V179_5 — A Conditional and Computational Program for Hard‑Edge PF3 Positivity of de Bruijn Moment Jensen Coefficients

Description This manuscript continues the finite‑order positivity program for hard‑edge Jensen‑Toeplitz minors associated with the de Bruijn moment sequence, focusing on the 3×3 solid Toeplitz layer (PF3). It does not claim a proof of the Riemann Hypothesis; instead, it develops a conditional PF3 framework, combining algebraic reduction, analytic roadmaps, certificate diagnostics, and new forward‑looking steps toward an unconditional PF3 theorem.

🔹 Main Contributions

• Normalized determinant identity reducing PF3 to a tractable inequality.

• Factorization into explicit rational geometry and analytic moment ratios.

• Tau‑Weak input proposed to control deviations of moment ratios and their discrete differences.

• Effective Tau‑Weak roadmap with tasks T0–T5 isolating analytic proof obligations.

• First effective saddle estimates proved, establishing location, curvature, and Gaussian scale bounds.

• Effective tail separation showing n ≥ 2 terms negligible compared to the main saddle contribution.

• Differentiable Laplace framework introduced to isolate the n = 1 contribution.

• Central‑region expansion and complement estimate for the n = 1 Laplace term.

• First differentiated Laplace correction and uniform differentiated estimates up to order four.

• Derivation of Tau‑Weak from differentiable Laplace bounds, including passage from Aq to Tq.

• Unified positivity framework covering bulk, left‑edge, and right‑edge regimes.

• Proposed saddle‑point route decomposed into conditional lemmas for amplitude control, Gaussian correction, and differentiated domination.

• Certificate architecture including finite rectangle checks, fixed‑small‑q diagnostics, fixed‑small‑h profiles, and transition‑band verification.

• Computational diagnostics validating finite regimes.

• High‑order Gamma diagnostics showing limitations of direct finite‑d all‑order scans.

• New in V179_5: A Machine Certificate Appendix for PF3, including certificate manifest, moment‑ball certificate, fixed‑small‑q half‑line certificate, finite rectangle certificate, fixed‑small‑h certificate, transition tiling certificate, verifier theorem, and current certificate status. This marks the first attempt to formalize machine‑verifiable certificate packages for PF3.

🔹 Structural Components

• Rational factor provides explicit hard‑edge geometry.

• Analytic difficulty concentrated entirely in moment ratios of de Bruijn moments.

• Bulk, left‑edge, and right‑edge positivity proven conditionally on Tau‑Weak.

• Saddle‑point route presented as a roadmap, with partial progress through differentiated Laplace estimates.

• Certificate diagnostics ensure finite regimes are rigorously verified.

• V179_5 extends the program by introducing machine‑certificate architecture, aiming to automate and formalize the verification process.

🔹 Finite Ladder Evidence

• Algebraic identities verified via determinant relations.

• Explicit rational estimates for coefficient factors.

• Saddle‑point analysis decomposed into proof obligations, with Tasks T0–T3 completed.

• Differentiable Laplace framework (Task T4) advanced with uniform estimates and derivation of Tau‑Weak.

• Diagnostic certificate plan ensures small‑q and small‑h families are checked.

• High‑order tests demonstrate finite‑order tractability but rule out naive all‑order extension.

• V179_5 adds machine‑certificate appendix, marking a new stage in the program.

🔹 Final Bottleneck Remaining analytic obligations include:

• Proof of Tau‑Weak moment‑ratio bounds.

• Rigorous computation of de Bruijn moments for finite certificates.

• Verification of transition‑band cases with interval arithmetic.

• Completion of analytic tasks T4–T5.

• Assembly of a minimal publishable certificate package for unconditional PF3.

• Implementation of machine‑certificate verification for PF3.

🔹 Conditional Main Theorem Assuming Tau‑Weak and the certificate architecture, all 3×3 solid Toeplitz minors are positive for valid ranges. Thus the hard‑edge Jensen coefficient sequence satisfies PF3 positivity conditionally.

🔹 Conclusion V179_5 consolidates the PF3 program into a conditional and computational framework, extends it with certificate architecture for unconditional PF3, and introduces a machine‑certificate appendix as the first step toward automated verification. It emphasizes finite‑order tractability, separates the analytic bottleneck into Tau‑Weak estimates, and introduces diagnostic checks to prevent overinterpretation.

👉 Key message: V179_5 demonstrates that PF3 positivity for Jensen coefficients can be conditionally reduced to Tau‑Weak moment‑ratio estimates plus finite certificate and computational diagnostics, while clarifying the limits of naive all‑order approaches. The new machine‑certificate appendix marks the next frontier: moving from conditional to unconditional PF3 with automated verification.

📩 Verification Note: If npz files or numerical audit data are required for independent verification, please request them by email at 24ping@naver.com.