V179_1 — 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 tests beyond PF3.

🔹 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_1: Finite certificate architecture for unconditional PF3 and PF4 diagnostic tests, marking the first exploration beyond PF3.

• Forward‑looking: Routes toward PF∞, analyzing fixed‑r ladders, uniformity challenges, and recommended research order.

🔹 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_1 extends the program by specifying certificate packages for unconditional PF3 and introducing PF4 diagnostics as the first test beyond PF3.

🔹 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_1 adds certificate architecture for unconditional PF3 and PF4 diagnostics, 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 minimal publishable certificate package for unconditional PF3.

• Development of PF4 diagnostics into a rigorous framework.

🔹 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_1 consolidates the PF3 program into a conditional and computational framework, extends it with certificate architecture for unconditional PF3, and introduces PF4 diagnostics as the first test beyond PF3. It emphasizes finite‑order tractability, separates the analytic bottleneck into Tau‑Weak estimates, and introduces diagnostic checks to prevent overinterpretation.

👉 Key message: V179_1 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 and sketching routes toward PF∞, with PF4 diagnostics marking the next frontier.

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