A registered result in its source context

CURVE.CODE.OUTERSPIN.01

Original ledger classification: 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).

Original claim text

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.

Original classification detail

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 as recorded

CURVE.CODE.2TORSION.01; ARF.SPINORCOMPILER.01; v784; v774; v752; v610; v611; ARF.ROOTCLASS.01