A registered result in its source context

CURVE.CODE.DOILY.01

Original ledger classification: Exact [E] incidence-operator identities and spectra (integer/Fraction, annihilator-certified) with the (2/3)^6 identification typed [H neu -> decidable] pending the six-step composition; the inner identification stays dead

Original claim text

CURVE.CODE.DOILY.01: the certified outer bridge sharpened into an incidence-operator statement with a spectral payoff, one probe (21/21 checks incl. the 3 must-fire controls, ~1 s; discovery probe curve_code_doily_probe.py, 2026-08-06, verdict DOILY-EXACT). THE EXACT IDENTITY: the K6 doily incidence N (15 duads x 15 synthemes on the six odd Arf refinements) satisfies NN^T = 3I + A_KG(6,2) = B + 2I entrywise, with B the certified v752/v774 symplectic incidence (synthemes <-> isotropic lines bijectively; row and column sums 3; two duads share a syntheme iff disjoint, then exactly one). THE SPECTRAL PAYOFF: spec(NN^T) = spec(N^T N) = {9^1, 4^9, 0^5} exact (annihilating polynomial + traces + rank over Q), so the normalized channel N/3 has singular values {1, 2/3, 0} -- THE RECOVERY BASE RATE 2/3 (v221) as the unique nontrivial singular value of the canonical duad-to-syntheme incidence; the identification with the corpus rate (2/3)^6 = 64/729 is TYPED [H neu -> decidable, not decided here]: it requires the six-step composition semantics (the hexagon). PETERSEN LOCALIZATION: at the q*-distinguished point the 15 duads split 5 + 10, the 10 internal duads form the Petersen graph, and syntheme -> internal pair is a BIJECTION onto E(Petersen) 15/15 -- Petersen vertices = code duads, Petersen edges = curve synthemes under the certified bridge. THE REFINED BRIDGE: every census bridge (rebuilt with the frozen BRIDGE-CANONICAL conditions, census 6) factors EXACTLY through N -- Delta(d) = {Gamma(s) : d in s} and the matrix identity N_code = P_Delta^T N_curve^T P_Gamma: the outer bridge conjugates the code incidence to the TRANSPOSE of the curve incidence, a DUALITY, not an isomorphism (control C3: 720 symplectic isomorphisms exist, 0 intertwine sigma with tau -- the inner route stays dead). THE ARF LAYER in doily coordinates: the switch set T6 = the six within-block duads of the q_Theta point-split, bijective onto the 6 odd theta characteristics -- the census-6 anchor. N/3 is doubly stochastic (typed CLASSICAL incidence channel, not CPTP; 1 + 9 + 5 spectral decomposition, rate 2/3 on the 9-dim window) -- measured, no descent claim. No matter semantics (ROOTCLASS-MIXED, v775); the inner identification stays CURVE-CODE-PARTIAL/dead. No marker move, NO RH claim. Python-only per GATE.WOLFRAM.02.

Original classification detail

DOILY-EXACT: NN^T = B + 2I entrywise; spec {9, 4^9, 0^5}; N/3 singular values {1, 2/3, 0} -- the recovery base rate as the unique nontrivial singular value, (2/3)^6 typed pending six-step composition; Petersen edges = synthemes 15/15; every census bridge factors exactly N_code = P_Delta^T N_curve^T P_Gamma -- a duality, not an isomorphism (inner route 0/720); Arf switch set T6 = the six within-block duads; N/3 doubly stochastic, typed classical not CPTP.

Dependencies as recorded

CURVE.CODE.OUTERSPIN.01; CURVE.CODE.2TORSION.01; v752; v774; v796; v221; v240; v425