Remark: Spectral matching and periodicity are explicit

This construction is exact for the stated commensurate spectrum. It does not encode an arbitrary prescribed Hamiltonian into the same finite additive constraint without an additional approximation or spectral-matching theorem. Its readings are periodic with period $2\pi /\omega$; an absolute winding number would require an additional classical interface.

The negative clock generator in (eq:clock-spectrum) is a mathematical constraint convention, not a thermodynamic claim. One may shift its spectrum upward and replace $K=0$ by a fixed total-energy constraint without changing the covariance or readout calculation.