{"id":"e1a4821c-5d3f-40bd-ab28-4b0c63fff630","arxiv_id":"2512.15427","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":4.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"For Gaussian random matrices with a rank-one mean, the large-N distribution of min-max normalized eigenvalues and the truncated factorization error depend only on J1/J0, via explicit formulas (9), (10), (18), (19).","lead":"This paper derives scaling laws and low-rank matrix-factorization error formulas for the min-max normalized eigenvalues of Gaussian random matrices, extending the authors' earlier approximate distribution. It shows that in the large-N limit these quantities depend only on the ratio of the noise to the mean coupling, and checks the formulas against small simulations.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Finite-N coupling error formula (16) inherits a spurious α-dependence from smearing the largest-eigenvalue atom in CDF (4); asymptotic plateau (18) survives, but finite-N claim is wrong.","rationale":"The paper's asymptotic claims—CDF (9)/(10) and coupling error (18)/(19)—are supported by the semicircle law and survive the flaw, so I do not want to overstate. But the central claim as presented includes finite-N formulas, and (16) is internally inconsistent with the definition of Δα. This is the weakest load-bearing point because it is a concrete mathematical error in the paper's own derivation, not just an unvalidated approximation. The reader's weakest_assumption correctly identifies the smeared largest-eigenvalue atom as the source. My concrete test would settle it by recomputing the integral with an atom at 1.","tokens_in":6880,"tokens_out":24042,"duration_ms":221103,"concrete_test":"Compute E[Δα] via Eq. (15) using the corrected CDF P(λ̂r, and inserting this into the coupling-error integral (15) produces the α-dependent term 16(α^3-r^3)/(3N(1-r)r^2) in Eq. (16) for α>r. But physically Δα=∑_{λ̂_i<α}λ̂_i^2 excludes λ̂1 for all α<1, so for α∈(r,1) the set of contributing eigenvalues is fixed (i=2..N) and the expectation must be constant in α. Thus Eq. (16) is not just approximate; it has the wrong α-dependence in that regime. The artifact disappears in the N→∞ limit (the smeared mass is 1/N, so the α^3 term vanishes), which is why the asymptotic plateau (18) is unaffected. However, the paper explicitly presents (16) as the finite-N analytical form, and the Fig. 3 caption says the theoretical line is computed from (16)/(17) while the text says (18)/(19) — an inconsistency that hides the defect. This is a genuine, locatable error in the paper's derivation, not a matter of consensus.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper studies the min-max normalized eigenvalues λ̂ = (λ - λ_N)/(λ_1 - λ_N) of a random matrix Q whose off-diagonal entries are i.i.d. N(μ, σ²) and whose diagonal entries are N(μ, 2σ²). Starting from an effective cumulative distribution function CDF proposed in the authors' earlier work [13], the paper derives (i) a scaling law stating that in the large-N limit the normalized empirical CDF depends only on the ratio J₁/J₀ = σ√N / μN, and (ii) closed-form expressions for the expected 'coupling error' when Q - λ_N I is approximated by a rank-k factorization. The main results are the CDF formulas (9), (10), the finite-N coupling-error formulas (16), (17), and their large-N limits (18), (19). Numerical experiments with N up to 500 and various J₁/J₀ are presented to support the theory.","tokens_in":7298,"tokens_out":17375,"duration_ms":160865,"significance":"If correct, the asymptotic formulas (9), (10), (18), (19) provide simple, parameter-light predictions for the spectrum and truncation error of a random matrix after min-max normalization, which could be useful for practitioners in matrix factorization and related ML models. The derivations from (4), (5) to (16)–(19) are explicit and the numerical experiments are reproducible. The main weakness is that the finite-N coupling-error formula (16) contains a qualitative error in the α > r regime, and the paper has an inconsistency between the text and the Fig. 3 caption about which formula is plotted. These issues affect the central finite-N claims, while the large-N plateau (18) appears sound.","major_comments":[{"comment":"The α>r branch of Eq. (16) has a spurious α-dependence. By definition, the largest min-max normalized eigenvalue is exactly λ̂₁ = 1. Therefore, for any α with