Common tiling complements for cyclic spectra of prime power or semiprime cardinality

Let B be a fixed complete spectrum in a finite cyclic group of arbitrary order. We prove that, if |B| is a prime power or a product of two primes, all ordinary sets having spectrum B admit one ordinary tiling complement. The proof uses the author's earlier descent theorem for odd primes that do not divide the cardinality. To handle an unrestricted power of two in the modulus, we reduce the complete Fourier matrix in the whole cyclotomic integer ring modulo two. An odd-weight equality argument lifts prime power spectra back to characteristic zero. For two distinct odd primes, a low-weight rigidity argument forces uniformity at the terminal prime layer. The remaining induction keeps an entire family of sources and shifted slice carriers together; the resulting translation invariance accounts for the carries when a complement is lifted.