{"id":"482f1ee8-3236-4986-a78d-8384176357bd","arxiv_id":"2603.28457","paper_version":2,"verdict":"UNVERDICTED","confidence":"LOW","novelty_score":6.0,"correctness_risk":"unknown","formal_verification":"none","parameter_count":1,"one_line_summary":"Three non-Hermitian edge universality classes show distinct repulsion, universal cubic small-spacing behaviour, and incomplete unfolding by complex spacing ratios at the edge.","lead":"This paper studies local edge statistics of three non-Hermitian random-matrix classes using complex spacing ratios and nearest-neighbour spacings. It matters because edge statistics govern spectral outliers and stability in open quantum and non-equilibrium systems.","discovery_kind":"extension","skeptic_critique":{"model":"grok-4.5","headline":"Correct full text of 2603.28457 is still missing; central edge-universality claims remain uncheckable beyond the abstract.","rationale":"The reader correctly diagnosed the manuscript mismatch and issued UNVERDICTED with low confidence on abstract-only grounds. The supplied “full text” is still the wrong paper, so no new technical content about spacing ratios, Coulomb-gas β, or edge unfolding can be examined. The representativeness assumption remains the softest link that can be stated from the abstract alone, but it cannot be elevated or demoted without the real manuscript. Hence the verdict stays UNVERDICTED and agreement with the reader is complete; no manufactured objection about the RMT mathematics is warranted until the correct source is available.","tokens_in":12432,"tokens_out":550,"duration_ms":16200,"concrete_test":"Retrieve the actual PDF/source of arXiv:2603.28457. Confirm that the analytic section on the conditional point process for class A and the edge numerics for AI† and AII† are present and that the reported complex-spacing-ratio moments and small-s NN tails quantitatively support the three-class edge claim and the incomplete-unfolding statement; if those sections are absent or the edge data collapse fails, the universality claim weakens.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The strongest claim (three generic non-Hermitian edge statistics, represented by Ginibre A / complex-symmetric AI† / complex self-dual AII†, with distinct complex spacing-ratio and NN distributions, incomplete edge unfolding, and universal cubic small-argument repulsion) cannot be stress-tested on the supplied manuscript. The CACHEABLE body is the unrelated HISA sparse-attention paper (arXiv:2603.28458), not the RMT work. Consequently every load-bearing technical step listed in the abstract—the conditional point process that simplifies finite-N complex spacing ratios in class A, the claim that an earlier uncontrolled bulk approximation becomes controlled at large N, the elliptic-Ginibre N=3 surmise, the numerical comparison of moments and NN distributions for all three ensembles versus 2D Poisson in bulk and edge, the effective 2D Coulomb-gas interpretation at different β, and the “indications” of incomplete edge unfolding—remains invisible. The weakest premise identified by the reader (that the three Gaussian ensembles faithfully represent the conjectured edge universality classes) therefore still rests solely on the abstract’s assertion that they are “the three simplest representatives,” with no derivations, figures, or error controls available to evaluate.","agreement_with_reader":"agree"},"referee_report":{"model":"grok-4.5","summary":"From the supplied abstract, the manuscript claims that the three conjectured non-Hermitian bulk universality classes extend to three generic local edge statistics, represented by the Gaussian complex Ginibre (A), complex-symmetric (AI†) and complex self-dual (AII†) ensembles. It reports (i) finite-N analytic results for complex spacing ratios in class A via a conditional point process, explaining why an earlier bulk approximation becomes controlled at large N, and a parameter-dependent N=3 surmise for the elliptic Ginibre ensemble that interpolates to the GUE; (ii) numerical comparisons of complex spacing ratios, their moments, and nearest-neighbour spacing distributions for all three ensembles against 2D Poisson, both in bulk and at the edge, interpreted via an effective 2D Coulomb gas at different β; (iii) indications that complex spacing ratios do not fully unfold edge statistics; and (iv) universal cubic small-argument repulsion of NN spacings in bulk and edge for all three classes. The body text supplied under this paper_id is, however, an unrelated systems paper on hierarchical sparse attention (HISA), so none of the analytic or numerical claims can be checked against derivations, figures, or tables.","tokens_in":12711,"tokens_out":939,"duration_ms":14921,"significance":"If the claims hold, the work would complete the bulk-to-edge extension of the three non-Hermitian local statistics and supply practical diagnostics (complex spacing ratios, NN distributions, Coulomb-gas β picture) used across non-Hermitian RMT and open quantum systems. The announced finite-N conditional point process, the controlled large-N justification of a prior approximation, and a parameter-dependent N=3 surmise would be concrete technical contributions. Those strengths cannot be credited from the present file, because the manuscript body does not contain the RMT analysis.","major_comments":[{"comment":"Manuscript identity mismatch: the title, abstract and paper_id (2603.28457, math-ph) describe non-Hermitian edge statistics, but the full text is the unrelated HISA sparse-attention paper (arXiv:2603.28458, cs.LG). Every load-bearing claim in the abstract—conditional point process, N=3 elliptic