Heat-Driven Descent and Re-Entrant Pólya-Frequency Obstructions for the Riemann Autocorrelation Kernel

Abstract

This paper studies finite-order total positivity for a heat-deformed autocorrelation constructed from the classical Riemann–Jacobi kernel. Although every kernel in the family remains positive definite, the results show that positive definiteness does not prevent the appearance of finite Pólya-frequency obstructions. At zero heat, a rigorous local obstruction of order eight is established, with negative Toeplitz minors occurring at arbitrarily small spacing. Certified heat-dependent configurations then exhibit a descent of the first negative leading minor from order eight through order two. A fixed order-eight determinant also undergoes re-entrant behavior, changing sign in three disjoint heat intervals. The analytic part determines the exact log-concavity threshold of the underlying heat-tilted kernel and derives a sufficient range in which the autocorrelation has the order-two Pólya-frequency property. The paper also proves failure of that property at a finite heat parameter and for all sufficiently large heat parameters. All finite sign determinations are established through outward-rounded interval quadrature, rigorous bounds for the truncated theta series, and exact rational determinant perturbation estimates. The results provide a heat-phase picture that separates positive definiteness, finite-order total positivity, and log-concavity for the Riemann autocorrelation kernel. Research methodology and AI assistance:This work was developed using the CARMA-Math research workflow, a cumulative AI-assisted mathematical research methodology using persistent research archives, literature and prior-art investigation, iterative proof exploration, and verification procedures. Generative AI (ChatGPT) was used extensively for mathematical exploration, proof development, computational reasoning, literature research, and manuscript preparation.