : { f : boundaryFramedSurj boundaryPackage_GammaA p.fst p.snd // sec7_frattiniTrivial_elementaryMStageCandidateLiftIndexProjectionFromRaw raw land f = ρ }) : Nonempty (Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionFiberRawDifferenceX1CoordinateKernelQuotientPointData cand raw land ρ x)

Former multiplicative Frattini-quotient X1 coordinate residual, now theorem-level from a concrete-kernel representative by quotienting modulo Φ(K).

theorem Q2Presentation.Induction.sec7_frattiniTrivial_elementaryMStageCandidateLiftIndexProjectionFiberRawDifferenceX1CoordinatePointData {p : FramedPair} {h : ¬ScalarTerminal p} {chief : FirstNonScalarChiefFactor p} {K : KernelMinimal chief.toNonScalarChiefFactor} {hR : Nat.card ↥(kernelFrattini K) = 1} {child : Sec7FrattiniTrivialElementaryChildData p} (cand : Sec7FrattiniTrivialElementaryMStageCandidateCocycleGroupData h chief K hR child) (raw : Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionRawMapData h chief K hR child) (land : Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionBoundaryLandingData raw) (ρ : boundaryFramedSurj boundaryPackage_GammaA child.lowerChild.fst child.lowerChild.snd) (x : { f : boundaryFramedSurj boundaryPackage_GammaA p.fst p.snd // sec7_frattiniTrivial_elementaryMStageCandidateLiftIndexProjectionFromRaw raw land f = ρ }) : Nonempty (Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionFiberRawDifferenceX1CoordinatePointData cand raw land ρ x)

Former pointwise X1 coordinate residual, now theorem-level by additivizing the multiplicative Frattini-quotient coordinate.

theorem Q2Presentation.Induction.sec7_frattiniTrivial_elementaryMStageCandidateLiftIndexProjectionFiberRawDifferenceX1CoordinateData {p : FramedPair} {h : ¬ScalarTerminal p} {chief : FirstNonScalarChiefFactor p} {K : KernelMinimal chief.toNonScalarChiefFactor} {hR : Nat.card ↥(kernelFrattini K) = 1} {child : Sec7FrattiniTrivialElementaryChildData p} (cand : Sec7FrattiniTrivialElementaryMStageCandidateCocycleGroupData h chief K hR child) (raw : Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionRawMapData h chief K hR child) (land : Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionBoundaryLandingData raw) : Nonempty (Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionFiberRawDifferenceX1CoordinateData cand raw land)

Former x₁ coordinate residual, now theorem-level from pointwise coordinate values over the projection fibre.

theorem Q2Presentation.Induction.sec7_frattiniTrivial_elementaryMStageCandidateLiftIndexProjectionFiberRawDifferenceCoordinateData {p : FramedPair} {h : ¬ScalarTerminal p} {chief : FirstNonScalarChiefFactor p} {K : KernelMinimal chief.toNonScalarChiefFactor} {hR : Nat.card ↥(kernelFrattini K) = 1} {child : Sec7FrattiniTrivialElementaryChildData p} (cand : Sec7FrattiniTrivialElementaryMStageCandidateCocycleGroupData h chief K hR child) (raw : Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionRawMapData h chief K hR child) (land : Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionBoundaryLandingData raw) : Nonempty (Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionFiberRawDifferenceCoordinateData cand raw land)

Former coordinate residual, now theorem-level from the four marked-generator coordinate constructors.

theorem Q2Presentation.Induction.sec7_frattiniTrivial_elementaryMStageCandidateLiftIndexProjectionFiberRawDifferenceMapData {p : FramedPair} {h : ¬ScalarTerminal p} {chief : FirstNonScalarChiefFactor p} {K : KernelMinimal chief.toNonScalarChiefFactor} {hR : Nat.card ↥(kernelFrattini K) = 1} {child : Sec7FrattiniTrivialElementaryChildData p} (cand : Sec7FrattiniTrivialElementaryMStageCandidateCocycleGroupData h chief K hR child) (raw : Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionRawMapData h chief K hR child) (land : Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionBoundaryLandingData raw) : Nonempty (Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionFiberRawDifferenceMapData cand raw land)

Former raw-difference residual, now theorem-level from the four coordinate extraction maps.

structure Q2Presentation.Induction.Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionFiberRawDifferenceD1ZeroData {p : FramedPair} {h : ¬ScalarTerminal p} {chief : FirstNonScalarChiefFactor p} {K : KernelMinimal chief.toNonScalarChiefFactor} {hR : Nat.card ↥(kernelFrattini K) = 1} {child : Sec7FrattiniTrivialElementaryChildData p} {cand : Sec7FrattiniTrivialElementaryMStageCandidateCocycleGroupData h chief K hR child} {raw : Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionRawMapData h chief K hR child} {land : Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionBoundaryLandingData raw} (diff : Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionFiberRawDifferenceMapData cand raw land) : Prop

Literal d¹ = 0 equation below the raw affine-difference cocycle-membership proof. This states the word-complex condition directly in the ambient four-coordinate module, before it is repackaged as membership in the kernel submodule Lifting.Z1.

• rawCocycle_d1_zero (ρ : boundaryFramedSurj boundaryPackage_GammaA child.lowerChild.fst child.lowerChild.snd) (x : { f : boundaryFramedSurj boundaryPackage_GammaA p.fst p.snd // sec7_frattiniTrivial_elementaryMStageCandidateLiftIndexProjectionFromRaw raw land f = ρ }) : (Lifting.d1 cand.Sinv cand.T cand.P) (diff.rawCocycleValue ρ x) = 0

  The extracted raw difference is killed by the evaluated candidate d¹.

structure Q2Presentation.Induction.Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionFiberRawDifferenceD1TameRowZeroData {p : FramedPair} {h : ¬ScalarTerminal p} {chief : FirstNonScalarChiefFactor p} {K : KernelMinimal chief.toNonScalarChiefFactor} {hR : Nat.card ↥(kernelFrattini K) = 1} {child : Sec7FrattiniTrivialElementaryChildData p} {cand : Sec7FrattiniTrivialElementaryMStageCandidateCocycleGroupData h chief K hR child} {raw : Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionRawMapData h chief K hR child} {land : Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionBoundaryLandingData raw} (diff : Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionFiberRawDifferenceMapData cand raw land) : Prop

Tame-row half of the literal d¹ = 0 equation.

• rawCocycle_d1_tame_zero (ρ : boundaryFramedSurj boundaryPackage_GammaA child.lowerChild.fst child.lowerChild.snd) (x : { f : boundaryFramedSurj boundaryPackage_GammaA p.fst p.snd // sec7_frattiniTrivial_elementaryMStageCandidateLiftIndexProjectionFromRaw raw land f = ρ }) : ((Lifting.d1 cand.Sinv cand.T cand.P) (diff.rawCocycleValue ρ x)).1 = 0

structure Q2Presentation.Induction.Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionFiberRawDifferenceD1WildRowZeroData {p : FramedPair} {h : ¬ScalarTerminal p} {chief : FirstNonScalarChiefFactor p} {K : KernelMinimal chief.toNonScalarChiefFactor} {hR : Nat.card ↥(kernelFrattini K) = 1} {child : Sec7FrattiniTrivialElementaryChildData p} {cand : Sec7FrattiniTrivialElementaryMStageCandidateCocycleGroupData h chief K hR child} {raw : Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionRawMapData h chief K hR child} {land : Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionBoundaryLandingData raw} (diff : Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionFiberRawDifferenceMapData cand raw land) : Prop

Wild-row half of the literal d¹ = 0 equation.

• rawCocycle_d1_wild_zero (ρ : boundaryFramedSurj boundaryPackage_GammaA child.lowerChild.fst child.lowerChild.snd) (x : { f : boundaryFramedSurj boundaryPackage_GammaA p.fst p.snd // sec7_frattiniTrivial_elementaryMStageCandidateLiftIndexProjectionFromRaw raw land f = ρ }) : ((Lifting.d1 cand.Sinv cand.T cand.P) (diff.rawCocycleValue ρ x)).2 = 0

structure Q2Presentation.Induction.Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionFiberRawDifferenceD1TameRowZeroPointData {p : FramedPair} {h : ¬ScalarTerminal p} {chief : FirstNonScalarChiefFactor p} {K : KernelMinimal chief.toNonScalarChiefFactor} {hR : Nat.card ↥(kernelFrattini K) = 1} {child : Sec7FrattiniTrivialElementaryChildData p} {cand : Sec7FrattiniTrivialElementaryMStageCandidateCocycleGroupData h chief K hR child} {raw : Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionRawMapData h chief K hR child} {land : Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionBoundaryLandingData raw} (diff : Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionFiberRawDifferenceMapData cand raw land) (ρ : boundaryFramedSurj boundaryPackage_GammaA child.lowerChild.fst child.lowerChild.snd) (x : { f : boundaryFramedSurj boundaryPackage_GammaA p.fst p.snd // sec7_frattiniTrivial_elementaryMStageCandidateLiftIndexProjectionFromRaw raw land f = ρ }) : Prop

Pointwise residual below the tame-row half of the literal d¹ = 0 equation.

It proves the selected row of Lifting.d1 vanishes for one raw affine-difference value; the former fibrewise row-zero packet is recovered theorem-level by assembling these pointwise proofs.

• rawCocycle_d1_tame_zero : ((Lifting.d1 cand.Sinv cand.T cand.P) (diff.rawCocycleValue ρ x)).1 = 0

structure Q2Presentation.Induction.Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionFiberRawDifferenceD1TameRowFormulaPointData {p : FramedPair} {h : ¬ScalarTerminal p} {chief : FirstNonScalarChiefFactor p} {K : KernelMinimal chief.toNonScalarChiefFactor} {hR : Nat.card ↥(kernelFrattini K) = 1} {child : Sec7FrattiniTrivialElementaryChildData p} {cand : Sec7FrattiniTrivialElementaryMStageCandidateCocycleGroupData h chief K hR child} {raw : Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionRawMapData h chief K hR child} {land : Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionBoundaryLandingData raw} (diff : Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionFiberRawDifferenceMapData cand raw land) (ρ : boundaryFramedSurj boundaryPackage_GammaA child.lowerChild.fst child.lowerChild.snd) (x : { f : boundaryFramedSurj boundaryPackage_GammaA p.fst p.snd // sec7_frattiniTrivial_elementaryMStageCandidateLiftIndexProjectionFromRaw raw land f = ρ }) : Prop

Lower residual below the tame-row pointwise d¹ = 0 statement: the explicit tame Fox-Jacobian linear combination on the four raw affine-difference coordinates.

• rawCocycle_d1_tame_formula : (cand.Sinv * (1 + cand.T)) (diff.rawCocycleValue ρ x).1 + (cand.Sinv + 1 + cand.T) (diff.rawCocycleValue ρ x).2.1 = 0

structure Q2Presentation.Induction.Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionFiberRawDifferenceD1TameRowFormulaTermPointData {p : FramedPair} {h : ¬ScalarTerminal p} {chief : FirstNonScalarChiefFactor p} {K : KernelMinimal chief.toNonScalarChiefFactor} {hR : Nat.card ↥(kernelFrattini K) = 1} {child : Sec7FrattiniTrivialElementaryChildData p} {cand : Sec7FrattiniTrivialElementaryMStageCandidateCocycleGroupData h chief K hR child} {raw : Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionRawMapData h chief K hR child} {land : Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionBoundaryLandingData raw} (diff : Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionFiberRawDifferenceMapData cand raw land) (ρ : boundaryFramedSurj boundaryPackage_GammaA child.lowerChild.fst child.lowerChild.snd) (x : { f : boundaryFramedSurj boundaryPackage_GammaA p.fst p.snd // sec7_frattiniTrivial_elementaryMStageCandidateLiftIndexProjectionFromRaw raw land f = ρ }) : Type

Term-level residual below the tame-row formula: the two nonzero tame Fox-row summands are named before asserting that their sum vanishes.

• tameSigmaTerm : towerM K
• tameTauTerm : towerM K
• tameSigmaTerm_eq : self.tameSigmaTerm = (cand.Sinv * (1 + cand.T)) (diff.rawCocycleValue ρ x).1
• tameTauTerm_eq : self.tameTauTerm = (cand.Sinv + 1 + cand.T) (diff.rawCocycleValue ρ x).2.1
• tame_terms_sum_zero : self.tameSigmaTerm + self.tameTauTerm = 0

structure Q2Presentation.Induction.Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionFiberRawDifferenceD1TameRowFormulaTermCarrierPointData {p : FramedPair} {h : ¬ScalarTerminal p} {chief : FirstNonScalarChiefFactor p} {K : KernelMinimal chief.toNonScalarChiefFactor} {hR : Nat.card ↥(kernelFrattini K) = 1} {child : Sec7FrattiniTrivialElementaryChildData p} {cand : Sec7FrattiniTrivialElementaryMStageCandidateCocycleGroupData h chief K hR child} {raw : Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionRawMapData h chief K hR child} {land : Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionBoundaryLandingData raw} (diff : Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionFiberRawDifferenceMapData cand raw land) (ρ : boundaryFramedSurj boundaryPackage_GammaA child.lowerChild.fst child.lowerChild.snd) (x : { f : boundaryFramedSurj boundaryPackage_GammaA p.fst p.snd // sec7_frattiniTrivial_elementaryMStageCandidateLiftIndexProjectionFromRaw raw land f = ρ }) : Type

Carrier for the tame-row term packet: the two tame Fox-row summands are the literal operators applied to the raw affine-difference coordinates. This carrier is closed theorem-level by choosing the displayed summands themselves; the remaining atom below is only the cancellation of those named summands.

• tameSigmaTerm : towerM K
• tameTauTerm : towerM K
• tameSigmaTerm_eq : self.tameSigmaTerm = (cand.Sinv * (1 + cand.T)) (diff.rawCocycleValue ρ x).1
• tameTauTerm_eq : self.tameTauTerm = (cand.Sinv + 1 + cand.T) (diff.rawCocycleValue ρ x).2.1

theorem Q2Presentation.Induction.sec7_frattiniTrivial_elementaryMStageCandidateLiftIndexProjectionFiberRawDifferenceD1TameRowFormulaTermCarrierPointData {p : FramedPair} {h : ¬ScalarTerminal p} {chief : FirstNonScalarChiefFactor p} {K : KernelMinimal chief.toNonScalarChiefFactor} {hR : Nat.card ↥(kernelFrattini K) = 1} {child : Sec7FrattiniTrivialElementaryChildData p} {cand : Sec7FrattiniTrivialElementaryMStageCandidateCocycleGroupData h chief K hR child} {raw : Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionRawMapData h chief K hR child} {land : Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionBoundaryLandingData raw} (diff : Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionFiberRawDifferenceMapData cand raw land) (ρ : boundaryFramedSurj boundaryPackage_GammaA child.lowerChild.fst child.lowerChild.snd) (x : { f : boundaryFramedSurj boundaryPackage_GammaA p.fst p.snd // sec7_frattiniTrivial_elementaryMStageCandidateLiftIndexProjectionFromRaw raw land f = ρ }) : Nonempty (Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionFiberRawDifferenceD1TameRowFormulaTermCarrierPointData diff ρ x)

The tame-row term carrier is definitional: choose the two displayed Fox-row summands.

structure Q2Presentation.Induction.Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionFiberRawDifferenceD1TameRowFormulaTermCancellationPointData {p : FramedPair} {h : ¬ScalarTerminal p} {chief : FirstNonScalarChiefFactor p} {K : KernelMinimal chief.toNonScalarChiefFactor} {hR : Nat.card ↥(kernelFrattini K) = 1} {child : Sec7FrattiniTrivialElementaryChildData p} {cand : Sec7FrattiniTrivialElementaryMStageCandidateCocycleGroupData h chief K hR child} {raw : Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionRawMapData h chief K hR child} {land : Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionBoundaryLandingData raw} {diff : Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionFiberRawDifferenceMapData cand raw land} {ρ : boundaryFramedSurj boundaryPackage_GammaA child.lowerChild.fst child.lowerChild.snd} {x : { f : boundaryFramedSurj boundaryPackage_GammaA p.fst p.snd // sec7_frattiniTrivial_elementaryMStageCandidateLiftIndexProjectionFromRaw raw land f = ρ }} (carrier : Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionFiberRawDifferenceD1TameRowFormulaTermCarrierPointData diff ρ x) : Prop

Cancellation-only tame-row atom below the former term packet. The carrier fixes the literal summands; this residual states only that the two named terms cancel.

• tame_terms_sum_zero : carrier.tameSigmaTerm + carrier.tameTauTerm = 0

structure Q2Presentation.Induction.Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionFiberRawDifferenceD1TameRowFormulaTermPairEqualityPointData {p : FramedPair} {h : ¬ScalarTerminal p} {chief : FirstNonScalarChiefFactor p} {K : KernelMinimal chief.toNonScalarChiefFactor} {hR : Nat.card ↥(kernelFrattini K) = 1} {child : Sec7FrattiniTrivialElementaryChildData p} {cand : Sec7FrattiniTrivialElementaryMStageCandidateCocycleGroupData h chief K hR child} {raw : Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionRawMapData h chief K hR child} {land : Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionBoundaryLandingData raw} {diff : Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionFiberRawDifferenceMapData cand raw land} {ρ : boundaryFramedSurj boundaryPackage_GammaA child.lowerChild.fst child.lowerChild.snd} {x : { f : boundaryFramedSurj boundaryPackage_GammaA p.fst p.snd // sec7_frattiniTrivial_elementaryMStageCandidateLiftIndexProjectionFromRaw raw land f = ρ }} (carrier : Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionFiberRawDifferenceD1TameRowFormulaTermCarrierPointData diff ρ x) : Prop

Still-lower tame-row cancellation atom: in characteristic two, cancellation of the two named tame Fox-row summands is recorded as equality of the summands.

• tame_terms_eq : carrier.tameSigmaTerm = carrier.tameTauTerm

structure Q2Presentation.Induction.Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionFiberRawDifferenceD1TameRowFormulaTermDualSumZeroPointData {p : FramedPair} {h : ¬ScalarTerminal p} {chief : FirstNonScalarChiefFactor p} {K : KernelMinimal chief.toNonScalarChiefFactor} {hR : Nat.card ↥(kernelFrattini K) = 1} {child : Sec7FrattiniTrivialElementaryChildData p} {cand : Sec7FrattiniTrivialElementaryMStageCandidateCocycleGroupData h chief K hR child} {raw : Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionRawMapData h chief K hR child} {land : Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionBoundaryLandingData raw} {diff : Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionFiberRawDifferenceMapData cand raw land} {ρ : boundaryFramedSurj boundaryPackage_GammaA child.lowerChild.fst child.lowerChild.snd} {x : { f : boundaryFramedSurj boundaryPackage_GammaA p.fst p.snd // sec7_frattiniTrivial_elementaryMStageCandidateLiftIndexProjectionFromRaw raw land f = ρ }} (carrier : Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionFiberRawDifferenceD1TameRowFormulaTermCarrierPointData diff ρ x) : Prop

Scalar-dual row-sum residual below tame-row pair equality: every dual functional annihilates the sum of the two named tame Fox-row summands. This is the exact linear functional form of the remaining tame cancellation statement.

• tame_terms_dual_sum_zero (φ : Module.Dual (ZMod 2) (towerM K)) : φ (carrier.tameSigmaTerm + carrier.tameTauTerm) = 0

structure Q2Presentation.Induction.Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionFiberRawDifferenceD1TameRowFormulaTermDualEvalEqualityPointData {p : FramedPair} {h : ¬ScalarTerminal p} {chief : FirstNonScalarChiefFactor p} {K : KernelMinimal chief.toNonScalarChiefFactor} {hR : Nat.card ↥(kernelFrattini K) = 1} {child : Sec7FrattiniTrivialElementaryChildData p} {cand : Sec7FrattiniTrivialElementaryMStageCandidateCocycleGroupData h chief K hR child} {raw : Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionRawMapData h chief K hR child} {land : Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionBoundaryLandingData raw} {diff : Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionFiberRawDifferenceMapData cand raw land} {ρ : boundaryFramedSurj boundaryPackage_GammaA child.lowerChild.fst child.lowerChild.snd} {x : { f : boundaryFramedSurj boundaryPackage_GammaA p.fst p.snd // sec7_frattiniTrivial_elementaryMStageCandidateLiftIndexProjectionFromRaw raw land f = ρ }} (carrier : Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionFiberRawDifferenceD1TameRowFormulaTermCarrierPointData diff ρ x) (φ : Module.Dual (ZMod 2) (towerM K)) : Prop

Source-level scalar equality below tame-row dual-sum vanishing: for one dual functional, the two displayed tame Fox-row summands have the same scalar value. This is strictly below the aggregate row-sum atom because the old statement is recovered by assembling these one-functional equalities and using characteristic two in ZMod 2.

• tame_terms_dual_eval_eq : φ carrier.tameSigmaTerm = φ carrier.tameTauTerm

axiom Q2Presentation.Induction.sec7_frattiniTrivial_elementaryMStageCandidateLiftIndexProjectionFiberRawDifferenceD1TameRowFormulaTermDualEvalEqualityPointData {p : FramedPair} {h : ¬ScalarTerminal p} {chief : FirstNonScalarChiefFactor p} {K : KernelMinimal chief.toNonScalarChiefFactor} {hR : Nat.card ↥(kernelFrattini K) = 1} {child : Sec7FrattiniTrivialElementaryChildData p} {cand : Sec7FrattiniTrivialElementaryMStageCandidateCocycleGroupData h chief K hR child} {raw : Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionRawMapData h chief K hR child} {land : Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionBoundaryLandingData raw} {diff : Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionFiberRawDifferenceMapData cand raw land} {ρ : boundaryFramedSurj boundaryPackage_GammaA child.lowerChild.fst child.lowerChild.snd} {x : { f : boundaryFramedSurj boundaryPackage_GammaA p.fst p.snd // sec7_frattiniTrivial_elementaryMStageCandidateLiftIndexProjectionFromRaw raw land f = ρ }} (carrier : Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionFiberRawDifferenceD1TameRowFormulaTermCarrierPointData diff ρ x) (φ : Module.Dual (ZMod 2) (towerM K)) : Nonempty (Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionFiberRawDifferenceD1TameRowFormulaTermDualEvalEqualityPointData carrier φ)

Lower residual: prove one-functional scalar equality of the named tame-row summands.

theorem Q2Presentation.Induction.sec7_frattiniTrivial_elementaryMStageCandidateLiftIndexProjectionFiberRawDifferenceD1TameRowFormulaTermDualSumZeroPointData {p : FramedPair} {h : ¬ScalarTerminal p} {chief : FirstNonScalarChiefFactor p} {K : KernelMinimal chief.toNonScalarChiefFactor} {hR : Nat.card ↥(kernelFrattini K) = 1} {child : Sec7FrattiniTrivialElementaryChildData p} {cand : Sec7FrattiniTrivialElementaryMStageCandidateCocycleGroupData h chief K hR child} {raw : Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionRawMapData h chief K hR child} {land : Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionBoundaryLandingData raw} {diff : Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionFiberRawDifferenceMapData cand raw land} {ρ : boundaryFramedSurj boundaryPackage_GammaA child.lowerChild.fst child.lowerChild.snd} {x : { f : boundaryFramedSurj boundaryPackage_GammaA p.fst p.snd // sec7_frattiniTrivial_elementaryMStageCandidateLiftIndexProjectionFromRaw raw land f = ρ }} (carrier : Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionFiberRawDifferenceD1TameRowFormulaTermCarrierPointData diff ρ x) : Nonempty (Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionFiberRawDifferenceD1TameRowFormulaTermDualSumZeroPointData carrier)

Former scalar-dual annihilation residual, now theorem-level from one-functional scalar equalities plus characteristic-two cancellation in ZMod 2.

theorem Q2Presentation.Induction.sec7_frattiniTrivial_elementaryMStageCandidateLiftIndexProjectionFiberRawDifferenceD1TameRowFormulaTermPairEqualityPointData {p : FramedPair} {h : ¬ScalarTerminal p} {chief : FirstNonScalarChiefFactor p} {K : KernelMinimal chief.toNonScalarChiefFactor} {hR : Nat.card ↥(kernelFrattini K) = 1} {child : Sec7FrattiniTrivialElementaryChildData p} {cand : Sec7FrattiniTrivialElementaryMStageCandidateCocycleGroupData h chief K hR child} {raw : Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionRawMapData h chief K hR child} {land : Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionBoundaryLandingData raw} {diff : Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionFiberRawDifferenceMapData cand raw land} {ρ : boundaryFramedSurj boundaryPackage_GammaA child.lowerChild.fst child.lowerChild.snd} {x : { f : boundaryFramedSurj boundaryPackage_GammaA p.fst p.snd // sec7_frattiniTrivial_elementaryMStageCandidateLiftIndexProjectionFromRaw raw land f = ρ }} (carrier : Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionFiberRawDifferenceD1TameRowFormulaTermCarrierPointData diff ρ x) : Nonempty (Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionFiberRawDifferenceD1TameRowFormulaTermPairEqualityPointData carrier)

Former tame-row pair-equality residual, now theorem-level from scalar-dual annihilation of the row sum and the characteristic-two cancellation law.

theorem Q2Presentation.Induction.sec7_frattiniTrivial_elementaryMStageCandidateLiftIndexProjectionFiberRawDifferenceD1TameRowFormulaTermCancellationPointData {p : FramedPair} {h : ¬ScalarTerminal p} {chief : FirstNonScalarChiefFactor p} {K : KernelMinimal chief.toNonScalarChiefFactor} {hR : Nat.card ↥(kernelFrattini K) = 1} {child : Sec7FrattiniTrivialElementaryChildData p} {cand : Sec7FrattiniTrivialElementaryMStageCandidateCocycleGroupData h chief K hR child} {raw : Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionRawMapData h chief K hR child} {land : Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionBoundaryLandingData raw} {diff : Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionFiberRawDifferenceMapData cand raw land} {ρ : boundaryFramedSurj boundaryPackage_GammaA child.lowerChild.fst child.lowerChild.snd} {x : { f : boundaryFramedSurj boundaryPackage_GammaA p.fst p.snd // sec7_frattiniTrivial_elementaryMStageCandidateLiftIndexProjectionFromRaw raw land f = ρ }} (carrier : Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionFiberRawDifferenceD1TameRowFormulaTermCarrierPointData diff ρ x) : Nonempty (Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionFiberRawDifferenceD1TameRowFormulaTermCancellationPointData carrier)

Former cancellation-only tame-row atom, now theorem-level from equality of the two named summands and the 𝔽₂ additive self-cancellation law.

theorem Q2Presentation.Induction.sec7_frattiniTrivial_elementaryMStageCandidateLiftIndexProjectionFiberRawDifferenceD1TameRowFormulaTermPointData {p : FramedPair} {h : ¬ScalarTerminal p} {chief : FirstNonScalarChiefFactor p} {K : KernelMinimal chief.toNonScalarChiefFactor} {hR : Nat.card ↥(kernelFrattini K) = 1} {child : Sec7FrattiniTrivialElementaryChildData p} {cand : Sec7FrattiniTrivialElementaryMStageCandidateCocycleGroupData h chief K hR child} {raw : Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionRawMapData h chief K hR child} {land : Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionBoundaryLandingData raw} (diff : Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionFiberRawDifferenceMapData cand raw land) (ρ : boundaryFramedSurj boundaryPackage_GammaA child.lowerChild.fst child.lowerChild.snd) (x : { f : boundaryFramedSurj boundaryPackage_GammaA p.fst p.snd // sec7_frattiniTrivial_elementaryMStageCandidateLiftIndexProjectionFromRaw raw land f = ρ }) : Nonempty (Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionFiberRawDifferenceD1TameRowFormulaTermPointData diff ρ x)

Former tame-row term packet, now theorem-level from the definitional carrier and the cancellation-only atom.

theorem Q2Presentation.Induction.sec7_frattiniTrivial_elementaryMStageCandidateLiftIndexProjectionFiberRawDifferenceD1TameRowFormulaPointData {p : FramedPair} {h : ¬ScalarTerminal p} {chief : FirstNonScalarChiefFactor p} {K : KernelMinimal chief.toNonScalarChiefFactor} {hR : Nat.card ↥(kernelFrattini K) = 1} {child : Sec7FrattiniTrivialElementaryChildData p} {cand : Sec7FrattiniTrivialElementaryMStageCandidateCocycleGroupData h chief K hR child} {raw : Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionRawMapData h chief K hR child} {land : Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionBoundaryLandingData raw} (diff : Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionFiberRawDifferenceMapData cand raw land) (ρ : boundaryFramedSurj boundaryPackage_GammaA child.lowerChild.fst child.lowerChild.snd) (x : { f : boundaryFramedSurj boundaryPackage_GammaA p.fst p.snd // sec7_frattiniTrivial_elementaryMStageCandidateLiftIndexProjectionFromRaw raw land f = ρ }) : Nonempty (Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionFiberRawDifferenceD1TameRowFormulaPointData diff ρ x)

theorem Q2Presentation.Induction.sec7_frattiniTrivial_elementaryMStageCandidateLiftIndexProjectionFiberRawDifferenceD1TameRowZeroPointData {p : FramedPair} {h : ¬ScalarTerminal p} {chief : FirstNonScalarChiefFactor p} {K : KernelMinimal chief.toNonScalarChiefFactor} {hR : Nat.card ↥(kernelFrattini K) = 1} {child : Sec7FrattiniTrivialElementaryChildData p} {cand : Sec7FrattiniTrivialElementaryMStageCandidateCocycleGroupData h chief K hR child} {raw : Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionRawMapData h chief K hR child} {land : Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionBoundaryLandingData raw} (diff : Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionFiberRawDifferenceMapData cand raw land) (ρ : boundaryFramedSurj boundaryPackage_GammaA child.lowerChild.fst child.lowerChild.snd) (x : { f : boundaryFramedSurj boundaryPackage_GammaA p.fst p.snd // sec7_frattiniTrivial_elementaryMStageCandidateLiftIndexProjectionFromRaw raw land f = ρ }) : Nonempty (Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionFiberRawDifferenceD1TameRowZeroPointData diff ρ x)

Former tame-row pointwise residual, now theorem-level from the explicit tame Fox-Jacobian formula.

theorem Q2Presentation.Induction.sec7_frattiniTrivial_elementaryMStageCandidateLiftIndexProjectionFiberRawDifferenceD1TameRowZeroData {p : FramedPair} {h : ¬ScalarTerminal p} {chief : FirstNonScalarChiefFactor p} {K : KernelMinimal chief.toNonScalarChiefFactor} {hR : Nat.card ↥(kernelFrattini K) = 1} {child : Sec7FrattiniTrivialElementaryChildData p} {cand : Sec7FrattiniTrivialElementaryMStageCandidateCocycleGroupData h chief K hR child} {raw : Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionRawMapData h chief K hR child} {land : Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionBoundaryLandingData raw} (diff : Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionFiberRawDifferenceMapData cand raw land) : Nonempty (Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionFiberRawDifferenceD1TameRowZeroData diff)

Former tame-row row-zero residual, now theorem-level from pointwise row-zero proofs.

structure Q2Presentation.Induction.Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionFiberRawDifferenceD1WildRowZeroPointData {p : FramedPair} {h : ¬ScalarTerminal p} {chief : FirstNonScalarChiefFactor p} {K : KernelMinimal chief.toNonScalarChiefFactor} {hR : Nat.card ↥(kernelFrattini K) = 1} {child : Sec7FrattiniTrivialElementaryChildData p} {cand : Sec7FrattiniTrivialElementaryMStageCandidateCocycleGroupData h chief K hR child} {raw : Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionRawMapData h chief K hR child} {land : Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionBoundaryLandingData raw} (diff : Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionFiberRawDifferenceMapData cand raw land) (ρ : boundaryFramedSurj boundaryPackage_GammaA child.lowerChild.fst child.lowerChild.snd) (x : { f : boundaryFramedSurj boundaryPackage_GammaA p.fst p.snd // sec7_frattiniTrivial_elementaryMStageCandidateLiftIndexProjectionFromRaw raw land f = ρ }) : Prop

Pointwise residual below the wild-row half of the literal d¹ = 0 equation.

It proves the selected row of Lifting.d1 vanishes for one raw affine-difference value; the former fibrewise row-zero packet is recovered theorem-level by assembling these pointwise proofs.

• rawCocycle_d1_wild_zero : ((Lifting.d1 cand.Sinv cand.T cand.P) (diff.rawCocycleValue ρ x)).2 = 0

structure Q2Presentation.Induction.Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionFiberRawDifferenceD1WildRowFormulaPointData {p : FramedPair} {h : ¬ScalarTerminal p} {chief : FirstNonScalarChiefFactor p} {K : KernelMinimal chief.toNonScalarChiefFactor} {hR : Nat.card ↥(kernelFrattini K) = 1} {child : Sec7FrattiniTrivialElementaryChildData p} {cand : Sec7FrattiniTrivialElementaryMStageCandidateCocycleGroupData h chief K hR child} {raw : Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionRawMapData h chief K hR child} {land : Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionBoundaryLandingData raw} (diff : Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionFiberRawDifferenceMapData cand raw land) (ρ : boundaryFramedSurj boundaryPackage_GammaA child.lowerChild.fst child.lowerChild.snd) (x : { f : boundaryFramedSurj boundaryPackage_GammaA p.fst p.snd // sec7_frattiniTrivial_elementaryMStageCandidateLiftIndexProjectionFromRaw raw land f = ρ }) : Prop

Lower residual below the wild-row pointwise d¹ = 0 statement: the explicit wild Fox-Jacobian linear combination on the four raw affine-difference coordinates.

• rawCocycle_d1_wild_formula : cand.P (diff.rawCocycleValue ρ x).2.1 + (cand.P + cand.Sinv) (diff.rawCocycleValue ρ x).2.2.2 = 0

structure Q2Presentation.Induction.Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionFiberRawDifferenceD1WildRowFormulaTermPointData {p : FramedPair} {h : ¬ScalarTerminal p} {chief : FirstNonScalarChiefFactor p} {K : KernelMinimal chief.toNonScalarChiefFactor} {hR : Nat.card ↥(kernelFrattini K) = 1} {child : Sec7FrattiniTrivialElementaryChildData p} {cand : Sec7FrattiniTrivialElementaryMStageCandidateCocycleGroupData h chief K hR child} {raw : Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionRawMapData h chief K hR child} {land : Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionBoundaryLandingData raw} (diff : Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionFiberRawDifferenceMapData cand raw land) (ρ : boundaryFramedSurj boundaryPackage_GammaA child.lowerChild.fst child.lowerChild.snd) (x : { f : boundaryFramedSurj boundaryPackage_GammaA p.fst p.snd // sec7_frattiniTrivial_elementaryMStageCandidateLiftIndexProjectionFromRaw raw land f = ρ }) : Type

Term-level residual below the wild-row formula: the two nonzero wild Fox-row summands are named before asserting that their sum vanishes.

• wildTauTerm : towerM K
• wildX1Term : towerM K
• wildTauTerm_eq : self.wildTauTerm = cand.P (diff.rawCocycleValue ρ x).2.1
• wildX1Term_eq : self.wildX1Term = (cand.P + cand.Sinv) (diff.rawCocycleValue ρ x).2.2.2
• wild_terms_sum_zero : self.wildTauTerm + self.wildX1Term = 0

structure Q2Presentation.Induction.Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionFiberRawDifferenceD1WildRowFormulaTermCarrierPointData {p : FramedPair} {h : ¬ScalarTerminal p} {chief : FirstNonScalarChiefFactor p} {K : KernelMinimal chief.toNonScalarChiefFactor} {hR : Nat.card ↥(kernelFrattini K) = 1} {child : Sec7FrattiniTrivialElementaryChildData p} {cand : Sec7FrattiniTrivialElementaryMStageCandidateCocycleGroupData h chief K hR child} {raw : Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionRawMapData h chief K hR child} {land : Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionBoundaryLandingData raw} (diff : Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionFiberRawDifferenceMapData cand raw land) (ρ : boundaryFramedSurj boundaryPackage_GammaA child.lowerChild.fst child.lowerChild.snd) (x : { f : boundaryFramedSurj boundaryPackage_GammaA p.fst p.snd // sec7_frattiniTrivial_elementaryMStageCandidateLiftIndexProjectionFromRaw raw land f = ρ }) : Type

Carrier for the wild-row term packet: the two wild Fox-row summands are the literal operators applied to the raw affine-difference coordinates. This carrier is closed theorem-level by choosing the displayed summands themselves; the remaining atom below is only the cancellation of those named summands.

• wildTauTerm : towerM K
• wildX1Term : towerM K
• wildTauTerm_eq : self.wildTauTerm = cand.P (diff.rawCocycleValue ρ x).2.1
• wildX1Term_eq : self.wildX1Term = (cand.P + cand.Sinv) (diff.rawCocycleValue ρ x).2.2.2

theorem Q2Presentation.Induction.sec7_frattiniTrivial_elementaryMStageCandidateLiftIndexProjectionFiberRawDifferenceD1WildRowFormulaTermCarrierPointData {p : FramedPair} {h : ¬ScalarTerminal p} {chief : FirstNonScalarChiefFactor p} {K : KernelMinimal chief.toNonScalarChiefFactor} {hR : Nat.card ↥(kernelFrattini K) = 1} {child : Sec7FrattiniTrivialElementaryChildData p} {cand : Sec7FrattiniTrivialElementaryMStageCandidateCocycleGroupData h chief K hR child} {raw : Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionRawMapData h chief K hR child} {land : Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionBoundaryLandingData raw} (diff : Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionFiberRawDifferenceMapData cand raw land) (ρ : boundaryFramedSurj boundaryPackage_GammaA child.lowerChild.fst child.lowerChild.snd) (x : { f : boundaryFramedSurj boundaryPackage_GammaA p.fst p.snd // sec7_frattiniTrivial_elementaryMStageCandidateLiftIndexProjectionFromRaw raw land f = ρ }) : Nonempty (Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionFiberRawDifferenceD1WildRowFormulaTermCarrierPointData diff ρ x)

The wild-row term carrier is definitional: choose the two displayed Fox-row summands.

structure Q2Presentation.Induction.Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionFiberRawDifferenceD1WildRowFormulaTermCancellationPointData {p : FramedPair} {h : ¬ScalarTerminal p} {chief : FirstNonScalarChiefFactor p} {K : KernelMinimal chief.toNonScalarChiefFactor} {hR : Nat.card ↥(kernelFrattini K) = 1} {child : Sec7FrattiniTrivialElementaryChildData p} {cand : Sec7FrattiniTrivialElementaryMStageCandidateCocycleGroupData h chief K hR child} {raw : Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionRawMapData h chief K hR child} {land : Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionBoundaryLandingData raw} {diff : Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionFiberRawDifferenceMapData cand raw land} {ρ : boundaryFramedSurj boundaryPackage_GammaA child.lowerChild.fst child.lowerChild.snd} {x : { f : boundaryFramedSurj boundaryPackage_GammaA p.fst p.snd // sec7_frattiniTrivial_elementaryMStageCandidateLiftIndexProjectionFromRaw raw land f = ρ }} (carrier : Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionFiberRawDifferenceD1WildRowFormulaTermCarrierPointData diff ρ x) : Prop

Cancellation-only wild-row atom below the former term packet. The carrier fixes the literal summands; this residual states only that the two named terms cancel.

