The Γ_A Gauss residue — assembly shell + the pinned-value seams

The final brick of the (83)-for-Γ_A lane: gaussZResidue_gammaA_{unramified,ramified} discharge the prop_8_9 ledger hypothesis hGaussZA at the pinned values ∓2^m, mirroring GaussZFinal.gaussZResidue_local_* with the Γ_A toolkit:

• the reduction gaussZ_reduction (generic) at Γ := GammaA, with Z¹-finiteness from GaussZCoordGammaA.finite_vcocycle_gammaA and the V^{C₀} = 0 freeness from hfix_of_simple_nt (hnt-only — no hfaith on the source side);
• the pinned-value seams sum_sign_QZeroBar_gammaA_{unramified,ramified}: ∑ sign(Q̄⁰) = ∓2^m over Z¹⧸B¹ — the A-4 core. Route (docs/orchestration/p16d6e4aA-a4-prep.md, the paper's Prop 6.5/6.9): reindex the quotient by the x₀-supported section (FoxHeisenberg.x0Supported, the paper's gauge; bijective onto H¹_w by the d¹-closed forms + card_H1w_gammaA); evaluate Q̄⁰ on the section through A-3's QZero_eq_relZPair_kappa0 by the κ⁰-ledger (the quadratic mirror of the banked mixed ledgers heisMarking_wildValue_z/_ramified — Prop 6.5's table: h₀ ↦ q(c) via classTwoIdentity, [d₀,z₀] ↦ B(c, U⁻¹c) ram / 0 split); identify with q̄ (unram, T = 1 ⟹ U = 1 collapse) / qDouble q̄ U (ram) and count via prop_6_9_unramified / lemma_6_8 clause 4 + gaussSum_eq_of_arf_eq.

Axioms: the shell is std-3; the seams proved as expected (word-side; the pins are proved).

File organisation. The proof is split into Kappa, Counts, Action, and Assembly, matching the A-4 dependency layers documented below. This umbrella preserves the original import path and public declaration names.

Paper-tag ledger (auto-generated by paperforge; do not edit)

• Prop 6.5 = ⟦prop-wordquadratic⟧