surmise, bulk/edge numerics for A/AI†/AII†, Coulomb-gas β interpretation, incomplete edge unfolding, cubic repulsion—is therefore uncheckable. The correct full text must be supplied before any scientific assessment is possible.","section":null},{"comment":"Abstract, choice of ensembles: the central universality claim rests on treating complex Ginibre, complex-symmetric and complex self-dual Gaussians as faithful representatives of the three edge classes. With no body text, there is no derivation, finite-N scaling, or comparison to other members of the same symmetry classes that would justify this representativeness. This premise remains an unchecked assumption.","section":null},{"comment":"Abstract, analytic part (class A): the claimed simplification of finite-N complex spacing ratios via a conditional point process, and the argument that an earlier uncontrolled bulk approximation becomes controlled at large N, cannot be verified without equations, error estimates or comparison to exact formulae. Same for the parameter-dependent N=3 elliptic-Ginibre surmise and its GUE limit.","section":null},{"comment":"Abstract, numerical part: comparisons of spacing-ratio moments and NN distributions to 2D Poisson in bulk and edge, the effective-β Coulomb-gas reading, and the “indications” of incomplete edge unfolding all require figures, sample sizes, unfolding procedure and error controls that are absent from the supplied file.","section":null}],"minor_comments":[{"comment":"Until the correct PDF is attached, presentation issues (notation for AI†/AII†, definition of the complex spacing ratio, edge vs bulk unfolding protocol) cannot be reviewed.","section":null}],"recommendation":"uncertain","confidential_remarks":"The cacheable full-text block is clearly the wrong arXiv paper (HISA / 2603.28458). This looks like a pipeline or packaging error rather than author misconduct, but it makes peer review impossible. Please request the correct PDF of 2603.28457 and re-assign; I would be willing to re-review the actual RMT manuscript. Recommendation is “uncertain” solely because the scientific content is missing, not because the abstract programme is intrinsically flawed."},"author_rebuttal":null,"desk_editor":{"model":"grok-4.5","letter":"The punchline is simple: we do not have the paper. The body that was supplied under 2603.28457 is the HISA sparse-attention manuscript (2603.28458). Everything below is therefore abstract-only for the RMT work, and confidence has to stay low.\n\nWhat the abstract actually promises is useful within non-Hermitian RMT. It takes the three bulk classes (Ginibre A, complex-symmetric AI†, complex self-dual AII†) and claims the same three classes organise local edge statistics. The analytic part for class A looks like the real novelty: a conditional point process that simplifies finite-N complex spacing ratios, an explanation of why an earlier uncontrolled bulk approximation works at large N, and a parameter-dependent N=3 surmise for the elliptic Ginibre ensemble that interpolates to the GUE surmise. The numerical part then compares complex spacing ratios, moments, and nearest-neighbour distributions for all three ensembles against 2D Poisson, both bulk and edge, interprets the different repulsion strengths via an effective 2D Coulomb gas at different β, flags incomplete unfolding by complex spacing ratios at the edge, and checks the universal cubic small-argument repulsion. That programme is standard and coherent for the subfield; if the derivations and figures are clean, it would be a solid characterisation paper.\n\nThe soft spots we can already see are structural, not invented. The universality claim rests on the assertion that these three Gaussian ensembles are faithful representatives of the three edge classes; that is stated, not demonstrated, in the abstract. “Indications” of incomplete edge unfolding is deliberately soft language. Free parameters (the elliptic deformation) and invented technical objects (the conditional point process, the N=3 surmise) are fine if they are derived carefully, but we cannot see the derivations. None of this is a load-bearing flaw yet; it is simply uncheckable.\n\nWho is this for? People already working on non-Hermitian spectral statistics who need practical edge diagnostics. It is not a Clay-tier result and does not open a new technology class. Until the correct PDF is attached, I would not bring it to reading group, would not cite it, and would not send it to referees. Once the real manuscript is available, the abstract is strong enough that a serious editor should send it out. Right now the only honest recommendation is: get the right file.","headline":"Wrong full text was cached for 2603.28457; only the abstract is usable, so the edge-universality claims cannot be checked and this is not ready for a real referee read yet.","tokens_in":13358,"tokens_out":582,"would_cite":false,"duration_ms":5686,"reading_group":"no","serious_thinker":"unclear","would_accept_peer_review":false},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["60B20","15B52","82B44"],"pacs":[],"model":"grok-4.5","headline":"Three distinct edge statistics govern non-Hermitian random matrices, with universal cubic repulsion at short range.","keywords":["non-Hermitian random matrices","edge statistics","complex spacing ratios","nearest-neighbour spacing distributions","Ginibre ensemble","universality classes","2D Coulomb gas","cubic repulsion"],"falsifier":"Compute complex spacing-ratio histograms and nearest-neighbour spacing distributions at the edge for a non-Gaussian matrix ensemble belonging to one of the three symmetry classes (or a different