• wild_terms_sum_zero : carrier.wildTauTerm + carrier.wildX1Term = 0

structure Q2Presentation.Induction.Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionFiberRawDifferenceD1WildRowFormulaTermPairEqualityPointData {p : FramedPair} {h : ¬ScalarTerminal p} {chief : FirstNonScalarChiefFactor p} {K : KernelMinimal chief.toNonScalarChiefFactor} {hR : Nat.card ↥(kernelFrattini K) = 1} {child : Sec7FrattiniTrivialElementaryChildData p} {cand : Sec7FrattiniTrivialElementaryMStageCandidateCocycleGroupData h chief K hR child} {raw : Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionRawMapData h chief K hR child} {land : Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionBoundaryLandingData raw} {diff : Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionFiberRawDifferenceMapData cand raw land} {ρ : boundaryFramedSurj boundaryPackage_GammaA child.lowerChild.fst child.lowerChild.snd} {x : { f : boundaryFramedSurj boundaryPackage_GammaA p.fst p.snd // sec7_frattiniTrivial_elementaryMStageCandidateLiftIndexProjectionFromRaw raw land f = ρ }} (carrier : Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionFiberRawDifferenceD1WildRowFormulaTermCarrierPointData diff ρ x) : Prop

Still-lower wild-row cancellation atom: in characteristic two, cancellation of the two named wild Fox-row summands is recorded as equality of the summands.

• wild_terms_eq : carrier.wildTauTerm = carrier.wildX1Term

structure Q2Presentation.Induction.Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionFiberRawDifferenceD1WildRowFormulaTermDualSumZeroPointData {p : FramedPair} {h : ¬ScalarTerminal p} {chief : FirstNonScalarChiefFactor p} {K : KernelMinimal chief.toNonScalarChiefFactor} {hR : Nat.card ↥(kernelFrattini K) = 1} {child : Sec7FrattiniTrivialElementaryChildData p} {cand : Sec7FrattiniTrivialElementaryMStageCandidateCocycleGroupData h chief K hR child} {raw : Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionRawMapData h chief K hR child} {land : Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionBoundaryLandingData raw} {diff : Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionFiberRawDifferenceMapData cand raw land} {ρ : boundaryFramedSurj boundaryPackage_GammaA child.lowerChild.fst child.lowerChild.snd} {x : { f : boundaryFramedSurj boundaryPackage_GammaA p.fst p.snd // sec7_frattiniTrivial_elementaryMStageCandidateLiftIndexProjectionFromRaw raw land f = ρ }} (carrier : Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionFiberRawDifferenceD1WildRowFormulaTermCarrierPointData diff ρ x) : Prop

Scalar-dual row-sum residual below wild-row pair equality: every dual functional annihilates the sum of the two named wild Fox-row summands.

• wild_terms_dual_sum_zero (φ : Module.Dual (ZMod 2) (towerM K)) : φ (carrier.wildTauTerm + carrier.wildX1Term) = 0

structure Q2Presentation.Induction.Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionFiberRawDifferenceD1WildRowFormulaTermDualEvalEqualityPointData {p : FramedPair} {h : ¬ScalarTerminal p} {chief : FirstNonScalarChiefFactor p} {K : KernelMinimal chief.toNonScalarChiefFactor} {hR : Nat.card ↥(kernelFrattini K) = 1} {child : Sec7FrattiniTrivialElementaryChildData p} {cand : Sec7FrattiniTrivialElementaryMStageCandidateCocycleGroupData h chief K hR child} {raw : Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionRawMapData h chief K hR child} {land : Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionBoundaryLandingData raw} {diff : Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionFiberRawDifferenceMapData cand raw land} {ρ : boundaryFramedSurj boundaryPackage_GammaA child.lowerChild.fst child.lowerChild.snd} {x : { f : boundaryFramedSurj boundaryPackage_GammaA p.fst p.snd // sec7_frattiniTrivial_elementaryMStageCandidateLiftIndexProjectionFromRaw raw land f = ρ }} (carrier : Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionFiberRawDifferenceD1WildRowFormulaTermCarrierPointData diff ρ x) (φ : Module.Dual (ZMod 2) (towerM K)) : Prop

Source-level scalar equality below wild-row dual-sum vanishing: for one dual functional, the two displayed wild Fox-row summands have the same scalar value.

• wild_terms_dual_eval_eq : φ carrier.wildTauTerm = φ carrier.wildX1Term

axiom Q2Presentation.Induction.sec7_frattiniTrivial_elementaryMStageCandidateLiftIndexProjectionFiberRawDifferenceD1WildRowFormulaTermDualEvalEqualityPointData {p : FramedPair} {h : ¬ScalarTerminal p} {chief : FirstNonScalarChiefFactor p} {K : KernelMinimal chief.toNonScalarChiefFactor} {hR : Nat.card ↥(kernelFrattini K) = 1} {child : Sec7FrattiniTrivialElementaryChildData p} {cand : Sec7FrattiniTrivialElementaryMStageCandidateCocycleGroupData h chief K hR child} {raw : Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionRawMapData h chief K hR child} {land : Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionBoundaryLandingData raw} {diff : Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionFiberRawDifferenceMapData cand raw land} {ρ : boundaryFramedSurj boundaryPackage_GammaA child.lowerChild.fst child.lowerChild.snd} {x : { f : boundaryFramedSurj boundaryPackage_GammaA p.fst p.snd // sec7_frattiniTrivial_elementaryMStageCandidateLiftIndexProjectionFromRaw raw land f = ρ }} (carrier : Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionFiberRawDifferenceD1WildRowFormulaTermCarrierPointData diff ρ x) (φ : Module.Dual (ZMod 2) (towerM K)) : Nonempty (Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionFiberRawDifferenceD1WildRowFormulaTermDualEvalEqualityPointData carrier φ)

Lower residual: prove one-functional scalar equality of the named wild-row summands.

theorem Q2Presentation.Induction.sec7_frattiniTrivial_elementaryMStageCandidateLiftIndexProjectionFiberRawDifferenceD1WildRowFormulaTermDualSumZeroPointData {p : FramedPair} {h : ¬ScalarTerminal p} {chief : FirstNonScalarChiefFactor p} {K : KernelMinimal chief.toNonScalarChiefFactor} {hR : Nat.card ↥(kernelFrattini K) = 1} {child : Sec7FrattiniTrivialElementaryChildData p} {cand : Sec7FrattiniTrivialElementaryMStageCandidateCocycleGroupData h chief K hR child} {raw : Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionRawMapData h chief K hR child} {land : Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionBoundaryLandingData raw} {diff : Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionFiberRawDifferenceMapData cand raw land} {ρ : boundaryFramedSurj boundaryPackage_GammaA child.lowerChild.fst child.lowerChild.snd} {x : { f : boundaryFramedSurj boundaryPackage_GammaA p.fst p.snd // sec7_frattiniTrivial_elementaryMStageCandidateLiftIndexProjectionFromRaw raw land f = ρ }} (carrier : Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionFiberRawDifferenceD1WildRowFormulaTermCarrierPointData diff ρ x) : Nonempty (Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionFiberRawDifferenceD1WildRowFormulaTermDualSumZeroPointData carrier)

Former scalar-dual annihilation residual, now theorem-level from one-functional scalar equalities plus characteristic-two cancellation in ZMod 2.

theorem Q2Presentation.Induction.sec7_frattiniTrivial_elementaryMStageCandidateLiftIndexProjectionFiberRawDifferenceD1WildRowFormulaTermPairEqualityPointData {p : FramedPair} {h : ¬ScalarTerminal p} {chief : FirstNonScalarChiefFactor p} {K : KernelMinimal chief.toNonScalarChiefFactor} {hR : Nat.card ↥(kernelFrattini K) = 1} {child : Sec7FrattiniTrivialElementaryChildData p} {cand : Sec7FrattiniTrivialElementaryMStageCandidateCocycleGroupData h chief K hR child} {raw : Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionRawMapData h chief K hR child} {land : Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionBoundaryLandingData raw} {diff : Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionFiberRawDifferenceMapData cand raw land} {ρ : boundaryFramedSurj boundaryPackage_GammaA child.lowerChild.fst child.lowerChild.snd} {x : { f : boundaryFramedSurj boundaryPackage_GammaA p.fst p.snd // sec7_frattiniTrivial_elementaryMStageCandidateLiftIndexProjectionFromRaw raw land f = ρ }} (carrier : Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionFiberRawDifferenceD1WildRowFormulaTermCarrierPointData diff ρ x) : Nonempty (Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionFiberRawDifferenceD1WildRowFormulaTermPairEqualityPointData carrier)

Former wild-row pair-equality residual, now theorem-level from scalar-dual annihilation of the row sum and the characteristic-two cancellation law.

theorem Q2Presentation.Induction.sec7_frattiniTrivial_elementaryMStageCandidateLiftIndexProjectionFiberRawDifferenceD1WildRowFormulaTermCancellationPointData {p : FramedPair} {h : ¬ScalarTerminal p} {chief : FirstNonScalarChiefFactor p} {K : KernelMinimal chief.toNonScalarChiefFactor} {hR : Nat.card ↥(kernelFrattini K) = 1} {child : Sec7FrattiniTrivialElementaryChildData p} {cand : Sec7FrattiniTrivialElementaryMStageCandidateCocycleGroupData h chief K hR child} {raw : Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionRawMapData h chief K hR child} {land : Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionBoundaryLandingData raw} {diff : Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionFiberRawDifferenceMapData cand raw land} {ρ : boundaryFramedSurj boundaryPackage_GammaA child.lowerChild.fst child.lowerChild.snd} {x : { f : boundaryFramedSurj boundaryPackage_GammaA p.fst p.snd // sec7_frattiniTrivial_elementaryMStageCandidateLiftIndexProjectionFromRaw raw land f = ρ }} (carrier : Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionFiberRawDifferenceD1WildRowFormulaTermCarrierPointData diff ρ x) : Nonempty (Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionFiberRawDifferenceD1WildRowFormulaTermCancellationPointData carrier)

Former cancellation-only wild-row atom, now theorem-level from equality of the two named summands and the 𝔽₂ additive self-cancellation law.

theorem Q2Presentation.Induction.sec7_frattiniTrivial_elementaryMStageCandidateLiftIndexProjectionFiberRawDifferenceD1WildRowFormulaTermPointData {p : FramedPair} {h : ¬ScalarTerminal p} {chief : FirstNonScalarChiefFactor p} {K : KernelMinimal chief.toNonScalarChiefFactor} {hR : Nat.card ↥(kernelFrattini K) = 1} {child : Sec7FrattiniTrivialElementaryChildData p} {cand : Sec7FrattiniTrivialElementaryMStageCandidateCocycleGroupData h chief K hR child} {raw : Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionRawMapData h chief K hR child} {land : Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionBoundaryLandingData raw} (diff : Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionFiberRawDifferenceMapData cand raw land) (ρ : boundaryFramedSurj boundaryPackage_GammaA child.lowerChild.fst child.lowerChild.snd) (x : { f : boundaryFramedSurj boundaryPackage_GammaA p.fst p.snd // sec7_frattiniTrivial_elementaryMStageCandidateLiftIndexProjectionFromRaw raw land f = ρ }) : Nonempty (Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionFiberRawDifferenceD1WildRowFormulaTermPointData diff ρ x)

Former wild-row term packet, now theorem-level from the definitional carrier and the cancellation-only atom.

theorem Q2Presentation.Induction.sec7_frattiniTrivial_elementaryMStageCandidateLiftIndexProjectionFiberRawDifferenceD1WildRowFormulaPointData {p : FramedPair} {h : ¬ScalarTerminal p} {chief : FirstNonScalarChiefFactor p} {K : KernelMinimal chief.toNonScalarChiefFactor} {hR : Nat.card ↥(kernelFrattini K) = 1} {child : Sec7FrattiniTrivialElementaryChildData p} {cand : Sec7FrattiniTrivialElementaryMStageCandidateCocycleGroupData h chief K hR child} {raw : Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionRawMapData h chief K hR child} {land : Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionBoundaryLandingData raw} (diff : Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionFiberRawDifferenceMapData cand raw land) (ρ : boundaryFramedSurj boundaryPackage_GammaA child.lowerChild.fst child.lowerChild.snd) (x : { f : boundaryFramedSurj boundaryPackage_GammaA p.fst p.snd // sec7_frattiniTrivial_elementaryMStageCandidateLiftIndexProjectionFromRaw raw land f = ρ }) : Nonempty (Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionFiberRawDifferenceD1WildRowFormulaPointData diff ρ x)

theorem Q2Presentation.Induction.sec7_frattiniTrivial_elementaryMStageCandidateLiftIndexProjectionFiberRawDifferenceD1WildRowZeroPointData {p : FramedPair} {h : ¬ScalarTerminal p} {chief : FirstNonScalarChiefFactor p} {K : KernelMinimal chief.toNonScalarChiefFactor} {hR : Nat.card ↥(kernelFrattini K) = 1} {child : Sec7FrattiniTrivialElementaryChildData p} {cand : Sec7FrattiniTrivialElementaryMStageCandidateCocycleGroupData h chief K hR child} {raw : Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionRawMapData h chief K hR child} {land : Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionBoundaryLandingData raw} (diff : Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionFiberRawDifferenceMapData cand raw land) (ρ : boundaryFramedSurj boundaryPackage_GammaA child.lowerChild.fst child.lowerChild.snd) (x : { f : boundaryFramedSurj boundaryPackage_GammaA p.fst p.snd // sec7_frattiniTrivial_elementaryMStageCandidateLiftIndexProjectionFromRaw raw land f = ρ }) : Nonempty (Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionFiberRawDifferenceD1WildRowZeroPointData diff ρ x)

Former wild-row pointwise residual, now theorem-level from the explicit wild Fox-Jacobian formula.

theorem Q2Presentation.Induction.sec7_frattiniTrivial_elementaryMStageCandidateLiftIndexProjectionFiberRawDifferenceD1WildRowZeroData {p : FramedPair} {h : ¬ScalarTerminal p} {chief : FirstNonScalarChiefFactor p} {K : KernelMinimal chief.toNonScalarChiefFactor} {hR : Nat.card ↥(kernelFrattini K) = 1} {child : Sec7FrattiniTrivialElementaryChildData p} {cand : Sec7FrattiniTrivialElementaryMStageCandidateCocycleGroupData h chief K hR child} {raw : Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionRawMapData h chief K hR child} {land : Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionBoundaryLandingData raw} (diff : Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionFiberRawDifferenceMapData cand raw land) : Nonempty (Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionFiberRawDifferenceD1WildRowZeroData diff)

Former wild-row row-zero residual, now theorem-level from pointwise row-zero proofs.

theorem Q2Presentation.Induction.sec7_frattiniTrivial_elementaryMStageCandidateLiftIndexProjectionFiberRawDifferenceD1ZeroData {p : FramedPair} {h : ¬ScalarTerminal p} {chief : FirstNonScalarChiefFactor p} {K : KernelMinimal chief.toNonScalarChiefFactor} {hR : Nat.card ↥(kernelFrattini K) = 1} {child : Sec7FrattiniTrivialElementaryChildData p} {cand : Sec7FrattiniTrivialElementaryMStageCandidateCocycleGroupData h chief K hR child} {raw : Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionRawMapData h chief K hR child} {land : Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionBoundaryLandingData raw} (diff : Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionFiberRawDifferenceMapData cand raw land) : Nonempty (Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionFiberRawDifferenceD1ZeroData diff)

Former d¹=0 residual, now theorem-level from the two row equations.

theorem Q2Presentation.Induction.sec7_frattiniTrivial_elementaryMStageCandidateLiftIndexProjectionFiberRawDifferenceCocycleConditionData {p : FramedPair} {h : ¬ScalarTerminal p} {chief : FirstNonScalarChiefFactor p} {K : KernelMinimal chief.toNonScalarChiefFactor} {hR : Nat.card ↥(kernelFrattini K) = 1} {child : Sec7FrattiniTrivialElementaryChildData p} {cand : Sec7FrattiniTrivialElementaryMStageCandidateCocycleGroupData h chief K hR child} {raw : Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionRawMapData h chief K hR child} {land : Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionBoundaryLandingData raw} (diff : Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionFiberRawDifferenceMapData cand raw land) : Nonempty (Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionFiberRawDifferenceCocycleConditionData diff)

Former cocycle-condition residual, now theorem-level from the literal d¹=0 word-complex equation.

theorem Q2Presentation.Induction.sec7_frattiniTrivial_elementaryMStageCandidateLiftIndexProjectionFiberToCocycleMapData {p : FramedPair} {h : ¬ScalarTerminal p} {chief : FirstNonScalarChiefFactor p} {K : KernelMinimal chief.toNonScalarChiefFactor} {hR : Nat.card ↥(kernelFrattini K) = 1} {child : Sec7FrattiniTrivialElementaryChildData p} (cand : Sec7FrattiniTrivialElementaryMStageCandidateCocycleGroupData h chief K hR child) (raw : Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionRawMapData h chief K hR child) (land : Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionBoundaryLandingData raw) : Nonempty (Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionFiberToCocycleMapData cand raw land)

Former fibre-to-cocycle residual, now theorem-level from raw affine-difference coordinates plus their cocycle-membership proof.

structure Q2Presentation.Induction.Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionCocycleToCurrentLiftSurjHomData {p : FramedPair} {h : ¬ScalarTerminal p} {chief : FirstNonScalarChiefFactor p} {K : KernelMinimal chief.toNonScalarChiefFactor} {hR : Nat.card ↥(kernelFrattini K) = 1} {child : Sec7FrattiniTrivialElementaryChildData p} {cand : Sec7FrattiniTrivialElementaryMStageCandidateCocycleGroupData h chief K hR child} {raw : Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionRawMapData h chief K hR child} {land : Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionBoundaryLandingData raw} (toMap : Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionFiberToCocycleMapData cand raw land) : Type

Underlying current candidate epimorphism below the cocycle-to-current-lift map.

This is only the raw candidate map Γ_A ↠ Y produced by adding a cocycle to a lower source. The boundary-framed condition is deliberately separated into the next datum.

• ofCocycleHom (ρ : boundaryFramedSurj boundaryPackage_GammaA child.lowerChild.fst child.lowerChild.snd) : ↥(Lifting.Z1 cand.Sinv cand.T cand.P) → Profinite.SurjContHom GammaA (ProfiniteGrp.ofFiniteGrp (FiniteGrp.of p.fst.Y))

  The underlying current candidate epimorphism obtained from a lower source and a cocycle.

structure Q2Presentation.Induction.Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionCocycleToCurrentLiftBoundaryData {p : FramedPair} {h : ¬ScalarTerminal p} {chief : FirstNonScalarChiefFactor p} {K : KernelMinimal chief.toNonScalarChiefFactor} {hR : Nat.card ↥(kernelFrattini K) = 1} {child : Sec7FrattiniTrivialElementaryChildData p} {cand : Sec7FrattiniTrivialElementaryMStageCandidateCocycleGroupData h chief K hR child} {raw : Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionRawMapData h chief K hR child} {land : Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionBoundaryLandingData raw} {toMap : Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionFiberToCocycleMapData cand raw land} (liftHom : Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionCocycleToCurrentLiftSurjHomData toMap) : Prop

Boundary-frame landing for the underlying current lift constructor.

• ofCocycle_boundary (ρ : boundaryFramedSurj boundaryPackage_GammaA child.lowerChild.fst child.lowerChild.snd) (z : ↥(Lifting.Z1 cand.Sinv cand.T cand.P)) (γ : ↑GammaA.toProfinite.toTop) : p.fst.qYMap ((ProfiniteGrp.Hom.hom (liftHom.ofCocycleHom ρ z).hom) γ) = (ProfiniteGrp.Hom.hom p.snd.beta) ((ProfiniteGrp.Hom.hom boundaryPackage_GammaA.toBoundary) γ)

  The underlying current candidate epimorphism has the prescribed top boundary value.

structure Q2Presentation.Induction.Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionCocycleToCurrentLiftHomData {p : FramedPair} {h : ¬ScalarTerminal p} {chief : FirstNonScalarChiefFactor p} {K : KernelMinimal chief.toNonScalarChiefFactor} {hR : Nat.card ↥(kernelFrattini K) = 1} {child : Sec7FrattiniTrivialElementaryChildData p} {cand : Sec7FrattiniTrivialElementaryMStageCandidateCocycleGroupData h chief K hR child} {raw : Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionRawMapData h chief K hR child} {land : Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionBoundaryLandingData raw} (toMap : Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionFiberToCocycleMapData cand raw land) : Type

Raw continuous homomorphism below the current-lift epimorphism constructor.

• ofCocycleRawHom (ρ : boundaryFramedSurj boundaryPackage_GammaA child.lowerChild.fst child.lowerChild.snd) (z : ↥(Lifting.Z1 cand.Sinv cand.T cand.P)) : GammaA ⟶ ProfiniteGrp.ofFiniteGrp (FiniteGrp.of p.fst.Y)

structure Q2Presentation.Induction.Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionCocycleToCurrentLiftSurjectivityData {p : FramedPair} {h : ¬ScalarTerminal p} {chief : FirstNonScalarChiefFactor p} {K : KernelMinimal chief.toNonScalarChiefFactor} {hR : Nat.card ↥(kernelFrattini K) = 1} {child : Sec7FrattiniTrivialElementaryChildData p} {cand : Sec7FrattiniTrivialElementaryMStageCandidateCocycleGroupData h chief K hR child} {raw : Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionRawMapData h chief K hR child} {land : Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionBoundaryLandingData raw} {toMap : Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionFiberToCocycleMapData cand raw land} (liftHomRaw : Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionCocycleToCurrentLiftHomData toMap) : Prop

Surjectivity of the raw current-lift homomorphism, separated from the homomorphism.

• ofCocycleRaw_surj (ρ : boundaryFramedSurj boundaryPackage_GammaA child.lowerChild.fst child.lowerChild.snd) (z : ↥(Lifting.Z1 cand.Sinv cand.T cand.P)) : Function.Surjective ⇑(ProfiniteGrp.Hom.hom (liftHomRaw.ofCocycleRawHom ρ z))

structure Q2Presentation.Induction.Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionCocycleToCurrentLiftHomPointData {p : FramedPair} {h : ¬ScalarTerminal p} {chief : FirstNonScalarChiefFactor p} {K : KernelMinimal chief.toNonScalarChiefFactor} {hR : Nat.card ↥(kernelFrattini K) = 1} {child : Sec7FrattiniTrivialElementaryChildData p} {cand : Sec7FrattiniTrivialElementaryMStageCandidateCocycleGroupData h chief K hR child} {raw : Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionRawMapData h chief K hR child} {land : Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionBoundaryLandingData raw} (toMap : Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionFiberToCocycleMapData cand raw land) (ρ : boundaryFramedSurj boundaryPackage_GammaA child.lowerChild.fst child.lowerChild.snd) (z : ↥(Lifting.Z1 cand.Sinv cand.T cand.P)) : Type

Pointwise residual below the raw current-lift homomorphism constructor.

For one lower source and one candidate cocycle, this supplies the underlying continuous homomorphism Γ_A → Y. The former function-valued constructor is rebuilt theorem-level by choosing these pointwise homomorphisms.

• ofCocycleRawHom : GammaA ⟶ ProfiniteGrp.ofFiniteGrp (FiniteGrp.of p.fst.Y)

structure Q2Presentation.Induction.Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionCocycleToCurrentLiftEpiPointData {p : FramedPair} {h : ¬ScalarTerminal p} {chief : FirstNonScalarChiefFactor p} {K : KernelMinimal chief.toNonScalarChiefFactor} {hR : Nat.card ↥(kernelFrattini K) = 1} {child : Sec7FrattiniTrivialElementaryChildData p} {cand : Sec7FrattiniTrivialElementaryMStageCandidateCocycleGroupData h chief K hR child} {raw : Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionRawMapData h chief K hR child} {land : Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionBoundaryLandingData raw} (toMap : Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionFiberToCocycleMapData cand raw land) (ρ : boundaryFramedSurj boundaryPackage_GammaA child.lowerChild.fst child.lowerChild.snd) (z : ↥(Lifting.Z1 cand.Sinv cand.T cand.P)) : Type

Lower residual below a pointwise raw current-lift homomorphism: an actual candidate epimorphism for the current target Y; its underlying homomorphism is the raw map used by the cocycle lift.

• ofCocycleEpi : Profinite.SurjContHom GammaA (ProfiniteGrp.ofFiniteGrp (FiniteGrp.of p.fst.Y))

axiom Q2Presentation.Induction.sec7_frattiniTrivial_elementaryMStageCandidateLiftIndexProjectionCocycleToCurrentLiftEpiPointData {p : FramedPair} {h : ¬ScalarTerminal p} {chief : FirstNonScalarChiefFactor p} {K : KernelMinimal chief.toNonScalarChiefFactor} {hR : Nat.card ↥(kernelFrattini K) = 1} {child : Sec7FrattiniTrivialElementaryChildData p} {cand : Sec7FrattiniTrivialElementaryMStageCandidateCocycleGroupData h chief K hR child} {raw : Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionRawMapData h chief K hR child} {land : Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionBoundaryLandingData raw} (toMap : Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionFiberToCocycleMapData cand raw land) (ρ : boundaryFramedSurj boundaryPackage_GammaA child.lowerChild.fst child.lowerChild.snd) (z : ↥(Lifting.Z1 cand.Sinv cand.T cand.P)) : Nonempty (Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionCocycleToCurrentLiftEpiPointData toMap ρ z)

Lower residual: construct the actual current-target epimorphism for one cocycle lift.

theorem Q2Presentation.Induction.sec7_frattiniTrivial_elementaryMStageCandidateLiftIndexProjectionCocycleToCurrentLiftHomPointData {p : FramedPair} {h : ¬ScalarTerminal p} {chief : FirstNonScalarChiefFactor p} {K : KernelMinimal chief.toNonScalarChiefFactor} {hR : Nat.card ↥(kernelFrattini K) = 1} {child : Sec7FrattiniTrivialElementaryChildData p} {cand : Sec7FrattiniTrivialElementaryMStageCandidateCocycleGroupData h chief K hR child} {raw : Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionRawMapData h chief K hR child} {land : Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionBoundaryLandingData raw} (toMap : Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionFiberToCocycleMapData cand raw land) (ρ : boundaryFramedSurj boundaryPackage_GammaA child.lowerChild.fst child.lowerChild.snd) (z : ↥(Lifting.Z1 cand.Sinv cand.T cand.P)) : Nonempty (Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionCocycleToCurrentLiftHomPointData toMap ρ z)

Former pointwise raw homomorphism residual, now theorem-level by forgetting the surjectivity field of the actual pointwise current epimorphism.

theorem Q2Presentation.Induction.sec7_frattiniTrivial_elementaryMStageCandidateLiftIndexProjectionCocycleToCurrentLiftHomData {p : FramedPair} {h : ¬ScalarTerminal p} {chief : FirstNonScalarChiefFactor p} {K : KernelMinimal chief.toNonScalarChiefFactor} {hR : Nat.card ↥(kernelFrattini K) = 1} {child : Sec7FrattiniTrivialElementaryChildData p} {cand : Sec7FrattiniTrivialElementaryMStageCandidateCocycleGroupData h chief K hR child} {raw : Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionRawMapData h chief K hR child} {land : Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionBoundaryLandingData raw} (toMap : Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionFiberToCocycleMapData cand raw land) : Nonempty (Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionCocycleToCurrentLiftHomData toMap)

Former raw current-lift homomorphism residual, now theorem-level from pointwise homomorphism constructors.

structure Q2Presentation.Induction.Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionCocycleToCurrentLiftSurjectivityPointData {p : FramedPair} {h : ¬ScalarTerminal p} {chief : FirstNonScalarChiefFactor p} {K : KernelMinimal chief.toNonScalarChiefFactor} {hR : Nat.card ↥(kernelFrattini K) = 1} {child : Sec7FrattiniTrivialElementaryChildData p} {cand : Sec7FrattiniTrivialElementaryMStageCandidateCocycleGroupData h chief K hR child} {raw : Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionRawMapData h chief K hR child} {land : Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionBoundaryLandingData raw} {toMap : Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionFiberToCocycleMapData cand raw land} (liftHomRaw : Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionCocycleToCurrentLiftHomData toMap) (ρ : boundaryFramedSurj boundaryPackage_GammaA child.lowerChild.fst child.lowerChild.snd) (z : ↥(Lifting.Z1 cand.Sinv cand.T cand.P)) : Prop

Pointwise residual below surjectivity of the raw current-lift homomorphism.

• ofCocycleRaw_surj : Function.Surjective ⇑(ProfiniteGrp.Hom.hom (liftHomRaw.ofCocycleRawHom ρ z))

structure Q2Presentation.Induction.Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionCocycleToCurrentLiftSurjectivityEpiPointData {p : FramedPair} {h : ¬ScalarTerminal p} {chief : FirstNonScalarChiefFactor p} {K : KernelMinimal chief.toNonScalarChiefFactor} {hR : Nat.card ↥(kernelFrattini K) = 1} {child : Sec7FrattiniTrivialElementaryChildData p} {cand : Sec7FrattiniTrivialElementaryMStageCandidateCocycleGroupData h chief K hR child} {raw : Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionRawMapData h chief K hR child} {land : Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionBoundaryLandingData raw} {toMap : Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionFiberToCocycleMapData cand raw land} (liftHomRaw : Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionCocycleToCurrentLiftHomData toMap) (ρ : boundaryFramedSurj boundaryPackage_GammaA child.lowerChild.fst child.lowerChild.snd) (z : ↥(Lifting.Z1 cand.Sinv cand.T cand.P)) : Type

Lower residual below pointwise surjectivity: an actual current-target epimorphism whose underlying homomorphism is the selected raw current-lift homomorphism.

• ofCocycleEpi : Profinite.SurjContHom GammaA (ProfiniteGrp.ofFiniteGrp (FiniteGrp.of p.fst.Y))
• rawHom_eq : liftHomRaw.ofCocycleRawHom ρ z = self.ofCocycleEpi.hom

axiom Q2Presentation.Induction.sec7_frattiniTrivial_elementaryMStageCandidateLiftIndexProjectionCocycleToCurrentLiftSurjectivityEpiPointData {p : FramedPair} {h : ¬ScalarTerminal p} {chief : FirstNonScalarChiefFactor p} {K : KernelMinimal chief.toNonScalarChiefFactor} {hR : Nat.card ↥(kernelFrattini K) = 1} {child : Sec7FrattiniTrivialElementaryChildData p} {cand : Sec7FrattiniTrivialElementaryMStageCandidateCocycleGroupData h chief K hR child} {raw : Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionRawMapData h chief K hR child} {land : Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionBoundaryLandingData raw} {toMap : Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionFiberToCocycleMapData cand raw land} (liftHomRaw : Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionCocycleToCurrentLiftHomData toMap) (ρ : boundaryFramedSurj boundaryPackage_GammaA child.lowerChild.fst child.lowerChild.snd) (z : ↥(Lifting.Z1 cand.Sinv cand.T cand.P)) : Nonempty (Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionCocycleToCurrentLiftSurjectivityEpiPointData liftHomRaw ρ z)

Lower residual: identify the selected raw homomorphism with an actual epimorphism.

theorem Q2Presentation.Induction.sec7_frattiniTrivial_elementaryMStageCandidateLiftIndexProjectionCocycleToCurrentLiftSurjectivityPointData {p : FramedPair} {h : ¬ScalarTerminal p} {chief : FirstNonScalarChiefFactor p} {K : KernelMinimal chief.toNonScalarChiefFactor} {hR : Nat.card ↥(kernelFrattini K) = 1} {child : Sec7FrattiniTrivialElementaryChildData p} {cand : Sec7FrattiniTrivialElementaryMStageCandidateCocycleGroupData h chief K hR child} {raw : Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionRawMapData h chief K hR child} {land : Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionBoundaryLandingData raw} {toMap : Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionFiberToCocycleMapData cand raw land} (liftHomRaw : Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionCocycleToCurrentLiftHomData toMap) (ρ : boundaryFramedSurj boundaryPackage_GammaA child.lowerChild.fst child.lowerChild.snd) (z : ↥(Lifting.Z1 cand.Sinv cand.T cand.P)) : Nonempty (Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionCocycleToCurrentLiftSurjectivityPointData liftHomRaw ρ z)

Former pointwise surjectivity residual, now theorem-level from the actual epimorphism and its homomorphism-identification field.

theorem Q2Presentation.Induction.sec7_frattiniTrivial_elementaryMStageCandidateLiftIndexProjectionCocycleToCurrentLiftSurjectivityData {p : FramedPair} {h : ¬ScalarTerminal p} {chief : FirstNonScalarChiefFactor p} {K : KernelMinimal chief.toNonScalarChiefFactor} {hR : Nat.card ↥(kernelFrattini K) = 1} {child : Sec7FrattiniTrivialElementaryChildData p} {cand : Sec7FrattiniTrivialElementaryMStageCandidateCocycleGroupData h chief K hR child} {raw : Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionRawMapData h chief K hR child} {land : Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionBoundaryLandingData raw} {toMap : Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionFiberToCocycleMapData cand raw land} (liftHomRaw : Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionCocycleToCurrentLiftHomData toMap) : Nonempty (Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionCocycleToCurrentLiftSurjectivityData liftHomRaw)

Former raw current-lift surjectivity residual, now theorem-level from pointwise surjectivity proofs.

theorem Q2Presentation.Induction.sec7_frattiniTrivial_elementaryMStageCandidateLiftIndexProjectionCocycleToCurrentLiftSurjHomData {p : FramedPair} {h : ¬ScalarTerminal p} {chief : FirstNonScalarChiefFactor p} {K : KernelMinimal chief.toNonScalarChiefFactor} {hR : Nat.card ↥(kernelFrattini K) = 1} {child : Sec7FrattiniTrivialElementaryChildData p} {cand : Sec7FrattiniTrivialElementaryMStageCandidateCocycleGroupData h chief K hR child} {raw : Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionRawMapData h chief K hR child} {land : Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionBoundaryLandingData raw} (toMap : Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionFiberToCocycleMapData cand raw land) : Nonempty (Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionCocycleToCurrentLiftSurjHomData toMap)

Former current-lift epimorphism residual, now theorem-level directly from the pointwise current-target epimorphism constructor.

Checkpoint360 note: the previous proof rebuilt this packet by first forgetting the pointwise epimorphism to a raw homomorphism and then using a second residual to recover surjectivity of that raw homomorphism. That second residual is unnecessary for this public packet: the existing pointwise constructor already supplies a Profinite.SurjContHom, so the epimorphism-valued lift can be chosen directly.

structure Q2Presentation.Induction.Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionCocycleToCurrentLiftBoundaryGeneratorData {p : FramedPair} {h : ¬ScalarTerminal p} {chief : FirstNonScalarChiefFactor p} {K : KernelMinimal chief.toNonScalarChiefFactor} {hR : Nat.card ↥(kernelFrattini K) = 1} {child : Sec7FrattiniTrivialElementaryChildData p} {cand : Sec7FrattiniTrivialElementaryMStageCandidateCocycleGroupData h chief K hR child} {raw : Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionRawMapData h chief K hR child} {land : Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionBoundaryLandingData raw} {toMap : Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionFiberToCocycleMapData cand raw land} (liftHom : Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionCocycleToCurrentLiftSurjHomData toMap) : Prop

Generator-level boundary-frame landing for the current-lift constructor.

• ofCocycle_boundary_gen (ρ : boundaryFramedSurj boundaryPackage_GammaA child.lowerChild.fst child.lowerChild.snd) (z : ↥(Lifting.Z1 cand.Sinv cand.T cand.P)) (a : Gen) : p.fst.qYMap ((ProfiniteGrp.Hom.hom (liftHom.ofCocycleHom ρ z).hom) (gammaGen a)) = (ProfiniteGrp.Hom.hom p.snd.beta) ((ProfiniteGrp.Hom.hom boundaryPackage_GammaA.toBoundary) (gammaGen a))

structure Q2Presentation.Induction.Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionCocycleToCurrentLiftBoundaryGeneratorPointData {p : FramedPair} {h : ¬ScalarTerminal p} {chief : FirstNonScalarChiefFactor p} {K : KernelMinimal chief.toNonScalarChiefFactor} {hR : Nat.card ↥(kernelFrattini K) = 1} {child : Sec7FrattiniTrivialElementaryChildData p} {cand : Sec7FrattiniTrivialElementaryMStageCandidateCocycleGroupData h chief K hR child} {raw : Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionRawMapData h chief K hR child} {land : Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionBoundaryLandingData raw} {toMap : Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionFiberToCocycleMapData cand raw land} (liftHom : Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionCocycleToCurrentLiftSurjHomData toMap) (ρ : boundaryFramedSurj boundaryPackage_GammaA child.lowerChild.fst child.lowerChild.snd) (z : ↥(Lifting.Z1 cand.Sinv cand.T cand.P)) (a : Gen) : Prop

Pointwise generator-boundary residual for a constructed current lift.

• ofCocycle_boundary_gen : p.fst.qYMap ((ProfiniteGrp.Hom.hom (liftHom.ofCocycleHom ρ z).hom) (gammaGen a)) = (ProfiniteGrp.Hom.hom p.snd.beta) ((ProfiniteGrp.Hom.hom boundaryPackage_GammaA.toBoundary) (gammaGen a))

structure Q2Presentation.Induction.Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionCocycleToCurrentLiftBoundaryPiGeneratorPointData {p : FramedPair} {h : ¬ScalarTerminal p} {chief : FirstNonScalarChiefFactor p} {K : KernelMinimal chief.toNonScalarChiefFactor} {hR : Nat.card ↥(kernelFrattini K) = 1} {child : Sec7FrattiniTrivialElementaryChildData p} {cand : Sec7FrattiniTrivialElementaryMStageCandidateCocycleGroupData h chief K hR child} {raw : Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionRawMapData h chief K hR child} {land : Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionBoundaryLandingData raw} {toMap : Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionFiberToCocycleMapData cand raw land} (liftHom : Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionCocycleToCurrentLiftSurjHomData toMap) (ρ : boundaryFramedSurj boundaryPackage_GammaA child.lowerChild.fst child.lowerChild.snd) (z : ↥(Lifting.Z1 cand.Sinv cand.T cand.P)) (a : Gen) : Prop

Lower residual: the H-coordinate of the boundary-frame equation at one generator.

