Route127 minimal-block pushout/determinant closer

Route127's public pushout frontier is the single coherent Section 7 atom

• minimalBlock_preheadActualCentralExtensionPushoutData.

Route126 already shows how to reconstruct that broad packet from the coherent Route82 scalar-pushout packet. This sidecar goes one step lower without editing the public route files: it obtains the Route82 packet from Route83's manuscript-shaped actual-scalar/coherent-determinant API, so the four Route82 residual providers disappear from this Route127 audit branch.

The resulting determinant provider has the same downstream shape as MinimalBlockPushoutRoute126Closers.minimalBlock_preheadDeterminantPackets_from_route126.

Route82 packet lowered through existing Route83 APIs

theorem Q2Presentation.Induction.MinimalBlockPushoutRoute127Closers.minimalBlock_preheadCentralPushoutPairingConstructionData_from_route127 {p : FramedPair} {chief : NonScalarChiefFactor p} (D : MinimalBlockPreheadTowerData p chief) : Nonempty (MinimalBlockPushoutRoute82Closers.MinimalBlockPreheadCentralPushoutPairingConstructionDataRoute82 D)

Route127 replacement for the Route82 pairing-construction atom.

This proves the same nonempty target as MinimalBlockPushoutRoute82Closers.minimalBlock_preheadCentralPushoutPairingConstructionData_route82, but the selected pairing is obtained from Route83's actual scalar pushout and coherent determinant packet rather than from the Route82 axiom.

theorem Q2Presentation.Induction.MinimalBlockPushoutRoute127Closers.minimalBlock_preheadCentralPushoutRoute82SplitPackets_from_route127 {p : FramedPair} {chief : NonScalarChiefFactor p} (D : MinimalBlockPreheadTowerData p chief) : ∃ (Phi : MinimalBlockPushoutRoute82Closers.MinimalBlockPreheadCentralPushoutPairingConstructionDataRoute82 D), Nonempty (MinimalBlockPushoutRoute82Closers.MinimalBlockPreheadCentralPushoutPairingNaturalityDataRoute82 Phi) ∧ ∃ (Ann : MinimalBlockPushoutRoute82Closers.MinimalBlockPreheadCentralPushoutAnnihilationDataRoute82 Phi), Nonempty (MinimalBlockPushoutRoute82Closers.MinimalBlockPreheadCentralPushoutDescendedNonsingularDataRoute82 Phi Ann)

The Route127 selected Route82 split packets, all tied to the same pairing.

This is the coherent replacement surface for the remaining Route82 leaves: naturality, annihilation on T, and descended nonsingularity are proved for the selected Phi produced by Route83.

theorem Q2Presentation.Induction.MinimalBlockPushoutRoute127Closers.minimalBlock_preheadPushoutRoute82Data_from_route127 {p : FramedPair} {chief : NonScalarChiefFactor p} (D : MinimalBlockPreheadTowerData p chief) : Nonempty (MinimalBlockPushoutRoute82Closers.MinimalBlockPreheadPushoutRoute82Data D)

Coherent Route82 packet for Route127, obtained without invoking the Route82 split-packet residual axioms.

Route126-shaped broad-packet and determinant providers

theorem Q2Presentation.Induction.MinimalBlockPushoutRoute127Closers.minimalBlock_preheadActualCentralExtensionPushoutData_from_route127 {p : FramedPair} {chief : NonScalarChiefFactor p} (D : MinimalBlockPreheadTowerData p chief) : Nonempty (MinimalBlockPreheadActualCentralExtensionPushoutData D)

Route127 reconstruction of the current broad public pushout atom.

The reconstruction uses Route126's theorem-level projection from a coherent Route82 packet, while the Route82 packet itself is supplied by Route83's lower actual-scalar/coherent-determinant APIs.

theorem Q2Presentation.Induction.MinimalBlockPushoutRoute127Closers.minimalBlock_preheadPushoutRouteLowerData_from_route127 {p : FramedPair} {chief : NonScalarChiefFactor p} (D : MinimalBlockPreheadTowerData p chief) : Nonempty (MinimalBlockPreheadPushoutRouteLowerData D)

Route127 lower pushout packet through the lowered Route82 packet.

theorem Q2Presentation.Induction.MinimalBlockPushoutRoute127Closers.minimalBlock_preheadDeterminantPackets_from_route127 {p : FramedPair} {chief : NonScalarChiefFactor p} (D : MinimalBlockPreheadTowerData p chief) : ∃ (F : MinimalBlockPreheadExtraspecialEquivariantFamilyData D), Nonempty (MinimalBlockPreheadExtraspecialAnnihilationData F) ∧ Nonempty (MinimalBlockPreheadExtraspecialRadicalData F)

Drop-in determinant-packet provider for the Route127 pushout line.

This has the same downstream conclusion as MinimalBlockPushoutRoute126Closers.minimalBlock_preheadDeterminantPackets_from_route126, but routes the Route82 atoms through the smaller Route83 actual-scalar plus coherent semantic determinant packets.

Current Route127 audit route with only the pushout provider swapped

theorem Q2Presentation.Induction.boundaryFramed_nonterminal_fourier_branch_currentIntegrated127_route127PushoutClosers (p : FramedPair) : ¬ScalarTerminal p → Nonempty (FourierBranchData p)

Route127 nonterminal Fourier branch with the pushout provider supplied by MinimalBlockPushoutRoute127Closers. The promoted Route127 files are not edited.

theorem Q2Presentation.Induction.boundaryFramed_recursive_branch_from_fourier_currentIntegrated127_route127PushoutClosers (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_currentIntegrated127_route127PushoutClosers (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_currentIntegrated127_route127PushoutClosers (p : FramedPair) : framedCountA p = framedCountQ p

theorem Q2Presentation.Induction.finite_surjection_counts_equal_currentIntegrated127_route127PushoutClosers (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)))

Route127 finite-surjection equality with only the pushout/determinant line routed through the lower Route127 closer in this file.