A finite reduction for spectral sets of fixed cardinality in cyclic groups

For every m-point spectral set in a finite cyclic group GN, we construct an m-point spectral set in a group GM, where M ∣ N and M ∣ ∏p≤m pm−1. The construction is determined by the source set in fixed Chinese remainder coordinates. It maps orthogonal character sets injectively to the smaller group, admits lifts in the other direction, and preserves both tiling and the existence of one tiling complement for all spectra. The main step deletes a nonbranching digit in a primary coordinate. A fiber identity for Ramanujan energies treats the character order at which the usual change-of-order formula does not apply.