r < α < 1, the set {i : λ̂_i < α} is exactly {2,...,N}, and the sum Σ_{λ̂_i<α} λ̂_i² cannot depend on α. Equation (4), however, spreads the 1/N mass of the largest eigenvalue uniformly over [r,1]. This smearing directly produces the term 16(α³-r³)/(3N(1-r)r²) in Eq. (16), which is an artifact. The correct finite-N expression in this regime should be constant, e.g. 5(N-2)/N under the same approximation. The large-N limit (18) survives because the spurious term is O(1/N), but Eq. (16) cannot be presented as the finite-N analytical result.","section":"§3, Eq. (16)"},{"comment":"The Fig. 3 caption states the black theoretical line is computed from Eq. (16) or (17), while the text says Eq. (18) or (19) is used. The plateau at value 5 for α>r, which the text explicitly associates with r, is consistent only with the asymptotic formula (18), not with (16). This inconsistency obscures the defect in Eq. (16). The authors must correct the caption/text mismatch and clearly state which formula is shown. If the data are compared with the asymptotic formulas (18)/(19), then finite-N formula (16) should not be presented as the verified finite-N prediction.","section":"§3, Fig. 3 caption vs. text"},{"comment":"Equation (3) gives λ₁ ≈ Nμ + σ²/μ and λ_N ≈ -2√Nσ, and the text implies these approximations lead to both (4) and (5). However, for σ > √Nμ (i.e. J₁ > J₀) the largest eigenvalue does not separate from the bulk; the correct approximation is λ₁ ≈ 2√Nσ, so that λ₁ - λ_N ≈ 4√Nσ. Equation (5) and the subsequent derivation of (17) implicitly use this latter value, not the λ₁ of Eq. (3). The paper should specify the regime of validity of Eq. (3) and explain how Eq. (5) is obtained. As written, the derivation is inconsistent and not self-contained.","section":"§2, Eq. (3) and Eqs. (4)–(5)"},{"comment":"The effective CDF (4) is imported from reference [13] without proof or error bounds. In particular, the treatment of the largest normalized eigenvalue (spread uniformly over [r,1] rather than an atom at x=1) is a modeling assumption that directly controls the finite-N coupling-error formula (16). Since this is a load-bearing premise for the finite-N claims, the authors should either provide a derivation or error analysis of (4), or explicitly restrict the claims to the large-N limit where the atom's location is immaterial.","section":"§2, Eq. (4)"}],"minor_comments":[{"comment":"The title has a typo: 'Mat rices' should be 'Matrices'.","section":"Title"},{"comment":"'varing N' should be 'varying N'.","section":"Fig. 2 caption"},{"comment":"The phrase 'with fixing J0 = 1' is grammatically awkward; suggest 'with J0 fixed to 1'.","section":"Fig. 3 caption"},{"comment":"There are several minor typos and grammatical issues (e.g., 'sufficiently', 'theoretically and experimentally verified'). A careful proofread is recommended.","section":"Various"},{"comment":"Reference [13] is an arXiv preprint; if a published version exists, it should be cited. Also, reference [35] may be updated if a journal version is available.","section":"References"}],"recommendation":"major_revision","confidential_remarks":"The asymptotic large-N results are plausible and potentially useful, and the numerical experiments provide reasonable support for them. However, the finite-N coupling-error formula (16) is demonstrably wrong in the α>r regime, and the paper contains an internal inconsistency between the text and Fig. 3 caption that obscures this. The authors should be asked to correct Eq. (16), clarify the status of Eq. (4), and align the Fig. 3 caption with the plotted formula before the paper can be considered for publication."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Two things you should know. First, the paper's headline asymptotic results — the scaling law (9),(10) and the large-N coupling-error plateau (18),(19) — are correct and well supported by the numerics. Second, the finite-N coupling-error formula (16) is not correct for α in (r,1): it inherits a spurious α-dependence from the way Eq. (4) smears the largest-eigenvalue atom. The stress-test note is right, and the flaw is locatable, not a matter of taste.\n\nWhat the paper does well: it takes the approximate CDF from the authors' prior work [13] and straightforwardly derives the consequences. The scaling-law observation (distribution depends only on J1/J0) is genuinely useful and the numerical verification (Figs. 1, 2) is clean. The algebra from (4),(5) to the integration formulas is internally consistent, and the asymptotic plateau at 5 is visible in the experiments.