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.
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.
A physical identification requires a domain of experiments, a realization, and an observation that distinguishes competing mechanisms. We construct these for the δ framework. Within finite causal quantum experiments, computable Choi data are equivalent to finite arithmetic programs producing valid instrument approximations at every requested accuracy. A separate pointer reader preserves the quantum posterior and supports adaptive composition. For the native two-atom census, the actual join update and common-mode retention select golden scaling and a signed contraction. Coherent return selects its minimal conserving completion; an independently calibrated spectral band then selects its constant interaction. A bright-mode construction and a stable short-time calibration procedure supply an explicit realization and uncertainty budget; measured duration and frequency supply the dimensional units. We derive a general endpoint identity between stored record energy and comparison imbalance, and distinguish the monotone fresh-record trajectory from coherent recurrence. The logical-to-physical bridge also has a sharp boundary: an explicit instance of the proposed completion generator has no effective value-reflecting physical reader or numerical comparison receipt. Sequence completion requires precisely the corresponding halting-oracle resource, not an unexplained scalar cost. This obstructs operational completion while leaving purely proof-language transport distinct. Native implementations of identical logical strength nevertheless carry different physical receipts. Joint covariance and the exact endpoint law state the valid relationship between those quantities. Finally, calibrated return and three-outcome tests distinguish the relevant phase and recording alternatives, with quantitative error bounds. First public preprint release, version 1.0. The files include the PDF and the main TeX file. AI-use disclosure: AI assistance was used for source analysis, mathematical exploration, manuscript drafting and revision, diagram preparation, and computational verification support. GPT-6 Astra reviews assessed arguments, scope, readability, and schematics. These were internal model reviews, not external peer review. Jonathan Washburn is the sole listed author and is responsible for the released manuscript.
A durable quantum record, the location of its coherence, and a uniquely actual outcome are different physical questions. We study their separation in a supplied finite-dimensional quantum theory. An explicit, time-independent apparatus couples a system–detector encoding to a controller, a distinct archive, and a finite returning source. A conditional energy penalty suppresses release of the original record without deleting its return coupling. An invariant reduction and a one-sided spectral-gap estimate yield a certified interval with archive occupation above 0.98438, controller-latched occupation above 0.99161, and filled-but-unlatched weight below 2.402 × 10^-7 in one stated design. The same record statistics admit different source-label overlaps and hence different coherence on a specified retained boundary. A conserved product of complementary exchanges accesses that coherence when the finite source is included. Given a calibrated multiregister reader, its finite-duration response separates the coherent apparatus from an independently added archive-gated reduction law, with the added process left active during readout. A conservative example retains a response gap above 0.01717 after separately stipulated one-percent error allowances. The contribution is the finite construction and its quantitative retention, access, and readout interfaces. Quantum carriers, Hamiltonian couplings, trace readout, source preparation, and the reader are supplied; local reader synthesis, empirical calibration, absolute rate selection, and unique-event actualization are not established. Paper code: CFQR-P1 (Complementarity-First Quantum Records — Paper 1). The associated bundle contains the technical supplement, editable LaTeX, numerical checking programs and rational certificates, and selected Lean sources with returned build and axiom-audit evidence. The 39 supporting declarations are partially formalized; eleven recorded type excerpts contain printing elisions. This is not end-to-end formalization, external human peer review, an experimental detection of fundamental collapse, or a derivation of unique-event actualization. Author-owned manuscripts, documentation and mathematical data are CC BY 4.0; author-owned software is MIT, subject to the bundle’s licence map and third-party notices.
We give a self-contained explicit counterexample for the existential unregularized bounded-region fermion question on printed page 371 of Oberwolfach Report 7/2025, with pointwise operator-norm topology distinguished from strong-operator topology on page 365. In dimension three, one fixed smooth nonnegative radial compactly supported pair potential and one fixed interaction ball give a fixed smooth normalized annihilation mode whose Heisenberg norm differences tend to the maximal value 2. Explicit packed finite-particle Slaters and times pi/(2N+1-kappa) witness the same vector-norm limit, a stronger vector asymptotic, and changing normal-state values tending to 4. Both diagonal-density and normal-ordered interactions are covered with ordered-pair coefficient 1 and no ultraviolet smearing. A fixed smooth-cutoff commutator, Slater second moment and two-sector Duhamel comparison give error O(N^(-1/6)+N^(-1/3)). The kinetic energy acts on all of R³. Continuity on every fixed vector and in every fixed normal state remains valid. Siebert's arXiv:2609.34402v1, submitted 28 September 2026 before the original 7 October candidate, supplies the earlier high-density cluster and localized phase mechanism. Its stated theorem concerns full-space difference potentials, but its proof directly adapts to this bounded symmetric cutoff kernel and yields the qualitative existential conclusion. Its Proposition 2.6 explicitly gives maximal norm 2 for the full-space constant-number reference. The present note gives an explicit free-Slater transfer to the fixed spatial cutoff. A separately checked audit-time energy bootstrap also recovers the cutoff maximal limit through the earlier localized phase mechanism. These cutoff implications are deductions made in the audit, not literal cutoff theorems attributed to the prior paper. No first solution, first quantitative refinement, historically certified novelty, or continued open status through October 2026 is claimed. The bounded primary-source priority audit records access and indexing limitations. This five-page unrefereed research note is accompanied by its identical authored PDF/LaTeX source, unchanged 26-file historical candidate and independently authored analytic, operator-algebra, computational, source/model, priority and priority-mapping audits. The fresh computational family has 2,668 exact finite predicates and 558 floating diagnostics; finite tests supplement the universal written proof, and numerical diagnostics are not rigorous interval enclosures or formal proof certificates. Historical optimized-assertion and inventory weaknesses are repaired by externally pinned closed regular-file validation and isolated nonoptimized checker execution, while all original bytes and the original 1/5 candidate count are preserved. Historical result outputs are not claimed byte-identical to current replays, and runtime source-byte checks are explicitly omitted. AI tools were used extensively in development, verification, adversarial review and writing. Independent AI-assisted review is not human or journal peer review; neither is claimed. No copied third-party source bodies are distributed. No all-potential, all-region, dimension-one/two, fixed-density/energy, thermodynamic, fixed-state-discontinuity or CAR-invariance result is asserted.
