Cyclic analytic functions, reciprocal families, and spectral zero data

Abstract

Fix a positive integer \(N\). Root of unity projection of the exponential produces residue-class sections that, for \(\lambda\ne0\), form a basis for \(y^{(N)}=\lambda^N y\). The signed sections for \(y^{(N)}=-y\) yield the denominator \({\mathsf D}_N\) and its reduced form \(\Delta_N\). We derive reciprocal coefficient polynomials, Bell and Newton recurrences, an additive parameter law, and residue section Appell and theta families. For \(N\geq2\), \(\Delta_N\) has order \(1/N\). We specialize classical two term zero theory for exponential sums to the regular exponent polygon. In this equal-weight family, conjugation symmetry combined with unique near-model localization places all sufficiently large zeros exactly on the symmetry rays, and direct expansion gives their leading displacement and derivative sign. The classical perimeter law yields zero-counting asymptotics with explicit leading constants and bounded error. These results give genus zero products, the standard diagonal Fredholm representation, zero-power-sum formulas for reciprocal and logarithmic coefficients, nearest-zero coefficient asymptotics, and meromorphic continuation of the symmetry-orbit zero zeta by comparison with the Hurwitz zeta function. Complex powers are interpreted either formally or through a fixed logarithm on a zero free domain. The paper treats only this cyclic analytic family. Arithmetic interpolation, general nonlinear trees, general resolvent theory, and abstract completion are reserved for Papers IX, III, VI, and X.