V167_3 — A Compact‑Support Certificate Framework for Weil Positivity: Finite Schur Certificates and a Paley‑Wiener Bottleneck

Description: This paper continues the compact‑support certificate program for Weil positivity in the logarithmic variable. It uses compactly supported translated bump functions, exact finite prime‑power sums, certified archimedean truncation bounds, and an endpoint/interior Schur decomposition. The framework ensures that finite numerical claims can be converted into auditable interval or rational certificates.

🔹 Certificate Architecture Finite claims are expressed as generalized eigenvalue or Schur complement bounds relative to the mass matrix, ensuring reproducibility and reviewer verification.

🔹 Structural Components

• Uniform Riesz stability of the bump basis

• Derivative‑frame bounds

• Compact archimedean tail estimate

• Recovery bridge from finite bump spaces to smooth test functions

• Conditional Schur certificate theorem

🔹 Normalization Audit A completed normalization audit is required to identify the working quadratic form exactly with a published Weil positivity criterion for the Riemann zeta function. Proposition 2.2 provides a first comparison, but the final identification remains an audit obligation.

🔹 Final Bottleneck The unresolved analytic problem is the Negative‑Well Non‑Concentration Principle: Paley‑Wiener functions must not concentrate Fourier mass on the negative wells of the truncated prime gamma field. This bottleneck is pinpointed as the precise location where the difficulty of the Riemann Hypothesis re‑enters the framework.

🔹 Pilot Computations and Direct‑Tail Previews Non‑certified runs for dimensions k = 64, 128, 256 suggest positivity after saturation, but these are reproducibility targets only. Appendix A introduces a direct‑tail certification protocol to sharpen archimedean tail analysis.

🔹 Reviewer‑Facing Obligations The framework specifies explicit certificate obligations: mass and Riesz certificates, endpoint leakage bounds, prime matrix enclosures, archimedean Toeplitz intervals, interior lower bounds, and Schur margins.

🔹 Conditional Main Theorem If all certificate obligations are met, model errors vanish, and the normalization audit identifies the quadratic form with a standard Weil positivity criterion, then finite positivity transfers to Weil positivity. Under this identification, the Riemann Hypothesis would follow.

🔹 Conclusion V167_3 does not prove RH. Instead, it refines the certificate architecture, adds direct‑tail certification protocols, and isolates the analytic bottleneck. A proof through this framework would require an asymptotic certificate ladder or an analytic theorem replacing it, together with a completed normalization audit.

👉 Key message: V167_3 advances the compact‑support certificate program by sharpening tail certification and reinforcing the audit requirements, while keeping the Paley‑Wiener bottleneck as the central unresolved analytic challenge on the path to the Riemann Hypothesis.