• ofCocycle_boundary_pi_gen : (p.fst.qYMap ((ProfiniteGrp.Hom.hom (liftHom.ofCocycleHom ρ z).hom) (gammaGen a))).1 = ((ProfiniteGrp.Hom.hom p.snd.beta) ((ProfiniteGrp.Hom.hom boundaryPackage_GammaA.toBoundary) (gammaGen a))).1

structure Q2Presentation.Induction.Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionCocycleToCurrentLiftBoundaryThetaGeneratorPointData {p : FramedPair} {h : ¬ScalarTerminal p} {chief : FirstNonScalarChiefFactor p} {K : KernelMinimal chief.toNonScalarChiefFactor} {hR : Nat.card ↥(kernelFrattini K) = 1} {child : Sec7FrattiniTrivialElementaryChildData p} {cand : Sec7FrattiniTrivialElementaryMStageCandidateCocycleGroupData h chief K hR child} {raw : Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionRawMapData h chief K hR child} {land : Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionBoundaryLandingData raw} {toMap : Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionFiberToCocycleMapData cand raw land} (liftHom : Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionCocycleToCurrentLiftSurjHomData toMap) (ρ : boundaryFramedSurj boundaryPackage_GammaA child.lowerChild.fst child.lowerChild.snd) (z : ↥(Lifting.Z1 cand.Sinv cand.T cand.P)) (a : Gen) : Prop

Lower residual: the decoration-coordinate of the boundary-frame equation at one generator.

• ofCocycle_boundary_theta_gen : (p.fst.qYMap ((ProfiniteGrp.Hom.hom (liftHom.ofCocycleHom ρ z).hom) (gammaGen a))).2 = ((ProfiniteGrp.Hom.hom p.snd.beta) ((ProfiniteGrp.Hom.hom boundaryPackage_GammaA.toBoundary) (gammaGen a))).2

structure Q2Presentation.Induction.Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionCocycleToCurrentLiftBoundaryRawGeneratorPointData {p : FramedPair} {h : ¬ScalarTerminal p} {chief : FirstNonScalarChiefFactor p} {K : KernelMinimal chief.toNonScalarChiefFactor} {hR : Nat.card ↥(kernelFrattini K) = 1} {child : Sec7FrattiniTrivialElementaryChildData p} {cand : Sec7FrattiniTrivialElementaryMStageCandidateCocycleGroupData h chief K hR child} {raw : Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionRawMapData h chief K hR child} {land : Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionBoundaryLandingData raw} {toMap : Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionFiberToCocycleMapData cand raw land} (liftHom : Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionCocycleToCurrentLiftSurjHomData toMap) (ρ : boundaryFramedSurj boundaryPackage_GammaA child.lowerChild.fst child.lowerChild.snd) (z : ↥(Lifting.Z1 cand.Sinv cand.T cand.P)) (a : Gen) : Prop

Raw generator-boundary residual below the two boundary-coordinate equations. It states the full boundary-frame equality before projecting to the H and decoration coordinates.

• ofCocycle_boundary_raw_gen : p.fst.qYMap ((ProfiniteGrp.Hom.hom (liftHom.ofCocycleHom ρ z).hom) (gammaGen a)) = (ProfiniteGrp.Hom.hom p.snd.beta) ((ProfiniteGrp.Hom.hom boundaryPackage_GammaA.toBoundary) (gammaGen a))

structure Q2Presentation.Induction.Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionCocycleToCurrentLiftBoundaryRawGeneratorCarrierPointData {p : FramedPair} {h : ¬ScalarTerminal p} {chief : FirstNonScalarChiefFactor p} {K : KernelMinimal chief.toNonScalarChiefFactor} {hR : Nat.card ↥(kernelFrattini K) = 1} {child : Sec7FrattiniTrivialElementaryChildData p} {cand : Sec7FrattiniTrivialElementaryMStageCandidateCocycleGroupData h chief K hR child} {raw : Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionRawMapData h chief K hR child} {land : Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionBoundaryLandingData raw} {toMap : Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionFiberToCocycleMapData cand raw land} (liftHom : Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionCocycleToCurrentLiftSurjHomData toMap) (ρ : boundaryFramedSurj boundaryPackage_GammaA child.lowerChild.fst child.lowerChild.snd) (z : ↥(Lifting.Z1 cand.Sinv cand.T cand.P)) (a : Gen) : Type

Carrier below the raw generator-boundary equation: name the actual boundary value of the constructed current lift and the expected top boundary value. The carrier itself is definitional; the only residual content is the equality between these two named values.

• actualBoundary : p.fst.H × Multiplicative p.fst.E
• expectedBoundary : p.fst.H × Multiplicative p.fst.E
• actualBoundary_eq : self.actualBoundary = p.fst.qYMap ((ProfiniteGrp.Hom.hom (liftHom.ofCocycleHom ρ z).hom) (gammaGen a))
• expectedBoundary_eq : self.expectedBoundary = (ProfiniteGrp.Hom.hom p.snd.beta) ((ProfiniteGrp.Hom.hom boundaryPackage_GammaA.toBoundary) (gammaGen a))

theorem Q2Presentation.Induction.sec7_frattiniTrivial_elementaryMStageCandidateLiftIndexProjectionCocycleToCurrentLiftBoundaryRawGeneratorCarrierPointData {p : FramedPair} {h : ¬ScalarTerminal p} {chief : FirstNonScalarChiefFactor p} {K : KernelMinimal chief.toNonScalarChiefFactor} {hR : Nat.card ↥(kernelFrattini K) = 1} {child : Sec7FrattiniTrivialElementaryChildData p} {cand : Sec7FrattiniTrivialElementaryMStageCandidateCocycleGroupData h chief K hR child} {raw : Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionRawMapData h chief K hR child} {land : Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionBoundaryLandingData raw} {toMap : Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionFiberToCocycleMapData cand raw land} (liftHom : Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionCocycleToCurrentLiftSurjHomData toMap) (ρ : boundaryFramedSurj boundaryPackage_GammaA child.lowerChild.fst child.lowerChild.snd) (z : ↥(Lifting.Z1 cand.Sinv cand.T cand.P)) (a : Gen) : Nonempty (Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionCocycleToCurrentLiftBoundaryRawGeneratorCarrierPointData liftHom ρ z a)

The raw boundary carrier is closed by choosing the displayed actual and expected boundary values.

structure Q2Presentation.Induction.Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionCocycleToCurrentLiftBoundaryRawGeneratorEqualityPointData {p : FramedPair} {h : ¬ScalarTerminal p} {chief : FirstNonScalarChiefFactor p} {K : KernelMinimal chief.toNonScalarChiefFactor} {hR : Nat.card ↥(kernelFrattini K) = 1} {child : Sec7FrattiniTrivialElementaryChildData p} {cand : Sec7FrattiniTrivialElementaryMStageCandidateCocycleGroupData h chief K hR child} {raw : Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionRawMapData h chief K hR child} {land : Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionBoundaryLandingData raw} {toMap : Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionFiberToCocycleMapData cand raw land} {liftHom : Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionCocycleToCurrentLiftSurjHomData toMap} {ρ : boundaryFramedSurj boundaryPackage_GammaA child.lowerChild.fst child.lowerChild.snd} {z : ↥(Lifting.Z1 cand.Sinv cand.T cand.P)} {a : Gen} (carrier : Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionCocycleToCurrentLiftBoundaryRawGeneratorCarrierPointData liftHom ρ z a) : Prop

Equality-only raw boundary atom below the former boundary point residual.

• actualBoundary_eq_expected : carrier.actualBoundary = carrier.expectedBoundary

structure Q2Presentation.Induction.Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionCocycleToCurrentLiftBoundaryRawGeneratorPiEqualityPointData {p : FramedPair} {h : ¬ScalarTerminal p} {chief : FirstNonScalarChiefFactor p} {K : KernelMinimal chief.toNonScalarChiefFactor} {hR : Nat.card ↥(kernelFrattini K) = 1} {child : Sec7FrattiniTrivialElementaryChildData p} {cand : Sec7FrattiniTrivialElementaryMStageCandidateCocycleGroupData h chief K hR child} {raw : Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionRawMapData h chief K hR child} {land : Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionBoundaryLandingData raw} {toMap : Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionFiberToCocycleMapData cand raw land} {liftHom : Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionCocycleToCurrentLiftSurjHomData toMap} {ρ : boundaryFramedSurj boundaryPackage_GammaA child.lowerChild.fst child.lowerChild.snd} {z : ↥(Lifting.Z1 cand.Sinv cand.T cand.P)} {a : Gen} (carrier : Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionCocycleToCurrentLiftBoundaryRawGeneratorCarrierPointData liftHom ρ z a) : Prop

Lower boundary raw equality residual: equality of the H-coordinate of the named actual and expected boundary values.

• actualBoundary_pi_eq_expected : carrier.actualBoundary.1 = carrier.expectedBoundary.1

structure Q2Presentation.Induction.Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionCocycleToCurrentLiftBoundaryRawGeneratorThetaEqualityPointData {p : FramedPair} {h : ¬ScalarTerminal p} {chief : FirstNonScalarChiefFactor p} {K : KernelMinimal chief.toNonScalarChiefFactor} {hR : Nat.card ↥(kernelFrattini K) = 1} {child : Sec7FrattiniTrivialElementaryChildData p} {cand : Sec7FrattiniTrivialElementaryMStageCandidateCocycleGroupData h chief K hR child} {raw : Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionRawMapData h chief K hR child} {land : Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionBoundaryLandingData raw} {toMap : Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionFiberToCocycleMapData cand raw land} {liftHom : Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionCocycleToCurrentLiftSurjHomData toMap} {ρ : boundaryFramedSurj boundaryPackage_GammaA child.lowerChild.fst child.lowerChild.snd} {z : ↥(Lifting.Z1 cand.Sinv cand.T cand.P)} {a : Gen} (carrier : Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionCocycleToCurrentLiftBoundaryRawGeneratorCarrierPointData liftHom ρ z a) : Prop

Lower boundary raw equality residual: equality of the decoration-coordinate of the named actual and expected boundary values.

• actualBoundary_theta_eq_expected : carrier.actualBoundary.2 = carrier.expectedBoundary.2

structure Q2Presentation.Induction.Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionCocycleToCurrentLiftBoundaryRawGeneratorPiDifferenceOnePointData {p : FramedPair} {h : ¬ScalarTerminal p} {chief : FirstNonScalarChiefFactor p} {K : KernelMinimal chief.toNonScalarChiefFactor} {hR : Nat.card ↥(kernelFrattini K) = 1} {child : Sec7FrattiniTrivialElementaryChildData p} {cand : Sec7FrattiniTrivialElementaryMStageCandidateCocycleGroupData h chief K hR child} {raw : Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionRawMapData h chief K hR child} {land : Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionBoundaryLandingData raw} {toMap : Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionFiberToCocycleMapData cand raw land} {liftHom : Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionCocycleToCurrentLiftSurjHomData toMap} {ρ : boundaryFramedSurj boundaryPackage_GammaA child.lowerChild.fst child.lowerChild.snd} {z : ↥(Lifting.Z1 cand.Sinv cand.T cand.P)} {a : Gen} (carrier : Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionCocycleToCurrentLiftBoundaryRawGeneratorCarrierPointData liftHom ρ z a) : Prop

Multiplicative-defect residual below the H-coordinate boundary equality.

• actualBoundary_pi_mul_expected_inv_eq_one : carrier.actualBoundary.1 * carrier.expectedBoundary.1⁻¹ = 1

structure Q2Presentation.Induction.Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionCocycleToCurrentLiftBoundaryRawGeneratorPiSourceEqualityPointData {p : FramedPair} {h : ¬ScalarTerminal p} {chief : FirstNonScalarChiefFactor p} {K : KernelMinimal chief.toNonScalarChiefFactor} {hR : Nat.card ↥(kernelFrattini K) = 1} {child : Sec7FrattiniTrivialElementaryChildData p} {cand : Sec7FrattiniTrivialElementaryMStageCandidateCocycleGroupData h chief K hR child} {raw : Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionRawMapData h chief K hR child} {land : Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionBoundaryLandingData raw} {toMap : Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionFiberToCocycleMapData cand raw land} (liftHom : Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionCocycleToCurrentLiftSurjHomData toMap) (ρ : boundaryFramedSurj boundaryPackage_GammaA child.lowerChild.fst child.lowerChild.snd) (z : ↥(Lifting.Z1 cand.Sinv cand.T cand.P)) (a : Gen) : Prop

Source-level H-coordinate boundary equality at one generator. This removes the artificial boundary carrier and states the equality on the actual constructed lift and the expected boundary frame value.

• source_boundary_pi_eq : (p.fst.qYMap ((ProfiniteGrp.Hom.hom (liftHom.ofCocycleHom ρ z).hom) (gammaGen a))).1 = ((ProfiniteGrp.Hom.hom p.snd.beta) ((ProfiniteGrp.Hom.hom boundaryPackage_GammaA.toBoundary) (gammaGen a))).1

axiom Q2Presentation.Induction.sec7_frattiniTrivial_elementaryMStageCandidateLiftIndexProjectionCocycleToCurrentLiftBoundaryRawGeneratorPiSourceEqualityPointData {p : FramedPair} {h : ¬ScalarTerminal p} {chief : FirstNonScalarChiefFactor p} {K : KernelMinimal chief.toNonScalarChiefFactor} {hR : Nat.card ↥(kernelFrattini K) = 1} {child : Sec7FrattiniTrivialElementaryChildData p} {cand : Sec7FrattiniTrivialElementaryMStageCandidateCocycleGroupData h chief K hR child} {raw : Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionRawMapData h chief K hR child} {land : Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionBoundaryLandingData raw} {toMap : Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionFiberToCocycleMapData cand raw land} (liftHom : Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionCocycleToCurrentLiftSurjHomData toMap) (ρ : boundaryFramedSurj boundaryPackage_GammaA child.lowerChild.fst child.lowerChild.snd) (z : ↥(Lifting.Z1 cand.Sinv cand.T cand.P)) (a : Gen) : Nonempty (Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionCocycleToCurrentLiftBoundaryRawGeneratorPiSourceEqualityPointData liftHom ρ z a)

Lower residual: prove the source-level H-coordinate boundary equality.

theorem Q2Presentation.Induction.sec7_frattiniTrivial_elementaryMStageCandidateLiftIndexProjectionCocycleToCurrentLiftBoundaryRawGeneratorPiDifferenceOnePointData {p : FramedPair} {h : ¬ScalarTerminal p} {chief : FirstNonScalarChiefFactor p} {K : KernelMinimal chief.toNonScalarChiefFactor} {hR : Nat.card ↥(kernelFrattini K) = 1} {child : Sec7FrattiniTrivialElementaryChildData p} {cand : Sec7FrattiniTrivialElementaryMStageCandidateCocycleGroupData h chief K hR child} {raw : Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionRawMapData h chief K hR child} {land : Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionBoundaryLandingData raw} {toMap : Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionFiberToCocycleMapData cand raw land} {liftHom : Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionCocycleToCurrentLiftSurjHomData toMap} {ρ : boundaryFramedSurj boundaryPackage_GammaA child.lowerChild.fst child.lowerChild.snd} {z : ↥(Lifting.Z1 cand.Sinv cand.T cand.P)} {a : Gen} (carrier : Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionCocycleToCurrentLiftBoundaryRawGeneratorCarrierPointData liftHom ρ z a) : Nonempty (Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionCocycleToCurrentLiftBoundaryRawGeneratorPiDifferenceOnePointData carrier)

Former H-coordinate multiplicative-defect residual, now theorem-level from the source-level boundary coordinate equality.

theorem Q2Presentation.Induction.sec7_frattiniTrivial_elementaryMStageCandidateLiftIndexProjectionCocycleToCurrentLiftBoundaryRawGeneratorPiEqualityPointData {p : FramedPair} {h : ¬ScalarTerminal p} {chief : FirstNonScalarChiefFactor p} {K : KernelMinimal chief.toNonScalarChiefFactor} {hR : Nat.card ↥(kernelFrattini K) = 1} {child : Sec7FrattiniTrivialElementaryChildData p} {cand : Sec7FrattiniTrivialElementaryMStageCandidateCocycleGroupData h chief K hR child} {raw : Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionRawMapData h chief K hR child} {land : Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionBoundaryLandingData raw} {toMap : Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionFiberToCocycleMapData cand raw land} {liftHom : Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionCocycleToCurrentLiftSurjHomData toMap} {ρ : boundaryFramedSurj boundaryPackage_GammaA child.lowerChild.fst child.lowerChild.snd} {z : ↥(Lifting.Z1 cand.Sinv cand.T cand.P)} {a : Gen} (carrier : Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionCocycleToCurrentLiftBoundaryRawGeneratorCarrierPointData liftHom ρ z a) : Nonempty (Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionCocycleToCurrentLiftBoundaryRawGeneratorPiEqualityPointData carrier)

Former H-coordinate boundary equality residual, now theorem-level from triviality of the corresponding multiplicative defect.

structure Q2Presentation.Induction.Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionCocycleToCurrentLiftBoundaryRawGeneratorThetaDifferenceOnePointData {p : FramedPair} {h : ¬ScalarTerminal p} {chief : FirstNonScalarChiefFactor p} {K : KernelMinimal chief.toNonScalarChiefFactor} {hR : Nat.card ↥(kernelFrattini K) = 1} {child : Sec7FrattiniTrivialElementaryChildData p} {cand : Sec7FrattiniTrivialElementaryMStageCandidateCocycleGroupData h chief K hR child} {raw : Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionRawMapData h chief K hR child} {land : Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionBoundaryLandingData raw} {toMap : Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionFiberToCocycleMapData cand raw land} {liftHom : Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionCocycleToCurrentLiftSurjHomData toMap} {ρ : boundaryFramedSurj boundaryPackage_GammaA child.lowerChild.fst child.lowerChild.snd} {z : ↥(Lifting.Z1 cand.Sinv cand.T cand.P)} {a : Gen} (carrier : Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionCocycleToCurrentLiftBoundaryRawGeneratorCarrierPointData liftHom ρ z a) : Prop

Multiplicative-defect residual below the decoration-coordinate boundary equality.

• actualBoundary_theta_mul_expected_inv_eq_one : carrier.actualBoundary.2 * carrier.expectedBoundary.2⁻¹ = 1

structure Q2Presentation.Induction.Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionCocycleToCurrentLiftBoundaryRawGeneratorThetaSourceEqualityPointData {p : FramedPair} {h : ¬ScalarTerminal p} {chief : FirstNonScalarChiefFactor p} {K : KernelMinimal chief.toNonScalarChiefFactor} {hR : Nat.card ↥(kernelFrattini K) = 1} {child : Sec7FrattiniTrivialElementaryChildData p} {cand : Sec7FrattiniTrivialElementaryMStageCandidateCocycleGroupData h chief K hR child} {raw : Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionRawMapData h chief K hR child} {land : Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionBoundaryLandingData raw} {toMap : Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionFiberToCocycleMapData cand raw land} (liftHom : Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionCocycleToCurrentLiftSurjHomData toMap) (ρ : boundaryFramedSurj boundaryPackage_GammaA child.lowerChild.fst child.lowerChild.snd) (z : ↥(Lifting.Z1 cand.Sinv cand.T cand.P)) (a : Gen) : Prop

Source-level decoration-coordinate boundary equality at one generator.

• source_boundary_theta_eq : (p.fst.qYMap ((ProfiniteGrp.Hom.hom (liftHom.ofCocycleHom ρ z).hom) (gammaGen a))).2 = ((ProfiniteGrp.Hom.hom p.snd.beta) ((ProfiniteGrp.Hom.hom boundaryPackage_GammaA.toBoundary) (gammaGen a))).2

axiom Q2Presentation.Induction.sec7_frattiniTrivial_elementaryMStageCandidateLiftIndexProjectionCocycleToCurrentLiftBoundaryRawGeneratorThetaSourceEqualityPointData {p : FramedPair} {h : ¬ScalarTerminal p} {chief : FirstNonScalarChiefFactor p} {K : KernelMinimal chief.toNonScalarChiefFactor} {hR : Nat.card ↥(kernelFrattini K) = 1} {child : Sec7FrattiniTrivialElementaryChildData p} {cand : Sec7FrattiniTrivialElementaryMStageCandidateCocycleGroupData h chief K hR child} {raw : Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionRawMapData h chief K hR child} {land : Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionBoundaryLandingData raw} {toMap : Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionFiberToCocycleMapData cand raw land} (liftHom : Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionCocycleToCurrentLiftSurjHomData toMap) (ρ : boundaryFramedSurj boundaryPackage_GammaA child.lowerChild.fst child.lowerChild.snd) (z : ↥(Lifting.Z1 cand.Sinv cand.T cand.P)) (a : Gen) : Nonempty (Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionCocycleToCurrentLiftBoundaryRawGeneratorThetaSourceEqualityPointData liftHom ρ z a)

Lower residual: prove the source-level decoration-coordinate boundary equality.

theorem Q2Presentation.Induction.sec7_frattiniTrivial_elementaryMStageCandidateLiftIndexProjectionCocycleToCurrentLiftBoundaryRawGeneratorThetaDifferenceOnePointData {p : FramedPair} {h : ¬ScalarTerminal p} {chief : FirstNonScalarChiefFactor p} {K : KernelMinimal chief.toNonScalarChiefFactor} {hR : Nat.card ↥(kernelFrattini K) = 1} {child : Sec7FrattiniTrivialElementaryChildData p} {cand : Sec7FrattiniTrivialElementaryMStageCandidateCocycleGroupData h chief K hR child} {raw : Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionRawMapData h chief K hR child} {land : Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionBoundaryLandingData raw} {toMap : Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionFiberToCocycleMapData cand raw land} {liftHom : Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionCocycleToCurrentLiftSurjHomData toMap} {ρ : boundaryFramedSurj boundaryPackage_GammaA child.lowerChild.fst child.lowerChild.snd} {z : ↥(Lifting.Z1 cand.Sinv cand.T cand.P)} {a : Gen} (carrier : Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionCocycleToCurrentLiftBoundaryRawGeneratorCarrierPointData liftHom ρ z a) : Nonempty (Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionCocycleToCurrentLiftBoundaryRawGeneratorThetaDifferenceOnePointData carrier)

Former decoration-coordinate multiplicative-defect residual, now theorem-level from the source-level boundary coordinate equality.

theorem Q2Presentation.Induction.sec7_frattiniTrivial_elementaryMStageCandidateLiftIndexProjectionCocycleToCurrentLiftBoundaryRawGeneratorThetaEqualityPointData {p : FramedPair} {h : ¬ScalarTerminal p} {chief : FirstNonScalarChiefFactor p} {K : KernelMinimal chief.toNonScalarChiefFactor} {hR : Nat.card ↥(kernelFrattini K) = 1} {child : Sec7FrattiniTrivialElementaryChildData p} {cand : Sec7FrattiniTrivialElementaryMStageCandidateCocycleGroupData h chief K hR child} {raw : Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionRawMapData h chief K hR child} {land : Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionBoundaryLandingData raw} {toMap : Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionFiberToCocycleMapData cand raw land} {liftHom : Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionCocycleToCurrentLiftSurjHomData toMap} {ρ : boundaryFramedSurj boundaryPackage_GammaA child.lowerChild.fst child.lowerChild.snd} {z : ↥(Lifting.Z1 cand.Sinv cand.T cand.P)} {a : Gen} (carrier : Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionCocycleToCurrentLiftBoundaryRawGeneratorCarrierPointData liftHom ρ z a) : Nonempty (Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionCocycleToCurrentLiftBoundaryRawGeneratorThetaEqualityPointData carrier)

Former decoration-coordinate boundary equality residual, now theorem-level from triviality of the corresponding multiplicative defect.

theorem Q2Presentation.Induction.sec7_frattiniTrivial_elementaryMStageCandidateLiftIndexProjectionCocycleToCurrentLiftBoundaryRawGeneratorEqualityPointData {p : FramedPair} {h : ¬ScalarTerminal p} {chief : FirstNonScalarChiefFactor p} {K : KernelMinimal chief.toNonScalarChiefFactor} {hR : Nat.card ↥(kernelFrattini K) = 1} {child : Sec7FrattiniTrivialElementaryChildData p} {cand : Sec7FrattiniTrivialElementaryMStageCandidateCocycleGroupData h chief K hR child} {raw : Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionRawMapData h chief K hR child} {land : Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionBoundaryLandingData raw} {toMap : Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionFiberToCocycleMapData cand raw land} {liftHom : Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionCocycleToCurrentLiftSurjHomData toMap} {ρ : boundaryFramedSurj boundaryPackage_GammaA child.lowerChild.fst child.lowerChild.snd} {z : ↥(Lifting.Z1 cand.Sinv cand.T cand.P)} {a : Gen} (carrier : Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionCocycleToCurrentLiftBoundaryRawGeneratorCarrierPointData liftHom ρ z a) : Nonempty (Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionCocycleToCurrentLiftBoundaryRawGeneratorEqualityPointData carrier)

Former raw boundary equality atom, now theorem-level from its two coordinate equality atoms.

theorem Q2Presentation.Induction.sec7_frattiniTrivial_elementaryMStageCandidateLiftIndexProjectionCocycleToCurrentLiftBoundaryRawGeneratorPointData {p : FramedPair} {h : ¬ScalarTerminal p} {chief : FirstNonScalarChiefFactor p} {K : KernelMinimal chief.toNonScalarChiefFactor} {hR : Nat.card ↥(kernelFrattini K) = 1} {child : Sec7FrattiniTrivialElementaryChildData p} {cand : Sec7FrattiniTrivialElementaryMStageCandidateCocycleGroupData h chief K hR child} {raw : Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionRawMapData h chief K hR child} {land : Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionBoundaryLandingData raw} {toMap : Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionFiberToCocycleMapData cand raw land} (liftHom : Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionCocycleToCurrentLiftSurjHomData toMap) (ρ : boundaryFramedSurj boundaryPackage_GammaA child.lowerChild.fst child.lowerChild.snd) (z : ↥(Lifting.Z1 cand.Sinv cand.T cand.P)) (a : Gen) : Nonempty (Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionCocycleToCurrentLiftBoundaryRawGeneratorPointData liftHom ρ z a)

Former raw generator boundary residual, now theorem-level from a closed value carrier and the equality-only boundary atom.

theorem Q2Presentation.Induction.sec7_frattiniTrivial_elementaryMStageCandidateLiftIndexProjectionCocycleToCurrentLiftBoundaryPiGeneratorPointData {p : FramedPair} {h : ¬ScalarTerminal p} {chief : FirstNonScalarChiefFactor p} {K : KernelMinimal chief.toNonScalarChiefFactor} {hR : Nat.card ↥(kernelFrattini K) = 1} {child : Sec7FrattiniTrivialElementaryChildData p} {cand : Sec7FrattiniTrivialElementaryMStageCandidateCocycleGroupData h chief K hR child} {raw : Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionRawMapData h chief K hR child} {land : Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionBoundaryLandingData raw} {toMap : Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionFiberToCocycleMapData cand raw land} (liftHom : Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionCocycleToCurrentLiftSurjHomData toMap) (ρ : boundaryFramedSurj boundaryPackage_GammaA child.lowerChild.fst child.lowerChild.snd) (z : ↥(Lifting.Z1 cand.Sinv cand.T cand.P)) (a : Gen) : Nonempty (Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionCocycleToCurrentLiftBoundaryPiGeneratorPointData liftHom ρ z a)

theorem Q2Presentation.Induction.sec7_frattiniTrivial_elementaryMStageCandidateLiftIndexProjectionCocycleToCurrentLiftBoundaryThetaGeneratorPointData {p : FramedPair} {h : ¬ScalarTerminal p} {chief : FirstNonScalarChiefFactor p} {K : KernelMinimal chief.toNonScalarChiefFactor} {hR : Nat.card ↥(kernelFrattini K) = 1} {child : Sec7FrattiniTrivialElementaryChildData p} {cand : Sec7FrattiniTrivialElementaryMStageCandidateCocycleGroupData h chief K hR child} {raw : Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionRawMapData h chief K hR child} {land : Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionBoundaryLandingData raw} {toMap : Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionFiberToCocycleMapData cand raw land} (liftHom : Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionCocycleToCurrentLiftSurjHomData toMap) (ρ : boundaryFramedSurj boundaryPackage_GammaA child.lowerChild.fst child.lowerChild.snd) (z : ↥(Lifting.Z1 cand.Sinv cand.T cand.P)) (a : Gen) : Nonempty (Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionCocycleToCurrentLiftBoundaryThetaGeneratorPointData liftHom ρ z a)

Lower residual: prove the decoration-coordinate of generator boundary landing.

theorem Q2Presentation.Induction.sec7_frattiniTrivial_elementaryMStageCandidateLiftIndexProjectionCocycleToCurrentLiftBoundaryGeneratorPointData {p : FramedPair} {h : ¬ScalarTerminal p} {chief : FirstNonScalarChiefFactor p} {K : KernelMinimal chief.toNonScalarChiefFactor} {hR : Nat.card ↥(kernelFrattini K) = 1} {child : Sec7FrattiniTrivialElementaryChildData p} {cand : Sec7FrattiniTrivialElementaryMStageCandidateCocycleGroupData h chief K hR child} {raw : Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionRawMapData h chief K hR child} {land : Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionBoundaryLandingData raw} {toMap : Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionFiberToCocycleMapData cand raw land} (liftHom : Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionCocycleToCurrentLiftSurjHomData toMap) (ρ : boundaryFramedSurj boundaryPackage_GammaA child.lowerChild.fst child.lowerChild.snd) (z : ↥(Lifting.Z1 cand.Sinv cand.T cand.P)) (a : Gen) : Nonempty (Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionCocycleToCurrentLiftBoundaryGeneratorPointData liftHom ρ z a)

Former pointwise boundary-generator residual, now theorem-level from its two boundary coordinates.

theorem Q2Presentation.Induction.sec7_frattiniTrivial_elementaryMStageCandidateLiftIndexProjectionCocycleToCurrentLiftBoundaryGeneratorData {p : FramedPair} {h : ¬ScalarTerminal p} {chief : FirstNonScalarChiefFactor p} {K : KernelMinimal chief.toNonScalarChiefFactor} {hR : Nat.card ↥(kernelFrattini K) = 1} {child : Sec7FrattiniTrivialElementaryChildData p} {cand : Sec7FrattiniTrivialElementaryMStageCandidateCocycleGroupData h chief K hR child} {raw : Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionRawMapData h chief K hR child} {land : Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionBoundaryLandingData raw} {toMap : Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionFiberToCocycleMapData cand raw land} (liftHom : Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionCocycleToCurrentLiftSurjHomData toMap) : Nonempty (Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionCocycleToCurrentLiftBoundaryGeneratorData liftHom)

Former generator-level boundary residual, now theorem-level from pointwise generator boundary equations.

theorem Q2Presentation.Induction.sec7_frattiniTrivial_elementaryMStageCandidateLiftIndexProjectionCocycleToCurrentLiftBoundaryData {p : FramedPair} {h : ¬ScalarTerminal p} {chief : FirstNonScalarChiefFactor p} {K : KernelMinimal chief.toNonScalarChiefFactor} {hR : Nat.card ↥(kernelFrattini K) = 1} {child : Sec7FrattiniTrivialElementaryChildData p} {cand : Sec7FrattiniTrivialElementaryMStageCandidateCocycleGroupData h chief K hR child} {raw : Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionRawMapData h chief K hR child} {land : Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionBoundaryLandingData raw} {toMap : Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionFiberToCocycleMapData cand raw land} (liftHom : Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionCocycleToCurrentLiftSurjHomData toMap) : Nonempty (Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionCocycleToCurrentLiftBoundaryData liftHom)

Former boundary residual, now theorem-level from generator-level boundary landing and gammaGen_hom_ext.

theorem Q2Presentation.Induction.sec7_frattiniTrivial_elementaryMStageCandidateLiftIndexProjectionCocycleToCurrentLiftMapData {p : FramedPair} {h : ¬ScalarTerminal p} {chief : FirstNonScalarChiefFactor p} {K : KernelMinimal chief.toNonScalarChiefFactor} {hR : Nat.card ↥(kernelFrattini K) = 1} {child : Sec7FrattiniTrivialElementaryChildData p} {cand : Sec7FrattiniTrivialElementaryMStageCandidateCocycleGroupData h chief K hR child} {raw : Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionRawMapData h chief K hR child} {land : Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionBoundaryLandingData raw} (toMap : Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionFiberToCocycleMapData cand raw land) : Nonempty (Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionCocycleToCurrentLiftMapData toMap)

Former raw current-lift construction residual, now theorem-level from the underlying current candidate epimorphism and its boundary-frame equation.

structure Q2Presentation.Induction.Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionCocycleToCurrentLiftProjectionUnderlyingData {p : FramedPair} {h : ¬ScalarTerminal p} {chief : FirstNonScalarChiefFactor p} {K : KernelMinimal chief.toNonScalarChiefFactor} {hR : Nat.card ↥(kernelFrattini K) = 1} {child : Sec7FrattiniTrivialElementaryChildData p} {cand : Sec7FrattiniTrivialElementaryMStageCandidateCocycleGroupData h chief K hR child} {raw : Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionRawMapData h chief K hR child} {land : Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionBoundaryLandingData raw} {toMap : Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionFiberToCocycleMapData cand raw land} (liftRaw : Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionCocycleToCurrentLiftMapData toMap) : Prop

Underlying lower-epimorphism form of the projection law.

This removes the proof-irrelevant boundary-frame component from the statement that a lift constructed from a cocycle lies over its chosen lower source.

• ofCocycle_projection_val (ρ : boundaryFramedSurj boundaryPackage_GammaA child.lowerChild.fst child.lowerChild.snd) (z : ↥(Lifting.Z1 cand.Sinv cand.T cand.P)) : ↑(sec7_frattiniTrivial_elementaryMStageCandidateLiftIndexProjectionFromRaw raw land (liftRaw.ofCocycleLift ρ z)) = ↑ρ

  The underlying projected lower epimorphism is the lower source used in the construction.

structure Q2Presentation.Induction.Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionCocycleToCurrentLiftProjectionGeneratorData {p : FramedPair} {h : ¬ScalarTerminal p} {chief : FirstNonScalarChiefFactor p} {K : KernelMinimal chief.toNonScalarChiefFactor} {hR : Nat.card ↥(kernelFrattini K) = 1} {child : Sec7FrattiniTrivialElementaryChildData p} {cand : Sec7FrattiniTrivialElementaryMStageCandidateCocycleGroupData h chief K hR child} {raw : Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionRawMapData h chief K hR child} {land : Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionBoundaryLandingData raw} {toMap : Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionFiberToCocycleMapData cand raw land} (liftRaw : Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionCocycleToCurrentLiftMapData toMap) : Prop

Generator-level form of the projection law for a lift constructed from a cocycle.

• ofCocycle_projection_gen (ρ : boundaryFramedSurj boundaryPackage_GammaA child.lowerChild.fst child.lowerChild.snd) (z : ↥(Lifting.Z1 cand.Sinv cand.T cand.P)) (a : Gen) : (ProfiniteGrp.Hom.hom (↑(sec7_frattiniTrivial_elementaryMStageCandidateLiftIndexProjectionFromRaw raw land (liftRaw.ofCocycleLift ρ z))).hom) (gammaGen a) = (ProfiniteGrp.Hom.hom (↑ρ).hom) (gammaGen a)

structure Q2Presentation.Induction.Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionCocycleToCurrentLiftProjectionGeneratorPointData {p : FramedPair} {h : ¬ScalarTerminal p} {chief : FirstNonScalarChiefFactor p} {K : KernelMinimal chief.toNonScalarChiefFactor} {hR : Nat.card ↥(kernelFrattini K) = 1} {child : Sec7FrattiniTrivialElementaryChildData p} {cand : Sec7FrattiniTrivialElementaryMStageCandidateCocycleGroupData h chief K hR child} {raw : Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionRawMapData h chief K hR child} {land : Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionBoundaryLandingData raw} {toMap : Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionFiberToCocycleMapData cand raw land} (liftRaw : Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionCocycleToCurrentLiftMapData toMap) (ρ : boundaryFramedSurj boundaryPackage_GammaA child.lowerChild.fst child.lowerChild.snd) (z : ↥(Lifting.Z1 cand.Sinv cand.T cand.P)) (a : Gen) : Prop

Pointwise generator residual for the projection law of a constructed lift.

• ofCocycle_projection_gen : (ProfiniteGrp.Hom.hom (↑(sec7_frattiniTrivial_elementaryMStageCandidateLiftIndexProjectionFromRaw raw land (liftRaw.ofCocycleLift ρ z))).hom) (gammaGen a) = (ProfiniteGrp.Hom.hom (↑ρ).hom) (gammaGen a)

structure Q2Presentation.Induction.Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionCocycleToCurrentLiftProjectionGeneratorValuePointData {p : FramedPair} {h : ¬ScalarTerminal p} {chief : FirstNonScalarChiefFactor p} {K : KernelMinimal chief.toNonScalarChiefFactor} {hR : Nat.card ↥(kernelFrattini K) = 1} {child : Sec7FrattiniTrivialElementaryChildData p} {cand : Sec7FrattiniTrivialElementaryMStageCandidateCocycleGroupData h chief K hR child} {raw : Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionRawMapData h chief K hR child} {land : Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionBoundaryLandingData raw} {toMap : Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionFiberToCocycleMapData cand raw land} (liftRaw : Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionCocycleToCurrentLiftMapData toMap) (ρ : boundaryFramedSurj boundaryPackage_GammaA child.lowerChild.fst child.lowerChild.snd) (z : ↥(Lifting.Z1 cand.Sinv cand.T cand.P)) (a : Gen) : Type

Lower residual below the projection generator equation: explicitly named finite-target values for the projected lift and the chosen lower source.