representative of the same class) and check whether they match the Ginibre, complex-symmetric or complex-self-dual edge statistics reported here; a clear mismatch would refute the claimed universality.","tokens_in":13323,"feed_emoji":"📐","tokens_out":926,"duration_ms":14620,"temperature":0.7,"pith_summary":"Non-Hermitian random matrices are conjectured to fall into three generic local bulk statistics; that conjecture has recently been extended to the spectral edge. This paper characterises those three edge classes through complex spacing ratios and nearest-neighbour spacing distributions, using the simplest Gaussian representatives: complex Ginibre, complex symmetric, and complex self-dual matrices. Analytically it simplifies finite-N expressions for the complex spacing ratio in the Ginibre class and supplies a parameter-dependent N=3 surmise that interpolates toward the Hermitian GUE. Numerically it maps bulk versus edge repulsion for all three ensembles against a two-dimensional Poisson process, interpreting the differences via an effective two-dimensional Coulomb gas at different inverse temperatures. The work finds that complex spacing ratios do not fully unfold local edge statistics, yet verifies that short-distance nearest-neighbour spacings show the same universal cubic repulsion in bulk and edge for every class.","feed_headline":"Three edge statistics rule non-Hermitian matrices","feed_subtitle":"Spacing ratios and nearest-neighbour laws confirm universal cubic repulsion at short range","key_machinery":"Complex spacing ratios together with nearest-neighbour spacing distributions, analysed via a conditional point process (analytic, class A) and an effective two-dimensional Coulomb-gas picture at different inverse temperatures β (numeric, all three classes).","core_discovery":"There exist three generic local edge statistics among non-Hermitian random-matrix symmetry classes, faithfully represented by the complex Ginibre, complex symmetric and complex self-dual Gaussian ensembles; these classes are distinguished by their complex spacing-ratio and nearest-neighbour distributions, share universal cubic repulsion at small argument, and are only incompletely unfolded by complex spacing ratios at the edge.","pith_inferences":["If the three edge classes are truly universal, the same cubic short-distance law and Coulomb-gas β hierarchy should appear in non-Gaussian and sparse non-Hermitian ensembles used in open quantum systems and neural-network Jacobians.","Incomplete unfolding by complex spacing ratios at the edge suggests that edge-specific normalisations (for example local mean density or radial scaling) may be needed before ratio statistics can be compared across ensembles.","The analytic conditional-point-process simplification for finite-N Ginibre ratios could be extended to the other two classes to test whether their edge statistics also admit closed finite-N expressions."],"forward_implications":["Local edge statistics of non-Hermitian matrices can be classified by the same three symmetry classes already used for the bulk.","An effective 2D Coulomb-gas description at different β accounts for the observed variation in edge repulsion across the three classes.","Complex spacing ratios, while convenient, leave residual global density effects at the edge and therefore do not fully unfold local edge statistics.","Short-distance cubic repulsion of nearest-neighbour spacings holds universally in both bulk and edge for all three classes.","A simple N=3 elliptic-Ginibre surmise already captures the crossover of the complex spacing ratio toward the Hermitian GUE limit."],"fun_headline_variants":["Three edge universality classes for non-Hermitian matrices","Complex spacing ratios mark triple edge statistics","Cubic repulsion unites three non-Hermitian edge classes","Ginibre ensembles reveal three local edge statistics","Incomplete unfolding at non-Hermitian edges by spacing ratios"],"cache_read_input_tokens":128,"weakest_assumption_plain":"The three chosen Gaussian ensembles are faithful representatives of the three conjectured edge universality classes for all non-Hermitian symmetry classes, so that results for them extend to the generic local edge statistics.","fun_headline_variants_meta":{"raw":{"variants":["Three edge universality classes for non-Hermitian matrices","Complex spacing ratios mark triple edge statistics","Cubic repulsion unites three non-Hermitian edge classes","Ginibre ensembles reveal three local edge statistics","Incomplete unfolding at non-Hermitian edges by spacing ratios"]},"model":"grok-4.5","effort":"low","cost_usd":0.00508,"raw_usage":{"total_tokens":1482,"prompt_tokens":857,"num_sources_used":0,"completion_tokens":75,"cost_in_usd_ticks":50800000,"prompt_tokens_details":{"text_tokens":857,"audio_tokens":0,"image_tokens":0,"cached_tokens":256},"completion_tokens_details":{"audio_tokens":0,"reasoning_tokens":550,"accepted_prediction_tokens":0,"rejected_prediction_tokens":0}},"tokens_in":857,"tokens_out":75,"duration_ms":5563,"temperature":1.0,"reasoning_tokens":550,"cache_read_input_tokens":256,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-07-13T16:18:40.865366+00:00","model_set":{"reader":"grok-4.5"},"falsifier":"Compute complex spacing-ratio histograms and nearest-neighbour spacing distributions at the edge for a non-Gaussian matrix ensemble belonging to one of the three symmetry classes (or a different representative of the same class) and check whether they match the Ginibre, complex-symmetric or complex-self-dual edge statistics reported here; a clear mismatch would refute the claimed universality.","supporting_citations":[],"review_version":1}