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: D3,q(d)>0(2≤q≤d−2). 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.