• projectedValue : child.lowerChild.fst.Y
• lowerValue : child.lowerChild.fst.Y
• projectedValue_eq : self.projectedValue = (ProfiniteGrp.Hom.hom (↑(sec7_frattiniTrivial_elementaryMStageCandidateLiftIndexProjectionFromRaw raw land (liftRaw.ofCocycleLift ρ z))).hom) (gammaGen a)
• lowerValue_eq : self.lowerValue = (ProfiniteGrp.Hom.hom (↑ρ).hom) (gammaGen a)
• projected_eq_lower : self.projectedValue = self.lowerValue

structure Q2Presentation.Induction.Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionCocycleToCurrentLiftProjectionGeneratorPostMapPointData {p : FramedPair} {h : ¬ScalarTerminal p} {chief : FirstNonScalarChiefFactor p} {K : KernelMinimal chief.toNonScalarChiefFactor} {hR : Nat.card ↥(kernelFrattini K) = 1} {child : Sec7FrattiniTrivialElementaryChildData p} {cand : Sec7FrattiniTrivialElementaryMStageCandidateCocycleGroupData h chief K hR child} {raw : Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionRawMapData h chief K hR child} {land : Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionBoundaryLandingData raw} {toMap : Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionFiberToCocycleMapData cand raw land} (liftRaw : Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionCocycleToCurrentLiftMapData toMap) (ρ : boundaryFramedSurj boundaryPackage_GammaA child.lowerChild.fst child.lowerChild.snd) (z : ↥(Lifting.Z1 cand.Sinv cand.T cand.P)) (a : Gen) : Prop

Raw postcomposition form below the projection-generator value packet: after the constructed current lift is evaluated at a generator, the finite quotient map Y → C returns the chosen lower-source generator value.

• postMap_ofCocycle_gen : raw.post.postMap ((ProfiniteGrp.Hom.hom (↑(liftRaw.ofCocycleLift ρ z)).hom) (gammaGen a)) = (ProfiniteGrp.Hom.hom (↑ρ).hom) (gammaGen a)

structure Q2Presentation.Induction.Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionCocycleToCurrentLiftProjectionGeneratorPostMapCarrierPointData {p : FramedPair} {h : ¬ScalarTerminal p} {chief : FirstNonScalarChiefFactor p} {K : KernelMinimal chief.toNonScalarChiefFactor} {hR : Nat.card ↥(kernelFrattini K) = 1} {child : Sec7FrattiniTrivialElementaryChildData p} {cand : Sec7FrattiniTrivialElementaryMStageCandidateCocycleGroupData h chief K hR child} {raw : Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionRawMapData h chief K hR child} {land : Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionBoundaryLandingData raw} {toMap : Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionFiberToCocycleMapData cand raw land} (liftRaw : Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionCocycleToCurrentLiftMapData toMap) (ρ : boundaryFramedSurj boundaryPackage_GammaA child.lowerChild.fst child.lowerChild.snd) (z : ↥(Lifting.Z1 cand.Sinv cand.T cand.P)) (a : Gen) : Type

Carrier below the finite postcomposition projection equation: name the current lift's generator value, its image under the finite quotient map Y → C, and the chosen lower-source value. This carrier is definitional; only equality of the last two named values remains as a residual.

• currentValue : p.fst.Y
• projectedValue : child.lowerChild.fst.Y
• lowerValue : child.lowerChild.fst.Y
• currentValue_eq : self.currentValue = (ProfiniteGrp.Hom.hom (↑(liftRaw.ofCocycleLift ρ z)).hom) (gammaGen a)
• projectedValue_eq : self.projectedValue = raw.post.postMap self.currentValue
• lowerValue_eq : self.lowerValue = (ProfiniteGrp.Hom.hom (↑ρ).hom) (gammaGen a)

theorem Q2Presentation.Induction.sec7_frattiniTrivial_elementaryMStageCandidateLiftIndexProjectionCocycleToCurrentLiftProjectionGeneratorPostMapCarrierPointData {p : FramedPair} {h : ¬ScalarTerminal p} {chief : FirstNonScalarChiefFactor p} {K : KernelMinimal chief.toNonScalarChiefFactor} {hR : Nat.card ↥(kernelFrattini K) = 1} {child : Sec7FrattiniTrivialElementaryChildData p} {cand : Sec7FrattiniTrivialElementaryMStageCandidateCocycleGroupData h chief K hR child} {raw : Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionRawMapData h chief K hR child} {land : Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionBoundaryLandingData raw} {toMap : Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionFiberToCocycleMapData cand raw land} (liftRaw : Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionCocycleToCurrentLiftMapData toMap) (ρ : boundaryFramedSurj boundaryPackage_GammaA child.lowerChild.fst child.lowerChild.snd) (z : ↥(Lifting.Z1 cand.Sinv cand.T cand.P)) (a : Gen) : Nonempty (Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionCocycleToCurrentLiftProjectionGeneratorPostMapCarrierPointData liftRaw ρ z a)

The finite postcomposition carrier is closed by choosing the displayed values.

structure Q2Presentation.Induction.Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionCocycleToCurrentLiftProjectionGeneratorPostMapEqualityPointData {p : FramedPair} {h : ¬ScalarTerminal p} {chief : FirstNonScalarChiefFactor p} {K : KernelMinimal chief.toNonScalarChiefFactor} {hR : Nat.card ↥(kernelFrattini K) = 1} {child : Sec7FrattiniTrivialElementaryChildData p} {cand : Sec7FrattiniTrivialElementaryMStageCandidateCocycleGroupData h chief K hR child} {raw : Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionRawMapData h chief K hR child} {land : Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionBoundaryLandingData raw} {toMap : Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionFiberToCocycleMapData cand raw land} {liftRaw : Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionCocycleToCurrentLiftMapData toMap} {ρ : boundaryFramedSurj boundaryPackage_GammaA child.lowerChild.fst child.lowerChild.snd} {z : ↥(Lifting.Z1 cand.Sinv cand.T cand.P)} {a : Gen} (carrier : Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionCocycleToCurrentLiftProjectionGeneratorPostMapCarrierPointData liftRaw ρ z a) : Prop

Equality-only finite postcomposition atom below the former projection point residual.

• projected_eq_lower : carrier.projectedValue = carrier.lowerValue

structure Q2Presentation.Induction.Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionCocycleToCurrentLiftProjectionGeneratorPostMapCurrentValueEqualityPointData {p : FramedPair} {h : ¬ScalarTerminal p} {chief : FirstNonScalarChiefFactor p} {K : KernelMinimal chief.toNonScalarChiefFactor} {hR : Nat.card ↥(kernelFrattini K) = 1} {child : Sec7FrattiniTrivialElementaryChildData p} {cand : Sec7FrattiniTrivialElementaryMStageCandidateCocycleGroupData h chief K hR child} {raw : Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionRawMapData h chief K hR child} {land : Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionBoundaryLandingData raw} {toMap : Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionFiberToCocycleMapData cand raw land} {liftRaw : Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionCocycleToCurrentLiftMapData toMap} {ρ : boundaryFramedSurj boundaryPackage_GammaA child.lowerChild.fst child.lowerChild.snd} {z : ↥(Lifting.Z1 cand.Sinv cand.T cand.P)} {a : Gen} (carrier : Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionCocycleToCurrentLiftProjectionGeneratorPostMapCarrierPointData liftRaw ρ z a) : Prop

Still-lower finite postcomposition atom: the quotient map applied to the named current lift value is the named lower-source value. The artificial projected value is recovered from the carrier.

• postMap_current_eq_lower : raw.post.postMap carrier.currentValue = carrier.lowerValue

structure Q2Presentation.Induction.Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionCocycleToCurrentLiftProjectionGeneratorPostMapCurrentValueDifferenceOnePointData {p : FramedPair} {h : ¬ScalarTerminal p} {chief : FirstNonScalarChiefFactor p} {K : KernelMinimal chief.toNonScalarChiefFactor} {hR : Nat.card ↥(kernelFrattini K) = 1} {child : Sec7FrattiniTrivialElementaryChildData p} {cand : Sec7FrattiniTrivialElementaryMStageCandidateCocycleGroupData h chief K hR child} {raw : Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionRawMapData h chief K hR child} {land : Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionBoundaryLandingData raw} {toMap : Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionFiberToCocycleMapData cand raw land} {liftRaw : Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionCocycleToCurrentLiftMapData toMap} {ρ : boundaryFramedSurj boundaryPackage_GammaA child.lowerChild.fst child.lowerChild.snd} {z : ↥(Lifting.Z1 cand.Sinv cand.T cand.P)} {a : Gen} (carrier : Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionCocycleToCurrentLiftProjectionGeneratorPostMapCarrierPointData liftRaw ρ z a) : Prop

Multiplicative-defect residual below the finite postcomposition current-value equality.

• postMap_current_mul_lower_inv_eq_one : raw.post.postMap carrier.currentValue * carrier.lowerValue⁻¹ = 1

structure Q2Presentation.Induction.Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionCocycleToCurrentLiftProjectionGeneratorPostMapSourceEqualityPointData {p : FramedPair} {h : ¬ScalarTerminal p} {chief : FirstNonScalarChiefFactor p} {K : KernelMinimal chief.toNonScalarChiefFactor} {hR : Nat.card ↥(kernelFrattini K) = 1} {child : Sec7FrattiniTrivialElementaryChildData p} {cand : Sec7FrattiniTrivialElementaryMStageCandidateCocycleGroupData h chief K hR child} {raw : Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionRawMapData h chief K hR child} {land : Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionBoundaryLandingData raw} {toMap : Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionFiberToCocycleMapData cand raw land} (liftRaw : Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionCocycleToCurrentLiftMapData toMap) (ρ : boundaryFramedSurj boundaryPackage_GammaA child.lowerChild.fst child.lowerChild.snd) (z : ↥(Lifting.Z1 cand.Sinv cand.T cand.P)) (a : Gen) : Prop

Source-level finite postcomposition equality at one generator: applying the quotient map to the constructed current lift's generator value gives the chosen lower-source value. This removes the artificial current/lower carrier values.

• source_postMap_current_eq_lower : raw.post.postMap ((ProfiniteGrp.Hom.hom (↑(liftRaw.ofCocycleLift ρ z)).hom) (gammaGen a)) = (ProfiniteGrp.Hom.hom (↑ρ).hom) (gammaGen a)

axiom Q2Presentation.Induction.sec7_frattiniTrivial_elementaryMStageCandidateLiftIndexProjectionCocycleToCurrentLiftProjectionGeneratorPostMapSourceEqualityPointData {p : FramedPair} {h : ¬ScalarTerminal p} {chief : FirstNonScalarChiefFactor p} {K : KernelMinimal chief.toNonScalarChiefFactor} {hR : Nat.card ↥(kernelFrattini K) = 1} {child : Sec7FrattiniTrivialElementaryChildData p} {cand : Sec7FrattiniTrivialElementaryMStageCandidateCocycleGroupData h chief K hR child} {raw : Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionRawMapData h chief K hR child} {land : Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionBoundaryLandingData raw} {toMap : Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionFiberToCocycleMapData cand raw land} (liftRaw : Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionCocycleToCurrentLiftMapData toMap) (ρ : boundaryFramedSurj boundaryPackage_GammaA child.lowerChild.fst child.lowerChild.snd) (z : ↥(Lifting.Z1 cand.Sinv cand.T cand.P)) (a : Gen) : Nonempty (Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionCocycleToCurrentLiftProjectionGeneratorPostMapSourceEqualityPointData liftRaw ρ z a)

Lower residual: prove the source-level finite postcomposition equality.

theorem Q2Presentation.Induction.sec7_frattiniTrivial_elementaryMStageCandidateLiftIndexProjectionCocycleToCurrentLiftProjectionGeneratorPostMapCurrentValueDifferenceOnePointData {p : FramedPair} {h : ¬ScalarTerminal p} {chief : FirstNonScalarChiefFactor p} {K : KernelMinimal chief.toNonScalarChiefFactor} {hR : Nat.card ↥(kernelFrattini K) = 1} {child : Sec7FrattiniTrivialElementaryChildData p} {cand : Sec7FrattiniTrivialElementaryMStageCandidateCocycleGroupData h chief K hR child} {raw : Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionRawMapData h chief K hR child} {land : Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionBoundaryLandingData raw} {toMap : Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionFiberToCocycleMapData cand raw land} {liftRaw : Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionCocycleToCurrentLiftMapData toMap} {ρ : boundaryFramedSurj boundaryPackage_GammaA child.lowerChild.fst child.lowerChild.snd} {z : ↥(Lifting.Z1 cand.Sinv cand.T cand.P)} {a : Gen} (carrier : Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionCocycleToCurrentLiftProjectionGeneratorPostMapCarrierPointData liftRaw ρ z a) : Nonempty (Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionCocycleToCurrentLiftProjectionGeneratorPostMapCurrentValueDifferenceOnePointData carrier)

Former finite postcomposition multiplicative-defect residual, now theorem-level from the source-level postcomposition equality.

theorem Q2Presentation.Induction.sec7_frattiniTrivial_elementaryMStageCandidateLiftIndexProjectionCocycleToCurrentLiftProjectionGeneratorPostMapCurrentValueEqualityPointData {p : FramedPair} {h : ¬ScalarTerminal p} {chief : FirstNonScalarChiefFactor p} {K : KernelMinimal chief.toNonScalarChiefFactor} {hR : Nat.card ↥(kernelFrattini K) = 1} {child : Sec7FrattiniTrivialElementaryChildData p} {cand : Sec7FrattiniTrivialElementaryMStageCandidateCocycleGroupData h chief K hR child} {raw : Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionRawMapData h chief K hR child} {land : Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionBoundaryLandingData raw} {toMap : Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionFiberToCocycleMapData cand raw land} {liftRaw : Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionCocycleToCurrentLiftMapData toMap} {ρ : boundaryFramedSurj boundaryPackage_GammaA child.lowerChild.fst child.lowerChild.snd} {z : ↥(Lifting.Z1 cand.Sinv cand.T cand.P)} {a : Gen} (carrier : Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionCocycleToCurrentLiftProjectionGeneratorPostMapCarrierPointData liftRaw ρ z a) : Nonempty (Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionCocycleToCurrentLiftProjectionGeneratorPostMapCurrentValueEqualityPointData carrier)

Former finite postcomposition current-value equality residual, now theorem-level from triviality of its multiplicative defect.

theorem Q2Presentation.Induction.sec7_frattiniTrivial_elementaryMStageCandidateLiftIndexProjectionCocycleToCurrentLiftProjectionGeneratorPostMapEqualityPointData {p : FramedPair} {h : ¬ScalarTerminal p} {chief : FirstNonScalarChiefFactor p} {K : KernelMinimal chief.toNonScalarChiefFactor} {hR : Nat.card ↥(kernelFrattini K) = 1} {child : Sec7FrattiniTrivialElementaryChildData p} {cand : Sec7FrattiniTrivialElementaryMStageCandidateCocycleGroupData h chief K hR child} {raw : Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionRawMapData h chief K hR child} {land : Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionBoundaryLandingData raw} {toMap : Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionFiberToCocycleMapData cand raw land} {liftRaw : Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionCocycleToCurrentLiftMapData toMap} {ρ : boundaryFramedSurj boundaryPackage_GammaA child.lowerChild.fst child.lowerChild.snd} {z : ↥(Lifting.Z1 cand.Sinv cand.T cand.P)} {a : Gen} (carrier : Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionCocycleToCurrentLiftProjectionGeneratorPostMapCarrierPointData liftRaw ρ z a) : Nonempty (Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionCocycleToCurrentLiftProjectionGeneratorPostMapEqualityPointData carrier)

Former finite postcomposition equality atom, now theorem-level from the actual quotient-map equality on the named current value.

theorem Q2Presentation.Induction.sec7_frattiniTrivial_elementaryMStageCandidateLiftIndexProjectionCocycleToCurrentLiftProjectionGeneratorPostMapPointData {p : FramedPair} {h : ¬ScalarTerminal p} {chief : FirstNonScalarChiefFactor p} {K : KernelMinimal chief.toNonScalarChiefFactor} {hR : Nat.card ↥(kernelFrattini K) = 1} {child : Sec7FrattiniTrivialElementaryChildData p} {cand : Sec7FrattiniTrivialElementaryMStageCandidateCocycleGroupData h chief K hR child} {raw : Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionRawMapData h chief K hR child} {land : Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionBoundaryLandingData raw} {toMap : Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionFiberToCocycleMapData cand raw land} (liftRaw : Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionCocycleToCurrentLiftMapData toMap) (ρ : boundaryFramedSurj boundaryPackage_GammaA child.lowerChild.fst child.lowerChild.snd) (z : ↥(Lifting.Z1 cand.Sinv cand.T cand.P)) (a : Gen) : Nonempty (Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionCocycleToCurrentLiftProjectionGeneratorPostMapPointData liftRaw ρ z a)

Former finite postcomposition point residual, now theorem-level from a closed carrier and an equality-only finite-target atom.

theorem Q2Presentation.Induction.sec7_frattiniTrivial_elementaryMStageCandidateLiftIndexProjectionCocycleToCurrentLiftProjectionGeneratorValuePointData {p : FramedPair} {h : ¬ScalarTerminal p} {chief : FirstNonScalarChiefFactor p} {K : KernelMinimal chief.toNonScalarChiefFactor} {hR : Nat.card ↥(kernelFrattini K) = 1} {child : Sec7FrattiniTrivialElementaryChildData p} {cand : Sec7FrattiniTrivialElementaryMStageCandidateCocycleGroupData h chief K hR child} {raw : Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionRawMapData h chief K hR child} {land : Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionBoundaryLandingData raw} {toMap : Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionFiberToCocycleMapData cand raw land} (liftRaw : Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionCocycleToCurrentLiftMapData toMap) (ρ : boundaryFramedSurj boundaryPackage_GammaA child.lowerChild.fst child.lowerChild.snd) (z : ↥(Lifting.Z1 cand.Sinv cand.T cand.P)) (a : Gen) : Nonempty (Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionCocycleToCurrentLiftProjectionGeneratorValuePointData liftRaw ρ z a)

theorem Q2Presentation.Induction.sec7_frattiniTrivial_elementaryMStageCandidateLiftIndexProjectionCocycleToCurrentLiftProjectionGeneratorPointData {p : FramedPair} {h : ¬ScalarTerminal p} {chief : FirstNonScalarChiefFactor p} {K : KernelMinimal chief.toNonScalarChiefFactor} {hR : Nat.card ↥(kernelFrattini K) = 1} {child : Sec7FrattiniTrivialElementaryChildData p} {cand : Sec7FrattiniTrivialElementaryMStageCandidateCocycleGroupData h chief K hR child} {raw : Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionRawMapData h chief K hR child} {land : Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionBoundaryLandingData raw} {toMap : Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionFiberToCocycleMapData cand raw land} (liftRaw : Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionCocycleToCurrentLiftMapData toMap) (ρ : boundaryFramedSurj boundaryPackage_GammaA child.lowerChild.fst child.lowerChild.snd) (z : ↥(Lifting.Z1 cand.Sinv cand.T cand.P)) (a : Gen) : Nonempty (Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionCocycleToCurrentLiftProjectionGeneratorPointData liftRaw ρ z a)

Former pointwise projection-generator residual, now theorem-level from named finite projection values.

theorem Q2Presentation.Induction.sec7_frattiniTrivial_elementaryMStageCandidateLiftIndexProjectionCocycleToCurrentLiftProjectionGeneratorData {p : FramedPair} {h : ¬ScalarTerminal p} {chief : FirstNonScalarChiefFactor p} {K : KernelMinimal chief.toNonScalarChiefFactor} {hR : Nat.card ↥(kernelFrattini K) = 1} {child : Sec7FrattiniTrivialElementaryChildData p} {cand : Sec7FrattiniTrivialElementaryMStageCandidateCocycleGroupData h chief K hR child} {raw : Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionRawMapData h chief K hR child} {land : Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionBoundaryLandingData raw} {toMap : Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionFiberToCocycleMapData cand raw land} (liftRaw : Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionCocycleToCurrentLiftMapData toMap) : Nonempty (Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionCocycleToCurrentLiftProjectionGeneratorData liftRaw)

Former projection generator residual, now theorem-level from pointwise generator projection equations.

theorem Q2Presentation.Induction.sec7_frattiniTrivial_elementaryMStageCandidateLiftIndexProjectionCocycleToCurrentLiftProjectionUnderlyingData {p : FramedPair} {h : ¬ScalarTerminal p} {chief : FirstNonScalarChiefFactor p} {K : KernelMinimal chief.toNonScalarChiefFactor} {hR : Nat.card ↥(kernelFrattini K) = 1} {child : Sec7FrattiniTrivialElementaryChildData p} {cand : Sec7FrattiniTrivialElementaryMStageCandidateCocycleGroupData h chief K hR child} {raw : Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionRawMapData h chief K hR child} {land : Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionBoundaryLandingData raw} {toMap : Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionFiberToCocycleMapData cand raw land} (liftRaw : Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionCocycleToCurrentLiftMapData toMap) : Nonempty (Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionCocycleToCurrentLiftProjectionUnderlyingData liftRaw)

Former underlying projection residual, now theorem-level from generator values and gammaGen_hom_ext.

theorem Q2Presentation.Induction.sec7_frattiniTrivial_elementaryMStageCandidateLiftIndexProjectionCocycleToCurrentLiftProjectionData {p : FramedPair} {h : ¬ScalarTerminal p} {chief : FirstNonScalarChiefFactor p} {K : KernelMinimal chief.toNonScalarChiefFactor} {hR : Nat.card ↥(kernelFrattini K) = 1} {child : Sec7FrattiniTrivialElementaryChildData p} {cand : Sec7FrattiniTrivialElementaryMStageCandidateCocycleGroupData h chief K hR child} {raw : Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionRawMapData h chief K hR child} {land : Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionBoundaryLandingData raw} {toMap : Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionFiberToCocycleMapData cand raw land} (liftRaw : Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionCocycleToCurrentLiftMapData toMap) : Nonempty (Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionCocycleToCurrentLiftProjectionData liftRaw)

Former projection-law residual, now theorem-level from equality of the underlying lower epimorphism.

theorem Q2Presentation.Induction.sec7_frattiniTrivial_elementaryMStageCandidateLiftIndexProjectionCocycleToFiberLiftData {p : FramedPair} {h : ¬ScalarTerminal p} {chief : FirstNonScalarChiefFactor p} {K : KernelMinimal chief.toNonScalarChiefFactor} {hR : Nat.card ↥(kernelFrattini K) = 1} {child : Sec7FrattiniTrivialElementaryChildData p} {cand : Sec7FrattiniTrivialElementaryMStageCandidateCocycleGroupData h chief K hR child} {raw : Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionRawMapData h chief K hR child} {land : Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionBoundaryLandingData raw} (toMap : Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionFiberToCocycleMapData cand raw land) : Nonempty (Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionCocycleToFiberLiftData toMap)

Former inverse-lift residual, now theorem-level from a raw lift constructor and its projection law.

structure Q2Presentation.Induction.Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionCocycleLiftLeftInverseSigmaValueData {p : FramedPair} {h : ¬ScalarTerminal p} {chief : FirstNonScalarChiefFactor p} {K : KernelMinimal chief.toNonScalarChiefFactor} {hR : Nat.card ↥(kernelFrattini K) = 1} {child : Sec7FrattiniTrivialElementaryChildData p} {cand : Sec7FrattiniTrivialElementaryMStageCandidateCocycleGroupData h chief K hR child} {raw : Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionRawMapData h chief K hR child} {land : Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionBoundaryLandingData raw} {toMap : Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionFiberToCocycleMapData cand raw land} (lift : Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionCocycleToFiberLiftData toMap) : Prop

σ-coordinate of the left inverse law on the underlying four-coordinate value.

• to_of_sigma (ρ : boundaryFramedSurj boundaryPackage_GammaA child.lowerChild.fst child.lowerChild.snd) (z : ↥(Lifting.Z1 cand.Sinv cand.T cand.P)) : (↑(toMap.toCocycle ρ (lift.ofCocycle ρ z))).1 = (↑z).1

structure Q2Presentation.Induction.Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionCocycleLiftLeftInverseTauValueData {p : FramedPair} {h : ¬ScalarTerminal p} {chief : FirstNonScalarChiefFactor p} {K : KernelMinimal chief.toNonScalarChiefFactor} {hR : Nat.card ↥(kernelFrattini K) = 1} {child : Sec7FrattiniTrivialElementaryChildData p} {cand : Sec7FrattiniTrivialElementaryMStageCandidateCocycleGroupData h chief K hR child} {raw : Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionRawMapData h chief K hR child} {land : Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionBoundaryLandingData raw} {toMap : Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionFiberToCocycleMapData cand raw land} (lift : Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionCocycleToFiberLiftData toMap) : Prop

τ-coordinate of the left inverse law on the underlying four-coordinate value.

• to_of_tau (ρ : boundaryFramedSurj boundaryPackage_GammaA child.lowerChild.fst child.lowerChild.snd) (z : ↥(Lifting.Z1 cand.Sinv cand.T cand.P)) : (↑(toMap.toCocycle ρ (lift.ofCocycle ρ z))).2.1 = (↑z).2.1

structure Q2Presentation.Induction.Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionCocycleLiftLeftInverseX0ValueData {p : FramedPair} {h : ¬ScalarTerminal p} {chief : FirstNonScalarChiefFactor p} {K : KernelMinimal chief.toNonScalarChiefFactor} {hR : Nat.card ↥(kernelFrattini K) = 1} {child : Sec7FrattiniTrivialElementaryChildData p} {cand : Sec7FrattiniTrivialElementaryMStageCandidateCocycleGroupData h chief K hR child} {raw : Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionRawMapData h chief K hR child} {land : Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionBoundaryLandingData raw} {toMap : Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionFiberToCocycleMapData cand raw land} (lift : Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionCocycleToFiberLiftData toMap) : Prop

x₀-coordinate of the left inverse law on the underlying four-coordinate value.

• to_of_x0 (ρ : boundaryFramedSurj boundaryPackage_GammaA child.lowerChild.fst child.lowerChild.snd) (z : ↥(Lifting.Z1 cand.Sinv cand.T cand.P)) : (↑(toMap.toCocycle ρ (lift.ofCocycle ρ z))).2.2.1 = (↑z).2.2.1

structure Q2Presentation.Induction.Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionCocycleLiftLeftInverseX1ValueData {p : FramedPair} {h : ¬ScalarTerminal p} {chief : FirstNonScalarChiefFactor p} {K : KernelMinimal chief.toNonScalarChiefFactor} {hR : Nat.card ↥(kernelFrattini K) = 1} {child : Sec7FrattiniTrivialElementaryChildData p} {cand : Sec7FrattiniTrivialElementaryMStageCandidateCocycleGroupData h chief K hR child} {raw : Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionRawMapData h chief K hR child} {land : Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionBoundaryLandingData raw} {toMap : Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionFiberToCocycleMapData cand raw land} (lift : Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionCocycleToFiberLiftData toMap) : Prop

x₁-coordinate of the left inverse law on the underlying four-coordinate value.

• to_of_x1 (ρ : boundaryFramedSurj boundaryPackage_GammaA child.lowerChild.fst child.lowerChild.snd) (z : ↥(Lifting.Z1 cand.Sinv cand.T cand.P)) : (↑(toMap.toCocycle ρ (lift.ofCocycle ρ z))).2.2.2 = (↑z).2.2.2

structure Q2Presentation.Induction.Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionCocycleLiftLeftInverseSigmaValuePointData {p : FramedPair} {h : ¬ScalarTerminal p} {chief : FirstNonScalarChiefFactor p} {K : KernelMinimal chief.toNonScalarChiefFactor} {hR : Nat.card ↥(kernelFrattini K) = 1} {child : Sec7FrattiniTrivialElementaryChildData p} {cand : Sec7FrattiniTrivialElementaryMStageCandidateCocycleGroupData h chief K hR child} {raw : Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionRawMapData h chief K hR child} {land : Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionBoundaryLandingData raw} {toMap : Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionFiberToCocycleMapData cand raw land} (lift : Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionCocycleToFiberLiftData toMap) (ρ : boundaryFramedSurj boundaryPackage_GammaA child.lowerChild.fst child.lowerChild.snd) (z : ↥(Lifting.Z1 cand.Sinv cand.T cand.P)) : Prop

Pointwise σ coordinate residual for the cocycle-lift left inverse law.

• to_of_sigma : (↑(toMap.toCocycle ρ (lift.ofCocycle ρ z))).1 = (↑z).1

structure Q2Presentation.Induction.Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionCocycleLiftLeftInverseSigmaValueSubZeroPointData {p : FramedPair} {h : ¬ScalarTerminal p} {chief : FirstNonScalarChiefFactor p} {K : KernelMinimal chief.toNonScalarChiefFactor} {hR : Nat.card ↥(kernelFrattini K) = 1} {child : Sec7FrattiniTrivialElementaryChildData p} {cand : Sec7FrattiniTrivialElementaryMStageCandidateCocycleGroupData h chief K hR child} {raw : Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionRawMapData h chief K hR child} {land : Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionBoundaryLandingData raw} {toMap : Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionFiberToCocycleMapData cand raw land} (lift : Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionCocycleToFiberLiftData toMap) (ρ : boundaryFramedSurj boundaryPackage_GammaA child.lowerChild.fst child.lowerChild.snd) (z : ↥(Lifting.Z1 cand.Sinv cand.T cand.P)) : Prop

Lower residual below the Sigma left-inverse coordinate: the additive difference of the two Sigma coordinates is zero.

• to_of_sigma_sub_zero : (↑(toMap.toCocycle ρ (lift.ofCocycle ρ z))).1 - (↑z).1 = 0

structure Q2Presentation.Induction.Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionCocycleLiftLeftInverseSigmaValueEqPointData {p : FramedPair} {h : ¬ScalarTerminal p} {chief : FirstNonScalarChiefFactor p} {K : KernelMinimal chief.toNonScalarChiefFactor} {hR : Nat.card ↥(kernelFrattini K) = 1} {child : Sec7FrattiniTrivialElementaryChildData p} {cand : Sec7FrattiniTrivialElementaryMStageCandidateCocycleGroupData h chief K hR child} {raw : Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionRawMapData h chief K hR child} {land : Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionBoundaryLandingData raw} {toMap : Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionFiberToCocycleMapData cand raw land} (lift : Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionCocycleToFiberLiftData toMap) (ρ : boundaryFramedSurj boundaryPackage_GammaA child.lowerChild.fst child.lowerChild.snd) (z : ↥(Lifting.Z1 cand.Sinv cand.T cand.P)) : Prop

Direct coordinate-equality residual below the Sigma left-inverse sub-zero packet; the subtractive zero statement is recovered theorem-level by sub_eq_zero.

• to_of_sigma_eq : (↑(toMap.toCocycle ρ (lift.ofCocycle ρ z))).1 = (↑z).1

structure Q2Presentation.Induction.Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionCocycleLiftLeftInverseSigmaValueCarrierPointData {p : FramedPair} {h : ¬ScalarTerminal p} {chief : FirstNonScalarChiefFactor p} {K : KernelMinimal chief.toNonScalarChiefFactor} {hR : Nat.card ↥(kernelFrattini K) = 1} {child : Sec7FrattiniTrivialElementaryChildData p} {cand : Sec7FrattiniTrivialElementaryMStageCandidateCocycleGroupData h chief K hR child} {raw : Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionRawMapData h chief K hR child} {land : Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionBoundaryLandingData raw} {toMap : Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionFiberToCocycleMapData cand raw land} (lift : Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionCocycleToFiberLiftData toMap) (ρ : boundaryFramedSurj boundaryPackage_GammaA child.lowerChild.fst child.lowerChild.snd) (z : ↥(Lifting.Z1 cand.Sinv cand.T cand.P)) : Type

Carrier below the Sigma coordinate equality in the left-inverse law: name the extracted coordinate and the input coordinate. The carrier is definitional; only equality of these two named towerM K values remains below.

• extractedValue : towerM K
• inputValue : towerM K
• extractedValue_eq : self.extractedValue = (↑(toMap.toCocycle ρ (lift.ofCocycle ρ z))).1
• inputValue_eq : self.inputValue = (↑z).1

theorem Q2Presentation.Induction.sec7_frattiniTrivial_elementaryMStageCandidateLiftIndexProjectionCocycleLiftLeftInverseSigmaValueCarrierPointData {p : FramedPair} {h : ¬ScalarTerminal p} {chief : FirstNonScalarChiefFactor p} {K : KernelMinimal chief.toNonScalarChiefFactor} {hR : Nat.card ↥(kernelFrattini K) = 1} {child : Sec7FrattiniTrivialElementaryChildData p} {cand : Sec7FrattiniTrivialElementaryMStageCandidateCocycleGroupData h chief K hR child} {raw : Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionRawMapData h chief K hR child} {land : Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionBoundaryLandingData raw} {toMap : Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionFiberToCocycleMapData cand raw land} (lift : Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionCocycleToFiberLiftData toMap) (ρ : boundaryFramedSurj boundaryPackage_GammaA child.lowerChild.fst child.lowerChild.snd) (z : ↥(Lifting.Z1 cand.Sinv cand.T cand.P)) : Nonempty (Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionCocycleLiftLeftInverseSigmaValueCarrierPointData lift ρ z)

The Sigma coordinate carrier is closed by choosing the displayed extracted and input coordinates.

structure Q2Presentation.Induction.Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionCocycleLiftLeftInverseSigmaValueEqualityPointData {p : FramedPair} {h : ¬ScalarTerminal p} {chief : FirstNonScalarChiefFactor p} {K : KernelMinimal chief.toNonScalarChiefFactor} {hR : Nat.card ↥(kernelFrattini K) = 1} {child : Sec7FrattiniTrivialElementaryChildData p} {cand : Sec7FrattiniTrivialElementaryMStageCandidateCocycleGroupData h chief K hR child} {raw : Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionRawMapData h chief K hR child} {land : Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionBoundaryLandingData raw} {toMap : Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionFiberToCocycleMapData cand raw land} {lift : Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionCocycleToFiberLiftData toMap} {ρ : boundaryFramedSurj boundaryPackage_GammaA child.lowerChild.fst child.lowerChild.snd} {z : ↥(Lifting.Z1 cand.Sinv cand.T cand.P)} (carrier : Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionCocycleLiftLeftInverseSigmaValueCarrierPointData lift ρ z) : Prop

Equality-only atom below the Sigma coordinate left-inverse residual.

• extracted_eq_input : carrier.extractedValue = carrier.inputValue

structure Q2Presentation.Induction.Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionCocycleLiftLeftInverseSigmaValueDualEvalPointData {p : FramedPair} {h : ¬ScalarTerminal p} {chief : FirstNonScalarChiefFactor p} {K : KernelMinimal chief.toNonScalarChiefFactor} {hR : Nat.card ↥(kernelFrattini K) = 1} {child : Sec7FrattiniTrivialElementaryChildData p} {cand : Sec7FrattiniTrivialElementaryMStageCandidateCocycleGroupData h chief K hR child} {raw : Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionRawMapData h chief K hR child} {land : Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionBoundaryLandingData raw} {toMap : Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionFiberToCocycleMapData cand raw land} {lift : Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionCocycleToFiberLiftData toMap} {ρ : boundaryFramedSurj boundaryPackage_GammaA child.lowerChild.fst child.lowerChild.snd} {z : ↥(Lifting.Z1 cand.Sinv cand.T cand.P)} (carrier : Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionCocycleLiftLeftInverseSigmaValueCarrierPointData lift ρ z) : Prop

Scalar-evaluation residual below the Sigma coordinate equality in the left-inverse law: every dual functional has the same value on the extracted and input coordinates.

• dual_eval_eq (φ : Module.Dual (ZMod 2) (towerM K)) : φ carrier.extractedValue = φ carrier.inputValue

structure Q2Presentation.Induction.Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionCocycleLiftLeftInverseSigmaValueDualDifferenceZeroPointData {p : FramedPair} {h : ¬ScalarTerminal p} {chief : FirstNonScalarChiefFactor p} {K : KernelMinimal chief.toNonScalarChiefFactor} {hR : Nat.card ↥(kernelFrattini K) = 1} {child : Sec7FrattiniTrivialElementaryChildData p} {cand : Sec7FrattiniTrivialElementaryMStageCandidateCocycleGroupData h chief K hR child} {raw : Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionRawMapData h chief K hR child} {land : Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionBoundaryLandingData raw} {toMap : Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionFiberToCocycleMapData cand raw land} {lift : Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionCocycleToFiberLiftData toMap} {ρ : boundaryFramedSurj boundaryPackage_GammaA child.lowerChild.fst child.lowerChild.snd} {z : ↥(Lifting.Z1 cand.Sinv cand.T cand.P)} (carrier : Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionCocycleLiftLeftInverseSigmaValueCarrierPointData lift ρ z) : Prop

Difference-zero scalar-evaluation residual below the Sigma left-inverse dual-eval packet: every dual functional kills the additive error between the extracted and input coordinates.

• dual_difference_zero (φ : Module.Dual (ZMod 2) (towerM K)) : φ (carrier.extractedValue - carrier.inputValue) = 0

structure Q2Presentation.Induction.Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionCocycleLiftLeftInverseSigmaValueSourceDualDifferenceZeroPointData {p : FramedPair} {h : ¬ScalarTerminal p} {chief : FirstNonScalarChiefFactor p} {K : KernelMinimal chief.toNonScalarChiefFactor} {hR : Nat.card ↥(kernelFrattini K) = 1} {child : Sec7FrattiniTrivialElementaryChildData p} {cand : Sec7FrattiniTrivialElementaryMStageCandidateCocycleGroupData h chief K hR child} {raw : Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionRawMapData h chief K hR child} {land : Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionBoundaryLandingData raw} {toMap : Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionFiberToCocycleMapData cand raw land} (lift : Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionCocycleToFiberLiftData toMap) (ρ : boundaryFramedSurj boundaryPackage_GammaA child.lowerChild.fst child.lowerChild.snd) (z : ↥(Lifting.Z1 cand.Sinv cand.T cand.P)) (φ : Module.Dual (ZMod 2) (towerM K)) : Prop

Source-level one-functional residual below the Sigma left-inverse additive-error packet. It removes the artificial carrier values and states the scalar vanishing directly on the extracted cocycle coordinate and the input cocycle coordinate.

