Certified Strong-Coupling Spectral Bounds for a Finite-Mode Moving-Mirror Hamiltonian

We study a periodic relativistic mirror with one internal harmonic oscillator coupled toa pair of scalar-field modes through a completed-square interaction. The field-mode cutoffretains arbitrarily many quanta, so the Hamiltonian remains infinite dimensional. We prove,at coupling g = 1 and with the normalized masses, frequency and circumference specifiedbelow, that its ground state is unique and lies in total-momentum sector J = 0. Its firstexcited energy has exact multiplicity two, with states in J = ±1. The complete spectral gapsatisfies the exact enclosures0.40611 < λ2 − λ1 < 0.41100.We obtain the bounds from a 175-dimensional free-energy projection, 425 explicitly generatedresidual states, an energy-weighted Schur complement and validated integer interval factoriza-tions. All discarded occupation states, including the infinitely many that never appear in theresidual matrix, are controlled by a coercive operator inequality. The paper gives the analyticargument before the computer-assisted certificate, then supplies the complete standard-libraryPython verifier in the appendix. No numerical ultraviolet-limit gap is asserted.