SEPTEMBER 22, 2026 · VERSION 1 · NS-57

Local integration audit

Disposition: no outstanding major or minor mathematical issue identified in this local audit. This is a self-review by the integrating agent, not an independent review. It does not certify the inherited v1.36/v1.53 inputs anew.

Mathematical checks

Bounded-perturbation detection. The translate combination is fixed before the window grows. Ordinary norm convergence and the complete operator residual imply q[g_a] → 0. Only the added operator B is bounded. No global closability, global form density, or uniform approximation coefficients are used.

Parity and real tests. The source lemma supplies dense reflected translate spans in both sectors. Real even and imaginary odd Fourier bumps provide real physical negative directions for the real even symbol. Real parts of approximants preserve the required limits. There is no substituted source-complement assumption.

Finite cosine symbol. Its long-interval mean is zero and its mean square is 2 ∑ δ_n². Distinct positive frequencies are sufficient even for prime powers with dependent logarithms. A nonzero nonnegative bounded symbol would contradict those two mean identities. The single-coefficient lower edge is −2|δ|.

First-zero continuity. Compression at prime entry is not norm-continuous. The proof instead uses a common rescaled logarithmic form domain, locally uniformly bounded strongly continuous perturbations, and compact form embedding. The Rayleigh upper bound and subsequential lower semicontinuity establish continuity. Small-window positivity and later negative compact tests then give an actual operator nullvector.

Support geometry. At the full first zero, translation of a shorter essential support would contradict positivity on a smaller interval. At a parity-sector first zero, reflection already makes the support symmetric, so no translation out of that sector is needed. Neither assertion demands a nonzero boundary trace.

Sensitivity. For a fixed coefficient pattern, both parity witnesses can be chosen before ε tends to zero. Their translate centers are bounded independently of ε. The complete v1.53 residual yields C exp(−cλ²); the bounded perturbation eventually contributes at most −εκ/2. The result is an upper bound in λ = exp(a), with pattern-dependent constants. It is not a computed threshold or an asymptotic equality.

Compensation and floor. Fixed bounded changes transfer finite cofinal floors in both directions. Necessity of B + cI ≥ 0 uses the near-radical lemma; necessity of RH uses the original v1.36 floor theorem. Under RH, density supplies the upper bound matching inf σ(B); if RH fails, bounded perturbations cannot stop the archived divergence to −∞.

Scope checks

The controls change prime coefficients and therefore do not preserve the original arithmetic radical identity. Their negative directions are not negative directions of the original Weil form. Exact positivity failure under finite changes does not obstruct a method proving a robust finite lower floor.

The integrated theorem references replace standalone label names; “first-zero parameter” is explicitly λ_* = exp(a_*). The original versioned standalone report and proof remain historical provenance. The private draft email is excluded from publication.

Actual validation

Twelve finite exact Maxima checks pass in the integration worktree. The complete manuscript builds to 291 pages with zero undefined or duplicate references and zero overfull boxes. The new proof pages are visually checked; the validation file records the inspected page numbers and source hashes. These checks are separate from the analytic arguments.

Next attempt

NS-58 asks whether the exact polarized arithmetic radical identities supply an independently signed exterior or boundary estimate for a hypothetical first-zero nullvector. Retain the full weak domain, both parities, both poles, and every prime shift. Test any proposed uniqueness lemma against the altered-weight controls. If the identity merely rewrites the null equation, stop that attempt without recording a new sign result.

API, the original uniform floor, G2 and RH remain open.