A registered result in its source context

SEAM.CODE.TYPEII.01

Original ledger classification: Exact [E] exhaustive finite equivalence chain at length 8 (all 308,993 subspaces) modulo the cited classical lattice-VOA/theta ingredients, each machine-verified in every instance used; the OPE-linearity scope note typed; R1 -> R1' [C] residual registered; NO P2 marker move (Python-only per GATE.WOLFRAM.02).

Original claim text

Type II (doubly-even self-dual) is FORCED from three physical seam axioms -- the direct attack on residual R1 of ARF.BOUNDARY.CODE.01, decided exhaustively (19/19 checks, ~5 s; discovery probe seam_code_typeii_probe.py, 2026-08-06, verdict TYPEII-FORCED). THE THREE EQUIVALENCE CENSUSES, exhaustive over ALL 308,993 subspaces of F2^8 of dim 0..4 (totals = the Gaussian binomials 1, 255, 10795, 97155, 200787; dim > 4 excluded by rank-nullity): (1) MUTUAL LOCALITY of the Construction-A boundary vertex words <=> C self-orthogonal, ZERO exceptions (braiding exponent <(c+2m)/sqrt2, (c'+2m')/sqrt2> = c.c'/2 + Z -- the lift-shift identity verified exhaustively; word-pair census all 65536 pairs; survivors per dim 1/28/165/280/135); (2) INTEGER CONFORMAL SPIN (all T sector phases i^{wt(c)} trivial, |c+2m|^2 == wt(c) mod 4 exhaustive; the eta^8 multiplier typed universal/code-independent) <=> C doubly even, ZERO exceptions (survivors 1/8/38/52/30); (3) HOLOMORPHY / INDEX ONE (|disc| = det(rescaled Gram) = 2^{8-2k} exact in Fractions; |disc| = 1 = single modular-closed sector) <=> C self-dual <=> k = 4 -- and EVERY non-self-dual doubly-even code has a word in C-perp \ C of weight != 0 mod 4, i.e. a discriminant sector of FRACTIONAL conformal weight on which the S-image cannot close (proved exhaustively; detailed witness: the [8,3] subcode of the boundary-selected C* with exact coset theta series to h <= 2, theta_{L*} != 2 theta_L). THE KILL EXCLUDED: the survivor set of (locality, integer spin, |disc| = 1) over ALL subspaces IS the Type II set = the 30 e8-hat copies (weight enumerator 1 + 14x^4 + x^8); no boundary-legal non-Type-II code exists at length 8. R1 -> R1': the universe premise 'the boundary realizes a doubly-even self-dual code' is REPLACED by the three physical axioms A1 mutual locality (trivial monodromy / bose statistics), A2 integer conformal spin (bosonic T-alignment), A3 holomorphy / index one (single modular-closed sector) -- each a standard physical demand, not a coding-theory premise; given A1-A3 the chain boundary axioms + deck + orientation bit -> unique code -> E8 -> F2^4 -> g_car = 5 stands with this single sharpened [C] residual. IMPORTED INGREDIENTS typed and machine-verified in every instance used (I1 Frenkel-Kac-Segal braiding factor, FLM 1988/Kac, Adamo-Moriwaki-Tanimoto arXiv:2407.18222; I2 h = |lambda|^2/2 lattice-VOA grading; I3 Conway-Sloane/Ebeling theta S-transform with L* = A(C-perp) verified); the OPE-linearity scope note (admissible word sets are linear codes) typed as a scope assumption. Controls: braiding monodromy -1 witness; T-phase witnesses i and -1; the [8,3] S-closure violation. Uniqueness (30 copies; the boundary data select ONE) CITED from v776, count echo verified. NO P2 marker move. No marker move, NO RH claim. Python-only per GATE.WOLFRAM.02. CONTRACT (registered; the sharpened [C] residual replacing R1 of ARF.BOUNDARY.CODE.01): the physical seam satisfies A1 mutual locality (trivial monodromy of admissible boundary vertex words), A2 integer conformal spin (bosonic T-alignment), A3 holomorphy/index one (a single modular-closed sector); given A1-A3, Type II is FORCED at length 8 (this module), e8-hat is unique (30 copies) and the v776 boundary data select one copy -> E8 -> F2^4 -> g_car = 5; kill: a seam construction violating A1-A3 while staying boundary-legal, or a boundary-legal non-Type-II code at length 8 (excluded exhaustively here).

Original classification detail

TYPEII-FORCED: exhaustively over all subspaces of F2^8 (dim <= 4), locality <=> self-orthogonal, integer spin <=> doubly even, S-closure/|disc| = 1 <=> self-dual; survivor set = exactly the 30 e8-hat copies; no boundary-legal non-Type-II code; R1 -> R1' = the three physical axioms A1-A3; imported ingredients classical, cited, instance-verified; NO P2 marker move; no RH statement.

Dependencies as recorded

ARF.BOUNDARY.CODE.01; AX.P2.01; ARF.SPINORCOMPILER.01; v776; v752; E8.GAUSSCODE.01