V178_9 — A Conditional and Computational Program for Hard‑Edge PF3 Positivity of de Bruijn Moment Jensen Coefficients

Description This manuscript develops a finite‑order positivity program for hard‑edge Jensen‑Toeplitz minors associated with the de Bruijn moment sequence. The focus is on the 3×3 solid Toeplitz layer (PF3). The work does not claim a proof of the Riemann Hypothesis; instead, it isolates a tractable PF3 mechanism, conditional on effective moment‑ratio estimates and finite certificate verification.

🔹 Main Contributions

• Normalized determinant identity reducing PF3 to a clean inequality.

• Factorization of coefficients 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 outlining analytic tasks (T0–T5) needed for a full proof.

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

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

• Differentiable Laplace framework introduced to isolate the n = 1 contribution and reduce the analytic bottleneck to a domination lemma.

• 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 Sturm diagnostics, fixed‑small‑h profiles, and transition‑band verification.

• Computational diagnostics introduced to validate finite regimes.

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

🔹 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 (effective saddle estimates, tail separation, Laplace framework).

• Certificate diagnostics ensure finite regimes are rigorously verified.

🔹 Finite Ladder Evidence

• Algebraic identities verified via determinant relations.

• Explicit rational estimates for coefficient factors in bulk and edge regimes.

• Saddle‑point analysis decomposed into proof obligations, with Tasks T0–T3 completed (lower bound, saddle location, curvature scale, tail separation).

• Differentiable Laplace framework (Task T4) set up, reducing the problem to a domination lemma.

• 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.

🔹 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 (differentiable Laplace expansion and derivation of Tau‑Weak).

🔹 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 V178_9 consolidates the PF3 program into a conditional and computational framework. It emphasizes finite‑order tractability, separates the analytic bottleneck into Tau‑Weak estimates, and introduces diagnostic checks to prevent overinterpretation. The manuscript makes clear that PF3 positivity is achievable under conditional inputs, while PF∞ remains unresolved.

👉 Key message: V178_9 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.

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