The p-Adic Digit Polygon Tower: Spectral Trees, Fermat Quotients, and the Hensel Equidistribution Machine

This paper develops the complete spectral theory of the p-adic digit polygon tower — the sequence of digit polygons for 1/p^n as n grows — for a prime p coprime to the base. The discrete Fourier frequencies of the digit sequence of 1/p^n are shown to organize into a p-ary spectral tree: the k_n = pⁿ⁻¹k frequencies decompose into n levels, with each new level activating spectral degrees of freedom invisible at all prior levels. The signed area decomposes additively along this tree into a base contribution pⁿ⁻¹A(1/p) and successive refinement areas R_m(p) contributed by frequencies first appearing at each tower level. The Hensel spectral identity establishes that the residue-orbit DFT at a level-m frequency is controlled by the p-adic digits encoding the successive Hensel lifts of the orbit in Z/p^mZ. The level-2 refinement area is evaluated in closed form via Dedekind sum reciprocity, with the Fermat quotient q_p(b) as the controlling parameter. Wieferich primes are characterized as spectral anomalies — dead branches in the spectral tree — and a generalized Wieferich condition is identified at each tower level. The exponential equidistribution theorem proves that the normalized area converges to the universal attractor −(b^2−1)/24 with exponentially small error O(n^2 (log p)^2 / pn/2). The paper concludes by constructing the projective limit polygon as a pro-finite object in Z_p^* whose spectral measure converges to Haar measure on the completed group.