Original ledger status
PS.NCG.ORIENT.01
PS.NCG.ORIENT.01
ORIENTABILITY + POINCARE DUALITY -- HONEST STRUCTURAL TREATMENT (2026-06-17): the two NCG obligations of v250 that do NOT close to a clean [E] on the simplified v252 representation. NO faked proof. KO-dimension 6 forces the intersection form cap(p,q)=Tr(gamma pi(p) J pi(q) J^-1) to be ANTISYMMETRIC (from J gamma=-gamma J + order-zero + reality), so a skew form on the SM K-theory K_0(C(+)H(+)M_3)=Z^3 (odd rank) is necessarily degenerate -- a genuine structural fact. (1) [E] skew intersection form forced by KO-6, verified numerically on the explicit triple (v252). (2) [E] rank/corank: rank 2, corank 1, null direction ~ (1,1,-1) in (C,H,M_3). (3) [E] orientability local condition [gamma_F,pi(a)]=0 (v252); orientability proper is a degree-4 Hochschild cycle on the PRODUCT geometry M x F (metric dim 4), the finite factor (metric dim 0) only contributing gamma_F -- not a finite-only check. (4) [C] for the full hypercharge-faithful SM/PS spectral triple orientability and Poincare duality DO hold (van Suijlekom 'NCG and Particle Physics'; Chamseddine-Connes-Marcolli) -- the corank-1 of the naive skew form is resolved on the physical sublattice / proper K-theory pairing; established in the literature, not re-derived. (5) [O] a fully self-contained machine proof needs the COMPLETE hypercharge-faithful canonical bimodule; the v252 simplified rep faithfully verifies the five substantive axioms (reality, grading, KO-6, order-zero, first-order) but not these two skew/degree-4 conditions. Honestly open.
Recorded status: Numerical (NCG; honest status) [E]/[C]/[O]
Dependencies, status detail and source
Canonical status: [E] KO-6 skew intersection form + rank2/corank1/null(1,1,-1) + orientability local condition; [C] canonical triple satisfies both (literature); [O] self-contained machine proof needs full canonical bimodule -- no faked closure
Dependencies: PS.DIRAC.02
Scripts: v256_orientability_poincare.py
Location: tfpt_research_contracts
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