V177_1 — A Conditional Hard‑Edge PF3 Program for Jensen Coefficients of the Riemann E‑Function

Description This manuscript formulates a conditional program for proving the PF3 hard‑edge layer of Jensen‑type coefficients associated with the Riemann E‑function. The purpose is not to prove the Riemann Hypothesis, but to isolate a first nontrivial positivity layer in the Pólya–Schur–Aissen–Schoenberg–Whitney total positivity framework. The focus is on the positivity of all 3×3 solid Toeplitz minors, reduced to effective estimates for de Bruijn moment ratios.

🔹 Main Contributions

• Normalized determinant identity for PF3 minors: N3,q=2v3−v4−(1−vq)2Eq.

• Factorization: xq=Pd,qTq, separating explicit rational geometry from analytic difficulty.

• Positivity criterion using finite differences and scale bounds.

• Tau‑Weak moment‑ratio theorem as the central analytic input.

• Saddle‑point Laplace route proposed for proving Tau‑Weak.

• Certificate architecture covering finite rectangular, fixed‑small‑q, fixed‑small‑h, and transition‑band regimes.

🔹 Structural Components

• Rational factor Pd,q supplies explicit hard‑edge geometry.

• Analytic difficulty concentrated entirely in moment ratios Tq=mq−1mq+1mq2.

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

• Certificate architecture ensures finite regimes are covered rigorously.

🔹 Finite Ladder Evidence

• Algebraic identities verified via Desnanot–Jacobi relations.

• Explicit rational estimates for Pd,q in bulk and edge regimes.

• Saddle‑point analysis outlines derivation of Tau‑Weak bounds.

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

🔹 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.

🔹 Conditional Main Theorem Assuming Tau‑Weak and the certificate architecture, all 3×3 solid Toeplitz minors are positive:

Thus the hard‑edge Jensen coefficient sequence satisfies PF3 positivity conditionally.

🔹 Conclusion V177_1 reduces PF3 positivity to effective control of moment ratios. The algebraic structure is explicit and clean; the analytic bottleneck is entirely in proving Tau‑Weak. With certificate architecture, finite regimes are covered, leaving the moment‑ratio theorem as the central analytic challenge.

👉 Key message: V177_1 demonstrates that PF3 positivity for Jensen coefficients can be conditionally reduced to Tau‑Weak moment‑ratio estimates plus finite certificate verification.

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