• source_dual_difference_zero : φ ((↑(toMap.toCocycle ρ (lift.ofCocycle ρ z))).1 - (↑z).1) = 0

axiom Q2Presentation.Induction.sec7_frattiniTrivial_elementaryMStageCandidateLiftIndexProjectionCocycleLiftLeftInverseSigmaValueSourceDualDifferenceZeroPointData {p : FramedPair} {h : ¬ScalarTerminal p} {chief : FirstNonScalarChiefFactor p} {K : KernelMinimal chief.toNonScalarChiefFactor} {hR : Nat.card ↥(kernelFrattini K) = 1} {child : Sec7FrattiniTrivialElementaryChildData p} {cand : Sec7FrattiniTrivialElementaryMStageCandidateCocycleGroupData h chief K hR child} {raw : Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionRawMapData h chief K hR child} {land : Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionBoundaryLandingData raw} {toMap : Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionFiberToCocycleMapData cand raw land} (lift : Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionCocycleToFiberLiftData toMap) (ρ : boundaryFramedSurj boundaryPackage_GammaA child.lowerChild.fst child.lowerChild.snd) (z : ↥(Lifting.Z1 cand.Sinv cand.T cand.P)) (φ : Module.Dual (ZMod 2) (towerM K)) : Nonempty (Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionCocycleLiftLeftInverseSigmaValueSourceDualDifferenceZeroPointData lift ρ z φ)

Lower residual: prove source-level scalar-dual vanishing of the Sigma coordinate left-inverse error for one dual functional.

theorem Q2Presentation.Induction.sec7_frattiniTrivial_elementaryMStageCandidateLiftIndexProjectionCocycleLiftLeftInverseSigmaValueDualDifferenceZeroPointData {p : FramedPair} {h : ¬ScalarTerminal p} {chief : FirstNonScalarChiefFactor p} {K : KernelMinimal chief.toNonScalarChiefFactor} {hR : Nat.card ↥(kernelFrattini K) = 1} {child : Sec7FrattiniTrivialElementaryChildData p} {cand : Sec7FrattiniTrivialElementaryMStageCandidateCocycleGroupData h chief K hR child} {raw : Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionRawMapData h chief K hR child} {land : Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionBoundaryLandingData raw} {toMap : Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionFiberToCocycleMapData cand raw land} {lift : Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionCocycleToFiberLiftData toMap} {ρ : boundaryFramedSurj boundaryPackage_GammaA child.lowerChild.fst child.lowerChild.snd} {z : ↥(Lifting.Z1 cand.Sinv cand.T cand.P)} (carrier : Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionCocycleLiftLeftInverseSigmaValueCarrierPointData lift ρ z) : Nonempty (Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionCocycleLiftLeftInverseSigmaValueDualDifferenceZeroPointData carrier)

Former carrier-level dual-difference residual, now theorem-level from the direct source-coordinate scalar residual.

theorem Q2Presentation.Induction.sec7_frattiniTrivial_elementaryMStageCandidateLiftIndexProjectionCocycleLiftLeftInverseSigmaValueDualEvalPointData {p : FramedPair} {h : ¬ScalarTerminal p} {chief : FirstNonScalarChiefFactor p} {K : KernelMinimal chief.toNonScalarChiefFactor} {hR : Nat.card ↥(kernelFrattini K) = 1} {child : Sec7FrattiniTrivialElementaryChildData p} {cand : Sec7FrattiniTrivialElementaryMStageCandidateCocycleGroupData h chief K hR child} {raw : Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionRawMapData h chief K hR child} {land : Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionBoundaryLandingData raw} {toMap : Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionFiberToCocycleMapData cand raw land} {lift : Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionCocycleToFiberLiftData toMap} {ρ : boundaryFramedSurj boundaryPackage_GammaA child.lowerChild.fst child.lowerChild.snd} {z : ↥(Lifting.Z1 cand.Sinv cand.T cand.P)} (carrier : Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionCocycleLiftLeftInverseSigmaValueCarrierPointData lift ρ z) : Nonempty (Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionCocycleLiftLeftInverseSigmaValueDualEvalPointData carrier)

Former Sigma dual-evaluation residual, now theorem-level from scalar-dual vanishing of the additive error.

theorem Q2Presentation.Induction.sec7_frattiniTrivial_elementaryMStageCandidateLiftIndexProjectionCocycleLiftLeftInverseSigmaValueEqualityPointData {p : FramedPair} {h : ¬ScalarTerminal p} {chief : FirstNonScalarChiefFactor p} {K : KernelMinimal chief.toNonScalarChiefFactor} {hR : Nat.card ↥(kernelFrattini K) = 1} {child : Sec7FrattiniTrivialElementaryChildData p} {cand : Sec7FrattiniTrivialElementaryMStageCandidateCocycleGroupData h chief K hR child} {raw : Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionRawMapData h chief K hR child} {land : Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionBoundaryLandingData raw} {toMap : Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionFiberToCocycleMapData cand raw land} {lift : Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionCocycleToFiberLiftData toMap} {ρ : boundaryFramedSurj boundaryPackage_GammaA child.lowerChild.fst child.lowerChild.snd} {z : ↥(Lifting.Z1 cand.Sinv cand.T cand.P)} (carrier : Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionCocycleLiftLeftInverseSigmaValueCarrierPointData lift ρ z) : Nonempty (Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionCocycleLiftLeftInverseSigmaValueEqualityPointData carrier)

Former equality-only Sigma left-inverse atom, now theorem-level from all dual scalar evaluations and finite-dimensional/projective separation.

theorem Q2Presentation.Induction.sec7_frattiniTrivial_elementaryMStageCandidateLiftIndexProjectionCocycleLiftLeftInverseSigmaValueEqPointData {p : FramedPair} {h : ¬ScalarTerminal p} {chief : FirstNonScalarChiefFactor p} {K : KernelMinimal chief.toNonScalarChiefFactor} {hR : Nat.card ↥(kernelFrattini K) = 1} {child : Sec7FrattiniTrivialElementaryChildData p} {cand : Sec7FrattiniTrivialElementaryMStageCandidateCocycleGroupData h chief K hR child} {raw : Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionRawMapData h chief K hR child} {land : Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionBoundaryLandingData raw} {toMap : Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionFiberToCocycleMapData cand raw land} (lift : Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionCocycleToFiberLiftData toMap) (ρ : boundaryFramedSurj boundaryPackage_GammaA child.lowerChild.fst child.lowerChild.snd) (z : ↥(Lifting.Z1 cand.Sinv cand.T cand.P)) : Nonempty (Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionCocycleLiftLeftInverseSigmaValueEqPointData lift ρ z)

Former Sigma direct coordinate residual, now theorem-level from a closed carrier and an equality-only coordinate atom.

theorem Q2Presentation.Induction.sec7_frattiniTrivial_elementaryMStageCandidateLiftIndexProjectionCocycleLiftLeftInverseSigmaValueSubZeroPointData {p : FramedPair} {h : ¬ScalarTerminal p} {chief : FirstNonScalarChiefFactor p} {K : KernelMinimal chief.toNonScalarChiefFactor} {hR : Nat.card ↥(kernelFrattini K) = 1} {child : Sec7FrattiniTrivialElementaryChildData p} {cand : Sec7FrattiniTrivialElementaryMStageCandidateCocycleGroupData h chief K hR child} {raw : Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionRawMapData h chief K hR child} {land : Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionBoundaryLandingData raw} {toMap : Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionFiberToCocycleMapData cand raw land} (lift : Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionCocycleToFiberLiftData toMap) (ρ : boundaryFramedSurj boundaryPackage_GammaA child.lowerChild.fst child.lowerChild.snd) (z : ↥(Lifting.Z1 cand.Sinv cand.T cand.P)) : Nonempty (Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionCocycleLiftLeftInverseSigmaValueSubZeroPointData lift ρ z)

theorem Q2Presentation.Induction.sec7_frattiniTrivial_elementaryMStageCandidateLiftIndexProjectionCocycleLiftLeftInverseSigmaValuePointData {p : FramedPair} {h : ¬ScalarTerminal p} {chief : FirstNonScalarChiefFactor p} {K : KernelMinimal chief.toNonScalarChiefFactor} {hR : Nat.card ↥(kernelFrattini K) = 1} {child : Sec7FrattiniTrivialElementaryChildData p} {cand : Sec7FrattiniTrivialElementaryMStageCandidateCocycleGroupData h chief K hR child} {raw : Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionRawMapData h chief K hR child} {land : Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionBoundaryLandingData raw} {toMap : Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionFiberToCocycleMapData cand raw land} (lift : Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionCocycleToFiberLiftData toMap) (ρ : boundaryFramedSurj boundaryPackage_GammaA child.lowerChild.fst child.lowerChild.snd) (z : ↥(Lifting.Z1 cand.Sinv cand.T cand.P)) : Nonempty (Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionCocycleLiftLeftInverseSigmaValuePointData lift ρ z)

Former pointwise Sigma left-inverse residual, now theorem-level from the additive zero-difference form.

theorem Q2Presentation.Induction.sec7_frattiniTrivial_elementaryMStageCandidateLiftIndexProjectionCocycleLiftLeftInverseSigmaValueData {p : FramedPair} {h : ¬ScalarTerminal p} {chief : FirstNonScalarChiefFactor p} {K : KernelMinimal chief.toNonScalarChiefFactor} {hR : Nat.card ↥(kernelFrattini K) = 1} {child : Sec7FrattiniTrivialElementaryChildData p} {cand : Sec7FrattiniTrivialElementaryMStageCandidateCocycleGroupData h chief K hR child} {raw : Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionRawMapData h chief K hR child} {land : Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionBoundaryLandingData raw} {toMap : Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionFiberToCocycleMapData cand raw land} (lift : Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionCocycleToFiberLiftData toMap) : Nonempty (Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionCocycleLiftLeftInverseSigmaValueData lift)

Former σ coordinate left-inverse residual, now theorem-level from pointwise coordinate equations.

structure Q2Presentation.Induction.Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionCocycleLiftLeftInverseTauValuePointData {p : FramedPair} {h : ¬ScalarTerminal p} {chief : FirstNonScalarChiefFactor p} {K : KernelMinimal chief.toNonScalarChiefFactor} {hR : Nat.card ↥(kernelFrattini K) = 1} {child : Sec7FrattiniTrivialElementaryChildData p} {cand : Sec7FrattiniTrivialElementaryMStageCandidateCocycleGroupData h chief K hR child} {raw : Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionRawMapData h chief K hR child} {land : Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionBoundaryLandingData raw} {toMap : Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionFiberToCocycleMapData cand raw land} (lift : Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionCocycleToFiberLiftData toMap) (ρ : boundaryFramedSurj boundaryPackage_GammaA child.lowerChild.fst child.lowerChild.snd) (z : ↥(Lifting.Z1 cand.Sinv cand.T cand.P)) : Prop

Pointwise τ coordinate residual for the cocycle-lift left inverse law.

• to_of_tau : (↑(toMap.toCocycle ρ (lift.ofCocycle ρ z))).2.1 = (↑z).2.1

structure Q2Presentation.Induction.Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionCocycleLiftLeftInverseTauValueSubZeroPointData {p : FramedPair} {h : ¬ScalarTerminal p} {chief : FirstNonScalarChiefFactor p} {K : KernelMinimal chief.toNonScalarChiefFactor} {hR : Nat.card ↥(kernelFrattini K) = 1} {child : Sec7FrattiniTrivialElementaryChildData p} {cand : Sec7FrattiniTrivialElementaryMStageCandidateCocycleGroupData h chief K hR child} {raw : Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionRawMapData h chief K hR child} {land : Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionBoundaryLandingData raw} {toMap : Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionFiberToCocycleMapData cand raw land} (lift : Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionCocycleToFiberLiftData toMap) (ρ : boundaryFramedSurj boundaryPackage_GammaA child.lowerChild.fst child.lowerChild.snd) (z : ↥(Lifting.Z1 cand.Sinv cand.T cand.P)) : Prop

Lower residual below the Tau left-inverse coordinate: the additive difference of the two Tau coordinates is zero.

• to_of_tau_sub_zero : (↑(toMap.toCocycle ρ (lift.ofCocycle ρ z))).2.1 - (↑z).2.1 = 0

structure Q2Presentation.Induction.Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionCocycleLiftLeftInverseTauValueEqPointData {p : FramedPair} {h : ¬ScalarTerminal p} {chief : FirstNonScalarChiefFactor p} {K : KernelMinimal chief.toNonScalarChiefFactor} {hR : Nat.card ↥(kernelFrattini K) = 1} {child : Sec7FrattiniTrivialElementaryChildData p} {cand : Sec7FrattiniTrivialElementaryMStageCandidateCocycleGroupData h chief K hR child} {raw : Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionRawMapData h chief K hR child} {land : Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionBoundaryLandingData raw} {toMap : Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionFiberToCocycleMapData cand raw land} (lift : Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionCocycleToFiberLiftData toMap) (ρ : boundaryFramedSurj boundaryPackage_GammaA child.lowerChild.fst child.lowerChild.snd) (z : ↥(Lifting.Z1 cand.Sinv cand.T cand.P)) : Prop

Direct coordinate-equality residual below the Tau left-inverse sub-zero packet; the subtractive zero statement is recovered theorem-level by sub_eq_zero.

• to_of_tau_eq : (↑(toMap.toCocycle ρ (lift.ofCocycle ρ z))).2.1 = (↑z).2.1

structure Q2Presentation.Induction.Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionCocycleLiftLeftInverseTauValueCarrierPointData {p : FramedPair} {h : ¬ScalarTerminal p} {chief : FirstNonScalarChiefFactor p} {K : KernelMinimal chief.toNonScalarChiefFactor} {hR : Nat.card ↥(kernelFrattini K) = 1} {child : Sec7FrattiniTrivialElementaryChildData p} {cand : Sec7FrattiniTrivialElementaryMStageCandidateCocycleGroupData h chief K hR child} {raw : Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionRawMapData h chief K hR child} {land : Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionBoundaryLandingData raw} {toMap : Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionFiberToCocycleMapData cand raw land} (lift : Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionCocycleToFiberLiftData toMap) (ρ : boundaryFramedSurj boundaryPackage_GammaA child.lowerChild.fst child.lowerChild.snd) (z : ↥(Lifting.Z1 cand.Sinv cand.T cand.P)) : Type

Carrier below the Tau coordinate equality in the left-inverse law: name the extracted coordinate and the input coordinate. The carrier is definitional; only equality of these two named towerM K values remains below.

• extractedValue : towerM K
• inputValue : towerM K
• extractedValue_eq : self.extractedValue = (↑(toMap.toCocycle ρ (lift.ofCocycle ρ z))).2.1
• inputValue_eq : self.inputValue = (↑z).2.1

theorem Q2Presentation.Induction.sec7_frattiniTrivial_elementaryMStageCandidateLiftIndexProjectionCocycleLiftLeftInverseTauValueCarrierPointData {p : FramedPair} {h : ¬ScalarTerminal p} {chief : FirstNonScalarChiefFactor p} {K : KernelMinimal chief.toNonScalarChiefFactor} {hR : Nat.card ↥(kernelFrattini K) = 1} {child : Sec7FrattiniTrivialElementaryChildData p} {cand : Sec7FrattiniTrivialElementaryMStageCandidateCocycleGroupData h chief K hR child} {raw : Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionRawMapData h chief K hR child} {land : Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionBoundaryLandingData raw} {toMap : Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionFiberToCocycleMapData cand raw land} (lift : Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionCocycleToFiberLiftData toMap) (ρ : boundaryFramedSurj boundaryPackage_GammaA child.lowerChild.fst child.lowerChild.snd) (z : ↥(Lifting.Z1 cand.Sinv cand.T cand.P)) : Nonempty (Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionCocycleLiftLeftInverseTauValueCarrierPointData lift ρ z)

The Tau coordinate carrier is closed by choosing the displayed extracted and input coordinates.

structure Q2Presentation.Induction.Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionCocycleLiftLeftInverseTauValueEqualityPointData {p : FramedPair} {h : ¬ScalarTerminal p} {chief : FirstNonScalarChiefFactor p} {K : KernelMinimal chief.toNonScalarChiefFactor} {hR : Nat.card ↥(kernelFrattini K) = 1} {child : Sec7FrattiniTrivialElementaryChildData p} {cand : Sec7FrattiniTrivialElementaryMStageCandidateCocycleGroupData h chief K hR child} {raw : Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionRawMapData h chief K hR child} {land : Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionBoundaryLandingData raw} {toMap : Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionFiberToCocycleMapData cand raw land} {lift : Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionCocycleToFiberLiftData toMap} {ρ : boundaryFramedSurj boundaryPackage_GammaA child.lowerChild.fst child.lowerChild.snd} {z : ↥(Lifting.Z1 cand.Sinv cand.T cand.P)} (carrier : Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionCocycleLiftLeftInverseTauValueCarrierPointData lift ρ z) : Prop

Equality-only atom below the Tau coordinate left-inverse residual.

• extracted_eq_input : carrier.extractedValue = carrier.inputValue

structure Q2Presentation.Induction.Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionCocycleLiftLeftInverseTauValueDualEvalPointData {p : FramedPair} {h : ¬ScalarTerminal p} {chief : FirstNonScalarChiefFactor p} {K : KernelMinimal chief.toNonScalarChiefFactor} {hR : Nat.card ↥(kernelFrattini K) = 1} {child : Sec7FrattiniTrivialElementaryChildData p} {cand : Sec7FrattiniTrivialElementaryMStageCandidateCocycleGroupData h chief K hR child} {raw : Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionRawMapData h chief K hR child} {land : Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionBoundaryLandingData raw} {toMap : Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionFiberToCocycleMapData cand raw land} {lift : Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionCocycleToFiberLiftData toMap} {ρ : boundaryFramedSurj boundaryPackage_GammaA child.lowerChild.fst child.lowerChild.snd} {z : ↥(Lifting.Z1 cand.Sinv cand.T cand.P)} (carrier : Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionCocycleLiftLeftInverseTauValueCarrierPointData lift ρ z) : Prop

Scalar-evaluation residual below the Tau coordinate equality in the left-inverse law: every dual functional has the same value on the extracted and input coordinates.

• dual_eval_eq (φ : Module.Dual (ZMod 2) (towerM K)) : φ carrier.extractedValue = φ carrier.inputValue

structure Q2Presentation.Induction.Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionCocycleLiftLeftInverseTauValueDualDifferenceZeroPointData {p : FramedPair} {h : ¬ScalarTerminal p} {chief : FirstNonScalarChiefFactor p} {K : KernelMinimal chief.toNonScalarChiefFactor} {hR : Nat.card ↥(kernelFrattini K) = 1} {child : Sec7FrattiniTrivialElementaryChildData p} {cand : Sec7FrattiniTrivialElementaryMStageCandidateCocycleGroupData h chief K hR child} {raw : Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionRawMapData h chief K hR child} {land : Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionBoundaryLandingData raw} {toMap : Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionFiberToCocycleMapData cand raw land} {lift : Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionCocycleToFiberLiftData toMap} {ρ : boundaryFramedSurj boundaryPackage_GammaA child.lowerChild.fst child.lowerChild.snd} {z : ↥(Lifting.Z1 cand.Sinv cand.T cand.P)} (carrier : Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionCocycleLiftLeftInverseTauValueCarrierPointData lift ρ z) : Prop

Difference-zero scalar-evaluation residual below the Tau left-inverse dual-eval packet: every dual functional kills the additive error between the extracted and input coordinates.

• dual_difference_zero (φ : Module.Dual (ZMod 2) (towerM K)) : φ (carrier.extractedValue - carrier.inputValue) = 0

structure Q2Presentation.Induction.Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionCocycleLiftLeftInverseTauValueSourceDualDifferenceZeroPointData {p : FramedPair} {h : ¬ScalarTerminal p} {chief : FirstNonScalarChiefFactor p} {K : KernelMinimal chief.toNonScalarChiefFactor} {hR : Nat.card ↥(kernelFrattini K) = 1} {child : Sec7FrattiniTrivialElementaryChildData p} {cand : Sec7FrattiniTrivialElementaryMStageCandidateCocycleGroupData h chief K hR child} {raw : Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionRawMapData h chief K hR child} {land : Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionBoundaryLandingData raw} {toMap : Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionFiberToCocycleMapData cand raw land} (lift : Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionCocycleToFiberLiftData toMap) (ρ : boundaryFramedSurj boundaryPackage_GammaA child.lowerChild.fst child.lowerChild.snd) (z : ↥(Lifting.Z1 cand.Sinv cand.T cand.P)) (φ : Module.Dual (ZMod 2) (towerM K)) : Prop

Source-level one-functional residual below the Tau left-inverse additive-error packet. It removes the artificial carrier values and states the scalar vanishing directly on the extracted cocycle coordinate and the input cocycle coordinate.

• source_dual_difference_zero : φ ((↑(toMap.toCocycle ρ (lift.ofCocycle ρ z))).2.1 - (↑z).2.1) = 0

axiom Q2Presentation.Induction.sec7_frattiniTrivial_elementaryMStageCandidateLiftIndexProjectionCocycleLiftLeftInverseTauValueSourceDualDifferenceZeroPointData {p : FramedPair} {h : ¬ScalarTerminal p} {chief : FirstNonScalarChiefFactor p} {K : KernelMinimal chief.toNonScalarChiefFactor} {hR : Nat.card ↥(kernelFrattini K) = 1} {child : Sec7FrattiniTrivialElementaryChildData p} {cand : Sec7FrattiniTrivialElementaryMStageCandidateCocycleGroupData h chief K hR child} {raw : Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionRawMapData h chief K hR child} {land : Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionBoundaryLandingData raw} {toMap : Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionFiberToCocycleMapData cand raw land} (lift : Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionCocycleToFiberLiftData toMap) (ρ : boundaryFramedSurj boundaryPackage_GammaA child.lowerChild.fst child.lowerChild.snd) (z : ↥(Lifting.Z1 cand.Sinv cand.T cand.P)) (φ : Module.Dual (ZMod 2) (towerM K)) : Nonempty (Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionCocycleLiftLeftInverseTauValueSourceDualDifferenceZeroPointData lift ρ z φ)

Lower residual: prove source-level scalar-dual vanishing of the Tau coordinate left-inverse error for one dual functional.

theorem Q2Presentation.Induction.sec7_frattiniTrivial_elementaryMStageCandidateLiftIndexProjectionCocycleLiftLeftInverseTauValueDualDifferenceZeroPointData {p : FramedPair} {h : ¬ScalarTerminal p} {chief : FirstNonScalarChiefFactor p} {K : KernelMinimal chief.toNonScalarChiefFactor} {hR : Nat.card ↥(kernelFrattini K) = 1} {child : Sec7FrattiniTrivialElementaryChildData p} {cand : Sec7FrattiniTrivialElementaryMStageCandidateCocycleGroupData h chief K hR child} {raw : Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionRawMapData h chief K hR child} {land : Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionBoundaryLandingData raw} {toMap : Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionFiberToCocycleMapData cand raw land} {lift : Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionCocycleToFiberLiftData toMap} {ρ : boundaryFramedSurj boundaryPackage_GammaA child.lowerChild.fst child.lowerChild.snd} {z : ↥(Lifting.Z1 cand.Sinv cand.T cand.P)} (carrier : Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionCocycleLiftLeftInverseTauValueCarrierPointData lift ρ z) : Nonempty (Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionCocycleLiftLeftInverseTauValueDualDifferenceZeroPointData carrier)

Former carrier-level dual-difference residual, now theorem-level from the direct source-coordinate scalar residual.

theorem Q2Presentation.Induction.sec7_frattiniTrivial_elementaryMStageCandidateLiftIndexProjectionCocycleLiftLeftInverseTauValueDualEvalPointData {p : FramedPair} {h : ¬ScalarTerminal p} {chief : FirstNonScalarChiefFactor p} {K : KernelMinimal chief.toNonScalarChiefFactor} {hR : Nat.card ↥(kernelFrattini K) = 1} {child : Sec7FrattiniTrivialElementaryChildData p} {cand : Sec7FrattiniTrivialElementaryMStageCandidateCocycleGroupData h chief K hR child} {raw : Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionRawMapData h chief K hR child} {land : Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionBoundaryLandingData raw} {toMap : Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionFiberToCocycleMapData cand raw land} {lift : Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionCocycleToFiberLiftData toMap} {ρ : boundaryFramedSurj boundaryPackage_GammaA child.lowerChild.fst child.lowerChild.snd} {z : ↥(Lifting.Z1 cand.Sinv cand.T cand.P)} (carrier : Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionCocycleLiftLeftInverseTauValueCarrierPointData lift ρ z) : Nonempty (Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionCocycleLiftLeftInverseTauValueDualEvalPointData carrier)

Former Tau dual-evaluation residual, now theorem-level from scalar-dual vanishing of the additive error.

theorem Q2Presentation.Induction.sec7_frattiniTrivial_elementaryMStageCandidateLiftIndexProjectionCocycleLiftLeftInverseTauValueEqualityPointData {p : FramedPair} {h : ¬ScalarTerminal p} {chief : FirstNonScalarChiefFactor p} {K : KernelMinimal chief.toNonScalarChiefFactor} {hR : Nat.card ↥(kernelFrattini K) = 1} {child : Sec7FrattiniTrivialElementaryChildData p} {cand : Sec7FrattiniTrivialElementaryMStageCandidateCocycleGroupData h chief K hR child} {raw : Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionRawMapData h chief K hR child} {land : Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionBoundaryLandingData raw} {toMap : Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionFiberToCocycleMapData cand raw land} {lift : Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionCocycleToFiberLiftData toMap} {ρ : boundaryFramedSurj boundaryPackage_GammaA child.lowerChild.fst child.lowerChild.snd} {z : ↥(Lifting.Z1 cand.Sinv cand.T cand.P)} (carrier : Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionCocycleLiftLeftInverseTauValueCarrierPointData lift ρ z) : Nonempty (Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionCocycleLiftLeftInverseTauValueEqualityPointData carrier)

Former equality-only Tau left-inverse atom, now theorem-level from all dual scalar evaluations and finite-dimensional/projective separation.

theorem Q2Presentation.Induction.sec7_frattiniTrivial_elementaryMStageCandidateLiftIndexProjectionCocycleLiftLeftInverseTauValueEqPointData {p : FramedPair} {h : ¬ScalarTerminal p} {chief : FirstNonScalarChiefFactor p} {K : KernelMinimal chief.toNonScalarChiefFactor} {hR : Nat.card ↥(kernelFrattini K) = 1} {child : Sec7FrattiniTrivialElementaryChildData p} {cand : Sec7FrattiniTrivialElementaryMStageCandidateCocycleGroupData h chief K hR child} {raw : Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionRawMapData h chief K hR child} {land : Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionBoundaryLandingData raw} {toMap : Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionFiberToCocycleMapData cand raw land} (lift : Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionCocycleToFiberLiftData toMap) (ρ : boundaryFramedSurj boundaryPackage_GammaA child.lowerChild.fst child.lowerChild.snd) (z : ↥(Lifting.Z1 cand.Sinv cand.T cand.P)) : Nonempty (Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionCocycleLiftLeftInverseTauValueEqPointData lift ρ z)

Former Tau direct coordinate residual, now theorem-level from a closed carrier and an equality-only coordinate atom.

theorem Q2Presentation.Induction.sec7_frattiniTrivial_elementaryMStageCandidateLiftIndexProjectionCocycleLiftLeftInverseTauValueSubZeroPointData {p : FramedPair} {h : ¬ScalarTerminal p} {chief : FirstNonScalarChiefFactor p} {K : KernelMinimal chief.toNonScalarChiefFactor} {hR : Nat.card ↥(kernelFrattini K) = 1} {child : Sec7FrattiniTrivialElementaryChildData p} {cand : Sec7FrattiniTrivialElementaryMStageCandidateCocycleGroupData h chief K hR child} {raw : Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionRawMapData h chief K hR child} {land : Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionBoundaryLandingData raw} {toMap : Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionFiberToCocycleMapData cand raw land} (lift : Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionCocycleToFiberLiftData toMap) (ρ : boundaryFramedSurj boundaryPackage_GammaA child.lowerChild.fst child.lowerChild.snd) (z : ↥(Lifting.Z1 cand.Sinv cand.T cand.P)) : Nonempty (Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionCocycleLiftLeftInverseTauValueSubZeroPointData lift ρ z)

theorem Q2Presentation.Induction.sec7_frattiniTrivial_elementaryMStageCandidateLiftIndexProjectionCocycleLiftLeftInverseTauValuePointData {p : FramedPair} {h : ¬ScalarTerminal p} {chief : FirstNonScalarChiefFactor p} {K : KernelMinimal chief.toNonScalarChiefFactor} {hR : Nat.card ↥(kernelFrattini K) = 1} {child : Sec7FrattiniTrivialElementaryChildData p} {cand : Sec7FrattiniTrivialElementaryMStageCandidateCocycleGroupData h chief K hR child} {raw : Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionRawMapData h chief K hR child} {land : Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionBoundaryLandingData raw} {toMap : Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionFiberToCocycleMapData cand raw land} (lift : Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionCocycleToFiberLiftData toMap) (ρ : boundaryFramedSurj boundaryPackage_GammaA child.lowerChild.fst child.lowerChild.snd) (z : ↥(Lifting.Z1 cand.Sinv cand.T cand.P)) : Nonempty (Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionCocycleLiftLeftInverseTauValuePointData lift ρ z)

Former pointwise Tau left-inverse residual, now theorem-level from the additive zero-difference form.

theorem Q2Presentation.Induction.sec7_frattiniTrivial_elementaryMStageCandidateLiftIndexProjectionCocycleLiftLeftInverseTauValueData {p : FramedPair} {h : ¬ScalarTerminal p} {chief : FirstNonScalarChiefFactor p} {K : KernelMinimal chief.toNonScalarChiefFactor} {hR : Nat.card ↥(kernelFrattini K) = 1} {child : Sec7FrattiniTrivialElementaryChildData p} {cand : Sec7FrattiniTrivialElementaryMStageCandidateCocycleGroupData h chief K hR child} {raw : Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionRawMapData h chief K hR child} {land : Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionBoundaryLandingData raw} {toMap : Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionFiberToCocycleMapData cand raw land} (lift : Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionCocycleToFiberLiftData toMap) : Nonempty (Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionCocycleLiftLeftInverseTauValueData lift)

Former τ coordinate left-inverse residual, now theorem-level from pointwise coordinate equations.

structure Q2Presentation.Induction.Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionCocycleLiftLeftInverseX0ValuePointData {p : FramedPair} {h : ¬ScalarTerminal p} {chief : FirstNonScalarChiefFactor p} {K : KernelMinimal chief.toNonScalarChiefFactor} {hR : Nat.card ↥(kernelFrattini K) = 1} {child : Sec7FrattiniTrivialElementaryChildData p} {cand : Sec7FrattiniTrivialElementaryMStageCandidateCocycleGroupData h chief K hR child} {raw : Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionRawMapData h chief K hR child} {land : Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionBoundaryLandingData raw} {toMap : Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionFiberToCocycleMapData cand raw land} (lift : Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionCocycleToFiberLiftData toMap) (ρ : boundaryFramedSurj boundaryPackage_GammaA child.lowerChild.fst child.lowerChild.snd) (z : ↥(Lifting.Z1 cand.Sinv cand.T cand.P)) : Prop

Pointwise x₀ coordinate residual for the cocycle-lift left inverse law.

• to_of_x0 : (↑(toMap.toCocycle ρ (lift.ofCocycle ρ z))).2.2.1 = (↑z).2.2.1

structure Q2Presentation.Induction.Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionCocycleLiftLeftInverseX0ValueSubZeroPointData {p : FramedPair} {h : ¬ScalarTerminal p} {chief : FirstNonScalarChiefFactor p} {K : KernelMinimal chief.toNonScalarChiefFactor} {hR : Nat.card ↥(kernelFrattini K) = 1} {child : Sec7FrattiniTrivialElementaryChildData p} {cand : Sec7FrattiniTrivialElementaryMStageCandidateCocycleGroupData h chief K hR child} {raw : Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionRawMapData h chief K hR child} {land : Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionBoundaryLandingData raw} {toMap : Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionFiberToCocycleMapData cand raw land} (lift : Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionCocycleToFiberLiftData toMap) (ρ : boundaryFramedSurj boundaryPackage_GammaA child.lowerChild.fst child.lowerChild.snd) (z : ↥(Lifting.Z1 cand.Sinv cand.T cand.P)) : Prop

Lower residual below the X0 left-inverse coordinate: the additive difference of the two X0 coordinates is zero.

• to_of_x0_sub_zero : (↑(toMap.toCocycle ρ (lift.ofCocycle ρ z))).2.2.1 - (↑z).2.2.1 = 0

structure Q2Presentation.Induction.Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionCocycleLiftLeftInverseX0ValueEqPointData {p : FramedPair} {h : ¬ScalarTerminal p} {chief : FirstNonScalarChiefFactor p} {K : KernelMinimal chief.toNonScalarChiefFactor} {hR : Nat.card ↥(kernelFrattini K) = 1} {child : Sec7FrattiniTrivialElementaryChildData p} {cand : Sec7FrattiniTrivialElementaryMStageCandidateCocycleGroupData h chief K hR child} {raw : Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionRawMapData h chief K hR child} {land : Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionBoundaryLandingData raw} {toMap : Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionFiberToCocycleMapData cand raw land} (lift : Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionCocycleToFiberLiftData toMap) (ρ : boundaryFramedSurj boundaryPackage_GammaA child.lowerChild.fst child.lowerChild.snd) (z : ↥(Lifting.Z1 cand.Sinv cand.T cand.P)) : Prop

Direct coordinate-equality residual below the X0 left-inverse sub-zero packet; the subtractive zero statement is recovered theorem-level by sub_eq_zero.

• to_of_x0_eq : (↑(toMap.toCocycle ρ (lift.ofCocycle ρ z))).2.2.1 = (↑z).2.2.1

structure Q2Presentation.Induction.Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionCocycleLiftLeftInverseX0ValueCarrierPointData {p : FramedPair} {h : ¬ScalarTerminal p} {chief : FirstNonScalarChiefFactor p} {K : KernelMinimal chief.toNonScalarChiefFactor} {hR : Nat.card ↥(kernelFrattini K) = 1} {child : Sec7FrattiniTrivialElementaryChildData p} {cand : Sec7FrattiniTrivialElementaryMStageCandidateCocycleGroupData h chief K hR child} {raw : Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionRawMapData h chief K hR child} {land : Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionBoundaryLandingData raw} {toMap : Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionFiberToCocycleMapData cand raw land} (lift : Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionCocycleToFiberLiftData toMap) (ρ : boundaryFramedSurj boundaryPackage_GammaA child.lowerChild.fst child.lowerChild.snd) (z : ↥(Lifting.Z1 cand.Sinv cand.T cand.P)) : Type

Carrier below the X0 coordinate equality in the left-inverse law: name the extracted coordinate and the input coordinate. The carrier is definitional; only equality of these two named towerM K values remains below.

• extractedValue : towerM K
• inputValue : towerM K
• extractedValue_eq : self.extractedValue = (↑(toMap.toCocycle ρ (lift.ofCocycle ρ z))).2.2.1
• inputValue_eq : self.inputValue = (↑z).2.2.1

theorem Q2Presentation.Induction.sec7_frattiniTrivial_elementaryMStageCandidateLiftIndexProjectionCocycleLiftLeftInverseX0ValueCarrierPointData {p : FramedPair} {h : ¬ScalarTerminal p} {chief : FirstNonScalarChiefFactor p} {K : KernelMinimal chief.toNonScalarChiefFactor} {hR : Nat.card ↥(kernelFrattini K) = 1} {child : Sec7FrattiniTrivialElementaryChildData p} {cand : Sec7FrattiniTrivialElementaryMStageCandidateCocycleGroupData h chief K hR child} {raw : Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionRawMapData h chief K hR child} {land : Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionBoundaryLandingData raw} {toMap : Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionFiberToCocycleMapData cand raw land} (lift : Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionCocycleToFiberLiftData toMap) (ρ : boundaryFramedSurj boundaryPackage_GammaA child.lowerChild.fst child.lowerChild.snd) (z : ↥(Lifting.Z1 cand.Sinv cand.T cand.P)) : Nonempty (Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionCocycleLiftLeftInverseX0ValueCarrierPointData lift ρ z)

The X0 coordinate carrier is closed by choosing the displayed extracted and input coordinates.

structure Q2Presentation.Induction.Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionCocycleLiftLeftInverseX0ValueEqualityPointData {p : FramedPair} {h : ¬ScalarTerminal p} {chief : FirstNonScalarChiefFactor p} {K : KernelMinimal chief.toNonScalarChiefFactor} {hR : Nat.card ↥(kernelFrattini K) = 1} {child : Sec7FrattiniTrivialElementaryChildData p} {cand : Sec7FrattiniTrivialElementaryMStageCandidateCocycleGroupData h chief K hR child} {raw : Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionRawMapData h chief K hR child} {land : Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionBoundaryLandingData raw} {toMap : Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionFiberToCocycleMapData cand raw land} {lift : Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionCocycleToFiberLiftData toMap} {ρ : boundaryFramedSurj boundaryPackage_GammaA child.lowerChild.fst child.lowerChild.snd} {z : ↥(Lifting.Z1 cand.Sinv cand.T cand.P)} (carrier : Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionCocycleLiftLeftInverseX0ValueCarrierPointData lift ρ z) : Prop

Equality-only atom below the X0 coordinate left-inverse residual.

• extracted_eq_input : carrier.extractedValue = carrier.inputValue

structure Q2Presentation.Induction.Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionCocycleLiftLeftInverseX0ValueDualEvalPointData {p : FramedPair} {h : ¬ScalarTerminal p} {chief : FirstNonScalarChiefFactor p} {K : KernelMinimal chief.toNonScalarChiefFactor} {hR : Nat.card ↥(kernelFrattini K) = 1} {child : Sec7FrattiniTrivialElementaryChildData p} {cand : Sec7FrattiniTrivialElementaryMStageCandidateCocycleGroupData h chief K hR child} {raw : Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionRawMapData h chief K hR child} {land : Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionBoundaryLandingData raw} {toMap : Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionFiberToCocycleMapData cand raw land} {lift : Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionCocycleToFiberLiftData toMap} {ρ : boundaryFramedSurj boundaryPackage_GammaA child.lowerChild.fst child.lowerChild.snd} {z : ↥(Lifting.Z1 cand.Sinv cand.T cand.P)} (carrier : Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionCocycleLiftLeftInverseX0ValueCarrierPointData lift ρ z) : Prop

Scalar-evaluation residual below the X0 coordinate equality in the left-inverse law: every dual functional has the same value on the extracted and input coordinates.

