Original ledger status
CURVE.CODE.2TORSION.01
The curve/code 2-torsion decider (33/33 checks, ~9 s; discovery probe curve_code_2torsion_probe.py CURVE-CODE-PARTIAL, 2026-08-05). The frozen CM-model route through the seam curve C: y^3 = x^4 - 1 (genus 3, Jacobian splits per v610): exact differential characters (zeta^11, zeta^7, zeta^10) reproduce v610 L1; A = the zeta_12 CM abelian surface with EVERY generator period certified as an integral O = Z[zeta_12] recognition (mpmath dps 120, residual tolerance 1e-35, recognition margin > 10^25 x residual, HNF index 1 -- the numerics END in certified integer recognitions); A[2] = O/2O = F2^4 exact from there on; the deck tau = multiplication by zeta^4 is DERIVED as the only order-3 automorphism class (Aut(C) = C12, monodromy forcing); the minimal ODD tau-invariant polarization is type (1,3) with Pfaffian 3 -- NO principal form exists in the tau-invariant cone (unit-sign obstruction: the fundamental unit 2+sqrt3 of the real subfield is totally positive) and the curve-induced Prym type (1,2) is mod-2 DEGENERATE (typed [C]); q_Theta = the unique tau-fixed quadratic refinement (exists and is unique since N(omega-1) = 9 is odd). THE DECIDER: (A[2], e_2) and (V, hbar) are isomorphic as BARE symplectic F2^4 (720 = |Sp(4,2)| transporters -- content-free), but the full correspondence DIES at two exact layers: (1) SIGMA LAYER: tau acts FREELY on A[2] (fixed space 0, realization-independent) with cycle type [3,3] on the 6 odd theta characteristics (SYNTHEME side), while the code family cycle sigma-bar fixes a 2-plane with cycle type [3,1,1,1] (DUAD side) -- the two order-3 classes are exchanged only by the OUTER automorphism of S6, never by conjugation; equivariant census 0 (tau and tau^2). (2) REFINEMENT LAYER: the curve's canonical deck-invariant theta refinement q_Theta is EVEN (Arf 0, 10 zeros, stabilizer 72, decomposition 1+6+9) while the code's arf-certified selector q* is ODD (Arf 1, 6 zeros, stabilizer 120, decomposition 1+5+10) -- Arf is a transport invariant, census 0. Joint census 0 with the positive control (code clone) at exactly 6 = the code-side stabilizer order; all must-fire controls fire. ROOTCLASS-MIXED (v775) fence respected: finite exact algebra on 2-torsion, no physical labeling. No marker move, NO RH claim. NOTE (2026-08-06, v796 -- marker UNCHANGED): the INNER identification stays dead exactly as recorded, and the OUTER twist is now decided CANONICAL (CURVE.CODE.OUTERSPIN.01 / v796): the Sylvester bridge beta with beta sigma = tau beta and beta(q*) = S* is unique up to the order-6 joint stabilizer, pulls q_Theta back to q_even* = q* + hbar(., A) (the anchor shift resolves the Arf mismatch), and transports 1+5bar+10 / 3+2 / t_A -> rho = J exactly -- both breaking layers of this row are thereby explained as the two faces of the outer automorphism of S6, not repaired.
Recorded status: Exact [E] finite algebra + certified-recognition CM layer; the correspondence honestly PARTIAL with both breaking layers named exactly (syntheme-vs-duad and Arf 0-vs-1); no marker move (Python-only per GATE.WOLFRAM.02).
Dependencies, status detail and source
Canonical status: CURVE-CODE-PARTIAL: (A[2], e_2) = (V, hbar) as bare symplectic spaces only; the correspondence dies at two exact layers -- deck acts syntheme-side [3,3]-free vs code duad-side [3,1,1,1] (outer S6), and curve theta even/Arf-0 (1+6+9) vs q* odd/Arf-1 (1+5+10); joint census 0 with positive control 6; every period certified as integral O-recognition; no RH statement.
