Finite Fourier Duality, Radial Values, and Arithmetic of a Mod Eight False Theta Quotient

Abstract

We study a family of mod-eight false-theta moment quotients and determine their radial behaviour at every root of unity. The main structural result establishes an exact finite Fourier closure for the associated periodic boundary sequences. This closure yields a completed functional equation and proves the nonvanishing of the relevant periodic Dirichlet values at all negative odd integers. Consequently, each quotient exhibits either universal normalized factorial asymptotics or a finite nonzero radial value. The finite values belong to cyclotomic fields, satisfy Galois covariance, and specialize at the principal root to signed odd Springer numbers. We also determine exact coefficient divisibility, construct an optimal coefficientwise two-adic interpolation, and identify the unique dominant zero of the common denominator. This zero gives an unconditional coefficient decomposition. Finally, using exact boundary estimates and Möbius inversion, we prove positivity of every exponent in the formal Euler transform of the denominator.