Continuous Dual Hahn Structure in the Riemann–Xi Center Transport: Sharp Supercongruences and Exact p-Adic Dynamics

Abstract

This preprint studies the arithmetic and finite-characteristic structure of the center transport associated with the standard Riemann–Jacobi kernel for the Riemann Xi function. The center transport is shown to admit an exact description in terms of a specific family of continuous dual Hahn polynomials. This identification makes it possible to analyze the characteristic polynomials of finite transport blocks over finite fields and over prime-power rings. The main result establishes a sharp uniform supercongruence for the characteristic polynomial at its nonzero roots modulo an odd prime. In particular, the relevant special values have exactly determined prime-adic valuation. This information is then used to control the lifting of the finite-field spectral data to higher prime powers. Combining the supercongruence with Hensel lifting and an exact analysis of the associated principal units, the paper determines the minimal signed period and the minimal preperiod of every single prime-sized transport block modulo arbitrary powers of an odd prime. The prime three is treated separately through an explicit local factorization. The results provide a direct connection between the continuous dual Hahn structure arising from the Riemann–Xi center expansion, sharp prime-power congruences, and exact finite prime-adic dynamics. Reproducibility code is included for independent verification of the supercongruence, the prime-power transport relation, and the minimality statements.