{"id":"c8108ae3-4e97-4b8b-a028-bb898f00c0b8","arxiv_id":"2606.03447","paper_version":1,"verdict":"UNVERDICTED","confidence":"LOW","novelty_score":7.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"Finite-N derivation of eigenvalue density in interpolating non-Hermitian ensemble reveals transitional edge regime at σ = 1 - κ N^{-1/2} conjectured to be universal.","lead":"Authors derive the joint eigenvalue-eigenvector distribution using Kac-Rice for a Gaussian ensemble interpolating between Ginibre and symmetric non-Hermitian matrices. They identify a new transitional scaling regime at the spectral edge when the interpolation parameter approaches the symmetric limit proportionally to the inverse square root of matrix size.","discovery_kind":"extension","skeptic_critique":{"model":"grok-4.3","headline":"Kac-Rice joint may not permit exact marginal eigenvalue density extraction without further approximations under the transitional scaling","rationale":"The reader's weakest assumption isolates precisely the step whose validity determines whether the transitional regime exists as stated. All other elements (bulk circular law recovery, fixed-σ edge behavior, numerical universality checks) are downstream of this step. No separate internal inconsistency or unjustified scaling choice appears in the claim structure.","tokens_in":1778,"tokens_out":338,"duration_ms":22864,"concrete_test":"From the explicit Kac-Rice joint in the manuscript, compute the marginal eigenvalue density at finite N; then evaluate its large-N edge behavior under σ = 1 − κ N^{-1/2} and verify whether the resulting density matches the claimed interpolating form exactly (without extra saddle-point or truncation steps). Also confirm recovery of the known Class A and AI† edge densities for fixed σ away from the transition.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central claim requires that the Kac-Rice formalism supplies an exact finite-N joint distribution of eigenvalue and normalized right eigenvector for the interpolating Gaussian ensemble, from which the marginal eigenvalue density follows exactly at finite N and whose edge asymptotics can then be taken under the specific scaling σ = 1 − κ N^{-1/2} to produce a new interpolating density. The load-bearing step is the passage from joint to marginal (integration over the eigenvector) and the subsequent N → ∞ analysis: if either step introduces uncontrolled approximations or hidden assumptions beyond the stated scaling, the transitional regime and its density do not follow directly from the joint distribution.","agreement_with_reader":"agree"},"referee_report":{"model":"grok-4.3","summary":"The manuscript employs the Kac-Rice formalism to derive the exact finite-N joint distribution of an eigenvalue and its normalized right eigenvector for a Gaussian ensemble interpolating between class A (Ginibre) and class AI† (complex symmetric) matrices. The marginal eigenvalue density is obtained at finite N and analyzed in the large-N limit, recovering the circular law in the bulk for all interpolation parameters σ and the class A edge density for fixed σ. A transitional regime is identified when σ = 1 - κ N^{-1/2}, yielding a new interpolating edge density conjectured to be universal, with numerical support provided.","tokens_in":1929,"tokens_out":562,"duration_ms":19439,"significance":"If the transitional regime and associated density are placed on a rigorous footing, the result would identify a new universality class for the edge eigenvalues of non-Hermitian matrices under this scaling of the interpolation parameter, providing a smooth interpolation between the known class A and AI† edge laws. The exact finite-N joint distribution via Kac-Rice and the recovery of the circular law and fixed-σ edge limits are clear strengths. The numerical evidence offered for universality is a positive feature, but the conjectural status of the transitional density limits the immediate significance.","major_comments":[{"comment":"Transitional regime (section following the definition of σ = 1 − κ N^{-1/2}): the new edge density is presented as a conjecture supported by numerics rather than derived from the finite-N marginal via explicit asymptotic analysis under the stated scaling. This is load-bearing for the central claim, since the bulk and fixed-σ results recover known laws while the transitional regime constitutes the novel contribution.","section":"Transitional regime analysis"},{"comment":"Finite-N marginal density (Kac-Rice section): although the joint distribution is asserted to be exact, the explicit integration over the normalized right-eigenvector component to obtain the marginal eigenvalue density is not shown in sufficient detail for the transitional scaling; without these steps it is unclear whether the subsequent N → ∞ limit is fully controlled by the scaling alone.","section":"Kac-Rice joint distribution"}],"minor_comments":[{"comment":"Figure captions for the numerical