A registered result in its source context
Original ledger classification: Numerical (NCG; honest status) [E]/[C]/[O]
Recorded statement: 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…
Statement, assumptions and sources
Paper context: Research contracts
The exact identifier, original classification and full claim are available below.
The source wording and classification below are preserved as recorded. A paper link, shared script or authored topic relation does not upgrade the claim or report a new proof run. Read later reviewed findings separately.
Read the exact registered claim and its dependencies
PS.NCG.ORIENT.01
Original ledger classification: Numerical (NCG; honest status) [E]/[C]/[O]
Original claim text
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.
Original classification detail
[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 as recorded
PS.DIRAC.02
Open the original ledger source, row 342Location recorded in the ledger
These document locations are recorded directly on the ledger row. They are document level navigation, rather than a newly inferred section citation.
- Research contracts
Sections that explicitly cite this claim
A literal claim identifier is present in these paper sections. The section and its premises determine what the citation supports.
- 2026-06-17 (Remaining NCG obligations discharged or honestly typed: inner fluctuations, spectral action, orientability/Poincar\'e, v254– v256)Research changelog; TFPT — Introduction and reading guide
- Front matter / supportTFPT — Introduction and reading guide; Verification reference catalog
Navigation through a shared verification script
These sections cite a script associated with the claim. This is a source navigation join, rather than an explicit citation of the claim or a theorem equivalence.
- Research Contract 1 — [formula in source]tfpt research contracts
- 2026-06-17 (Orientability + Poincar\'e duality: the honest, literature-confirmed status — strict fail, weakened hold, v257)Research changelog; TFPT — Introduction and reading guide
- Research Contract 1 — [formula in source]tfpt research contracts
- Front matter / supportResearch contracts
Registered verification modules
- v256_orientability_poincare.pyRegistered verification module · registry
Read the description recorded in the module registry
PS.NCG.ORIENT.01 — orientability and Poincaré duality treated honestly (no faked proof). KO-dim 6 forces the intersection form cap(p,q)=Tr(γ π(p) J π(q) J⁻¹) to be antisymmetric. [E] skew form forced by Jγ=−γJ + order-zero + reality (verified); [E] on the SM K-theory Z³ the skew form has rank 2, corank 1, null ~ (1,1,−1) — the structural obstruction; [E] orientability local condition [γ_F,π(a)]=0, full orientability is a degree-4 cycle on M×F; [C] the canonical hypercharge-faithful SM/PS triple satisfies both (van Suijlekom; CCM) — literature; [O] a self-contained machine proof needs the complete canonical bimodule.
A registered module is a source reference. Its presence does not assert that the module was executed in this snapshot.