This paper gives a self-contained treatment of the spectral structure of digit polygons along the p-adic tower, developing the fine frequency geometry that underlies the prime-power attractor theorem. For an odd prime p coprime to the base b, the digit polygon of 1/pⁿ is the cyclic polygon whose vertices are consecutive pairs of digits in the base-b expansion of 1/pⁿ, with period equal to the multiplicative order of b modulo pⁿ. Five exact results are proved. First, the frequency indices at level n admit a canonical stratification by p-adic valuation into levels, where the level-1 frequencies are inherited from the base period k and the higher-level frequencies are new at each step of the tower. The partition structure and exact cardinalities of each frequency stratum are determined. Second, the block-average projection onto the period-k subspace is identified exactly as the spectral projection onto the level-0 and level-1 frequencies, yielding an exact orthogonal decomposition of the area into an inherited block-average contribution and an orthogonal refinement contribution. Third, an exact Hensel-layer decomposition of the residue discrete Fourier transform is proved: a level-m frequency receives contributions from the p-adic digit layers of the residue orbit only at layer indices at or above a threshold determined by the Wieferich index and the frequency level, so the first contributing layer is rigidly determined by the arithmetic of the tower. Fourth, in the non-Wieferich case where the Wieferich index equals 1, the first new p-adic digit layer is computed explicitly in terms of the Fermat quotient, showing that on each residue class modulo k the first lifted digit is an affine progression modulo p. Fifth, a self-contained proof of the prime-power attractor theorem is given: the normalized area density along the p-adic tower converges to negative one-half the digit variance, equal to - (b²-1) /24, with an explicit error of order p^ (- (n-t) ). Two natural conjectures about the tower are proved false. The naive scaling law claiming that the area of 1/pⁿ equals p^ (n-1) times the area of 1/p plus a correction is refuted by a concrete numerical counterexample in base 10 at p=7. The spectral measures built from the uncentered digits cannot converge projectively to Haar measure on the inverse-limit frequency group because the zero-frequency atom survives with positive limiting mass. After centering the digit sequences, both obstructions disappear and the centered Haar convergence problem is identified as a well-posed open problem. All results are proved for arbitrary base b and odd prime p coprime to b.
Kevin Fathi (Sun,) studied this question.