Cyclic-Shift Energies: Minimal State and Cyclotomic Gaps

A cyclotomic divisibility criterion decides equality in an integer-occupancy spectral bound. At prime periods, algebraic norms yield a quantitative positive gap for nonflat spectra. The bound is attained at exactly five occupancies in a complete 210-state example. Minimal linear state and retained-potential identities supply the representation framework. The principal result is the quantitative cyclotomic spectral gap, its equality obstruction and the attained five-cycle certificate. The Fourier diagonalization, positive-semidefinite rank argument, cyclotomic irreducibility and field-norm integrality are established ingredients and receive attribution. The record includes the mathematical manuscript with full proofs, exact verifier and certificates, a native Blender scene and its source-derived 3840 × 2705 figure.