A registered result in its source context

QGEO.TWISTOPE.01

Original ledger classification: Exact [E] weights/crossing + [E-float] two-/four-point and OPE + [O] typed open items + [X-slice] honest RP negative (Python-only per GATE.WOLFRAM.02).

Original claim text

The twist-field/OPE round (one module, 25 checks, ~1 s; discovery probe orbifold_twist_ope_probe.py 25/25, verdict TWIST-OPE-COHERENT-RP-OPEN). The interacting/CFT-level slice of the Z3 orbifold residue (v623/v628): h_sigma = 1/36 from THREE independent exact routes (v628 Casimir coefficient = 2(h - c/24); zeta-function mode sum -zeta(-1,nu)/2; bosonization pair h(1/2-eta)+h(1/2+eta) = eta^2/2), with the convention traps resolved exactly (eta(1-eta)/2 = the Ramond-deficit identity); lattice dictionary Delta = eta^2 = 1/9 = 4 h_sigma; the lattice twist two-point (U(1) string determinants on the half-filled NS seam double; equal-time covariance = half the v519 seam kernel, dev 8e-16) hits 2 Delta = 2/9 at 0.06% (N -> 1536) and 0.11% (within-ring); Ising control (eta = 1/2) at 0.02%; the four-point: exact lattice crossing (gap rotation = w -> 1-w, |G4| invariant at 1.4e-14: symmetry-protected), conformal collapse onto w (< 1%), the exact closed form A^2 [w(1-w)]^{-2 Delta} at < 0.1% after N^{-2/3} extrapolation (deviation identified as the Fisher-Hartwig artifact, ratio 0.637 vs 2^{-2/3}); OPE data: cross-channel block (1-w)^{-2/9} at 0.19%, c1 = 2 Delta = 2/9 model-bound at 0.19% (the exact vertex value 2 eta^2), identity normalization C -> 1 at 0.07%; HONEST OPEN [O]: the model-free c1 extractors bracket 2/9 only within +-15%; the Majorana-level block split not fixed by G4 alone. REFLECTION POSITIVITY DIFFERENTIATED: all mirror Grams Hermitian at machine precision; POSITIVE CONTROL: the real Z2 string class exactly PSD at every N; HONEST NEGATIVE [X-slice]: the naive antiunitary Z3 pairing fails positivity by an N-STABLE amount (-4.5e-3 charged; indefinite already in the continuum) -- the positivity-carrying pairing (parafermionic Klein twist, Jaffe-Pedrocchi-type) is a named OPEN item. [2026-08-02 XLIII: the named open item is RESOLVED at the lattice OS level -- v645: the positivity-carrying pairing IS the parafermionic Klein twist (mirror composed with charge conjugation, omega^{k q} zero-mode phases, eta = (1,1) unique among the 36 scanned pairs), all charged blocks real symmetric and PSD, N-stable 48..384, both axes, full mixed identity+twist+dipole OS Gram PSD; the naive pairing keeps its -4.5e-3 violation as the must-fail control; continuum OS reconstruction/modularity stay typed open; GATE.QGEO does not move; see QGEO.KLEINRP.01.] [2026-08-02 XLV: the orbifold front is COMPLETE at the lattice level -- modular data (v650: nine Z[g,h] vs |theta/eta|^2 < 1%, S measured at N^{-4}, T exact as the Z6 Dehn shift, deck^3 = -1 as exact Z6 trace arithmetic), assembly (v651: H^2(Z3,U(1)) = 0 enumerated, exactly one seam assembly B, T kills the naive Z3 exactly), Arf/GSO hardening (v652: the 6^6 classification leaves {B, C3}, the one-Fock-space pin forces B, the mu6 ladder measures the C3-excised states directly), and the bond-defect premise as a lattice theorem (v653); the [C] remainder is the continuum CFT statement + the seam-identification premise; no marker moves.]

Original classification detail

TWIST-OPE-COHERENT-RP-OPEN: the interacting slice stands at the abelian vertex level (weights 1/36, exponent 2/9, crossing, closed four-point, OPE data); RP validated in the real Z2 class, the complex Z3 pairing typed open (parafermionic Klein twist); GATE.QGEO does not move.

Dependencies as recorded

QGEO.ORBCAS.01; QGEO.COVERSEAM.01; QGEO.SEAMID.01; v519; v622; v623; v628