SEPTEMBER 22, 2026 · VERSION 1 · NS-58

Local integration audit

Local self-review only. No independent review is claimed. The analytic conclusions use the archived v1.35–v1.36 radical construction and complete fixed-window realization; those inputs are not reproved or independently certified here.

Sharp restrictions. The compact piecewise smooth zero extension has Fourier decay O(1/|ξ|). Its logarithmic multiplier is in L², giving membership in the operator domain. No zero trace or H¹ regularity of a candidate nullvector is inferred.

Exterior convergence. The continuous tail grows at most logarithmically near each endpoint; this is square integrable. The entire prime series is uniformly absolutely convergent on the fixed window using a summable Λ(n)/n^5/2 majorant. Both pole moments are absolutely convergent. Cauchy–Schwarz justifies the full pairing with every form-domain vector.

Conjugation and converse. D_f(ψ) = q(f, conjugate(ψ)|_I) is linear in ψ, supported on the closed window, and bounded by its C¹ seminorm. Reality and evenness of φ identify its convolution with the tested pairing. The Fourier transform of a compactly supported distribution is smooth on the real axis. A nonzero entire φ transform has isolated real zeros, so the product identity forces D_f = 0. Only then is the known local form core used.

Parity. Reflection commutes with the complete operator. Reflected translates test the same equation in each sector. The even exterior moment vanishes for odd tests; the remaining odd pole is negative.

Forced controls. The split φ_t = E_au_t + r_t gives A^δu_t = −R^δ_a,t + P_aB_δφ_t. Dropping the last term would be incorrect. A nonzero finite cosine multiplier has a zero set of measure zero, hence its whole-line L² operator is injective.

Compactness scope. The squared Hilbert–Schmidt norm of the forcing map is 2a‖B_δφ‖². This excludes a positive lower bound on the entire infinite-dimensional fixed-window parity space. It does not exclude one on a particular finite-dimensional nullspace. The unitary Fourier convention contributes the displayed factor 2π in the profile norm formula.

Outcome. No outstanding mathematical issue was identified in this local integration audit. The result is an exact equivalence and a control defect identity, with no independent arithmetic sign estimate. The proof does not assume or establish an actual first zero. NS-58 stops as specified; API, the uniform floor, G2 and RH remain open.

Inherited PDF link repair. The older manually tagged RBC equation used a numbered equation environment, producing four duplicate PDF destination warnings. Switching that single display to the starred environment preserves the printed tag and formula and removes all four warnings. The final complete build has 294 pages, zero undefined or multiply defined references, and no PDF destination warnings.

Build, exact-check, source-hash and visual-inspection results are recorded in validation.json. Finite algebra does not certify the analytic proofs, and no earlier numerical gate is replayed.