Finite-Characteristic Dynamics of Riemann Xi Center Polynomials and Splitting in Q(i)

Abstract

This preprint studies the finite-characteristic dynamics of the center-polynomial family associated with the even derivatives of the standard Jacobi-theta kernel for the Riemann Xi function. Using Romik’s centered expansion of the Jacobi theta constant, the paper places the center polynomials in an integral Hurwitz basis and derives an explicit tridiagonal transport operator governing their coefficients. It proves an exact recovery theorem showing that the arithmetic information carried by Romik’s integer sequence can be reconstructed from two consecutive center polynomials. The transport operator is then analyzed over finite fields. A universal spectral law is obtained for every odd prime, together with explicit periodicity relations for finite transport coordinates. For primes that are inert in the Gaussian number field, known prime-power divisibility results for Romik’s sequence are combined with a weighted escape argument to prove eventual signed periodicity of the entire center-polynomial family modulo every prime power. For split primes, recurring nonvanishing of the arithmetic coefficients forces the polynomial degrees to become arbitrarily large, ruling out eventual periodicity. As a consequence, the paper gives a finite-characteristic characterization of Gaussian splitting: an odd prime is inert precisely when the full sequence of Riemann Xi center polynomials is eventually periodic modulo each of its prime powers. The results concern the dynamics of the derived Xi center-polynomial family and are distinct from both the previously established local arithmetic of these polynomials and the known congruence theory of the underlying Romik coefficients.