Spectral Sector Decomposition in the Rubik's Cube Representation

This paper studies averaging operators in finite-dimensional unitary representations, using the 228-dimensional Rubik's Cube cubie representation as a computational testbed. It separates block-spectral reduction from the arithmetic explanation of the resulting eigenvalues. For the standard inverse-closed family of 18 face-turn generators, the registered block spectra form six numerical eigenvalue layers matched to 1, 8/9, 7/9, 2/3, 5/9, and 1/3, with multiplicities summing to 228. The corner-permutation and edge-permutation blocks admit analytic combinatorial reductions. The corner-orientation block admits a symmetry-guided computational reduction, while its observed accidental degeneracy remains a computational observation. The edge-orientation assignment remains a numerical-representation observation. The exact results include inverse-closure Hermiticity, an eigenspace compression-trace identity, block compatibility and spectral union, and conditional rationality criteria. Partition integrality is established as a sufficient certificate format, but the registered canonical face partition does not satisfy its integer hypothesis and is not used to prove canonical Rubik rationality. A broken-face generator family supplies a finite negative control with values numerically matched to Q(sqrt(5)). Source repository: https://github.com/dooven-prime/rime-lite Paper I of the Representation-Induced Mechanics and Evolution (RIME) program. Independent researcher; no institutional affiliation.