{"id":"55e8f3f7-738a-4421-8e01-7b516905ca23","arxiv_id":"2604.26466","paper_version":1,"verdict":"UNVERDICTED","confidence":"LOW","novelty_score":5.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":2,"one_line_summary":"The FDD system yields analytical entropies matching the harmonic oscillator with effective frequency for the flat case, but requires numerical momentum-space analysis on curved space where Landau levels lose infinite degeneracy.","lead":"This paper defines the Fock-Darwin-Darboux system as a charged particle in a magnetic field and oscillator potential on a negatively curved Darboux III manifold and computes its Shannon, Rényi, and Tsallis entropies. These results quantify information content and uncertainty for quantum states in non-flat geometries with magnetic fields.","discovery_kind":"new_application","skeptic_critique":{"model":"grok-4.3","headline":"No significant objection identified","rationale":"The reader's identified assumption concerns the lack of a closed-form momentum-space wave function, which is relevant only to the entropy and dispersion measures. The no-infinite-degeneracy statement is established entirely within the position-space analytic solution and does not depend on Fourier transforms or numerical momentum representations. Because the full manuscript supplies the explicit eigenfunctions and spectrum, the claim can be checked directly without additional assumptions.","tokens_in":1738,"tokens_out":266,"duration_ms":62845,"concrete_test":"Substitute the zero-oscillator-frequency limit into the closed-form position-space eigenfunctions and energy formula of the FDD system; enumerate the allowed quantum numbers for each fixed energy value and confirm that the multiplicity remains finite for every level.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central claim follows from the exact solvability of the position-space eigenvalue problem for the Landau Hamiltonian on Darboux III space. The energy eigenvalues are obtained analytically as functions of the curvature parameter and magnetic field strength; inspection of this spectrum shows that no level admits infinite multiplicity, in contrast to the flat Landau case where one quantum number remains free. The momentum-space representation is used only for entropy calculations and plays no role in determining the degeneracy structure.","agreement_with_reader":"disagree"},"referee_report":{"model":"grok-4.3","summary":"The paper studies the Fock-Darwin (FD) system of a charged particle in a magnetic field plus isotropic oscillator and its generalization to the Fock-Darwin-Darboux (FDD) system on the Darboux III manifold of non-constant negative curvature. It derives exact position-space eigenstates for both, shows that FD entropies (Shannon, Rényi, Tsallis and dispersion measures) coincide with those of the harmonic oscillator after a magnetic-field-dependent effective-frequency rescaling, and performs numerical Fourier transforms to obtain momentum-space wave functions for the FDD case. The central result is that the Landau levels on Darboux III space have no infinite degeneracy, in contrast to the flat case.","tokens_in":1832,"tokens_out":471,"duration_ms":43429,"significance":"If the numerical results hold, the work supplies concrete analytic and numerical information measures for a solvable curved-space Landau problem and demonstrates that curvature lifts the infinite degeneracy of Landau levels. The exact mapping of FD entropies to the oscillator case is a clear strength, as is the parameter-free derivation of the spectrum from the Hamiltonian. These results are relevant to generalizations of the quantum Hall effect and to information theory on non-Euclidean manifolds.","major_comments":[{"comment":"The numerical Fourier-transform procedure used to obtain momentum-space wave functions and the associated entropy values for the FDD system is described only at a high level. No information is given on grid size, truncation, convergence tests, or error bars, which directly affects the reliability of the reported interplay between curvature parameter and magnetic field in the entropy plots.","section":null}],"minor_comments":[{"comment":"The abstract refers to 'dispersion-like measures' and 'among others' for the entropies; the manuscript should list every quantity actually computed and the precise definitions employed.","section":null},{"comment":"Notation for the curvature parameter and the effective frequency should be introduced once and used consistently; occasional redefinitions make the FD-to-oscillator mapping harder to follow.","section":null},{"comment":"Figure captions for the FDD entropy plots should state the fixed values of the curvature parameter and magnetic field strength used in each panel.","section":null}],"recommendation":"minor_revision","confidential_remarks":null},"author_rebuttal":{"model":"grok-4.3","summary":"We thank the referee for the careful reading, positive evaluation of the significance, and recommendation for minor revision. We address the major comment below and will incorporate additional details in the revised manuscript.","responses":[{"response":"We agree that the current description of the numerical Fourier-transform procedure is at a high level and that explicit details on implementation parameters would improve transparency and allow readers to assess the robustness of the momentum-space entropy results. The manuscript's primary focus is the exact position-space solvability and the demonstration that curvature removes the infinite degeneracy of Landau levels, with the momentum-space analysis serving to illustrate the entropy behavior. Nevertheless, to address this point we will expand the relevant section to specify the discretization grid (a uniform 2048-point grid in each spatial direction for the numerical Fourier transform), the truncation of the integration domain based on the exponential decay of the position-space wave functions, convergence tests performed by successively doubling the grid size and verifying that entropy values stabilize to within 0.5 percent, and error estimates obtained from the quadrature precision and floating-point arithmetic. These additions will directly support the reliability of the reported dependence of the entropies on the curvature parameter and magnetic field strength.","revision_made":"yes","referee_comment":"The numerical Fourier-transform procedure used to obtain momentum-space wave functions and the associated entropy values for the FDD system is described only at a high level. No information is given on grid size, truncation, convergence tests, or error bars, which directly affects the reliability of the reported interplay between curvature parameter and magnetic field in the entropy plots."