A Self‑Contained Finite‑Regime Certificate for a Local 3×3 PF3‑Tail Minor

Abstract This paper assembles and audits a certificate package for positivity of a local three‑by‑three PF3 tail minor arising from normalized moment ratios associated with the de Bruijn–Pólya kernel. The target local statement is that the normalized minor remains positive for all indices q≥2, with finite parameter values d≥q+2, and with the limiting hard‑edge endpoint treated separately.

The proof is split into three regimes:

• Regime I (2 ≤ q < 200): direct moment and Bernstein certification.

• Regime II (200 ≤ q < 1000): finite Bernstein certification.

• Regime III (q ≥ 1000): analytic certified‑envelope argument using ECSR (exponential saddle/cumulant remainder) bounds.

The finite regimes are now self‑contained: the moments m0,…,m1001 are enclosed by rigorous Arb/FLINT interval quadrature using acb.integral, explicit n‑tail majorants, and an explicit [8,∞) tail bound. Arb interval propagation from these moment enclosures yields positive Bernstein coefficients with zero failures. The large‑q curvature envelope is no longer a separate hypothesis; it is recorded as a lemma, conditional only on the same ECSR budget used elsewhere in the large‑q proof.

Numerical constants are reported conservatively:

• For 2 ≤ q < 200: Nq(t)≥2.13×10−7.

• For 200 ≤ q < 1000: Nq(t)≥1.68/q3.

• For q ≥ 1000: Nq(t)≥0.45/q3.

This is only a local 3×3 minor certificate. It is not a proof of the Riemann Hypothesis, global PF3, higher‑order minors, M5/uniformity, infinite‑tail closure, or the full de Bruijn program.

Keywords PF3 tail minor; de Bruijn–Pólya kernel; Arb/FLINT quadrature; Bernstein positivity; exponential saddle remainder; curvature envelope; interval arithmetic; reproducibility

Highlights

• Rigorous Arb/FLINT quadrature replaces diagnostic moment generation.

• Finite regimes (2 ≤ q < 1000) are now unconditional and self‑contained.

• Large‑q regime remains conditional on the ECSR budget.

• Curvature envelope verified as a lemma, not a hypothesis.

• All Bernstein coefficients positive with zero failures.

• SHA256 checksums and reproducibility logs included.

Conclusion This manuscript records a completed finite‑regime certificate with explicit boundaries. The constants are rounded conservatively, and the finite regimes are now fully self‑contained. The only remaining external input is the ECSR budget for the large‑q branch. The certificate is therefore a useful completed unit for the local 3×3 PF3 program, but it should not be advertised as a global total‑positivity result.