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/pⁿ as n grows — for a prime p coprime to the base. The discrete Fourier frequencies of the digit sequence of 1/pⁿ are shown to organize into a p-ary spectral tree: the kₙ = p^n−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 p^n−1A (1/p) and successive refinement areas Rₘ (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/pᵐZ. The level-2 refinement area is evaluated in closed form via Dedekind sum reciprocity, with the Fermat quotient qₚ (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 − (b²−1) /24 with exponentially small error O (n² (log p) ² / p^n/2). The paper concludes by constructing the projective limit polygon as a pro-finite object in Zₚ^* whose spectral measure converges to Haar measure on the completed group.