This is a preprint of the manuscript: "Prime Polygons, Midpoint Averaging, and the Exceptional Pentagon" Abstract:The midpoint averaging operator on a regular n-gon replaces each vertex by the midpoint of its two neighbors. Its eigenvalues are cos(2πk/n) for k = 0, 1, …, n−1. We establish a spectral dichotomy between prime and composite polygons. By Niven’s theorem, the only rational values taken by cosine at rational multiples of π are 0, ±1/2, and ±1; consequently, for prime p ≥ 5, every nontrivial eigenvalue is irrational and no Fourier mode is annihilated in finite time. For composite n, rational eigenvalues or zero modes can appear whenever n is divisible by 2, 3, 4, or 6. We prove that the regular pentagon is the smallest polygon with at least two nontrivial modes and a completely irrational, nondegenerate spectrum. Its two nontrivial eigenvalues have magnitudes in the ratio φ², where φ is the golden ratio—a consequence, not a hypothesis. Finally, we observe that successive prime polygons introduce algebraic numbers of strictly increasing degree, and that the circle—their common limit—requires the transcendental constant π, connecting polygon geometry to the classical Euler product. The manuscript has been submitted to a peer-reviewed journal and is currently under review. Keywords: regular polygon, midpoint operator, circulant matrix, Niven’s theorem, cyclotomic field, golden ratio, prime number, transcendence
Zhendong Wang (Sun,) studied this question.