histograms in the transitional regime should overlay the conjectured analytic density (if available) or state the precise range of κ and N values used.","section":"Numerical figures"},{"comment":"Notation: the interpolation parameter σ is introduced in the abstract and ensemble definition; a single consolidated definition early in the text would improve readability.","section":"Introduction/ensemble definition"}],"recommendation":"major_revision","confidential_remarks":null},"author_rebuttal":{"model":"grok-4.3","summary":"We thank the referee for their careful reading and constructive comments. We address each major comment below.","responses":[{"response":"We agree that the transitional edge density is presented as a conjecture rather than a fully derived result. The exact finite-N joint distribution and marginal eigenvalue density are obtained via the Kac-Rice formalism for any σ. The bulk circular law and fixed-σ edge law follow from rigorous asymptotic analysis of this expression. However, the explicit large-N asymptotics under the scaling σ = 1 − κ N^{-1/2} involve substantial technical challenges in controlling the integrals, which we have not completed. The conjectured density is supported by numerical evidence and provides a smooth interpolation between known classes. We will revise the manuscript to more explicitly state the conjectural status and the origin of the scaling, but we do not claim a complete derivation.","revision_made":"partial","referee_comment":"[Transitional regime analysis] Transitional regime (section following the definition of σ = 1 − κ N^{-1/2}): the new edge density is presented as a conjecture supported by numerics rather than derived from the finite-N marginal via explicit asymptotic analysis under the stated scaling. This is load-bearing for the central claim, since the bulk and fixed-σ results recover known laws while the transitional regime constitutes the novel contribution."},{"response":"The joint distribution of eigenvalue and normalized right eigenvector is derived exactly via Kac-Rice for finite N and any σ ∈ [0,1]. The marginal eigenvalue density is obtained by integrating out the eigenvector component. The manuscript outlines these steps, but we acknowledge that additional explicit details on the integration, particularly in preparation for the transitional scaling, would improve clarity. We will add these steps (possibly in an appendix) in the revised manuscript to show how the finite-N marginal is controlled before taking N → ∞.","revision_made":"yes","referee_comment":"[Kac-Rice joint distribution] Finite-N marginal density (Kac-Rice section): although the joint distribution is asserted to be exact, the explicit integration over the normalized right-eigenvector component to obtain the marginal eigenvalue density is not shown in sufficient detail for the transitional scaling; without these steps it is unclear whether the subsequent N → ∞ limit is fully controlled by the scaling alone."}],"tokens_in":1471,"tokens_out":523,"duration_ms":21559,"standing_objections":["Explicit asymptotic derivation of the transitional edge density under the scaling σ = 1 − κ N^{-1/2}"]},"desk_editor":{"model":"grok-4.3","letter":"The new piece here is the transitional regime: when sigma scales as 1 minus kappa over sqrt(N), the edge eigenvalue density interpolates between the known class A and AI dagger behaviors. They recover the circular law in the bulk for any fixed sigma and the class A edge for fixed sigma away from 1, which checks out against prior work.\n\nThe finite-N joint distribution from Kac-Rice looks like a clean starting point for this ensemble, and pulling the marginal eigenvalue density from it at finite N is a reasonable step. The numerics for the transitional density are presented as supporting evidence for a universality conjecture.\n\nThe soft spot is that the transitional density itself is stated as a conjecture rather than a derived result. The abstract does not show the explicit integration over the eigenvector or the asymptotic analysis under the specific scaling, so it is not clear how much of the marginal extraction is exact versus approximated before taking N to infinity. If that step introduces uncontrolled terms, the interpolating form does not follow directly.\n\nThis is for readers already working in non-Hermitian random matrix theory who care about symmetry transitions and edge statistics. A serious referee could check the Kac-Rice application, verify the marginal derivation, and assess whether the numerics are sufficient to motivate the conjecture or if a proof is needed.