A registered result in its source context

CURVE.CODE.2TORSION.01

Original ledger classification: 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).

Original claim text

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.

Original classification detail

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 as recorded

v610; v611; v752; v753; ARF.SPINORCOMPILER.01; ARF.ROOTCLASS.01