Strict Positivity of the Fourth Toda–Hankel Determinant for the Riemann Xi Kernel

This preprint establishes the strict positivity of the fourth derivative Hankel determinant associated with the standard Jacobi-theta kernel in the Fourier representation of the Riemann xi-function. The result holds on the entire nonnegative half-line and advances the continuous Toda–Hankel hierarchy beyond the previously established lower-order sign properties. The proof is computer-assisted but exact. On the compact region, a symmetry-preserving finite theta core is combined with a high-order Taylor approximation, exact Bernstein-basis positivity certificates, and rigorous derivative-error bounds. On the unbounded region, the first theta summand provides an explicit positive determinant margin, while the remaining theta tail is controlled through exact first-order estimates and a uniform nonlinear perturbation bound. All decisive finite sign checks are reduced to integer, rational, or exact symbolic comparisons. The accompanying reproducibility package contains the certificate data, deterministic reconstruction programs, and an independently implemented verifier. It permits both direct verification of the supplied certificates and reconstruction of the finite certificates from the defining formulas. As a consequence of the Toda identity and the previously known lower-order signs, the result also yields a new strict logarithmic concavity property for the preceding determinant in the hierarchy. The theorem is a kernel inequality within the classical Laguerre–Turán program surrounding the Riemann hypothesis. It does not prove the Riemann hypothesis. 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.