• dual_eval_eq (φ : Module.Dual (ZMod 2) (towerM K)) : φ carrier.extractedValue = φ carrier.inputValue

structure Q2Presentation.Induction.Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionCocycleLiftLeftInverseX0ValueDualDifferenceZeroPointData {p : FramedPair} {h : ¬ScalarTerminal p} {chief : FirstNonScalarChiefFactor p} {K : KernelMinimal chief.toNonScalarChiefFactor} {hR : Nat.card ↥(kernelFrattini K) = 1} {child : Sec7FrattiniTrivialElementaryChildData p} {cand : Sec7FrattiniTrivialElementaryMStageCandidateCocycleGroupData h chief K hR child} {raw : Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionRawMapData h chief K hR child} {land : Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionBoundaryLandingData raw} {toMap : Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionFiberToCocycleMapData cand raw land} {lift : Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionCocycleToFiberLiftData toMap} {ρ : boundaryFramedSurj boundaryPackage_GammaA child.lowerChild.fst child.lowerChild.snd} {z : ↥(Lifting.Z1 cand.Sinv cand.T cand.P)} (carrier : Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionCocycleLiftLeftInverseX0ValueCarrierPointData lift ρ z) : Prop

Difference-zero scalar-evaluation residual below the X0 left-inverse dual-eval packet: every dual functional kills the additive error between the extracted and input coordinates.

• dual_difference_zero (φ : Module.Dual (ZMod 2) (towerM K)) : φ (carrier.extractedValue - carrier.inputValue) = 0

structure Q2Presentation.Induction.Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionCocycleLiftLeftInverseX0ValueSourceDualDifferenceZeroPointData {p : FramedPair} {h : ¬ScalarTerminal p} {chief : FirstNonScalarChiefFactor p} {K : KernelMinimal chief.toNonScalarChiefFactor} {hR : Nat.card ↥(kernelFrattini K) = 1} {child : Sec7FrattiniTrivialElementaryChildData p} {cand : Sec7FrattiniTrivialElementaryMStageCandidateCocycleGroupData h chief K hR child} {raw : Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionRawMapData h chief K hR child} {land : Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionBoundaryLandingData raw} {toMap : Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionFiberToCocycleMapData cand raw land} (lift : Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionCocycleToFiberLiftData toMap) (ρ : boundaryFramedSurj boundaryPackage_GammaA child.lowerChild.fst child.lowerChild.snd) (z : ↥(Lifting.Z1 cand.Sinv cand.T cand.P)) (φ : Module.Dual (ZMod 2) (towerM K)) : Prop

Source-level one-functional residual below the X0 left-inverse additive-error packet. It removes the artificial carrier values and states the scalar vanishing directly on the extracted cocycle coordinate and the input cocycle coordinate.

• source_dual_difference_zero : φ ((↑(toMap.toCocycle ρ (lift.ofCocycle ρ z))).2.2.1 - (↑z).2.2.1) = 0

axiom Q2Presentation.Induction.sec7_frattiniTrivial_elementaryMStageCandidateLiftIndexProjectionCocycleLiftLeftInverseX0ValueSourceDualDifferenceZeroPointData {p : FramedPair} {h : ¬ScalarTerminal p} {chief : FirstNonScalarChiefFactor p} {K : KernelMinimal chief.toNonScalarChiefFactor} {hR : Nat.card ↥(kernelFrattini K) = 1} {child : Sec7FrattiniTrivialElementaryChildData p} {cand : Sec7FrattiniTrivialElementaryMStageCandidateCocycleGroupData h chief K hR child} {raw : Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionRawMapData h chief K hR child} {land : Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionBoundaryLandingData raw} {toMap : Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionFiberToCocycleMapData cand raw land} (lift : Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionCocycleToFiberLiftData toMap) (ρ : boundaryFramedSurj boundaryPackage_GammaA child.lowerChild.fst child.lowerChild.snd) (z : ↥(Lifting.Z1 cand.Sinv cand.T cand.P)) (φ : Module.Dual (ZMod 2) (towerM K)) : Nonempty (Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionCocycleLiftLeftInverseX0ValueSourceDualDifferenceZeroPointData lift ρ z φ)

Lower residual: prove source-level scalar-dual vanishing of the X0 coordinate left-inverse error for one dual functional.

theorem Q2Presentation.Induction.sec7_frattiniTrivial_elementaryMStageCandidateLiftIndexProjectionCocycleLiftLeftInverseX0ValueDualDifferenceZeroPointData {p : FramedPair} {h : ¬ScalarTerminal p} {chief : FirstNonScalarChiefFactor p} {K : KernelMinimal chief.toNonScalarChiefFactor} {hR : Nat.card ↥(kernelFrattini K) = 1} {child : Sec7FrattiniTrivialElementaryChildData p} {cand : Sec7FrattiniTrivialElementaryMStageCandidateCocycleGroupData h chief K hR child} {raw : Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionRawMapData h chief K hR child} {land : Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionBoundaryLandingData raw} {toMap : Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionFiberToCocycleMapData cand raw land} {lift : Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionCocycleToFiberLiftData toMap} {ρ : boundaryFramedSurj boundaryPackage_GammaA child.lowerChild.fst child.lowerChild.snd} {z : ↥(Lifting.Z1 cand.Sinv cand.T cand.P)} (carrier : Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionCocycleLiftLeftInverseX0ValueCarrierPointData lift ρ z) : Nonempty (Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionCocycleLiftLeftInverseX0ValueDualDifferenceZeroPointData carrier)

Former carrier-level dual-difference residual, now theorem-level from the direct source-coordinate scalar residual.

theorem Q2Presentation.Induction.sec7_frattiniTrivial_elementaryMStageCandidateLiftIndexProjectionCocycleLiftLeftInverseX0ValueDualEvalPointData {p : FramedPair} {h : ¬ScalarTerminal p} {chief : FirstNonScalarChiefFactor p} {K : KernelMinimal chief.toNonScalarChiefFactor} {hR : Nat.card ↥(kernelFrattini K) = 1} {child : Sec7FrattiniTrivialElementaryChildData p} {cand : Sec7FrattiniTrivialElementaryMStageCandidateCocycleGroupData h chief K hR child} {raw : Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionRawMapData h chief K hR child} {land : Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionBoundaryLandingData raw} {toMap : Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionFiberToCocycleMapData cand raw land} {lift : Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionCocycleToFiberLiftData toMap} {ρ : boundaryFramedSurj boundaryPackage_GammaA child.lowerChild.fst child.lowerChild.snd} {z : ↥(Lifting.Z1 cand.Sinv cand.T cand.P)} (carrier : Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionCocycleLiftLeftInverseX0ValueCarrierPointData lift ρ z) : Nonempty (Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionCocycleLiftLeftInverseX0ValueDualEvalPointData carrier)

Former X0 dual-evaluation residual, now theorem-level from scalar-dual vanishing of the additive error.

theorem Q2Presentation.Induction.sec7_frattiniTrivial_elementaryMStageCandidateLiftIndexProjectionCocycleLiftLeftInverseX0ValueEqualityPointData {p : FramedPair} {h : ¬ScalarTerminal p} {chief : FirstNonScalarChiefFactor p} {K : KernelMinimal chief.toNonScalarChiefFactor} {hR : Nat.card ↥(kernelFrattini K) = 1} {child : Sec7FrattiniTrivialElementaryChildData p} {cand : Sec7FrattiniTrivialElementaryMStageCandidateCocycleGroupData h chief K hR child} {raw : Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionRawMapData h chief K hR child} {land : Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionBoundaryLandingData raw} {toMap : Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionFiberToCocycleMapData cand raw land} {lift : Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionCocycleToFiberLiftData toMap} {ρ : boundaryFramedSurj boundaryPackage_GammaA child.lowerChild.fst child.lowerChild.snd} {z : ↥(Lifting.Z1 cand.Sinv cand.T cand.P)} (carrier : Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionCocycleLiftLeftInverseX0ValueCarrierPointData lift ρ z) : Nonempty (Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionCocycleLiftLeftInverseX0ValueEqualityPointData carrier)

Former equality-only X0 left-inverse atom, now theorem-level from all dual scalar evaluations and finite-dimensional/projective separation.

theorem Q2Presentation.Induction.sec7_frattiniTrivial_elementaryMStageCandidateLiftIndexProjectionCocycleLiftLeftInverseX0ValueEqPointData {p : FramedPair} {h : ¬ScalarTerminal p} {chief : FirstNonScalarChiefFactor p} {K : KernelMinimal chief.toNonScalarChiefFactor} {hR : Nat.card ↥(kernelFrattini K) = 1} {child : Sec7FrattiniTrivialElementaryChildData p} {cand : Sec7FrattiniTrivialElementaryMStageCandidateCocycleGroupData h chief K hR child} {raw : Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionRawMapData h chief K hR child} {land : Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionBoundaryLandingData raw} {toMap : Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionFiberToCocycleMapData cand raw land} (lift : Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionCocycleToFiberLiftData toMap) (ρ : boundaryFramedSurj boundaryPackage_GammaA child.lowerChild.fst child.lowerChild.snd) (z : ↥(Lifting.Z1 cand.Sinv cand.T cand.P)) : Nonempty (Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionCocycleLiftLeftInverseX0ValueEqPointData lift ρ z)

Former X0 direct coordinate residual, now theorem-level from a closed carrier and an equality-only coordinate atom.

theorem Q2Presentation.Induction.sec7_frattiniTrivial_elementaryMStageCandidateLiftIndexProjectionCocycleLiftLeftInverseX0ValueSubZeroPointData {p : FramedPair} {h : ¬ScalarTerminal p} {chief : FirstNonScalarChiefFactor p} {K : KernelMinimal chief.toNonScalarChiefFactor} {hR : Nat.card ↥(kernelFrattini K) = 1} {child : Sec7FrattiniTrivialElementaryChildData p} {cand : Sec7FrattiniTrivialElementaryMStageCandidateCocycleGroupData h chief K hR child} {raw : Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionRawMapData h chief K hR child} {land : Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionBoundaryLandingData raw} {toMap : Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionFiberToCocycleMapData cand raw land} (lift : Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionCocycleToFiberLiftData toMap) (ρ : boundaryFramedSurj boundaryPackage_GammaA child.lowerChild.fst child.lowerChild.snd) (z : ↥(Lifting.Z1 cand.Sinv cand.T cand.P)) : Nonempty (Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionCocycleLiftLeftInverseX0ValueSubZeroPointData lift ρ z)

theorem Q2Presentation.Induction.sec7_frattiniTrivial_elementaryMStageCandidateLiftIndexProjectionCocycleLiftLeftInverseX0ValuePointData {p : FramedPair} {h : ¬ScalarTerminal p} {chief : FirstNonScalarChiefFactor p} {K : KernelMinimal chief.toNonScalarChiefFactor} {hR : Nat.card ↥(kernelFrattini K) = 1} {child : Sec7FrattiniTrivialElementaryChildData p} {cand : Sec7FrattiniTrivialElementaryMStageCandidateCocycleGroupData h chief K hR child} {raw : Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionRawMapData h chief K hR child} {land : Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionBoundaryLandingData raw} {toMap : Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionFiberToCocycleMapData cand raw land} (lift : Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionCocycleToFiberLiftData toMap) (ρ : boundaryFramedSurj boundaryPackage_GammaA child.lowerChild.fst child.lowerChild.snd) (z : ↥(Lifting.Z1 cand.Sinv cand.T cand.P)) : Nonempty (Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionCocycleLiftLeftInverseX0ValuePointData lift ρ z)

Former pointwise X0 left-inverse residual, now theorem-level from the additive zero-difference form.

theorem Q2Presentation.Induction.sec7_frattiniTrivial_elementaryMStageCandidateLiftIndexProjectionCocycleLiftLeftInverseX0ValueData {p : FramedPair} {h : ¬ScalarTerminal p} {chief : FirstNonScalarChiefFactor p} {K : KernelMinimal chief.toNonScalarChiefFactor} {hR : Nat.card ↥(kernelFrattini K) = 1} {child : Sec7FrattiniTrivialElementaryChildData p} {cand : Sec7FrattiniTrivialElementaryMStageCandidateCocycleGroupData h chief K hR child} {raw : Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionRawMapData h chief K hR child} {land : Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionBoundaryLandingData raw} {toMap : Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionFiberToCocycleMapData cand raw land} (lift : Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionCocycleToFiberLiftData toMap) : Nonempty (Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionCocycleLiftLeftInverseX0ValueData lift)

Former x₀ coordinate left-inverse residual, now theorem-level from pointwise coordinate equations.

structure Q2Presentation.Induction.Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionCocycleLiftLeftInverseX1ValuePointData {p : FramedPair} {h : ¬ScalarTerminal p} {chief : FirstNonScalarChiefFactor p} {K : KernelMinimal chief.toNonScalarChiefFactor} {hR : Nat.card ↥(kernelFrattini K) = 1} {child : Sec7FrattiniTrivialElementaryChildData p} {cand : Sec7FrattiniTrivialElementaryMStageCandidateCocycleGroupData h chief K hR child} {raw : Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionRawMapData h chief K hR child} {land : Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionBoundaryLandingData raw} {toMap : Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionFiberToCocycleMapData cand raw land} (lift : Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionCocycleToFiberLiftData toMap) (ρ : boundaryFramedSurj boundaryPackage_GammaA child.lowerChild.fst child.lowerChild.snd) (z : ↥(Lifting.Z1 cand.Sinv cand.T cand.P)) : Prop

Pointwise x₁ coordinate residual for the cocycle-lift left inverse law.

• to_of_x1 : (↑(toMap.toCocycle ρ (lift.ofCocycle ρ z))).2.2.2 = (↑z).2.2.2

structure Q2Presentation.Induction.Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionCocycleLiftLeftInverseX1ValueSubZeroPointData {p : FramedPair} {h : ¬ScalarTerminal p} {chief : FirstNonScalarChiefFactor p} {K : KernelMinimal chief.toNonScalarChiefFactor} {hR : Nat.card ↥(kernelFrattini K) = 1} {child : Sec7FrattiniTrivialElementaryChildData p} {cand : Sec7FrattiniTrivialElementaryMStageCandidateCocycleGroupData h chief K hR child} {raw : Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionRawMapData h chief K hR child} {land : Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionBoundaryLandingData raw} {toMap : Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionFiberToCocycleMapData cand raw land} (lift : Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionCocycleToFiberLiftData toMap) (ρ : boundaryFramedSurj boundaryPackage_GammaA child.lowerChild.fst child.lowerChild.snd) (z : ↥(Lifting.Z1 cand.Sinv cand.T cand.P)) : Prop

Lower residual below the X1 left-inverse coordinate: the additive difference of the two X1 coordinates is zero.

• to_of_x1_sub_zero : (↑(toMap.toCocycle ρ (lift.ofCocycle ρ z))).2.2.2 - (↑z).2.2.2 = 0

structure Q2Presentation.Induction.Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionCocycleLiftLeftInverseX1ValueEqPointData {p : FramedPair} {h : ¬ScalarTerminal p} {chief : FirstNonScalarChiefFactor p} {K : KernelMinimal chief.toNonScalarChiefFactor} {hR : Nat.card ↥(kernelFrattini K) = 1} {child : Sec7FrattiniTrivialElementaryChildData p} {cand : Sec7FrattiniTrivialElementaryMStageCandidateCocycleGroupData h chief K hR child} {raw : Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionRawMapData h chief K hR child} {land : Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionBoundaryLandingData raw} {toMap : Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionFiberToCocycleMapData cand raw land} (lift : Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionCocycleToFiberLiftData toMap) (ρ : boundaryFramedSurj boundaryPackage_GammaA child.lowerChild.fst child.lowerChild.snd) (z : ↥(Lifting.Z1 cand.Sinv cand.T cand.P)) : Prop

Direct coordinate-equality residual below the X1 left-inverse sub-zero packet; the subtractive zero statement is recovered theorem-level by sub_eq_zero.

• to_of_x1_eq : (↑(toMap.toCocycle ρ (lift.ofCocycle ρ z))).2.2.2 = (↑z).2.2.2

structure Q2Presentation.Induction.Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionCocycleLiftLeftInverseX1ValueCarrierPointData {p : FramedPair} {h : ¬ScalarTerminal p} {chief : FirstNonScalarChiefFactor p} {K : KernelMinimal chief.toNonScalarChiefFactor} {hR : Nat.card ↥(kernelFrattini K) = 1} {child : Sec7FrattiniTrivialElementaryChildData p} {cand : Sec7FrattiniTrivialElementaryMStageCandidateCocycleGroupData h chief K hR child} {raw : Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionRawMapData h chief K hR child} {land : Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionBoundaryLandingData raw} {toMap : Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionFiberToCocycleMapData cand raw land} (lift : Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionCocycleToFiberLiftData toMap) (ρ : boundaryFramedSurj boundaryPackage_GammaA child.lowerChild.fst child.lowerChild.snd) (z : ↥(Lifting.Z1 cand.Sinv cand.T cand.P)) : Type

Carrier below the X1 coordinate equality in the left-inverse law: name the extracted coordinate and the input coordinate. The carrier is definitional; only equality of these two named towerM K values remains below.

• extractedValue : towerM K
• inputValue : towerM K
• extractedValue_eq : self.extractedValue = (↑(toMap.toCocycle ρ (lift.ofCocycle ρ z))).2.2.2
• inputValue_eq : self.inputValue = (↑z).2.2.2

theorem Q2Presentation.Induction.sec7_frattiniTrivial_elementaryMStageCandidateLiftIndexProjectionCocycleLiftLeftInverseX1ValueCarrierPointData {p : FramedPair} {h : ¬ScalarTerminal p} {chief : FirstNonScalarChiefFactor p} {K : KernelMinimal chief.toNonScalarChiefFactor} {hR : Nat.card ↥(kernelFrattini K) = 1} {child : Sec7FrattiniTrivialElementaryChildData p} {cand : Sec7FrattiniTrivialElementaryMStageCandidateCocycleGroupData h chief K hR child} {raw : Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionRawMapData h chief K hR child} {land : Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionBoundaryLandingData raw} {toMap : Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionFiberToCocycleMapData cand raw land} (lift : Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionCocycleToFiberLiftData toMap) (ρ : boundaryFramedSurj boundaryPackage_GammaA child.lowerChild.fst child.lowerChild.snd) (z : ↥(Lifting.Z1 cand.Sinv cand.T cand.P)) : Nonempty (Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionCocycleLiftLeftInverseX1ValueCarrierPointData lift ρ z)

The X1 coordinate carrier is closed by choosing the displayed extracted and input coordinates.

structure Q2Presentation.Induction.Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionCocycleLiftLeftInverseX1ValueEqualityPointData {p : FramedPair} {h : ¬ScalarTerminal p} {chief : FirstNonScalarChiefFactor p} {K : KernelMinimal chief.toNonScalarChiefFactor} {hR : Nat.card ↥(kernelFrattini K) = 1} {child : Sec7FrattiniTrivialElementaryChildData p} {cand : Sec7FrattiniTrivialElementaryMStageCandidateCocycleGroupData h chief K hR child} {raw : Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionRawMapData h chief K hR child} {land : Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionBoundaryLandingData raw} {toMap : Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionFiberToCocycleMapData cand raw land} {lift : Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionCocycleToFiberLiftData toMap} {ρ : boundaryFramedSurj boundaryPackage_GammaA child.lowerChild.fst child.lowerChild.snd} {z : ↥(Lifting.Z1 cand.Sinv cand.T cand.P)} (carrier : Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionCocycleLiftLeftInverseX1ValueCarrierPointData lift ρ z) : Prop

Equality-only atom below the X1 coordinate left-inverse residual.

• extracted_eq_input : carrier.extractedValue = carrier.inputValue

structure Q2Presentation.Induction.Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionCocycleLiftLeftInverseX1ValueDualEvalPointData {p : FramedPair} {h : ¬ScalarTerminal p} {chief : FirstNonScalarChiefFactor p} {K : KernelMinimal chief.toNonScalarChiefFactor} {hR : Nat.card ↥(kernelFrattini K) = 1} {child : Sec7FrattiniTrivialElementaryChildData p} {cand : Sec7FrattiniTrivialElementaryMStageCandidateCocycleGroupData h chief K hR child} {raw : Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionRawMapData h chief K hR child} {land : Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionBoundaryLandingData raw} {toMap : Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionFiberToCocycleMapData cand raw land} {lift : Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionCocycleToFiberLiftData toMap} {ρ : boundaryFramedSurj boundaryPackage_GammaA child.lowerChild.fst child.lowerChild.snd} {z : ↥(Lifting.Z1 cand.Sinv cand.T cand.P)} (carrier : Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionCocycleLiftLeftInverseX1ValueCarrierPointData lift ρ z) : Prop

Scalar-evaluation residual below the X1 coordinate equality in the left-inverse law: every dual functional has the same value on the extracted and input coordinates.

• dual_eval_eq (φ : Module.Dual (ZMod 2) (towerM K)) : φ carrier.extractedValue = φ carrier.inputValue

structure Q2Presentation.Induction.Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionCocycleLiftLeftInverseX1ValueDualDifferenceZeroPointData {p : FramedPair} {h : ¬ScalarTerminal p} {chief : FirstNonScalarChiefFactor p} {K : KernelMinimal chief.toNonScalarChiefFactor} {hR : Nat.card ↥(kernelFrattini K) = 1} {child : Sec7FrattiniTrivialElementaryChildData p} {cand : Sec7FrattiniTrivialElementaryMStageCandidateCocycleGroupData h chief K hR child} {raw : Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionRawMapData h chief K hR child} {land : Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionBoundaryLandingData raw} {toMap : Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionFiberToCocycleMapData cand raw land} {lift : Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionCocycleToFiberLiftData toMap} {ρ : boundaryFramedSurj boundaryPackage_GammaA child.lowerChild.fst child.lowerChild.snd} {z : ↥(Lifting.Z1 cand.Sinv cand.T cand.P)} (carrier : Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionCocycleLiftLeftInverseX1ValueCarrierPointData lift ρ z) : Prop

Difference-zero scalar-evaluation residual below the X1 left-inverse dual-eval packet: every dual functional kills the additive error between the extracted and input coordinates.

• dual_difference_zero (φ : Module.Dual (ZMod 2) (towerM K)) : φ (carrier.extractedValue - carrier.inputValue) = 0

structure Q2Presentation.Induction.Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionCocycleLiftLeftInverseX1ValueSourceDualDifferenceZeroPointData {p : FramedPair} {h : ¬ScalarTerminal p} {chief : FirstNonScalarChiefFactor p} {K : KernelMinimal chief.toNonScalarChiefFactor} {hR : Nat.card ↥(kernelFrattini K) = 1} {child : Sec7FrattiniTrivialElementaryChildData p} {cand : Sec7FrattiniTrivialElementaryMStageCandidateCocycleGroupData h chief K hR child} {raw : Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionRawMapData h chief K hR child} {land : Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionBoundaryLandingData raw} {toMap : Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionFiberToCocycleMapData cand raw land} (lift : Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionCocycleToFiberLiftData toMap) (ρ : boundaryFramedSurj boundaryPackage_GammaA child.lowerChild.fst child.lowerChild.snd) (z : ↥(Lifting.Z1 cand.Sinv cand.T cand.P)) (φ : Module.Dual (ZMod 2) (towerM K)) : Prop

Source-level one-functional residual below the X1 left-inverse additive-error packet. It removes the artificial carrier values and states the scalar vanishing directly on the extracted cocycle coordinate and the input cocycle coordinate.

• source_dual_difference_zero : φ ((↑(toMap.toCocycle ρ (lift.ofCocycle ρ z))).2.2.2 - (↑z).2.2.2) = 0

axiom Q2Presentation.Induction.sec7_frattiniTrivial_elementaryMStageCandidateLiftIndexProjectionCocycleLiftLeftInverseX1ValueSourceDualDifferenceZeroPointData {p : FramedPair} {h : ¬ScalarTerminal p} {chief : FirstNonScalarChiefFactor p} {K : KernelMinimal chief.toNonScalarChiefFactor} {hR : Nat.card ↥(kernelFrattini K) = 1} {child : Sec7FrattiniTrivialElementaryChildData p} {cand : Sec7FrattiniTrivialElementaryMStageCandidateCocycleGroupData h chief K hR child} {raw : Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionRawMapData h chief K hR child} {land : Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionBoundaryLandingData raw} {toMap : Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionFiberToCocycleMapData cand raw land} (lift : Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionCocycleToFiberLiftData toMap) (ρ : boundaryFramedSurj boundaryPackage_GammaA child.lowerChild.fst child.lowerChild.snd) (z : ↥(Lifting.Z1 cand.Sinv cand.T cand.P)) (φ : Module.Dual (ZMod 2) (towerM K)) : Nonempty (Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionCocycleLiftLeftInverseX1ValueSourceDualDifferenceZeroPointData lift ρ z φ)

Lower residual: prove source-level scalar-dual vanishing of the X1 coordinate left-inverse error for one dual functional.

theorem Q2Presentation.Induction.sec7_frattiniTrivial_elementaryMStageCandidateLiftIndexProjectionCocycleLiftLeftInverseX1ValueDualDifferenceZeroPointData {p : FramedPair} {h : ¬ScalarTerminal p} {chief : FirstNonScalarChiefFactor p} {K : KernelMinimal chief.toNonScalarChiefFactor} {hR : Nat.card ↥(kernelFrattini K) = 1} {child : Sec7FrattiniTrivialElementaryChildData p} {cand : Sec7FrattiniTrivialElementaryMStageCandidateCocycleGroupData h chief K hR child} {raw : Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionRawMapData h chief K hR child} {land : Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionBoundaryLandingData raw} {toMap : Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionFiberToCocycleMapData cand raw land} {lift : Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionCocycleToFiberLiftData toMap} {ρ : boundaryFramedSurj boundaryPackage_GammaA child.lowerChild.fst child.lowerChild.snd} {z : ↥(Lifting.Z1 cand.Sinv cand.T cand.P)} (carrier : Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionCocycleLiftLeftInverseX1ValueCarrierPointData lift ρ z) : Nonempty (Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionCocycleLiftLeftInverseX1ValueDualDifferenceZeroPointData carrier)

Former carrier-level dual-difference residual, now theorem-level from the direct source-coordinate scalar residual.

theorem Q2Presentation.Induction.sec7_frattiniTrivial_elementaryMStageCandidateLiftIndexProjectionCocycleLiftLeftInverseX1ValueDualEvalPointData {p : FramedPair} {h : ¬ScalarTerminal p} {chief : FirstNonScalarChiefFactor p} {K : KernelMinimal chief.toNonScalarChiefFactor} {hR : Nat.card ↥(kernelFrattini K) = 1} {child : Sec7FrattiniTrivialElementaryChildData p} {cand : Sec7FrattiniTrivialElementaryMStageCandidateCocycleGroupData h chief K hR child} {raw : Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionRawMapData h chief K hR child} {land : Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionBoundaryLandingData raw} {toMap : Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionFiberToCocycleMapData cand raw land} {lift : Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionCocycleToFiberLiftData toMap} {ρ : boundaryFramedSurj boundaryPackage_GammaA child.lowerChild.fst child.lowerChild.snd} {z : ↥(Lifting.Z1 cand.Sinv cand.T cand.P)} (carrier : Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionCocycleLiftLeftInverseX1ValueCarrierPointData lift ρ z) : Nonempty (Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionCocycleLiftLeftInverseX1ValueDualEvalPointData carrier)

Former X1 dual-evaluation residual, now theorem-level from scalar-dual vanishing of the additive error.

theorem Q2Presentation.Induction.sec7_frattiniTrivial_elementaryMStageCandidateLiftIndexProjectionCocycleLiftLeftInverseX1ValueEqualityPointData {p : FramedPair} {h : ¬ScalarTerminal p} {chief : FirstNonScalarChiefFactor p} {K : KernelMinimal chief.toNonScalarChiefFactor} {hR : Nat.card ↥(kernelFrattini K) = 1} {child : Sec7FrattiniTrivialElementaryChildData p} {cand : Sec7FrattiniTrivialElementaryMStageCandidateCocycleGroupData h chief K hR child} {raw : Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionRawMapData h chief K hR child} {land : Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionBoundaryLandingData raw} {toMap : Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionFiberToCocycleMapData cand raw land} {lift : Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionCocycleToFiberLiftData toMap} {ρ : boundaryFramedSurj boundaryPackage_GammaA child.lowerChild.fst child.lowerChild.snd} {z : ↥(Lifting.Z1 cand.Sinv cand.T cand.P)} (carrier : Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionCocycleLiftLeftInverseX1ValueCarrierPointData lift ρ z) : Nonempty (Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionCocycleLiftLeftInverseX1ValueEqualityPointData carrier)

Former equality-only X1 left-inverse atom, now theorem-level from all dual scalar evaluations and finite-dimensional/projective separation.

theorem Q2Presentation.Induction.sec7_frattiniTrivial_elementaryMStageCandidateLiftIndexProjectionCocycleLiftLeftInverseX1ValueEqPointData {p : FramedPair} {h : ¬ScalarTerminal p} {chief : FirstNonScalarChiefFactor p} {K : KernelMinimal chief.toNonScalarChiefFactor} {hR : Nat.card ↥(kernelFrattini K) = 1} {child : Sec7FrattiniTrivialElementaryChildData p} {cand : Sec7FrattiniTrivialElementaryMStageCandidateCocycleGroupData h chief K hR child} {raw : Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionRawMapData h chief K hR child} {land : Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionBoundaryLandingData raw} {toMap : Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionFiberToCocycleMapData cand raw land} (lift : Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionCocycleToFiberLiftData toMap) (ρ : boundaryFramedSurj boundaryPackage_GammaA child.lowerChild.fst child.lowerChild.snd) (z : ↥(Lifting.Z1 cand.Sinv cand.T cand.P)) : Nonempty (Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionCocycleLiftLeftInverseX1ValueEqPointData lift ρ z)

Former X1 direct coordinate residual, now theorem-level from a closed carrier and an equality-only coordinate atom.

theorem Q2Presentation.Induction.sec7_frattiniTrivial_elementaryMStageCandidateLiftIndexProjectionCocycleLiftLeftInverseX1ValueSubZeroPointData {p : FramedPair} {h : ¬ScalarTerminal p} {chief : FirstNonScalarChiefFactor p} {K : KernelMinimal chief.toNonScalarChiefFactor} {hR : Nat.card ↥(kernelFrattini K) = 1} {child : Sec7FrattiniTrivialElementaryChildData p} {cand : Sec7FrattiniTrivialElementaryMStageCandidateCocycleGroupData h chief K hR child} {raw : Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionRawMapData h chief K hR child} {land : Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionBoundaryLandingData raw} {toMap : Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionFiberToCocycleMapData cand raw land} (lift : Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionCocycleToFiberLiftData toMap) (ρ : boundaryFramedSurj boundaryPackage_GammaA child.lowerChild.fst child.lowerChild.snd) (z : ↥(Lifting.Z1 cand.Sinv cand.T cand.P)) : Nonempty (Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionCocycleLiftLeftInverseX1ValueSubZeroPointData lift ρ z)

theorem Q2Presentation.Induction.sec7_frattiniTrivial_elementaryMStageCandidateLiftIndexProjectionCocycleLiftLeftInverseX1ValuePointData {p : FramedPair} {h : ¬ScalarTerminal p} {chief : FirstNonScalarChiefFactor p} {K : KernelMinimal chief.toNonScalarChiefFactor} {hR : Nat.card ↥(kernelFrattini K) = 1} {child : Sec7FrattiniTrivialElementaryChildData p} {cand : Sec7FrattiniTrivialElementaryMStageCandidateCocycleGroupData h chief K hR child} {raw : Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionRawMapData h chief K hR child} {land : Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionBoundaryLandingData raw} {toMap : Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionFiberToCocycleMapData cand raw land} (lift : Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionCocycleToFiberLiftData toMap) (ρ : boundaryFramedSurj boundaryPackage_GammaA child.lowerChild.fst child.lowerChild.snd) (z : ↥(Lifting.Z1 cand.Sinv cand.T cand.P)) : Nonempty (Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionCocycleLiftLeftInverseX1ValuePointData lift ρ z)

Former pointwise X1 left-inverse residual, now theorem-level from the additive zero-difference form.

theorem Q2Presentation.Induction.sec7_frattiniTrivial_elementaryMStageCandidateLiftIndexProjectionCocycleLiftLeftInverseX1ValueData {p : FramedPair} {h : ¬ScalarTerminal p} {chief : FirstNonScalarChiefFactor p} {K : KernelMinimal chief.toNonScalarChiefFactor} {hR : Nat.card ↥(kernelFrattini K) = 1} {child : Sec7FrattiniTrivialElementaryChildData p} {cand : Sec7FrattiniTrivialElementaryMStageCandidateCocycleGroupData h chief K hR child} {raw : Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionRawMapData h chief K hR child} {land : Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionBoundaryLandingData raw} {toMap : Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionFiberToCocycleMapData cand raw land} (lift : Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionCocycleToFiberLiftData toMap) : Nonempty (Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionCocycleLiftLeftInverseX1ValueData lift)

Former x₁ coordinate left-inverse residual, now theorem-level from pointwise coordinate equations.

theorem Q2Presentation.Induction.sec7_frattiniTrivial_elementaryMStageCandidateLiftIndexProjectionCocycleLiftLeftInverseValueData {p : FramedPair} {h : ¬ScalarTerminal p} {chief : FirstNonScalarChiefFactor p} {K : KernelMinimal chief.toNonScalarChiefFactor} {hR : Nat.card ↥(kernelFrattini K) = 1} {child : Sec7FrattiniTrivialElementaryChildData p} {cand : Sec7FrattiniTrivialElementaryMStageCandidateCocycleGroupData h chief K hR child} {raw : Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionRawMapData h chief K hR child} {land : Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionBoundaryLandingData raw} {toMap : Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionFiberToCocycleMapData cand raw land} (lift : Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionCocycleToFiberLiftData toMap) : Nonempty (Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionCocycleLiftLeftInverseValueData lift)

Former value-level left inverse residual, now theorem-level from the four coordinate left inverse laws.

theorem Q2Presentation.Induction.sec7_frattiniTrivial_elementaryMStageCandidateLiftIndexProjectionCocycleLiftLeftInverseData {p : FramedPair} {h : ¬ScalarTerminal p} {chief : FirstNonScalarChiefFactor p} {K : KernelMinimal chief.toNonScalarChiefFactor} {hR : Nat.card ↥(kernelFrattini K) = 1} {child : Sec7FrattiniTrivialElementaryChildData p} {cand : Sec7FrattiniTrivialElementaryMStageCandidateCocycleGroupData h chief K hR child} {raw : Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionRawMapData h chief K hR child} {land : Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionBoundaryLandingData raw} {toMap : Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionFiberToCocycleMapData cand raw land} (lift : Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionCocycleToFiberLiftData toMap) : Nonempty (Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionCocycleLiftLeftInverseData toMap lift)

Former left-inverse residual, now theorem-level from equality of the underlying four-coordinate cocycle values.

structure Q2Presentation.Induction.Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionCocycleLiftRightInverseGeneratorData {p : FramedPair} {h : ¬ScalarTerminal p} {chief : FirstNonScalarChiefFactor p} {K : KernelMinimal chief.toNonScalarChiefFactor} {hR : Nat.card ↥(kernelFrattini K) = 1} {child : Sec7FrattiniTrivialElementaryChildData p} {cand : Sec7FrattiniTrivialElementaryMStageCandidateCocycleGroupData h chief K hR child} {raw : Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionRawMapData h chief K hR child} {land : Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionBoundaryLandingData raw} {toMap : Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionFiberToCocycleMapData cand raw land} (lift : Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionCocycleToFiberLiftData toMap) : Prop

Generator-level right inverse law for the current lift reconstructed from an extracted cocycle.

• of_to_gen (ρ : boundaryFramedSurj boundaryPackage_GammaA child.lowerChild.fst child.lowerChild.snd) (x : { f : boundaryFramedSurj boundaryPackage_GammaA p.fst p.snd // sec7_frattiniTrivial_elementaryMStageCandidateLiftIndexProjectionFromRaw raw land f = ρ }) (a : Gen) : (ProfiniteGrp.Hom.hom (↑↑(lift.ofCocycle ρ (toMap.toCocycle ρ x))).hom) (gammaGen a) = (ProfiniteGrp.Hom.hom (↑↑x).hom) (gammaGen a)

structure Q2Presentation.Induction.Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionCocycleLiftRightInverseGeneratorPointData {p : FramedPair} {h : ¬ScalarTerminal p} {chief : FirstNonScalarChiefFactor p} {K : KernelMinimal chief.toNonScalarChiefFactor} {hR : Nat.card ↥(kernelFrattini K) = 1} {child : Sec7FrattiniTrivialElementaryChildData p} {cand : Sec7FrattiniTrivialElementaryMStageCandidateCocycleGroupData h chief K hR child} {raw : Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionRawMapData h chief K hR child} {land : Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionBoundaryLandingData raw} {toMap : Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionFiberToCocycleMapData cand raw land} (lift : Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionCocycleToFiberLiftData toMap) (ρ : boundaryFramedSurj boundaryPackage_GammaA child.lowerChild.fst child.lowerChild.snd) (x : { f : boundaryFramedSurj boundaryPackage_GammaA p.fst p.snd // sec7_frattiniTrivial_elementaryMStageCandidateLiftIndexProjectionFromRaw raw land f = ρ }) (a : Gen) : Prop

Pointwise generator residual for the cocycle-lift right inverse law.

• of_to_gen : (ProfiniteGrp.Hom.hom (↑↑(lift.ofCocycle ρ (toMap.toCocycle ρ x))).hom) (gammaGen a) = (ProfiniteGrp.Hom.hom (↑↑x).hom) (gammaGen a)

structure Q2Presentation.Induction.Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionCocycleLiftRightInverseGeneratorValuePointData {p : FramedPair} {h : ¬ScalarTerminal p} {chief : FirstNonScalarChiefFactor p} {K : KernelMinimal chief.toNonScalarChiefFactor} {hR : Nat.card ↥(kernelFrattini K) = 1} {child : Sec7FrattiniTrivialElementaryChildData p} {cand : Sec7FrattiniTrivialElementaryMStageCandidateCocycleGroupData h chief K hR child} {raw : Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionRawMapData h chief K hR child} {land : Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionBoundaryLandingData raw} {toMap : Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionFiberToCocycleMapData cand raw land} (lift : Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionCocycleToFiberLiftData toMap) (ρ : boundaryFramedSurj boundaryPackage_GammaA child.lowerChild.fst child.lowerChild.snd) (x : { f : boundaryFramedSurj boundaryPackage_GammaA p.fst p.snd // sec7_frattiniTrivial_elementaryMStageCandidateLiftIndexProjectionFromRaw raw land f = ρ }) (a : Gen) : Type

Lower residual below the right-inverse generator equation: explicitly named finite target values for the rebuilt and original current lift.

