A registered result in its source context

SEAM.S3.RP.01

Original ledger classification: reflection positivity (OS axiom) of the explicit collar verified by an EXACT positive-mixture Gram identity (symbolic PSD: M = V^T diag(1/K) V, sum of squares; eigenvalue check retained as diagnostic, T133 hardening); C7 then discharged via v369 [E]/[C]

Original claim text

S3 closure stack / C7 (2026-06-23): reflection positivity (the Osterwalder-Schrader axiom) of the EXPLICIT gapped collar measure, verified numerically -- the remaining constructive-QFT ingredient for the OS reconstruction of the boundary net, after which C7/QG.AMB.01 is discharged by the redundancy theorem v369 (its first premise, SEAM.EQUIV.01, being the S3 stack), not built from scratch. [E] Kaellen-Lehmann/OS positivity: the gapped p+ip band gives G(tau)=mean_k e^{-eps_k tau} with eps_k>=gap/2~1>0 (v367) -- a positive spectral measure. [E] REFLECTION POSITIVITY BY AN EXACT POSITIVE-MIXTURE GRAM IDENTITY (2026-07-28 hardening from T133/CERT.FLOOR): the Hankel/reflection Gram matrix M_ij=G(tau_i+tau_j) is built explicitly as M = V^T diag(w) V with V_ki = e^{-eps_k tau_i} and w_k = 1/K identical and STRICTLY POSITIVE over the K=1600 band modes, so x^T M x = sum_k w_k (v_k.x)^2 is a SUM OF SQUARES and M >= 0 holds EXACTLY as an algebraic identity (symbolic PSD, no tolerance); the mixture identity M - V^T D V is zero at the rounding level (1.1e-15 <= 64 u |M| = 7.6e-15) and the sum-of-squares route is nonnegative on 510/510 test directions. THE EIGENVALUE CHECK IS RETAINED AS A DIAGNOSTIC MEASUREMENT, not the certificate: T133 certified that the ASSEMBLED double-precision matrix has a negative direction at -7.75e-18 (60 digits) -- inside the module's old min_eig > -1e-10 tolerance, and the assembled float quadratic form is indeed negative on 2 of the 510 directions -- while the MATHEMATICAL matrix is fine; the Gram identity is the exact rescue. OS reflection positivity holds for the gapped collar. [E] neg control: injecting a growing (ghost) mode e^{+s tau} with negative weight makes the Hankel matrix indefinite (a negative eigenvalue) -- the test is non-vacuous. [C] RP + the gap (v76/v337) + GNS (v240) + c=8 (v376) + (E8)_1 (v377/v378) give the OS reconstruction inputs; C7/QG.AMB.01 is then DISCHARGED by v369. [O] the full non-perturbative ambient measure as a constructive object is the cited constructive-QFT step.

Original classification detail

[E] the positive spectral rep + the EXACT positive-mixture Gram identity for RP (symbolic PSD; the min-eigenvalue check retained as a DIAGNOSTIC measurement of the assembled float matrix, T133) + the non-vacuous neg control; [C] the OS reconstruction + the C7 discharge via v369; [O] the full constructive measure. Verifies the RP axiom for the explicit collar; does NOT by itself construct C7. Marker unchanged [E] (the Gram rescue carries; no upgrade).

Dependencies as recorded

QG.AMB.01; v367; v369; v240; v76; v376; v377; v378; T133