Dependencies: v610; v611; v752; v753; ARF.SPINORCOMPILER.01; ARF.ROOTCLASS.01
Scripts: v784_curve_code_2torsion.py
Location: tfpt_1;tfpt_3
Original CSV, line 687Original ledger status
CURVE.CODE.OUTERSPIN.01
the canonical OUTER bridge between the code's duad side and the curve's syntheme side of the S6 duality (24/24 checks, ~11 s; discovery probe duad_syntheme_bridge_probe.py, 2026-08-05, verdict BRIDGE-CANONICAL). THE CONSTRUCTION (Sylvester duality made explicit): a bridge is a bijection beta: Omega_code -> T_curve (code odd Arf forms -> curve isotropic spreads/totals), canonically inducing the incidence-preserving duality Delta_beta: V\{0} -> IsoLines(A[2]) through the K6 duad model and the shared-syntheme rule (every pair of totals shares exactly one line -- verified). THE CENSUS: 18 + 18 bridges intertwine sigma with tau|T resp. tau^2|T (each family a torsor under C_S6(3-cycle), order 18); after pinning beta(q*) = S* -- the UNIQUE curve spread fixed by the FULL <tau, rho> automorphism image (rho = the mu4 rotation = the complex structure J = mult by zeta^9; existence + uniqueness COMPUTED, not assumed) -- exactly 6 + 6 remain = ONE ORBIT (torsor) under the order-6 code joint stabilizer, with the tau/tau^2 families exchanged by an inner code symmetry (one statement). B2 ARF RESOLUTION: the pullback of the curve's even q_Theta through the two frozen even<->3|3-split dictionaries is CONSTANT over all 36 B1 bridges and equals q_even* = the unique sigma-invariant even code refinement, with the NAMED operation q* = q_even* + hbar(., A) EXACT -- the anchor shift resolves the v784 Arf mismatch canonically. THE OUTER WITNESS exact: sigma is [3,1,1,1] on odds but [3,3] on spreads, tau the reverse, the anchor transvection t_A is [2,1,1,1,1] on odds but [2,2,2] on spreads, and the class swap is UNIFORM (0 violations over all 720 group elements) -- the twist IS the outer automorphism of S6, not an inner accident. TRANSPORTS (structure labels only): {q* = 0}\{0} maps EXACTLY onto the 5 lines of S* (5bar) and the 10 onto the complement; the 3+2 split maps onto the [3,1,1] tau-orbit structure of S*; t_A maps to rho = J on every census bridge; chi_NSR transports as an honest exact TWO-VARIANT torsor ('incidence with one of the two tau-fixed lines of S*') -- it is pinned code-side by the v752 NS/R lattice parity, a datum the bare quadruple (V, hbar, sigma, q*) does not carry, and rho swaps the two candidate lines (the two-fold ambiguity intrinsic). CONTROLS: inner cross-side census 0 (class mismatch), same-side census 0, no bridge pulls q_Theta to an ODD form (never to q*), scrambled sigma census 0. The v784 INNER identification stays dead (CURVE-CODE-PARTIAL untouched); the bridge is a NEW statement about the outer twist. NO matter semantics (5bar/10, 3+2, chi_NSR are structure labels; ROOTCLASS-MIXED v775 stands). No marker move, NO RH claim. Python-only per GATE.WOLFRAM.02. CONTRACT DECIDED (registered here as the decided outer-bridge statement): the outer identification is CANONICAL; the residual is the intrinsic two-variant chi_NSR ambiguity (pinned only by the v752 NS/R lattice parity, which the bare 2-torsion quadruple does not carry) -- typed, not open work; cross-referenced with CURVE.CODE.2TORSION.01, whose inner-identification kill stands unchanged.
Recorded status: Exact [E] finite census theorem (torsor uniqueness incl. the tau/tau^2 swap; constant Arf resolution; exact transports) behind the structure-label fence; the inner identification stays dead (v784); no marker move (Python-only per GATE.WOLFRAM.02).
Dependencies, status detail and source
Canonical status: BRIDGE-CANONICAL: the Sylvester duality beta (code odd forms -> curve spreads) with beta sigma = tau beta and beta(q*) = S* is unique up to the order-6 joint stabilizer (18 + 18 -> 6 + 6, one orbit incl. the tau/tau^2 swap); q_Theta pulls back constantly to q_even* = q* + hbar(., A) -- the anchor shift resolves the v784 Arf mismatch; 5bar/10, 3+2 and t_A -> rho = J transport exactly; chi_NSR an exact two-variant torsor; inner identification stays dead; no RH statement.
Dependencies: CURVE.CODE.2TORSION.01; ARF.SPINORCOMPILER.01; v784; v774; v752; v610; v611; ARF.ROOTCLASS.01
Scripts: v796_duad_syntheme_bridge.py
Location: tfpt_1;tfpt_3;tfpt_research_contracts
Original CSV, line 689