Integrated current route, twenty-first refinement

Route21 keeps the route20 assembly shape but swaps in all four route21 sidecars:

• target K/R through TargetKmodRRoute21ActualProofs;
• candidate scalar/Frattini through CandidateScalarFrattiniRoute21Proofs;
• Section 7 kernel/head through MinimalBlockKernelConcreteRoute21Proofs;
• pushout/determinant through MinimalBlockPushoutRoute21Proofs.

The point of this route is not yet a net axiom-count reduction. It replaces the seven route20 non-local leaves by more manuscript-shaped field-level packets.

theorem Q2Presentation.Induction.minimalBlock_groupTowerData_from_currentIntegrated21 {p : FramedPair} (chief : NonScalarChiefFactor p) : Nonempty (MinimalBlockGroupTowerData p chief)

Route21 group tower: the kernel/head route20 pair replaced by route21's same-tower field-level packet.

theorem Q2Presentation.Induction.centralCover_realization_proven_fine_from_selfGroupTowerCurrentIntegrated21 {p : FramedPair} {chief : NonScalarChiefFactor p} (G : MinimalBlockGroupTowerData p chief) (h : ¬ScalarTerminal p) (Fdet : MinimalBlockPreheadExtraspecialEquivariantFamilyData G.preheadTowerData) (A : MinimalBlockPreheadExtraspecialAnnihilationData Fdet) (Rdet : MinimalBlockPreheadExtraspecialRadicalData Fdet) : Nonempty (CoverRealization (G.toMinimalBlock h Fdet A Rdet))

Route21 cover realization: target route21 projection/action, candidate route21 Frattini partition, kernel route21, and route21 actual central-extension pushout data.

theorem Q2Presentation.Induction.boundaryFramed_nonterminal_fourier_branch_currentIntegrated21 (p : FramedPair) : ¬ScalarTerminal p → Nonempty (FourierBranchData p)

Route21 nonterminal branch.

theorem Q2Presentation.Induction.boundaryFramed_recursive_branch_from_fourier_currentIntegrated21 (p : FramedPair) (h : ¬ScalarTerminal p) : ∃ (children : List FramedPair) (combine : List Cardinal.{0} → Cardinal.{0}), (∀ c ∈ children, framedMeasure c < framedMeasure p) ∧ framedCountA p = combine (List.map framedCountA children) ∧ framedCountQ p = combine (List.map framedCountQ children)

theorem Q2Presentation.Induction.boundaryFramed_recursion_step_currentIntegrated21 (p : FramedPair) : framedCountA p = framedCountQ p ∨ ∃ (children : List FramedPair) (combine : List Cardinal.{0} → Cardinal.{0}), (∀ c ∈ children, framedMeasure c < framedMeasure p) ∧ framedCountA p = combine (List.map framedCountA children) ∧ framedCountQ p = combine (List.map framedCountQ children)

theorem Q2Presentation.Induction.boundaryFramed_counts_equal_currentIntegrated21 (p : FramedPair) : framedCountA p = framedCountQ p

theorem Q2Presentation.Induction.finite_surjection_counts_equal_currentIntegrated21 (G : Type) [Group G] [Finite G] [Nonempty G] : Cardinal.mk (Profinite.SurjContHom GammaA (ProfiniteGrp.ofFiniteGrp (FiniteGrp.of G))) = Cardinal.mk (Profinite.SurjContHom GQ2Profinite (ProfiniteGrp.ofFiniteGrp (FiniteGrp.of G)))

Route21 finite-surjection equality.