• rebuiltValue : p.fst.Y
• originalValue : p.fst.Y
• rebuiltValue_eq : self.rebuiltValue = (ProfiniteGrp.Hom.hom (↑↑(lift.ofCocycle ρ (toMap.toCocycle ρ x))).hom) (gammaGen a)
• originalValue_eq : self.originalValue = (ProfiniteGrp.Hom.hom (↑↑x).hom) (gammaGen a)
• rebuilt_eq_original : self.rebuiltValue = self.originalValue

structure Q2Presentation.Induction.Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionCocycleLiftRightInverseGeneratorEqPointData {p : FramedPair} {h : ¬ScalarTerminal p} {chief : FirstNonScalarChiefFactor p} {K : KernelMinimal chief.toNonScalarChiefFactor} {hR : Nat.card ↥(kernelFrattini K) = 1} {child : Sec7FrattiniTrivialElementaryChildData p} {cand : Sec7FrattiniTrivialElementaryMStageCandidateCocycleGroupData h chief K hR child} {raw : Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionRawMapData h chief K hR child} {land : Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionBoundaryLandingData raw} {toMap : Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionFiberToCocycleMapData cand raw land} (lift : Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionCocycleToFiberLiftData toMap) (ρ : boundaryFramedSurj boundaryPackage_GammaA child.lowerChild.fst child.lowerChild.snd) (x : { f : boundaryFramedSurj boundaryPackage_GammaA p.fst p.snd // sec7_frattiniTrivial_elementaryMStageCandidateLiftIndexProjectionFromRaw raw land f = ρ }) (a : Gen) : Prop

Direct generator-equality residual below the right-inverse value packet; the artificially named rebuilt/original finite values are recovered theorem-level.

• rebuilt_eq_original_gen : (ProfiniteGrp.Hom.hom (↑↑(lift.ofCocycle ρ (toMap.toCocycle ρ x))).hom) (gammaGen a) = (ProfiniteGrp.Hom.hom (↑↑x).hom) (gammaGen a)

structure Q2Presentation.Induction.Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionCocycleLiftRightInverseGeneratorCarrierPointData {p : FramedPair} {h : ¬ScalarTerminal p} {chief : FirstNonScalarChiefFactor p} {K : KernelMinimal chief.toNonScalarChiefFactor} {hR : Nat.card ↥(kernelFrattini K) = 1} {child : Sec7FrattiniTrivialElementaryChildData p} {cand : Sec7FrattiniTrivialElementaryMStageCandidateCocycleGroupData h chief K hR child} {raw : Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionRawMapData h chief K hR child} {land : Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionBoundaryLandingData raw} {toMap : Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionFiberToCocycleMapData cand raw land} (lift : Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionCocycleToFiberLiftData toMap) (ρ : boundaryFramedSurj boundaryPackage_GammaA child.lowerChild.fst child.lowerChild.snd) (x : { f : boundaryFramedSurj boundaryPackage_GammaA p.fst p.snd // sec7_frattiniTrivial_elementaryMStageCandidateLiftIndexProjectionFromRaw raw land f = ρ }) (a : Gen) : Type

Carrier below the generator equality in the right-inverse law: name the rebuilt current-lift generator value and the original current-lift generator value. The carrier is definitional; only equality of those named values remains below.

• rebuiltValue : p.fst.Y
• originalValue : p.fst.Y
• rebuiltValue_eq : self.rebuiltValue = (ProfiniteGrp.Hom.hom (↑↑(lift.ofCocycle ρ (toMap.toCocycle ρ x))).hom) (gammaGen a)
• originalValue_eq : self.originalValue = (ProfiniteGrp.Hom.hom (↑↑x).hom) (gammaGen a)

theorem Q2Presentation.Induction.sec7_frattiniTrivial_elementaryMStageCandidateLiftIndexProjectionCocycleLiftRightInverseGeneratorCarrierPointData {p : FramedPair} {h : ¬ScalarTerminal p} {chief : FirstNonScalarChiefFactor p} {K : KernelMinimal chief.toNonScalarChiefFactor} {hR : Nat.card ↥(kernelFrattini K) = 1} {child : Sec7FrattiniTrivialElementaryChildData p} {cand : Sec7FrattiniTrivialElementaryMStageCandidateCocycleGroupData h chief K hR child} {raw : Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionRawMapData h chief K hR child} {land : Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionBoundaryLandingData raw} {toMap : Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionFiberToCocycleMapData cand raw land} (lift : Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionCocycleToFiberLiftData toMap) (ρ : boundaryFramedSurj boundaryPackage_GammaA child.lowerChild.fst child.lowerChild.snd) (x : { f : boundaryFramedSurj boundaryPackage_GammaA p.fst p.snd // sec7_frattiniTrivial_elementaryMStageCandidateLiftIndexProjectionFromRaw raw land f = ρ }) (a : Gen) : Nonempty (Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionCocycleLiftRightInverseGeneratorCarrierPointData lift ρ x a)

The right-inverse generator carrier is closed by choosing the displayed rebuilt and original values.

structure Q2Presentation.Induction.Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionCocycleLiftRightInverseGeneratorEqualityPointData {p : FramedPair} {h : ¬ScalarTerminal p} {chief : FirstNonScalarChiefFactor p} {K : KernelMinimal chief.toNonScalarChiefFactor} {hR : Nat.card ↥(kernelFrattini K) = 1} {child : Sec7FrattiniTrivialElementaryChildData p} {cand : Sec7FrattiniTrivialElementaryMStageCandidateCocycleGroupData h chief K hR child} {raw : Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionRawMapData h chief K hR child} {land : Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionBoundaryLandingData raw} {toMap : Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionFiberToCocycleMapData cand raw land} {lift : Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionCocycleToFiberLiftData toMap} {ρ : boundaryFramedSurj boundaryPackage_GammaA child.lowerChild.fst child.lowerChild.snd} {x : { f : boundaryFramedSurj boundaryPackage_GammaA p.fst p.snd // sec7_frattiniTrivial_elementaryMStageCandidateLiftIndexProjectionFromRaw raw land f = ρ }} {a : Gen} (carrier : Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionCocycleLiftRightInverseGeneratorCarrierPointData lift ρ x a) : Prop

Equality-only atom below the right-inverse generator residual.

• rebuilt_eq_original : carrier.rebuiltValue = carrier.originalValue

structure Q2Presentation.Induction.Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionCocycleLiftRightInverseGeneratorDifferenceOnePointData {p : FramedPair} {h : ¬ScalarTerminal p} {chief : FirstNonScalarChiefFactor p} {K : KernelMinimal chief.toNonScalarChiefFactor} {hR : Nat.card ↥(kernelFrattini K) = 1} {child : Sec7FrattiniTrivialElementaryChildData p} {cand : Sec7FrattiniTrivialElementaryMStageCandidateCocycleGroupData h chief K hR child} {raw : Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionRawMapData h chief K hR child} {land : Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionBoundaryLandingData raw} {toMap : Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionFiberToCocycleMapData cand raw land} {lift : Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionCocycleToFiberLiftData toMap} {ρ : boundaryFramedSurj boundaryPackage_GammaA child.lowerChild.fst child.lowerChild.snd} {x : { f : boundaryFramedSurj boundaryPackage_GammaA p.fst p.snd // sec7_frattiniTrivial_elementaryMStageCandidateLiftIndexProjectionFromRaw raw land f = ρ }} {a : Gen} (carrier : Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionCocycleLiftRightInverseGeneratorCarrierPointData lift ρ x a) : Prop

Still-lower right-inverse generator atom: the multiplicative difference between the named rebuilt and original finite-target values is trivial.

• rebuilt_mul_original_inv_eq_one : carrier.rebuiltValue * carrier.originalValue⁻¹ = 1

structure Q2Presentation.Induction.Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionCocycleLiftRightInverseGeneratorDefectKernelFrattiniPointData {p : FramedPair} {h : ¬ScalarTerminal p} {chief : FirstNonScalarChiefFactor p} {K : KernelMinimal chief.toNonScalarChiefFactor} {hR : Nat.card ↥(kernelFrattini K) = 1} {child : Sec7FrattiniTrivialElementaryChildData p} {cand : Sec7FrattiniTrivialElementaryMStageCandidateCocycleGroupData h chief K hR child} {raw : Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionRawMapData h chief K hR child} {land : Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionBoundaryLandingData raw} {toMap : Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionFiberToCocycleMapData cand raw land} {lift : Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionCocycleToFiberLiftData toMap} {ρ : boundaryFramedSurj boundaryPackage_GammaA child.lowerChild.fst child.lowerChild.snd} {x : { f : boundaryFramedSurj boundaryPackage_GammaA p.fst p.snd // sec7_frattiniTrivial_elementaryMStageCandidateLiftIndexProjectionFromRaw raw land f = ρ }} {a : Gen} (carrier : Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionCocycleLiftRightInverseGeneratorCarrierPointData lift ρ x a) : Prop

Kernel-Frattini residual below the right-inverse generator difference-one statement. It records that the rebuilt/original generator defect lies in the Frattini kernel R; under the active elementary hypothesis Nat.card R = 1, this defect is forced to be trivial theorem-level.

• rebuilt_mul_original_inv_mem_kernelFrattini : carrier.rebuiltValue * carrier.originalValue⁻¹ ∈ kernelFrattini K

structure Q2Presentation.Induction.Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionCocycleLiftRightInverseGeneratorDefectInternalFrattiniRepresentativePointData {p : FramedPair} {h : ¬ScalarTerminal p} {chief : FirstNonScalarChiefFactor p} {K : KernelMinimal chief.toNonScalarChiefFactor} {hR : Nat.card ↥(kernelFrattini K) = 1} {child : Sec7FrattiniTrivialElementaryChildData p} {cand : Sec7FrattiniTrivialElementaryMStageCandidateCocycleGroupData h chief K hR child} {raw : Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionRawMapData h chief K hR child} {land : Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionBoundaryLandingData raw} {toMap : Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionFiberToCocycleMapData cand raw land} {lift : Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionCocycleToFiberLiftData toMap} {ρ : boundaryFramedSurj boundaryPackage_GammaA child.lowerChild.fst child.lowerChild.snd} {x : { f : boundaryFramedSurj boundaryPackage_GammaA p.fst p.snd // sec7_frattiniTrivial_elementaryMStageCandidateLiftIndexProjectionFromRaw raw land f = ρ }} {a : Gen} (carrier : Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionCocycleLiftRightInverseGeneratorCarrierPointData lift ρ x a) : Type

Internal Frattini representative below the right-inverse generator defect. Instead of asking directly for ambient membership in kernelFrattini K = map K.subtype (Φ K), this records an explicit element of the abstract kernel Frattini subgroup whose image is the rebuilt/original defect.

• defectInK : ↥K.Ksub
• defectInK_mem_frattini : self.defectInK ∈ frattini ↥K.Ksub
• defectInK_eq : ↑self.defectInK = carrier.rebuiltValue * carrier.originalValue⁻¹

axiom Q2Presentation.Induction.sec7_frattiniTrivial_elementaryMStageCandidateLiftIndexProjectionCocycleLiftRightInverseGeneratorDefectInternalFrattiniRepresentativePointData {p : FramedPair} {h : ¬ScalarTerminal p} {chief : FirstNonScalarChiefFactor p} {K : KernelMinimal chief.toNonScalarChiefFactor} {hR : Nat.card ↥(kernelFrattini K) = 1} {child : Sec7FrattiniTrivialElementaryChildData p} {cand : Sec7FrattiniTrivialElementaryMStageCandidateCocycleGroupData h chief K hR child} {raw : Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionRawMapData h chief K hR child} {land : Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionBoundaryLandingData raw} {toMap : Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionFiberToCocycleMapData cand raw land} {lift : Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionCocycleToFiberLiftData toMap} {ρ : boundaryFramedSurj boundaryPackage_GammaA child.lowerChild.fst child.lowerChild.snd} {x : { f : boundaryFramedSurj boundaryPackage_GammaA p.fst p.snd // sec7_frattiniTrivial_elementaryMStageCandidateLiftIndexProjectionFromRaw raw land f = ρ }} {a : Gen} (carrier : Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionCocycleLiftRightInverseGeneratorCarrierPointData lift ρ x a) : Nonempty (Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionCocycleLiftRightInverseGeneratorDefectInternalFrattiniRepresentativePointData carrier)

Lower residual: produce an internal Frattini representative for the right-inverse generator defect.

theorem Q2Presentation.Induction.sec7_frattiniTrivial_elementaryMStageCandidateLiftIndexProjectionCocycleLiftRightInverseGeneratorDefectKernelFrattiniPointData {p : FramedPair} {h : ¬ScalarTerminal p} {chief : FirstNonScalarChiefFactor p} {K : KernelMinimal chief.toNonScalarChiefFactor} {hR : Nat.card ↥(kernelFrattini K) = 1} {child : Sec7FrattiniTrivialElementaryChildData p} {cand : Sec7FrattiniTrivialElementaryMStageCandidateCocycleGroupData h chief K hR child} {raw : Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionRawMapData h chief K hR child} {land : Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionBoundaryLandingData raw} {toMap : Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionFiberToCocycleMapData cand raw land} {lift : Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionCocycleToFiberLiftData toMap} {ρ : boundaryFramedSurj boundaryPackage_GammaA child.lowerChild.fst child.lowerChild.snd} {x : { f : boundaryFramedSurj boundaryPackage_GammaA p.fst p.snd // sec7_frattiniTrivial_elementaryMStageCandidateLiftIndexProjectionFromRaw raw land f = ρ }} {a : Gen} (carrier : Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionCocycleLiftRightInverseGeneratorCarrierPointData lift ρ x a) : Nonempty (Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionCocycleLiftRightInverseGeneratorDefectKernelFrattiniPointData carrier)

Former ambient kernelFrattini membership residual, now theorem-level from an internal Φ(K) representative.

theorem Q2Presentation.Induction.sec7_frattiniTrivial_elementaryMStageCandidateLiftIndexProjectionCocycleLiftRightInverseGeneratorDifferenceOnePointData {p : FramedPair} {h : ¬ScalarTerminal p} {chief : FirstNonScalarChiefFactor p} {K : KernelMinimal chief.toNonScalarChiefFactor} {hR : Nat.card ↥(kernelFrattini K) = 1} {child : Sec7FrattiniTrivialElementaryChildData p} {cand : Sec7FrattiniTrivialElementaryMStageCandidateCocycleGroupData h chief K hR child} {raw : Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionRawMapData h chief K hR child} {land : Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionBoundaryLandingData raw} {toMap : Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionFiberToCocycleMapData cand raw land} {lift : Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionCocycleToFiberLiftData toMap} {ρ : boundaryFramedSurj boundaryPackage_GammaA child.lowerChild.fst child.lowerChild.snd} {x : { f : boundaryFramedSurj boundaryPackage_GammaA p.fst p.snd // sec7_frattiniTrivial_elementaryMStageCandidateLiftIndexProjectionFromRaw raw land f = ρ }} {a : Gen} (carrier : Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionCocycleLiftRightInverseGeneratorCarrierPointData lift ρ x a) : Nonempty (Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionCocycleLiftRightInverseGeneratorDifferenceOnePointData carrier)

Former right-inverse generator difference-one residual, now theorem-level from membership of the defect in the trivial Frattini kernel R.

theorem Q2Presentation.Induction.sec7_frattiniTrivial_elementaryMStageCandidateLiftIndexProjectionCocycleLiftRightInverseGeneratorEqualityPointData {p : FramedPair} {h : ¬ScalarTerminal p} {chief : FirstNonScalarChiefFactor p} {K : KernelMinimal chief.toNonScalarChiefFactor} {hR : Nat.card ↥(kernelFrattini K) = 1} {child : Sec7FrattiniTrivialElementaryChildData p} {cand : Sec7FrattiniTrivialElementaryMStageCandidateCocycleGroupData h chief K hR child} {raw : Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionRawMapData h chief K hR child} {land : Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionBoundaryLandingData raw} {toMap : Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionFiberToCocycleMapData cand raw land} {lift : Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionCocycleToFiberLiftData toMap} {ρ : boundaryFramedSurj boundaryPackage_GammaA child.lowerChild.fst child.lowerChild.snd} {x : { f : boundaryFramedSurj boundaryPackage_GammaA p.fst p.snd // sec7_frattiniTrivial_elementaryMStageCandidateLiftIndexProjectionFromRaw raw land f = ρ }} {a : Gen} (carrier : Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionCocycleLiftRightInverseGeneratorCarrierPointData lift ρ x a) : Nonempty (Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionCocycleLiftRightInverseGeneratorEqualityPointData carrier)

Former equality-only right-inverse generator atom, now theorem-level from triviality of the rebuilt/original multiplicative difference.

theorem Q2Presentation.Induction.sec7_frattiniTrivial_elementaryMStageCandidateLiftIndexProjectionCocycleLiftRightInverseGeneratorEqPointData {p : FramedPair} {h : ¬ScalarTerminal p} {chief : FirstNonScalarChiefFactor p} {K : KernelMinimal chief.toNonScalarChiefFactor} {hR : Nat.card ↥(kernelFrattini K) = 1} {child : Sec7FrattiniTrivialElementaryChildData p} {cand : Sec7FrattiniTrivialElementaryMStageCandidateCocycleGroupData h chief K hR child} {raw : Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionRawMapData h chief K hR child} {land : Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionBoundaryLandingData raw} {toMap : Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionFiberToCocycleMapData cand raw land} (lift : Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionCocycleToFiberLiftData toMap) (ρ : boundaryFramedSurj boundaryPackage_GammaA child.lowerChild.fst child.lowerChild.snd) (x : { f : boundaryFramedSurj boundaryPackage_GammaA p.fst p.snd // sec7_frattiniTrivial_elementaryMStageCandidateLiftIndexProjectionFromRaw raw land f = ρ }) (a : Gen) : Nonempty (Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionCocycleLiftRightInverseGeneratorEqPointData lift ρ x a)

Former direct right-inverse generator residual, now theorem-level from a closed carrier and an equality-only generator-value atom.

theorem Q2Presentation.Induction.sec7_frattiniTrivial_elementaryMStageCandidateLiftIndexProjectionCocycleLiftRightInverseGeneratorValuePointData {p : FramedPair} {h : ¬ScalarTerminal p} {chief : FirstNonScalarChiefFactor p} {K : KernelMinimal chief.toNonScalarChiefFactor} {hR : Nat.card ↥(kernelFrattini K) = 1} {child : Sec7FrattiniTrivialElementaryChildData p} {cand : Sec7FrattiniTrivialElementaryMStageCandidateCocycleGroupData h chief K hR child} {raw : Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionRawMapData h chief K hR child} {land : Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionBoundaryLandingData raw} {toMap : Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionFiberToCocycleMapData cand raw land} (lift : Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionCocycleToFiberLiftData toMap) (ρ : boundaryFramedSurj boundaryPackage_GammaA child.lowerChild.fst child.lowerChild.snd) (x : { f : boundaryFramedSurj boundaryPackage_GammaA p.fst p.snd // sec7_frattiniTrivial_elementaryMStageCandidateLiftIndexProjectionFromRaw raw land f = ρ }) (a : Gen) : Nonempty (Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionCocycleLiftRightInverseGeneratorValuePointData lift ρ x a)

theorem Q2Presentation.Induction.sec7_frattiniTrivial_elementaryMStageCandidateLiftIndexProjectionCocycleLiftRightInverseGeneratorPointData {p : FramedPair} {h : ¬ScalarTerminal p} {chief : FirstNonScalarChiefFactor p} {K : KernelMinimal chief.toNonScalarChiefFactor} {hR : Nat.card ↥(kernelFrattini K) = 1} {child : Sec7FrattiniTrivialElementaryChildData p} {cand : Sec7FrattiniTrivialElementaryMStageCandidateCocycleGroupData h chief K hR child} {raw : Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionRawMapData h chief K hR child} {land : Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionBoundaryLandingData raw} {toMap : Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionFiberToCocycleMapData cand raw land} (lift : Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionCocycleToFiberLiftData toMap) (ρ : boundaryFramedSurj boundaryPackage_GammaA child.lowerChild.fst child.lowerChild.snd) (x : { f : boundaryFramedSurj boundaryPackage_GammaA p.fst p.snd // sec7_frattiniTrivial_elementaryMStageCandidateLiftIndexProjectionFromRaw raw land f = ρ }) (a : Gen) : Nonempty (Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionCocycleLiftRightInverseGeneratorPointData lift ρ x a)

Former pointwise right-inverse generator residual, now theorem-level from named finite source values.

theorem Q2Presentation.Induction.sec7_frattiniTrivial_elementaryMStageCandidateLiftIndexProjectionCocycleLiftRightInverseGeneratorData {p : FramedPair} {h : ¬ScalarTerminal p} {chief : FirstNonScalarChiefFactor p} {K : KernelMinimal chief.toNonScalarChiefFactor} {hR : Nat.card ↥(kernelFrattini K) = 1} {child : Sec7FrattiniTrivialElementaryChildData p} {cand : Sec7FrattiniTrivialElementaryMStageCandidateCocycleGroupData h chief K hR child} {raw : Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionRawMapData h chief K hR child} {land : Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionBoundaryLandingData raw} {toMap : Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionFiberToCocycleMapData cand raw land} (lift : Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionCocycleToFiberLiftData toMap) : Nonempty (Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionCocycleLiftRightInverseGeneratorData lift)

Former right-inverse generator residual, now theorem-level from pointwise generator equalities.

theorem Q2Presentation.Induction.sec7_frattiniTrivial_elementaryMStageCandidateLiftIndexProjectionCocycleLiftRightInverseUnderlyingData {p : FramedPair} {h : ¬ScalarTerminal p} {chief : FirstNonScalarChiefFactor p} {K : KernelMinimal chief.toNonScalarChiefFactor} {hR : Nat.card ↥(kernelFrattini K) = 1} {child : Sec7FrattiniTrivialElementaryChildData p} {cand : Sec7FrattiniTrivialElementaryMStageCandidateCocycleGroupData h chief K hR child} {raw : Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionRawMapData h chief K hR child} {land : Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionBoundaryLandingData raw} {toMap : Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionFiberToCocycleMapData cand raw land} (lift : Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionCocycleToFiberLiftData toMap) : Nonempty (Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionCocycleLiftRightInverseUnderlyingData lift)

Former right-inverse residual, now theorem-level from generator-level equality and gammaGen_hom_ext.

theorem Q2Presentation.Induction.sec7_frattiniTrivial_elementaryMStageCandidateLiftIndexProjectionCocycleLiftRightInverseData {p : FramedPair} {h : ¬ScalarTerminal p} {chief : FirstNonScalarChiefFactor p} {K : KernelMinimal chief.toNonScalarChiefFactor} {hR : Nat.card ↥(kernelFrattini K) = 1} {child : Sec7FrattiniTrivialElementaryChildData p} {cand : Sec7FrattiniTrivialElementaryMStageCandidateCocycleGroupData h chief K hR child} {raw : Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionRawMapData h chief K hR child} {land : Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionBoundaryLandingData raw} {toMap : Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionFiberToCocycleMapData cand raw land} (lift : Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionCocycleToFiberLiftData toMap) : Nonempty (Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionCocycleLiftRightInverseData toMap lift)

Former right-inverse residual, now theorem-level from equality of the underlying current candidate lifts.

theorem Q2Presentation.Induction.sec7_frattiniTrivial_elementaryMStageCandidateLiftIndexProjectionCocycleFiberEquivData {p : FramedPair} (h : ¬ScalarTerminal p) (chief : FirstNonScalarChiefFactor p) (K : KernelMinimal chief.toNonScalarChiefFactor) (hR : Nat.card ↥(kernelFrattini K) = 1) (child : Sec7FrattiniTrivialElementaryChildData p) (mult : Sec7FrattiniTrivialElementaryMStageMultiplicityData child) : Nonempty (Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionCocycleFiberEquivData h chief K hR child mult)

Former candidate lift-index realization input, now theorem-level from the separated projection map, landing proof, fibre-to-cocycle map, inverse lift map, and two inverse laws.

theorem Q2Presentation.Induction.sec7_frattiniTrivial_elementaryMStageCandidateLiftIndexProjectionCocycleCardData {p : FramedPair} {h : ¬ScalarTerminal p} {chief : FirstNonScalarChiefFactor p} {K : KernelMinimal chief.toNonScalarChiefFactor} {hR : Nat.card ↥(kernelFrattini K) = 1} {child : Sec7FrattiniTrivialElementaryChildData p} {mult : Sec7FrattiniTrivialElementaryMStageMultiplicityData child} (hdim : mult.dimM = sec7MStageDim K) (fib : Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionCocycleFiberEquivData h chief K hR child mult) : Nonempty (Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionCocycleCardData fib)

Former cocycle-cardinality residual, now theorem-level from the candidate cocycle cardinality formula and the canonical dimension normalization mult.dimM = sec7MStageDim K (discharged by rfl at the canonical multiplicity packet sec7MStageCanonicalMultiplicityData).

theorem Q2Presentation.Induction.sec7_frattiniTrivial_elementaryMStageCandidateLiftIndexProjectionFiberCardData {p : FramedPair} (h : ¬ScalarTerminal p) (chief : FirstNonScalarChiefFactor p) (K : KernelMinimal chief.toNonScalarChiefFactor) (hR : Nat.card ↥(kernelFrattini K) = 1) (child : Sec7FrattiniTrivialElementaryChildData p) (mult : Sec7FrattiniTrivialElementaryMStageMultiplicityData child) (hdim : mult.dimM = sec7MStageDim K) : Nonempty (Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexProjectionFiberCardData child mult)

Former fibrewise projection/cardinality residual, now theorem-level from the literal cocycle-fibre equivalence and the common candidate cocycle cardinality.

theorem Q2Presentation.Induction.sec7_frattiniTrivial_elementaryMStageCandidateLiftIndexCanonicalFinFiberSigmaCardData {p : FramedPair} (h : ¬ScalarTerminal p) (chief : FirstNonScalarChiefFactor p) (K : KernelMinimal chief.toNonScalarChiefFactor) (hR : Nat.card ↥(kernelFrattini K) = 1) (child : Sec7FrattiniTrivialElementaryChildData p) (mult : Sec7FrattiniTrivialElementaryMStageMultiplicityData child) (hdim : mult.dimM = sec7MStageDim K) : Nonempty (Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexCanonicalFinFiberSigmaCardData child mult)

Former cardinality-only sigma-fibre residual, now theorem-level from the fibrewise projection/cardinality datum.

theorem Q2Presentation.Induction.sec7_frattiniTrivial_elementaryMStageCandidateLiftIndexCanonicalFinFiberSigmaEquivData {p : FramedPair} (h : ¬ScalarTerminal p) (chief : FirstNonScalarChiefFactor p) (K : KernelMinimal chief.toNonScalarChiefFactor) (hR : Nat.card ↥(kernelFrattini K) = 1) (child : Sec7FrattiniTrivialElementaryChildData p) (mult : Sec7FrattiniTrivialElementaryMStageMultiplicityData child) (hdim : mult.dimM = sec7MStageDim K) : Nonempty (Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexCanonicalFinFiberSigmaEquivData child mult)

Former sigma-fibre candidate lift-index residual, now theorem-level from the cardinality-only sigma-fibre residual and finite-cardinality choice.

theorem Q2Presentation.Induction.sec7_frattiniTrivial_elementaryMStageCandidateLiftIndexCanonicalFinFiberTotalEquivData {p : FramedPair} (h : ¬ScalarTerminal p) (chief : FirstNonScalarChiefFactor p) (K : KernelMinimal chief.toNonScalarChiefFactor) (hR : Nat.card ↥(kernelFrattini K) = 1) (child : Sec7FrattiniTrivialElementaryChildData p) (mult : Sec7FrattiniTrivialElementaryMStageMultiplicityData child) (hdim : mult.dimM = sec7MStageDim K) : Nonempty (Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexCanonicalFinFiberTotalEquivData child mult)

Former product-form candidate lift-index atom, now theorem-level from the sigma-fibre index and the canonical constant-fibre sigma/product equivalence.

theorem Q2Presentation.Induction.sec7_frattiniTrivial_elementaryMStageCandidateLiftIndexCanonicalFinFiberEquivData {p : FramedPair} (h : ¬ScalarTerminal p) (chief : FirstNonScalarChiefFactor p) (K : KernelMinimal chief.toNonScalarChiefFactor) (hR : Nat.card ↥(kernelFrattini K) = 1) (child : Sec7FrattiniTrivialElementaryChildData p) (mult : Sec7FrattiniTrivialElementaryMStageMultiplicityData child) (hdim : mult.dimM = sec7MStageDim K) : Nonempty (Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexCanonicalFinFiberEquivData child mult)

Former canonical Fin-fibre lift-index atom, now theorem-level from the total product bijection plus general finiteness of candidate boundary-framed surjection sets.

theorem Q2Presentation.Induction.sec7_frattiniTrivial_elementaryMStageCandidateLiftIndexCanonicalFiberEquivData {p : FramedPair} (h : ¬ScalarTerminal p) (chief : FirstNonScalarChiefFactor p) (K : KernelMinimal chief.toNonScalarChiefFactor) (hR : Nat.card ↥(kernelFrattini K) = 1) (child : Sec7FrattiniTrivialElementaryChildData p) (mult : Sec7FrattiniTrivialElementaryMStageMultiplicityData child) (hdim : mult.dimM = sec7MStageDim K) : Nonempty (Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexCanonicalFiberEquivData child mult)

Former arbitrary-fibre canonical lift-index datum, now theorem-level from the canonical Fin liftMultiplicity fibre.

theorem Q2Presentation.Induction.sec7_frattiniTrivial_elementaryMStageCandidateLiftIndexFiberEquivData {p : FramedPair} (h : ¬ScalarTerminal p) (chief : FirstNonScalarChiefFactor p) (K : KernelMinimal chief.toNonScalarChiefFactor) (hR : Nat.card ↥(kernelFrattini K) = 1) (child : Sec7FrattiniTrivialElementaryChildData p) (mult : Sec7FrattiniTrivialElementaryMStageMultiplicityData child) (hdim : mult.dimM = sec7MStageDim K) : Nonempty (Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexFiberEquivData child mult)

Former candidate lower-index/fibre packet, now theorem-level from the canonical product decomposition over the actual lower-child candidate surjection set.

theorem Q2Presentation.Induction.sec7_frattiniTrivial_elementaryMStageCandidateLiftIndexEquivData {p : FramedPair} (h : ¬ScalarTerminal p) (chief : FirstNonScalarChiefFactor p) (K : KernelMinimal chief.toNonScalarChiefFactor) (hR : Nat.card ↥(kernelFrattini K) = 1) (child : Sec7FrattiniTrivialElementaryChildData p) (mult : Sec7FrattiniTrivialElementaryMStageMultiplicityData child) (hdim : mult.dimM = sec7MStageDim K) : Nonempty (Sec7FrattiniTrivialElementaryMStageCandidateLiftIndexEquivData child mult)

Former candidate lift-index equivalence residual, now theorem-level from the manuscript product decomposition: lower lift index times elementary cocycle fibre.

theorem Q2Presentation.Induction.sec7_frattiniTrivial_elementaryMStageCandidateSourceIndexData {p : FramedPair} (h : ¬ScalarTerminal p) (chief : FirstNonScalarChiefFactor p) (K : KernelMinimal chief.toNonScalarChiefFactor) (hR : Nat.card ↥(kernelFrattini K) = 1) (child : Sec7FrattiniTrivialElementaryChildData p) (mult : Sec7FrattiniTrivialElementaryMStageMultiplicityData child) (hdim : mult.dimM = sec7MStageDim K) : Nonempty (Sec7FrattiniTrivialElementaryMStageCandidateSourceIndexData child mult)

Former candidate source-index residual, now theorem-level from an actual lift-index bijection.

structure Q2Presentation.Induction.Sec7FrattiniTrivialElementaryMStageLocalLiftIndexFiberEquivData {p : FramedPair} (D : Sec7FrattiniTrivialElementaryChildData p) (mult : Sec7FrattiniTrivialElementaryMStageMultiplicityData D) : Type 1

Product/fibre form of the local elementary lift index.

• BaseQ : Type

  Local lower-child lift index.

• finBaseQ : Fintype self.BaseQ
• FiberQ : Type

  Local elementary cocycle fibre index.

• finFiberQ : Fintype self.FiberQ
• lower_equiv : boundaryFramedSurj boundaryPackage_GQ2 D.lowerChild.fst D.lowerChild.snd ≃ self.BaseQ

  The lower-child local boundary-framed surjection set is indexed by BaseQ.

• fiber_card : Nat.card self.FiberQ = mult.liftMultiplicity

  The elementary fibre cardinality is the common lift multiplicity.

• total_equiv : boundaryFramedSurj boundaryPackage_GQ2 p.fst p.snd ≃ self.BaseQ × self.FiberQ

  All local framed lifts at p are indexed by lower lift plus cocycle fibre.

structure Q2Presentation.Induction.Sec7FrattiniTrivialElementaryMStageLocalLiftIndexProjectionFiberCardData {p : FramedPair} (D : Sec7FrattiniTrivialElementaryChildData p) (mult : Sec7FrattiniTrivialElementaryMStageMultiplicityData D) : Type

Local lower-projection/fibre-cardinality form of the elementary M-stage lift index.

This is the local analogue of the candidate projection packet: a framed local lift at p has a lower-child shadow, and every fibre over a lower local lift has the common cardinality liftMultiplicity = |Z¹|. The product bijection used by eq:Mstage is recovered theorem-level from this fibre-cardinality statement.

• finTotalQ : Fintype (boundaryFramedSurj boundaryPackage_GQ2 p.fst p.snd)

  The total local framed lift set is finite at this elementary stage.

• finLowerQ : Fintype (boundaryFramedSurj boundaryPackage_GQ2 D.lowerChild.fst D.lowerChild.snd)

  The local lower-child framed lift set is finite at this elementary stage.

• liftProjection : boundaryFramedSurj boundaryPackage_GQ2 p.fst p.snd → boundaryFramedSurj boundaryPackage_GQ2 D.lowerChild.fst D.lowerChild.snd

  Projection from a local lift of p to its lower-child local lift.

• liftProjection_fiber_card (ρ : boundaryFramedSurj boundaryPackage_GQ2 D.lowerChild.fst D.lowerChild.snd) : Nat.card { f : boundaryFramedSurj boundaryPackage_GQ2 p.fst p.snd // self.liftProjection f = ρ } = mult.liftMultiplicity

  Every projection fibre has the common elementary cocycle cardinality.

structure Q2Presentation.Induction.Sec7FrattiniTrivialElementaryMStageLocalLiftIndexCanonicalFinFiberSigmaCardData {p : FramedPair} (D : Sec7FrattiniTrivialElementaryChildData p) (mult : Sec7FrattiniTrivialElementaryMStageMultiplicityData D) : Type

Cardinality-only sigma-fibre form of the local lift index.

• finTotalQ : Fintype (boundaryFramedSurj boundaryPackage_GQ2 p.fst p.snd)

  The total local framed lift set is finite.

• finLowerQ : Fintype (boundaryFramedSurj boundaryPackage_GQ2 D.lowerChild.fst D.lowerChild.snd)

  The local lower-child framed lift set is finite.

• total_sigma_card : Nat.card (boundaryFramedSurj boundaryPackage_GQ2 p.fst p.snd) = Nat.card ((_ : boundaryFramedSurj boundaryPackage_GQ2 D.lowerChild.fst D.lowerChild.snd) × Fin mult.liftMultiplicity)

  The total local lift set has the cardinality of the constant-fibre sigma index.

structure Q2Presentation.Induction.Sec7FrattiniTrivialElementaryMStageLocalLiftIndexCanonicalFinFiberSigmaEquivData {p : FramedPair} (D : Sec7FrattiniTrivialElementaryChildData p) (mult : Sec7FrattiniTrivialElementaryMStageMultiplicityData D) : Type

Sigma-fibre bijective form of the local lift index.

• finLowerQ : Fintype (boundaryFramedSurj boundaryPackage_GQ2 D.lowerChild.fst D.lowerChild.snd)

  The local lower-child framed lift set is finite.

• total_sigma_equiv : boundaryFramedSurj boundaryPackage_GQ2 p.fst p.snd ≃ (_ : boundaryFramedSurj boundaryPackage_GQ2 D.lowerChild.fst D.lowerChild.snd) × Fin mult.liftMultiplicity

  All local framed lifts at p are a sigma family with constant fibre Fin liftMultiplicity over lower-child local lifts.

axiom Q2Presentation.Induction.sec7_frattiniTrivial_elementaryMStageLocalLiftIndexProjectionFiberCardData {p : FramedPair} (h : ¬ScalarTerminal p) (chief : FirstNonScalarChiefFactor p) (K : KernelMinimal chief.toNonScalarChiefFactor) (hR : Nat.card ↥(kernelFrattini K) = 1) (child : Sec7FrattiniTrivialElementaryChildData p) (mult : Sec7FrattiniTrivialElementaryMStageMultiplicityData child) (hdim : mult.dimM = sec7MStageDim K) : Nonempty (Sec7FrattiniTrivialElementaryMStageLocalLiftIndexProjectionFiberCardData child mult)

Local lift-index source input, lowered to a projection with constant fibre cardinal.

