Quantitative Spectral Gaps and the Thermodynamic-Limit Obstruction for a Renormalized Moving-Mirror Hamiltonian

For a confined quantum system, uniqueness of its lowest-energy state does not by itselftell us how large the separation to the first excitation is. The question becomes harderwhen a moving mirror interacts with a bosonic field and high-frequency regulators areremoved. This article considers a specified scalar-renormalized moving-mirror Hamiltonianon a periodic ring. We first derive a quantitative comparison between ultraviolet andfinite-cutoff eigenvalues, with the quadratic-form assumptions needed for that comparisonstated explicitly. We then give two reproducible rational lower-gap certificates in thenormalized ring model: a cutoff-transfer bound greater than 0.04 at coupling g = 1/5, anda direct signed-form bound greater than 0.0025 at g = 1/4. These statements depend onthe companion ultraviolet form construction. A separate variational argument applies toarbitrary finite coupling and any ring circumference L. It proves that the full-system gap isat most √M 2 + (2π/L)2 −M , hence at most 2π2/(M L2). Therefore the finite-ring spectralgap cannot stay uniformly positive as L grows. We distinguish this necessary gap collapsefrom the existence of a thermodynamic-limit operator or state, which remains unprovedfor the infrared-singular infinite-line model. In particular, we do not claim a numericalultraviolet gap at strong coupling g = 1, or a theorem about fixed-total-momentum fiberson the infinite line.