Integrated current route, fifth refinement

This sidecar is the P1 integration step from the four-turn Pro finalization plan. Route4 is already the strongest kernel-module / extraspecial route. The only change here is that the target decoration-kernel input is routed through TargetKmodRSameTowerBridgeProofs, which removes the two route4 bridge residual names from the final closure and exposes the older same-tower actual quotient and ambient-action packets.

theorem Q2Presentation.Induction.centralCover_realization_proven_fine_from_selfGroupTowerCurrentIntegrated5 {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))

Cover realization through the route5 target same-tower bridge proofs.

theorem Q2Presentation.Induction.boundaryFramed_nonterminal_fourier_branch_currentIntegrated5 (p : FramedPair) : ¬ScalarTerminal p → Nonempty (FourierBranchData p)

Route5 nonterminal branch: route4's minimal-block route with the P1 target same-tower bridge replacement.

theorem Q2Presentation.Induction.boundaryFramed_recursive_branch_from_fourier_currentIntegrated5 (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_currentIntegrated5 (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_currentIntegrated5 (p : FramedPair) : framedCountA p = framedCountQ p

theorem Q2Presentation.Induction.finite_surjection_counts_equal_currentIntegrated5 (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)))

Route5 finite-surjection equality.