\n\nI would send it to peer review.","headline":"The paper derives a finite-N joint eigenvalue-eigenvector distribution via Kac-Rice for an interpolating ensemble and identifies a new transitional edge scaling, but the key density is a conjecture backed only by numerics.","tokens_in":2412,"tokens_out":360,"would_cite":false,"duration_ms":11252,"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":"Scaling the interpolation parameter as one minus kappa over square root N produces a new transitional edge density for eigenvalues that connects two non-Hermitian classes.","keywords":["non-Hermitian random matrices","eigenvalue density","universality classes","interpolation parameter","edge behavior","Kac-Rice formalism","transitional regime","circular law"],"falsifier":"Large-N numerical sampling of a non-Gaussian interpolating ensemble with the parameter fixed at one minus kappa over square root N should display the predicted transitional edge density.","tokens_in":2676,"feed_emoji":"","tokens_out":641,"duration_ms":16341,"temperature":0.7,"pith_summary":"The paper applies the Kac-Rice formalism to derive the exact finite-N joint distribution of an eigenvalue and its normalized right eigenvector in a Gaussian ensemble interpolating between complex Ginibre matrices and complex symmetric matrices. From the marginal eigenvalue density it recovers the circular law in the bulk for any interpolation strength and the class A edge density for fixed strength. A special scaling regime for the interpolation parameter reveals a new transitional edge density that smoothly interpolates the known behaviors of the two classes. The authors conjecture this transitional regime is universal and support it with numerical checks on non-Gaussian matrices.","feed_headline":"Scaled parameter produces transitional edge density between matrix classes","feed_subtitle":"When sigma equals one minus kappa N to the minus one half the eigenvalue edge density interpolates the known laws of the two classes.","key_machinery":"The Kac-Rice formalism applied to the interpolating ensemble, which supplies the exact finite-N joint distribution of eigenvalue and normalized right eigenvector from which the marginal density follows.","core_discovery":"In the interpolating Gaussian ensemble the marginal eigenvalue density follows the circular law in the bulk independent of the parameter. At the edge, fixed values of the parameter yield the density of class A, while the scaling sigma equals one minus kappa N to the minus one half produces a new density that interpolates between class A and class AI dagger edge behaviors. This transitional regime is conjectured to hold for non-Gaussian matrices.","pith_inferences":["Comparable transitional scalings may appear when other pairs of non-Hermitian ensembles are interpolated.","The new density supplies a concrete model for systems whose symmetry is broken at a rate tied to matrix size.","Exact analytic expressions for the transitional density could be extracted from the same Kac-Rice starting point."],"forward_implications":["The bulk spectrum is the circular law for every fixed value of the interpolation parameter.","Edge eigenvalues obey class A statistics unless the parameter is tuned to the special N-dependent scaling.","The transitional edge density provides a continuous connection between the two previously known edge laws.","Numerical evidence indicates the transitional density persists for non-Gaussian matrices."],"fun_headline_variants":["Transitional edge density between non-Hermitian classes A and AI dagger","Sigma scaling produces interpolating eigenvalue edge density","Bulk circular law holds across all interpolation parameters","Edge density follows class A for fixed sigma transitional at scaled values","Transitional regime conjectured for non-Gaussian matrices"],"cache_read_input_tokens":2112,"weakest_assumption_plain":"The Kac-Rice formalism supplies the exact joint distribution of eigenvalue and normalized right eigenvector for the Gaussian interpolating ensemble at finite N.","fun_headline_variants_meta":{"raw":{"variants":["Transitional edge density between non-Hermitian classes A and AI dagger","Sigma scaling produces interpolating eigenvalue edge density","Bulk circular law holds across all interpolation parameters","Edge density follows class A for fixed sigma transitional at scaled values","Transitional regime conjectured for non-Gaussian matrices"]},"model":"grok-4.3","cost_usd":0.007459,"raw_usage":{"total_tokens":3437,"prompt_tokens":692,"num_sources_used":0,"completion_tokens":68,"cost_in_usd_ticks":74587000,"prompt_tokens_details":{"text_tokens":692,"audio_tokens":0,"image_tokens":0,"cached_tokens":256},"completion_tokens_details":{"audio_tokens":0,"reasoning_tokens":2677,"accepted_prediction_tokens":0,"rejected_prediction_tokens":0}},"tokens_in":692,"tokens_out":68,"duration_ms":17603,"temperature":1.0,"reasoning_tokens":2677,"cache_read_input_tokens":256,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-06-28T08:05:58.585266+00:00","model_set":{"reader":"grok-4.3"},"falsifier":"Large-N numerical sampling of a non-Gaussian interpolating ensemble with the parameter fixed at one minus kappa over square root N should display the predicted transitional edge density.","supporting_citations":[],"review_version":1}