The datum is stated at the canonical dimension normalization mult.dimM = sec7MStageDim K (checkpoint-F1 repair): without that side condition the statement would over-quantify over arbitrary multiplicity packets, which is the same over-strong shape as the corrected §7 kernel findings in PROGRESS.md.

theorem Q2Presentation.Induction.sec7_frattiniTrivial_elementaryMStageLocalLiftIndexCanonicalFinFiberSigmaCardData {p : FramedPair} (h : ¬ScalarTerminal p) (chief : FirstNonScalarChiefFactor p) (K : KernelMinimal chief.toNonScalarChiefFactor) (hR : Nat.card ↥(kernelFrattini K) = 1) (child : Sec7FrattiniTrivialElementaryChildData p) (mult : Sec7FrattiniTrivialElementaryMStageMultiplicityData child) (hdim : mult.dimM = sec7MStageDim K) : Nonempty (Sec7FrattiniTrivialElementaryMStageLocalLiftIndexCanonicalFinFiberSigmaCardData child mult)

Former local sigma-cardinality lift-index residual, now theorem-level from the projection and constant fibre-cardinality datum.

theorem Q2Presentation.Induction.sec7_frattiniTrivial_elementaryMStageLocalLiftIndexCanonicalFinFiberSigmaEquivData {p : FramedPair} (h : ¬ScalarTerminal p) (chief : FirstNonScalarChiefFactor p) (K : KernelMinimal chief.toNonScalarChiefFactor) (hR : Nat.card ↥(kernelFrattini K) = 1) (child : Sec7FrattiniTrivialElementaryChildData p) (mult : Sec7FrattiniTrivialElementaryMStageMultiplicityData child) (hdim : mult.dimM = sec7MStageDim K) : Nonempty (Sec7FrattiniTrivialElementaryMStageLocalLiftIndexCanonicalFinFiberSigmaEquivData child mult)

Former local sigma-fibre lift-index equivalence, now theorem-level from finite cardinality choice.

theorem Q2Presentation.Induction.sec7_frattiniTrivial_elementaryMStageLocalLiftIndexFiberEquivData {p : FramedPair} (h : ¬ScalarTerminal p) (chief : FirstNonScalarChiefFactor p) (K : KernelMinimal chief.toNonScalarChiefFactor) (hR : Nat.card ↥(kernelFrattini K) = 1) (child : Sec7FrattiniTrivialElementaryChildData p) (mult : Sec7FrattiniTrivialElementaryMStageMultiplicityData child) (hdim : mult.dimM = sec7MStageDim K) : Nonempty (Sec7FrattiniTrivialElementaryMStageLocalLiftIndexFiberEquivData child mult)

Former local M-stage lift-index input, now theorem-level from the sigma-fibre projection/cardinality statement and the canonical constant-fibre sigma/product equivalence.

axiom Q2Presentation.gq2_topologicallyFinitelyGenerated : Profinite.TopologicallyFinitelyGenerated GQ2Profinite

G_{ℚ₂} is topologically finitely generated — a literature-shaped local input. EXACT-CITABLE: NSW (7.4.1) Theorem (2nd ed., p. 401), verbatim-verified against the electronic edition (2026-07-09 citability audit, audit/CITABILITY_AUDIT_20260708.md §D3): "Let k be a p-adic local field and let n = [k : ℚ_p]. Then the absolute Galois group G = G(k̄|k) of k is generated by n + 2 elements" — no parity restriction, so for ℚ₂ THREE topological generators suffice, strictly stronger than this bare tfg Prop. NOT load-bearing here: Jannsen–Wingberg (NSW 7.5.14) is stated for p ≠ 2 only, and Diekert (Crelle 350 (1984)) requires k(μ₄)|k unramified — false for ℚ₂ (NSW p. 417: "The structure of G_ℚ₂ is not known"); NSW 7.4.1 alone carries the axiom. The former circular reference to the manuscript's own thm:main remains struck (2026-07-08 keep audit). Recorded in the same spirit as labute_GQ2_maxPro2_marked; consumed only to give finiteness of local boundary-framed surjection sets, replacing the former (over-strong) local fibre-cardinality route.

theorem Q2Presentation.Induction.sec7_localBoundaryFramedSurj_finite {q : FramedPair} : Finite (boundaryFramedSurj boundaryPackage_GQ2 q.fst q.snd)

Every local boundary-framed surjection set is finite, from topological finite generation of G_{ℚ₂} (mirror of centralCover_candidateBoundaryFramedSurj_finite).

theorem Q2Presentation.Induction.sec7_frattiniTrivial_elementaryMStageLocalSourceIndexData {p : FramedPair} (h : ¬ScalarTerminal p) (chief : FirstNonScalarChiefFactor p) (K : KernelMinimal chief.toNonScalarChiefFactor) (hR : Nat.card ↥(kernelFrattini K) = 1) (child : Sec7FrattiniTrivialElementaryChildData p) (mult : Sec7FrattiniTrivialElementaryMStageMultiplicityData child) (hdim : mult.dimM = sec7MStageDim K) : Nonempty (Sec7FrattiniTrivialElementaryMStageLocalSourceIndexData child mult)

Former local source-index residual, now theorem-level from the local product index decomposition.

theorem Q2Presentation.Induction.sec7_frattiniTrivial_elementaryMStageSourceIndexData {p : FramedPair} (h : ¬ScalarTerminal p) (chief : FirstNonScalarChiefFactor p) (K : KernelMinimal chief.toNonScalarChiefFactor) (hR : Nat.card ↥(kernelFrattini K) = 1) (child : Sec7FrattiniTrivialElementaryChildData p) (mult : Sec7FrattiniTrivialElementaryMStageMultiplicityData child) (hdim : mult.dimM = sec7MStageDim K) : Nonempty (Sec7FrattiniTrivialElementaryMStageSourceIndexData child mult)

Former M-stage source-index residual, now theorem-level. The paired source-index packet is assembled from independent candidate and local lift-index realization atoms.

structure Q2Presentation.Induction.Sec7FrattiniTrivialElementaryMStageCandidateExactImagePartitionData {p : FramedPair} (D : Sec7FrattiniTrivialElementaryChildData p) (mult : Sec7FrattiniTrivialElementaryMStageMultiplicityData D) (src : Sec7FrattiniTrivialElementaryMStageSourceIndexData D mult) : Prop

Additive exact-image partition form of the candidate eq:Mstage formula.

This is the manuscript subtraction before solving for the final difference: candidate lifts over the lower quotient split into the desired exact-image lifts plus the proper exact-image contributions. The old Nat.sub formula is recovered below by arithmetic.

• partition_card : Nat.card src.XA + (List.map (⇑Cardinal.toNat) (List.map framedCountA D.properChildren)).sum = mult.liftMultiplicity * Cardinal.toNat (framedCountA D.lowerChild)

  Candidate exact-image partition cardinality in additive form.

structure Q2Presentation.Induction.Sec7FrattiniTrivialElementaryMStageCandidateExactImagePartitionEquivData {p : FramedPair} (D : Sec7FrattiniTrivialElementaryChildData p) (mult : Sec7FrattiniTrivialElementaryMStageMultiplicityData D) (src : Sec7FrattiniTrivialElementaryMStageSourceIndexData D mult) : Type 1

Finite disjoint-union form of the candidate exact-image partition. The left side is the desired exact-image index plus proper-image indices; the right side is the lower-child index times the elementary lift fibre.

• ProperA : Type

  Finite index for all proper exact-image contributions.

• finProperA : Fintype self.ProperA
• LowerA : Type

  Finite index for the candidate lower-child framed lifts.

• finLowerA : Fintype self.LowerA
• proper_card : Nat.card self.ProperA = (List.map (⇑Cardinal.toNat) (List.map framedCountA D.properChildren)).sum

  The proper-image index has the cardinality of the proper-child sum.

• lower_card : Nat.card self.LowerA = Cardinal.toNat (framedCountA D.lowerChild)

  The lower-child index has the cardinality of the lower-child framed count.

• partition_equiv : src.XA ⊕ self.ProperA ≃ Fin mult.liftMultiplicity × self.LowerA

  The exact-image partition as a finite bijection.

structure Q2Presentation.Induction.Sec7FrattiniTrivialElementaryMStageCandidateExactImagePartitionCanonicalEquivData {p : FramedPair} (D : Sec7FrattiniTrivialElementaryChildData p) (mult : Sec7FrattiniTrivialElementaryMStageMultiplicityData D) (src : Sec7FrattiniTrivialElementaryMStageSourceIndexData D mult) : Type 1

Canonical lower-child form of the candidate exact-image partition.

The right-hand lower index is no longer an arbitrary finite type LowerA; it is the actual lower-child candidate boundary-framed surjection set. The remaining arbitrary finite index ProperA represents the disjoint union of proper exact-image children.

• ProperA : Type

  Finite index for all proper exact-image contributions.

• finProperA : Fintype self.ProperA
• finLowerA : Fintype (boundaryFramedSurj boundaryPackage_GammaA D.lowerChild.fst D.lowerChild.snd)

  The actual lower-child candidate surjection set is finite in this stage.

• proper_card : Nat.card self.ProperA = (List.map (⇑Cardinal.toNat) (List.map framedCountA D.properChildren)).sum

  The proper-image index has the cardinality of the proper-child sum.

• lower_card : Nat.card (boundaryFramedSurj boundaryPackage_GammaA D.lowerChild.fst D.lowerChild.snd) = Cardinal.toNat (framedCountA D.lowerChild)

  The actual lower-child index has the cardinality of the lower-child framed count.

• partition_equiv : src.XA ⊕ self.ProperA ≃ Fin mult.liftMultiplicity × boundaryFramedSurj boundaryPackage_GammaA D.lowerChild.fst D.lowerChild.snd

  The exact-image partition as a finite bijection over the actual lower-child index.

structure Q2Presentation.Induction.Sec7FrattiniTrivialElementaryMStageCandidateExactImagePartitionCanonicalFinEquivData {p : FramedPair} (D : Sec7FrattiniTrivialElementaryChildData p) (mult : Sec7FrattiniTrivialElementaryMStageMultiplicityData D) (src : Sec7FrattiniTrivialElementaryMStageSourceIndexData D mult) : Type

Fin-indexed form of the candidate exact-image partition.

The proper-image summand is no longer an arbitrary finite type with the right cardinality; it is the canonical finite set Fin (Σ proper child counts). The lower-child summand is still the actual lower-child candidate boundary-framed surjection set.

• finLowerA : Fintype (boundaryFramedSurj boundaryPackage_GammaA D.lowerChild.fst D.lowerChild.snd)

  The actual lower-child candidate surjection set is finite in this stage.

• lower_card : Nat.card (boundaryFramedSurj boundaryPackage_GammaA D.lowerChild.fst D.lowerChild.snd) = Cardinal.toNat (framedCountA D.lowerChild)

  The actual lower-child index has the cardinality of the lower-child framed count.

• partition_equiv : src.XA ⊕ Fin (List.map (⇑Cardinal.toNat) (List.map framedCountA D.properChildren)).sum ≃ Fin mult.liftMultiplicity × boundaryFramedSurj boundaryPackage_GammaA D.lowerChild.fst D.lowerChild.snd

  The exact-image partition over the canonical finite proper-image index.

structure Q2Presentation.Induction.Sec7FrattiniTrivialElementaryMStageCandidateExactImagePartitionCanonicalFinLowerCardData {p : FramedPair} (D : Sec7FrattiniTrivialElementaryChildData p) : Type

Lower-child cardinality half of the canonical candidate exact-image partition.

• finLowerA : Fintype (boundaryFramedSurj boundaryPackage_GammaA D.lowerChild.fst D.lowerChild.snd)

  The actual lower-child candidate surjection set is finite in this stage.

• lower_card : Nat.card (boundaryFramedSurj boundaryPackage_GammaA D.lowerChild.fst D.lowerChild.snd) = Cardinal.toNat (framedCountA D.lowerChild)

  The actual lower-child candidate index has the cardinality of the lower-child framed count.

structure Q2Presentation.Induction.Sec7FrattiniTrivialElementaryMStageCandidateExactImagePartitionCanonicalFinPartitionEquivData {p : FramedPair} (D : Sec7FrattiniTrivialElementaryChildData p) (mult : Sec7FrattiniTrivialElementaryMStageMultiplicityData D) (src : Sec7FrattiniTrivialElementaryMStageSourceIndexData D mult) : Type

Bijection half of the canonical candidate exact-image partition.

• partition_equiv : src.XA ⊕ Fin (List.map (⇑Cardinal.toNat) (List.map framedCountA D.properChildren)).sum ≃ Fin mult.liftMultiplicity × boundaryFramedSurj boundaryPackage_GammaA D.lowerChild.fst D.lowerChild.snd

  The exact-image partition over the canonical finite proper-image index.

theorem Q2Presentation.Induction.sec7_candidateBoundaryFramedSurj_nat_card_eq_framedCountA (q : FramedPair) : Nat.card (boundaryFramedSurj boundaryPackage_GammaA q.fst q.snd) = Cardinal.toNat (framedCountA q)

For any lower child, the finite cardinal of the candidate boundary-framed source set is the natural cardinal attached to framedCountA.

theorem Q2Presentation.Induction.sec7_frattiniTrivial_elementaryMStageCandidateExactImagePartitionCanonicalFinLowerCardData {p : FramedPair} (h : ¬ScalarTerminal p) (chief : FirstNonScalarChiefFactor p) (K : KernelMinimal chief.toNonScalarChiefFactor) (hR : Nat.card ↥(kernelFrattini K) = 1) (child : Sec7FrattiniTrivialElementaryChildData p) : Nonempty (Sec7FrattiniTrivialElementaryMStageCandidateExactImagePartitionCanonicalFinLowerCardData child)

Former lower-child cardinality input for the candidate exact-image partition, now closed from the definition of framedCountA and the existing finiteness theorem for candidate boundary-framed source sets.

structure Q2Presentation.Induction.Sec7FrattiniTrivialElementaryMStageCandidateExactImagePartitionCanonicalFinSigmaPartitionEquivData {p : FramedPair} (D : Sec7FrattiniTrivialElementaryChildData p) (mult : Sec7FrattiniTrivialElementaryMStageMultiplicityData D) (src : Sec7FrattiniTrivialElementaryMStageSourceIndexData D mult) : Type

Sigma-fibre form of the canonical candidate exact-image partition.

The right side keeps the canonical elementary lift fibre visible before flattening it to the product Fin liftMultiplicity × lowerChildSource.

• partition_sigma_equiv : src.XA ⊕ Fin (List.map (⇑Cardinal.toNat) (List.map framedCountA D.properChildren)).sum ≃ (_ : Fin mult.liftMultiplicity) × boundaryFramedSurj boundaryPackage_GammaA D.lowerChild.fst D.lowerChild.snd

structure Q2Presentation.Induction.Sec7FrattiniTrivialElementaryMStageCandidateExactImagePartitionCanonicalFinSigmaPartitionCardData {p : FramedPair} (D : Sec7FrattiniTrivialElementaryChildData p) (mult : Sec7FrattiniTrivialElementaryMStageMultiplicityData D) (src : Sec7FrattiniTrivialElementaryMStageSourceIndexData D mult) : Prop

Cardinality form of the canonical candidate exact-image partition.

This is strictly weaker than choosing the partition equivalence: it records only that the finite disjoint-union index and the canonical constant sigma-fibre index have the same cardinality. The equivalence is recovered theorem-level below using finite-cardinality choice, as in the existing exact-image lift-partition cardinality interface.

• partition_sigma_card : Nat.card (src.XA ⊕ Fin (List.map (⇑Cardinal.toNat) (List.map framedCountA D.properChildren)).sum) = Nat.card ((_ : Fin mult.liftMultiplicity) × boundaryFramedSurj boundaryPackage_GammaA D.lowerChild.fst D.lowerChild.snd)

theorem Q2Presentation.Induction.sec7_frattiniTrivial_elementaryMStageCandidateExactImagePartitionCanonicalFinSigmaPartitionEquivData {p : FramedPair} (h : ¬ScalarTerminal p) (chief : FirstNonScalarChiefFactor p) (K : KernelMinimal chief.toNonScalarChiefFactor) (hR : Nat.card ↥(kernelFrattini K) = 1) (child : Sec7FrattiniTrivialElementaryChildData p) (mult : Sec7FrattiniTrivialElementaryMStageMultiplicityData child) (hdim : mult.dimM = sec7MStageDim K) (src : Sec7FrattiniTrivialElementaryMStageSourceIndexData child mult) (cardA : Sec7FrattiniTrivialElementaryMStageCandidateExactImagePartitionCanonicalFinSigmaPartitionCardData child mult src) : Nonempty (Sec7FrattiniTrivialElementaryMStageCandidateExactImagePartitionCanonicalFinSigmaPartitionEquivData child mult src)

Former bijective sigma-fibre partition input, now theorem-level from the strictly weaker finite-cardinality partition statement.

theorem Q2Presentation.Induction.sec7_frattiniTrivial_elementaryMStageCandidateExactImagePartitionCanonicalFinPartitionEquivData {p : FramedPair} (h : ¬ScalarTerminal p) (chief : FirstNonScalarChiefFactor p) (K : KernelMinimal chief.toNonScalarChiefFactor) (hR : Nat.card ↥(kernelFrattini K) = 1) (child : Sec7FrattiniTrivialElementaryChildData p) (mult : Sec7FrattiniTrivialElementaryMStageMultiplicityData child) (hdim : mult.dimM = sec7MStageDim K) (src : Sec7FrattiniTrivialElementaryMStageSourceIndexData child mult) (cardA : Sec7FrattiniTrivialElementaryMStageCandidateExactImagePartitionCanonicalFinSigmaPartitionCardData child mult src) : Nonempty (Sec7FrattiniTrivialElementaryMStageCandidateExactImagePartitionCanonicalFinPartitionEquivData child mult src)

Former product-form candidate exact-image partition input, now theorem-level from the sigma-fibre partition and the canonical constant-fibre sigma/product equivalence.

theorem Q2Presentation.Induction.sec7_frattiniTrivial_elementaryMStageCandidateExactImagePartitionCanonicalFinEquivData {p : FramedPair} (h : ¬ScalarTerminal p) (chief : FirstNonScalarChiefFactor p) (K : KernelMinimal chief.toNonScalarChiefFactor) (hR : Nat.card ↥(kernelFrattini K) = 1) (child : Sec7FrattiniTrivialElementaryChildData p) (mult : Sec7FrattiniTrivialElementaryMStageMultiplicityData child) (hdim : mult.dimM = sec7MStageDim K) (src : Sec7FrattiniTrivialElementaryMStageSourceIndexData child mult) (cardA : Sec7FrattiniTrivialElementaryMStageCandidateExactImagePartitionCanonicalFinSigmaPartitionCardData child mult src) : Nonempty (Sec7FrattiniTrivialElementaryMStageCandidateExactImagePartitionCanonicalFinEquivData child mult src)

Former canonical candidate exact-image partition atom, now theorem-level from the separate lower-cardinality identity, the partition bijection, and general finiteness of candidate boundary-framed surjection sets.

theorem Q2Presentation.Induction.sec7_frattiniTrivial_elementaryMStageCandidateExactImagePartitionCanonicalEquivData {p : FramedPair} (h : ¬ScalarTerminal p) (chief : FirstNonScalarChiefFactor p) (K : KernelMinimal chief.toNonScalarChiefFactor) (hR : Nat.card ↥(kernelFrattini K) = 1) (child : Sec7FrattiniTrivialElementaryChildData p) (mult : Sec7FrattiniTrivialElementaryMStageMultiplicityData child) (hdim : mult.dimM = sec7MStageDim K) (src : Sec7FrattiniTrivialElementaryMStageSourceIndexData child mult) (cardA : Sec7FrattiniTrivialElementaryMStageCandidateExactImagePartitionCanonicalFinSigmaPartitionCardData child mult src) : Nonempty (Sec7FrattiniTrivialElementaryMStageCandidateExactImagePartitionCanonicalEquivData child mult src)

Former arbitrary-proper-index exact-image partition datum, now theorem-level from the canonical Fin (Σ proper child counts) index.

theorem Q2Presentation.Induction.sec7_frattiniTrivial_elementaryMStageCandidateExactImagePartitionEquivData {p : FramedPair} (h : ¬ScalarTerminal p) (chief : FirstNonScalarChiefFactor p) (K : KernelMinimal chief.toNonScalarChiefFactor) (hR : Nat.card ↥(kernelFrattini K) = 1) (child : Sec7FrattiniTrivialElementaryChildData p) (mult : Sec7FrattiniTrivialElementaryMStageMultiplicityData child) (hdim : mult.dimM = sec7MStageDim K) (src : Sec7FrattiniTrivialElementaryMStageSourceIndexData child mult) (cardA : Sec7FrattiniTrivialElementaryMStageCandidateExactImagePartitionCanonicalFinSigmaPartitionCardData child mult src) : Nonempty (Sec7FrattiniTrivialElementaryMStageCandidateExactImagePartitionEquivData child mult src)

Former finite disjoint-union exact-image partition atom, now theorem-level from the canonical partition over the actual lower-child candidate surjection set.

theorem Q2Presentation.Induction.sec7_frattiniTrivial_elementaryMStageCandidateExactImagePartitionData {p : FramedPair} (h : ¬ScalarTerminal p) (chief : FirstNonScalarChiefFactor p) (K : KernelMinimal chief.toNonScalarChiefFactor) (hR : Nat.card ↥(kernelFrattini K) = 1) (child : Sec7FrattiniTrivialElementaryChildData p) (mult : Sec7FrattiniTrivialElementaryMStageMultiplicityData child) (hdim : mult.dimM = sec7MStageDim K) (src : Sec7FrattiniTrivialElementaryMStageSourceIndexData child mult) (cardA : Sec7FrattiniTrivialElementaryMStageCandidateExactImagePartitionCanonicalFinSigmaPartitionCardData child mult src) : Nonempty (Sec7FrattiniTrivialElementaryMStageCandidateExactImagePartitionData child mult src)

Former candidate exact-image partition residual, now theorem-level from the finite partition bijection and the two child-index cardinality identifications.

theorem Q2Presentation.Induction.sec7_frattiniTrivial_elementaryMStageCandidateExactCountData {p : FramedPair} (h : ¬ScalarTerminal p) (chief : FirstNonScalarChiefFactor p) (K : KernelMinimal chief.toNonScalarChiefFactor) (hR : Nat.card ↥(kernelFrattini K) = 1) (child : Sec7FrattiniTrivialElementaryChildData p) (mult : Sec7FrattiniTrivialElementaryMStageMultiplicityData child) (hdim : mult.dimM = sec7MStageDim K) (src : Sec7FrattiniTrivialElementaryMStageSourceIndexData child mult) (cardA : Sec7FrattiniTrivialElementaryMStageCandidateExactImagePartitionCanonicalFinSigmaPartitionCardData child mult src) : Nonempty (Sec7FrattiniTrivialElementaryMStageCandidateExactCountData child mult src)

Former candidate exact-count residual, now theorem-level from the additive exact-image partition cardinality.

structure Q2Presentation.Induction.Sec7FrattiniTrivialElementaryMStageLocalExactImagePartitionEquivData {p : FramedPair} (D : Sec7FrattiniTrivialElementaryChildData p) (mult : Sec7FrattiniTrivialElementaryMStageMultiplicityData D) (src : Sec7FrattiniTrivialElementaryMStageSourceIndexData D mult) : Type 1

Finite disjoint-union form of the local exact-image partition.

• ProperQ : Type

  Finite index for all local proper exact-image contributions.

• finProperQ : Fintype self.ProperQ
• LowerQ : Type

  Finite index for the local lower-child framed lifts.

• finLowerQ : Fintype self.LowerQ
• proper_card : Nat.card self.ProperQ = (List.map (⇑Cardinal.toNat) (List.map framedCountQ D.properChildren)).sum

  The proper-image index has the cardinality of the local proper-child sum.

• lower_card : Nat.card self.LowerQ = Cardinal.toNat (framedCountQ D.lowerChild)

  The lower-child index has the cardinality of the local lower-child framed count.

• partition_equiv : src.XQ ⊕ self.ProperQ ≃ Fin mult.liftMultiplicity × self.LowerQ

  The local exact-image partition as a finite bijection.

structure Q2Presentation.Induction.Sec7FrattiniTrivialElementaryMStageLocalExactImagePartitionCanonicalFinLowerCardData {p : FramedPair} (D : Sec7FrattiniTrivialElementaryChildData p) : Type

Lower-child cardinality half of the local exact-image partition.

This is the local analogue of the candidate lower-cardinality theorem, but it is kept as a semantic residual because the final local source side does not currently expose a general finiteness theorem for all boundaryFramedSurj boundaryPackage_GQ2 sets.

• finLowerQ : Fintype (boundaryFramedSurj boundaryPackage_GQ2 D.lowerChild.fst D.lowerChild.snd)

  The actual local lower-child boundary-framed surjection set is finite.

• lower_card : Nat.card (boundaryFramedSurj boundaryPackage_GQ2 D.lowerChild.fst D.lowerChild.snd) = Cardinal.toNat (framedCountQ D.lowerChild)

  The actual local lower-child index has the cardinality of the lower-child framed local count.

structure Q2Presentation.Induction.Sec7FrattiniTrivialElementaryMStageLocalExactImagePartitionCanonicalFinSigmaPartitionCardData {p : FramedPair} (D : Sec7FrattiniTrivialElementaryChildData p) (mult : Sec7FrattiniTrivialElementaryMStageMultiplicityData D) (src : Sec7FrattiniTrivialElementaryMStageSourceIndexData D mult) : Prop

Sigma-fibre cardinality form of the local exact-image partition.

The proper-image summand is the canonical finite set whose cardinality is the sum of the proper-child local counts; the lower-child summand is the actual local lower-child boundary-framed surjection set.

• partition_sigma_card : Nat.card (src.XQ ⊕ Fin (List.map (⇑Cardinal.toNat) (List.map framedCountQ D.properChildren)).sum) = Nat.card ((_ : Fin mult.liftMultiplicity) × boundaryFramedSurj boundaryPackage_GQ2 D.lowerChild.fst D.lowerChild.snd)

structure Q2Presentation.Induction.Sec7FrattiniTrivialElementaryMStageLocalExactImagePartitionCanonicalFinSigmaPartitionEquivData {p : FramedPair} (D : Sec7FrattiniTrivialElementaryChildData p) (mult : Sec7FrattiniTrivialElementaryMStageMultiplicityData D) (src : Sec7FrattiniTrivialElementaryMStageSourceIndexData D mult) : Type

Bijection form of the local exact-image partition before flattening the constant lower-child sigma fibre.

• partition_sigma_equiv : src.XQ ⊕ Fin (List.map (⇑Cardinal.toNat) (List.map framedCountQ D.properChildren)).sum ≃ (_ : Fin mult.liftMultiplicity) × boundaryFramedSurj boundaryPackage_GQ2 D.lowerChild.fst D.lowerChild.snd

structure Q2Presentation.Induction.Sec7FrattiniTrivialElementaryMStageLocalExactImagePartitionCanonicalFinEquivData {p : FramedPair} (D : Sec7FrattiniTrivialElementaryChildData p) (mult : Sec7FrattiniTrivialElementaryMStageMultiplicityData D) (src : Sec7FrattiniTrivialElementaryMStageSourceIndexData D mult) : Type

Product-form canonical local exact-image partition over the actual lower-child local source set.

• finLowerQ : Fintype (boundaryFramedSurj boundaryPackage_GQ2 D.lowerChild.fst D.lowerChild.snd)

  The actual local lower-child boundary-framed surjection set is finite.

• lower_card : Nat.card (boundaryFramedSurj boundaryPackage_GQ2 D.lowerChild.fst D.lowerChild.snd) = Cardinal.toNat (framedCountQ D.lowerChild)

  The lower-child local source set has the framed lower-child cardinality.

• partition_equiv : src.XQ ⊕ Fin (List.map (⇑Cardinal.toNat) (List.map framedCountQ D.properChildren)).sum ≃ Fin mult.liftMultiplicity × boundaryFramedSurj boundaryPackage_GQ2 D.lowerChild.fst D.lowerChild.snd

  The exact-image partition over the canonical finite proper-image index.

theorem Q2Presentation.Induction.sec7_frattiniTrivial_elementaryMStageLocalExactImagePartitionCanonicalFinLowerCardData {p : FramedPair} (h : ¬ScalarTerminal p) (chief : FirstNonScalarChiefFactor p) (K : KernelMinimal chief.toNonScalarChiefFactor) (hR : Nat.card ↥(kernelFrattini K) = 1) (child : Sec7FrattiniTrivialElementaryChildData p) : Nonempty (Sec7FrattiniTrivialElementaryMStageLocalExactImagePartitionCanonicalFinLowerCardData child)

Local lower-child finite/cardinality input for the elementary exact-image partition.

The local projection/fibre packet already contains the only nontrivial finiteness input needed here: finiteness of the lower local boundary-framed source set. Once that Fintype instance is available, the cardinality equality is just the definition framedCountQ q = #boundaryFramedSurj boundaryPackage_GQ2 q.1 q.2 followed by Cardinal.mk_toNat_eq_card.

theorem Q2Presentation.Induction.sec7_frattiniTrivial_elementaryMStageLocalExactImagePartitionCanonicalFinSigmaPartitionEquivData {p : FramedPair} (h : ¬ScalarTerminal p) (chief : FirstNonScalarChiefFactor p) (K : KernelMinimal chief.toNonScalarChiefFactor) (hR : Nat.card ↥(kernelFrattini K) = 1) (child : Sec7FrattiniTrivialElementaryChildData p) (mult : Sec7FrattiniTrivialElementaryMStageMultiplicityData child) (hdim : mult.dimM = sec7MStageDim K) (src : Sec7FrattiniTrivialElementaryMStageSourceIndexData child mult) (cardQ : Sec7FrattiniTrivialElementaryMStageLocalExactImagePartitionCanonicalFinSigmaPartitionCardData child mult src) : Nonempty (Sec7FrattiniTrivialElementaryMStageLocalExactImagePartitionCanonicalFinSigmaPartitionEquivData child mult src)

Former local exact-image sigma-partition equivalence, now theorem-level from the finite cardinality statement and the lower-child finiteness/cardinality residual.

theorem Q2Presentation.Induction.sec7_frattiniTrivial_elementaryMStageLocalExactImagePartitionCanonicalFinEquivData {p : FramedPair} (h : ¬ScalarTerminal p) (chief : FirstNonScalarChiefFactor p) (K : KernelMinimal chief.toNonScalarChiefFactor) (hR : Nat.card ↥(kernelFrattini K) = 1) (child : Sec7FrattiniTrivialElementaryChildData p) (mult : Sec7FrattiniTrivialElementaryMStageMultiplicityData child) (hdim : mult.dimM = sec7MStageDim K) (src : Sec7FrattiniTrivialElementaryMStageSourceIndexData child mult) (cardQ : Sec7FrattiniTrivialElementaryMStageLocalExactImagePartitionCanonicalFinSigmaPartitionCardData child mult src) : Nonempty (Sec7FrattiniTrivialElementaryMStageLocalExactImagePartitionCanonicalFinEquivData child mult src)

Former product-form local exact-image partition, now theorem-level from the sigma-fibre partition and the canonical constant-fibre sigma/product equivalence.

theorem Q2Presentation.Induction.sec7_frattiniTrivial_elementaryMStageLocalExactImagePartitionEquivData {p : FramedPair} (h : ¬ScalarTerminal p) (chief : FirstNonScalarChiefFactor p) (K : KernelMinimal chief.toNonScalarChiefFactor) (hR : Nat.card ↥(kernelFrattini K) = 1) (child : Sec7FrattiniTrivialElementaryChildData p) (mult : Sec7FrattiniTrivialElementaryMStageMultiplicityData child) (hdim : mult.dimM = sec7MStageDim K) (src : Sec7FrattiniTrivialElementaryMStageSourceIndexData child mult) (cardQ : Sec7FrattiniTrivialElementaryMStageLocalExactImagePartitionCanonicalFinSigmaPartitionCardData child mult src) : Nonempty (Sec7FrattiniTrivialElementaryMStageLocalExactImagePartitionEquivData child mult src)

Former local exact-image partition residual, now theorem-level from the canonical finite proper-image index and the actual local lower-child source set.

theorem Q2Presentation.Induction.sec7_frattiniTrivial_elementaryMStageLocalExactCountData {p : FramedPair} (h : ¬ScalarTerminal p) (chief : FirstNonScalarChiefFactor p) (K : KernelMinimal chief.toNonScalarChiefFactor) (hR : Nat.card ↥(kernelFrattini K) = 1) (child : Sec7FrattiniTrivialElementaryChildData p) (mult : Sec7FrattiniTrivialElementaryMStageMultiplicityData child) (hdim : mult.dimM = sec7MStageDim K) (src : Sec7FrattiniTrivialElementaryMStageSourceIndexData child mult) (cardQ : Sec7FrattiniTrivialElementaryMStageLocalExactImagePartitionCanonicalFinSigmaPartitionCardData child mult src) : Nonempty (Sec7FrattiniTrivialElementaryMStageLocalExactCountData child mult src)

Former local exact-count residual, now theorem-level from the local finite partition bijection.

axiom Q2Presentation.Induction.sec7_frattiniTrivial_elementaryMStagePartitionPacket {p : FramedPair} (h : ¬ScalarTerminal p) (chief : FirstNonScalarChiefFactor p) (K : KernelMinimal chief.toNonScalarChiefFactor) (hR : Nat.card ↥(kernelFrattini K) = 1) : Nonempty ((child : Sec7FrattiniTrivialElementaryChildData p) ×' (src : Sec7FrattiniTrivialElementaryMStageSourceIndexData child (sec7MStageCanonicalMultiplicityData K child)) ×' Sec7FrattiniTrivialElementaryMStageCandidateExactImagePartitionCanonicalFinSigmaPartitionCardData child (sec7MStageCanonicalMultiplicityData K child) src ×' Sec7FrattiniTrivialElementaryMStageLocalExactImagePartitionCanonicalFinSigmaPartitionCardData child (sec7MStageCanonicalMultiplicityData K child) src)

Fused eq:Mstage partition packet (soundness repair #15). Existence-form over the manuscript-constructed elementary child: replaces the two former ∀-child partition axioms, which were falsifiable at junk properChildren/lowerChild instantiations (PROGRESS.md). Asserts lem:elementarystage's two count identities exactly at a manuscript child with its canonical multiplicity and realized source indexes.

theorem Q2Presentation.Induction.sec7_frattiniTrivial_elementaryMStageExactCountData {p : FramedPair} (h : ¬ScalarTerminal p) (chief : FirstNonScalarChiefFactor p) (K : KernelMinimal chief.toNonScalarChiefFactor) (hR : Nat.card ↥(kernelFrattini K) = 1) (child : Sec7FrattiniTrivialElementaryChildData p) (mult : Sec7FrattiniTrivialElementaryMStageMultiplicityData child) (hdim : mult.dimM = sec7MStageDim K) (src : Sec7FrattiniTrivialElementaryMStageSourceIndexData child mult) (cardA : Sec7FrattiniTrivialElementaryMStageCandidateExactImagePartitionCanonicalFinSigmaPartitionCardData child mult src) (cardQ : Sec7FrattiniTrivialElementaryMStageLocalExactImagePartitionCanonicalFinSigmaPartitionCardData child mult src) : Nonempty (Sec7FrattiniTrivialElementaryMStageExactCountData child mult src)

Former M-stage exact-count residual, now theorem-level. The paired exact-image subtraction packet is assembled from its candidate and local eq:Mstage halves.

theorem Q2Presentation.Induction.sec7_frattiniTrivial_elementaryMStageNormalizedCountData {p : FramedPair} (h : ¬ScalarTerminal p) (chief : FirstNonScalarChiefFactor p) (K : KernelMinimal chief.toNonScalarChiefFactor) (hR : Nat.card ↥(kernelFrattini K) = 1) (child : Sec7FrattiniTrivialElementaryChildData p) (src : Sec7FrattiniTrivialElementaryMStageSourceIndexData child (sec7MStageCanonicalMultiplicityData K child)) (cardA : Sec7FrattiniTrivialElementaryMStageCandidateExactImagePartitionCanonicalFinSigmaPartitionCardData child (sec7MStageCanonicalMultiplicityData K child) src) (cardQ : Sec7FrattiniTrivialElementaryMStageLocalExactImagePartitionCanonicalFinSigmaPartitionCardData child (sec7MStageCanonicalMultiplicityData K child) src) : Nonempty (Sec7FrattiniTrivialElementaryMStageNormalizedCountData child)

Former M-stage normalized-count residual, now theorem-level. The public packet is assembled from the three manuscript mechanisms in lem:elementarystage / eq:Mstage: common lift multiplicity, source lift-index realization, and exact-image subtraction.

theorem Q2Presentation.Induction.sec7_frattiniTrivial_elementaryStageNormalizedCountData {p : FramedPair} (h : ¬ScalarTerminal p) (chief : FirstNonScalarChiefFactor p) (K : KernelMinimal chief.toNonScalarChiefFactor) (hR : Nat.card ↥(kernelFrattini K) = 1) (child : Sec7FrattiniTrivialElementaryChildData p) (src : Sec7FrattiniTrivialElementaryMStageSourceIndexData child (sec7MStageCanonicalMultiplicityData K child)) (cardA : Sec7FrattiniTrivialElementaryMStageCandidateExactImagePartitionCanonicalFinSigmaPartitionCardData child (sec7MStageCanonicalMultiplicityData K child) src) (cardQ : Sec7FrattiniTrivialElementaryMStageLocalExactImagePartitionCanonicalFinSigmaPartitionCardData child (sec7MStageCanonicalMultiplicityData K child) src) : Nonempty (Sec7FrattiniTrivialElementaryNormalizedCountData child)

Former normalized source-count residual, now theorem-level. The old normalized-count packet is assembled from the concrete eq:Mstage count packet by using the common functional z_M · e(C) - Σ_J e(J) as partialN.

theorem Q2Presentation.Induction.sec7_frattiniTrivial_elementaryStageSplitData {p : FramedPair} (h : ¬ScalarTerminal p) (chief : FirstNonScalarChiefFactor p) (K : KernelMinimal chief.toNonScalarChiefFactor) (hR : Nat.card ↥(kernelFrattini K) = 1) : Nonempty (Sec7FrattiniTrivialElementaryStageSplitData p)

Split residual (R = 1, elementary stage), now assembled theorem-level from the two halves. The child-geometry half is supplied by sec7_frattiniTrivial_elementaryChildData; the remaining source-count half is the separate normalized-count residual above.

theorem Q2Presentation.Induction.sec7_frattiniTrivial_elementaryStageCoreData {p : FramedPair} (h : ¬ScalarTerminal p) (chief : FirstNonScalarChiefFactor p) (K : KernelMinimal chief.toNonScalarChiefFactor) (hR : Nat.card ↥(kernelFrattini K) = 1) : Nonempty (Sec7FrattiniTrivialElementaryStageCoreData p)

Former smaller residual, now assembled theorem-level. The previous core packet is recovered from the split elementary-stage packet by converting the vacuous zero-obstruction fibres to the full lift index.

theorem Q2Presentation.Induction.sec7_frattiniTrivial_elementaryStageData {p : FramedPair} (h : ¬ScalarTerminal p) (chief : FirstNonScalarChiefFactor p) (K : KernelMinimal chief.toNonScalarChiefFactor) (hR : Nat.card ↥(kernelFrattini K) = 1) : Nonempty (Sec7FrattiniTrivialElementaryStageData p)

Lowered residual (R = 1, elementary stage). The previous elementary-stage packet is now a theorem assembled from the smaller core packet. The formerly residual all-character partial identities are discharged by Subsingleton.elim on Module.Dual (ZMod 2) PUnit.

theorem Q2Presentation.Induction.sec7_fourierBranch_of_frattiniTrivial {p : FramedPair} (h : ¬ScalarTerminal p) (chief : FirstNonScalarChiefFactor p) (K : KernelMinimal chief.toNonScalarChiefFactor) (hR : Nat.card ↥(kernelFrattini K) = 1) : Nonempty (FourierBranchData p)

The old branch statement, now proved from the lowered elementary-stage residual.