Spectral reflection in Rabi–Stark models and finite-count readout near collapse

We prove an exact reflection of the energy–splitting spectral curves for a class of Rabi–Stark Hamiltonians. Under common-domain, semiboundedness, and compact-resolvent assumptions, it preserves eigenvalue order and multiplicity without requiring the bosonic terms to commute. Its fixed line recovers known one-photon degeneracies. For two-photon coupling, reflected simple states are orthogonal and their marginal trace distances are explicit. A quantitative contact limit then yields universal probabilities for fixed detected counts near collapse. At known fixed efficiency, one- and two-count proportions distinguish the ground-state branches despite unknown source magnitude and missed acquisitions: consistency is possible precisely when M√Ω → ∞. With independent Poisson background of unknown positive mean and a positive missed-acquisition fraction, counts zero through four and overflow suffice, with threshold MΩ → ∞. Two exact minors establish the readout's rank at every interior efficiency and identify the leading blind point when count four is omitted. Here M is the total preparation budget and Ω ↓ 0 is the collapse limit. Generative-AI disclosure and author responsibility. ChatGPT (GPT-6 Astra Pro) was used for mathematical exploration, proof checking, drafting, and preparation of verification programs. The author is responsible for the final manuscript. 2020 Mathematics Subject Classification: Primary 81Q10; Secondary 81V80, 62F03. Files: the compiled manuscript Rabi_Stark_Spectral_Reflection_v1.pdf (14 pages) and the source archive Rabi_Stark_Spectral_Reflection_v1_source.zip, which contains the self-contained LaTeX source main.tex (bibliography included; compiles with pdfLaTeX) and the ancillary directory anc/: Python verification and readout programs, the stored original-Hamiltonian probability records behind the tables, the outputs of the review checks, a README, and SHA256SUMS.txt for package integrity. Running python anc/verify.py (Python 3.10 or later; dependencies in anc/requirements.txt) reruns the exact symbolic spin and determinant checks, independent coefficient sums, tensor-product spectral reflection tests, and the readout on the stored records; anc/verify_general_reflection.py checks the general reflection on 44 spectral pairs, and anc/reproduce.py recomputes the original-Hamiltonian count laws. Numerical checks are floating-point diagnostics, not interval certificates.