Ramified Prime-Step Descent for Spectral Pairs in Finite Cyclic Groups

Let N = q^e R, where q ≥ 3 is prime, e ≥ 2, and gcd(q,R) = 1. We prove automatic prime-step descent through the ramified q-primary factor. If (A, Λ) is an m-point spectral pair in Z_N and q does not divide m, then reduction modulo N/q is injective on both sets and the two reductions form an m-point spectral pair in Z_{N/q}. The proof is local along the q-adic residue filtration. Complementary source and spectrum projections commute. Inside each complementary parent cell, the next pair of digit decompositions splits into true intersections and copies of the regular representation of Z_q. A surviving regular copy gives the rational eigenvalue m/q to an unnormalized ambient Gram matrix with cyclotomic-integer entries. Algebraic integrality therefore forces q to divide m. When q does not divide m, every regular part vanishes, giving one-step-over commutation and residue-class Fourier zeros. A Galois automorphism on the coprime R-component converts these zeros into the orthogonality relations required after projection. Together with the previously established coprime endpoint, this removes every omitted odd prime power one copy at a time from a named spectral non-tile. In particular, the odd radical of the modulus of a minimal named counterexample must divide its cardinality.