We study the second temporal Rényi entropy S₂ = −log Tr ρ_A² of a heavy primary state in a 2d CFT, for an interval of temporal extent Δt and spatial extent Δφ. For BTZ microstates we define a gravitational model F_grav, built from the Gaussian-ensemble average over heavy OPE data: a sum of exact finite-c Virasoro vacuum blocks, continued to the Lorentzian sheet, in which the integer-spin projection (a sum over Dehn-twist images) is truncated to five images. Within F_grav, Tr ρ_A² has zeros at real (Δt, Δφ) near the transition between the same-sheet and winding saddles. All 18 zeros found for 46 ≤ c ≤ 241 pass a numerical validation by a degree argument. For the zeros at c = 46 and c = 61 we give computer-assisted proofs. Interval arithmetic with analytic bounds on every omitted series term shows that F_grav has Brouwer degree −1 on a disk of radius 0.01 around each. Analytic bounds on all omitted images show that on each disk the complete image sum differs from F_grav by less than 1/160 of a rigorous lower bound on |F_grav| at the boundary, so it also has a zero there. Assuming mainly that only two saddles compete near the transition, that their large-c expansion is uniform there and that their relative phase is monotonic, the zeros follow a Lee-Yang law. We test it numerically: it predicts their number in a fixed window and their positions out of sample, to within 0.0033 in Δφ. The spacing of consecutive zeros, which requires only |Tr ρ_A²|, gives the modulus of the mean slope of the relative action phase between them to within 2%. Persistence of the zeros in individual microstates requires a bound on the deviation of the microstate correlator from F_grav, or at c = 46 and c = 61 from the complete image sum, which we do not establish. Files. Robles_Blanquer_Fisher_zeros_BTZ.pdf is the manuscript. ancillary_files.zip contains its ancillary files (the folder anc/). certificate_c46/ and certificate_c61/ contain the programs, the coefficient data and the stored interval certificates of the computer-assisted proofs of Theorem 1 (c = 46) and Theorem 2 (c = 61), with checkers, pinned requirements and SHA-256 manifests. independent_checks/ contains independent checks, which are not inputs to the proofs. The other files are the numerical pipeline of the paper, the series coefficients, the validation records and the figure scripts. README.md lists the script of each step, table, figure and check. To check the stored certificates, use Python 3.12 with the pinned requirements of each folder (python-flint 0.9.0, mpmath 1.4.1, numpy 2.3.5, scipy 1.17.0). From the folder where the archive is unpacked, run:cd anc/certificate_c46/F6_boundary && python work/check_certificate.pycd anc/certificate_c46/image_sum && python work/check_image_tail.pycd anc/certificate_c61 && python work/check_image_tail.py 61 This preprint has not been peer reviewed. The computations, the code, the computer-assisted proofs and the manuscript were developed with substantial assistance from AI systems, under the author's direction. The author takes full responsibility for the content.
We have shown that the smallest possible singel-qubit critical coupling strength of the N-qubit two-photon Rabi model is only 1/N times that of the two-photon Rabi model. The spectral collapse can thus occur at a more attainable value of the critical coupling. For both of the two-qubit and three-qubit cases, we have also rigorously demonstrated that at the critical coupling the system not only has a set of discrete eigenenergies but also a continuous energy spectrum. The discrete eigenenergy spectrum can be derived via a simple one-to-one mapping to the bound state problem of a particle of variable effective mass in the presence of a finite potential well and a nonlocal potential. The energy difference of each qubit, which specifies both the depth of the finite potential well and the strength of the nonlocal potential, determines the number of bound states available, implying that the extent of the incomplete spectral collapse can be monitored in a straightforward manner.
Abstract The two-photon quantum Rabi model is analyzed using the Z 4 -symmetry and su ( 1 , 1 ) -algebra. We derive the G -function whose zeros give the exact eigenvalues of the Hamiltonian. The singularity structure of the G -function allows to draw conclusions about the distribution of these eigenvalues along the real axis and we derive the spectral collapse phenomenon at critical coupling g c , found numerically before. The spectrum at g c consists of a discrete and a continuous part: the ground state is always separated from the continuum by a finite excitation gap, ruling out a quantum phase transition in the usual sense. For large qubit splitting, also other low lying states split off from the continuum. However, perturbation theory predicts the vanishing of the gap to all orders, demonstrating its non-perturbative nature. We corroborate this result with a variational calculation for the ground state.