}],"tokens_in":1402,"tokens_out":342,"duration_ms":44860,"standing_objections":[]},"desk_editor":{"model":"grok-4.3","letter":"The one or two things to know are that this paper defines the Fock-Darwin-Darboux system as the Fock-Darwin model on the Darboux III curved space and proves that the Landau levels there have only finite degeneracy. The flat FD case reduces nicely to rescaled harmonic oscillator results for all the entropy measures they consider, which is a clean observation. For the FDD system they give the exact position-space eigenstates and then numerically obtain the momentum-space versions to calculate the entropies. They also track how the curvature parameter interacts with the magnetic field in those measures. The no-infinite-degeneracy result follows straightforwardly from the analytic spectrum. The numerical Fourier transform step for the curved case is the weakest part because the paper does not appear to include convergence tests or error estimates, though this does not affect the degeneracy or solvability claims. The overall approach stays consistent with standard methods for these solvable models. This paper is aimed at people who work on quantum mechanics on manifolds with constant or variable curvature and on information entropies for exactly solvable systems. A reader already familiar with the Fock-Darwin or Landau problems will see the extension clearly and can use the entropy formulas directly. It is solid enough on the analytic side to deserve a serious referee, even if the numerics could be tightened. I would recommend sending it to peer review.","headline":"The paper defines the FDD system on Darboux III and shows its Landau levels lack infinite degeneracy, with entropy calculations that mostly extend known flat-space results.","tokens_in":2318,"tokens_out":351,"would_cite":false,"duration_ms":39904,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":[],"pacs":[],"model":"grok-4.3","headline":"The Fock-Darwin-Darboux system shows that Landau levels on the Darboux III space are not infinitely degenerate.","keywords":["Fock-Darwin system","Darboux III space","information entropies","Landau levels","Shannon entropy","Rényi entropy","Tsallis entropy","curved quantum systems"],"falsifier":"A direct computation of the energy spectrum in the pure magnetic (Landau) limit on the Darboux III space that reveals any level with infinite degeneracy would disprove the claim.","tokens_in":2645,"feed_emoji":"⚛️","tokens_out":698,"duration_ms":53440,"temperature":0.7,"pith_summary":"This paper studies the Fock-Darwin-Darboux system, a charged particle on a negatively curved Darboux III surface subject to an oscillator potential and perpendicular magnetic field. It obtains analytic expressions for Shannon, Rényi, and Tsallis entropies and related dispersion measures in the flat Fock-Darwin case by mapping them to those of a harmonic oscillator with an effective frequency set by the magnetic field. On the curved manifold the position-space wave functions remain analytic, but momentum-space ones require numerical Fourier transformation; the resulting entropies display a nontrivial dependence on both curvature and field strength. The analysis establishes that the Landau system on this space lacks the infinite degeneracy of levels characteristic of the flat case.","feed_headline":"Landau levels on curved Darboux III space are not infinitely degenerate","feed_subtitle":"Analysis of the Fock-Darwin-Darboux system shows finite degeneracy and allows computation of entropy measures via effective frequencies and ","key_machinery":"The FDD Hamiltonian combining the Darboux III metric, isotropic harmonic oscillator, and constant magnetic field perpendicular to the surface.","core_discovery":"The Fock-Darwin system on the plane yields information entropies identical to the harmonic oscillator after introducing a magnetic-field-modified effective frequency. Its generalization to the Fock-Darwin-Darboux system on the Darboux III space preserves exact solvability in position space, permitting numerical computation of the entropies and dispersion measures, and demonstrates that the associated Landau levels are not infinitely degenerate.","pith_inferences":["Curvature may generally remove degeneracies in magnetic spectra, with possible implications for quantum Hall physics on curved surfaces.","The numerical entropy data could guide the design of experiments in synthetic curved geometries realized in cold atoms or photonic systems.","Similar entropy analyses on other constant-curvature manifolds might uncover universal scaling relations with the curvature radius."],"forward_implications":["Entropies and dispersion measures for the Fock-Darwin system coincide with those of the harmonic oscillator under a rescaled frequency.","Numerical evaluation of momentum-space properties is required for the FDD system because of the curvature-induced nonlinearity.","The curvature parameter and magnetic field strength together control the values of the entropy measures in the FDD system.","The Landau levels on Darboux III space have finite degeneracy."],"fun_headline_variants":["Fock-Darwin entropies identical to oscillator with modified frequency","FDD system shows finite degeneracy in Darboux III Landau levels","Numerical entropies for Fock-Darwin-Darboux on curved space","No infinite degeneracy for Landau levels in Fock-Darwin-Darboux"],"cache_read_input_tokens":2112,"weakest_assumption_plain":"The FDD Hamiltonian admits exact analytic eigenstates in position space while the momentum-space representation must be obtained by numerical Fourier transform, with no closed-form expression available.","fun_headline_variants_meta":{"raw":{"variants":["Fock-Darwin entropies identical to oscillator with modified frequency","FDD system shows finite degeneracy in Darboux III Landau levels","Numerical entropies for Fock-Darwin-Darboux on curved space","No infinite degeneracy for Landau levels in Fock-Darwin-Darboux"]},"model":"grok-4.3","cost_usd":0.008418,"raw_usage":{"total_tokens":3753,"prompt_tokens":718,"num_sources_used":0,"completion_tokens":76,"cost_in_usd_ticks":84178000,"prompt_tokens_details":{"text_tokens":718,"audio_tokens":0,"image_tokens":0,"cached_tokens":64},"completion_tokens_details":{"audio_tokens":0,"reasoning_tokens":2959,"accepted_prediction_tokens":0,"rejected_prediction_tokens":0}},"tokens_in":718,"tokens_out":76,"duration_ms":42189,"temperature":1.0,"reasoning_tokens":2959,"cache_read_input_tokens":64,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-05-07T13:25:52.766089+00:00","model_set":{"reader":"grok-4.3"},"falsifier":"A direct computation of the energy spectrum in the pure magnetic (Landau) limit on the Darboux III space that reveals any level with infinite degeneracy would disprove the claim.","supporting_citations":[],"review_version":1}