\n\nSoft spots, in order. The main one is Eq. (16). In (4), for x>r the CDF grows linearly from 1−1/N to 1, i.e. the 1/N mass of λ̂1=1 is spread uniformly over [r,1]. Plugging that into (15) gives the α^3 term in (16). But physically λ̂1=1, so for any α<1 the set {λ̂_i<α} excludes λ1 and the sum is constant on (r,1). The α-dependence in (16) is an artifact; the plateau (18) survives because the smeared mass is O(1/N). So (16) should not be presented as a finite-N prediction. There is also a minor inconsistency: Eq. (3) uses λ1≈Nμ+σ^2/μ, while Eq. (5) (J1>J0) effectively uses λ1−λN≈4√Nσ; those two approximations don't connect, and the paper doesn't clarify which is in force. Also, the Fig. 3 caption says the theory line is from (16)/(17) while the text says (18)/(19); for J1=0.1,0.3 those differ at finite N, so the reader can't tell which was actually plotted.\n\nBottom line: the asymptotic results are worth having, the finite-N claim needs a correction. The paper deserves a serious referee, but I'd ask the authors to revise (16) or at least restrict it to the asymptotic regime and fix the inconsistencies. Practitioners using low-rank factorization of noisy matrices should use the plateau formulas, not (16), for finite N. I'd bring it to a reading group as a lesson in finite-size artifacts in continuum approximations, but it's not a must-read.","headline":"The asymptotic scaling/plateau results hold up; the finite-N coupling-error formula (16) is wrong for α∈(r,1) due to a smeared largest-eigenvalue atom — worth peer review after revision.","tokens_in":7705,"tokens_out":3936,"would_cite":false,"duration_ms":38992,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["60B20","15B52"],"pacs":[],"model":"deepseek-v4-flash","headline":"For a Gaussian random matrix, the distribution of min-max normalized eigenvalues and the low-rank coupling error depend, in the large-N limit, only on the ratio J1/J0, and the paper derives explicit closed-form formulas for both.","keywords":["random matrix theory","min-max normalization","eigenvalue distribution","Wigner semicircle law","Gaussian random matrix","matrix factorization","coupling error","scaling law"],"falsifier":"Measure the empirical cumulative distribution of min-max normalized eigenvalues for a single Gaussian matrix of size N≈100 with J1/J0=0.3, and look near x=1: Eq. (4) predicts a linear rise with slope 1/((1−r)N) throughout (r,1), whereas the true distribution should be flat until a jump of size 1/N at x=1; whichever shape appears decides whether the continuum CDF is reliable.","tokens_in":6801,"feed_emoji":"🎲","tokens_out":9795,"duration_ms":88700,"temperature":0.7,"pith_summary":"Most data-processing pipelines normalize inputs before using them, and eigenvalues are no exception. This paper studies the eigenvalues of a Gaussian random matrix after min-max scaling, so that the smallest eigenvalue becomes 0 and the largest becomes 1. It claims that, once the matrix is large, the whole distribution of these normalized eigenvalues and the residual error of a low-rank factorization of Q − λ_N I depend only on the ratio of the two coupling parameters J1/J0, not on their absolute values. The paper derives explicit formulas — a cumulative distribution with a semicircle-shaped body and a plateau, and a coupling error that saturates at the constant 5 once the truncation threshold passes the point r — and verifies them numerically for matrices up to 500×500. If correct, this gives a closed-form way to anticipate the truncation error from the ratio alone, without diagonalizing the matrix.","feed_headline":"One ratio sets the eigenvalue spectrum and truncation error","feed_subtitle":"Gaussian matrices with the same J1/J0 give the same normalized spectrum and the same low-rank factorization error.","key_machinery":"The central object is the approximate effective cumulative distribution P(λ̂ J0 to a universal semicircle integral. From the same distribution it evaluates the expected coupling error of factoring Q − λ_N I as VV^T, obtaining a formula that plateaus at 5 once the threshold α passes r. Both results depend on the matrix only through J1/J0 — a property absent for unnormalized eigenvalues.","pith_inferences":["Because Eq. (4) spreads the largest eigenvalue's mass uniformly over (r,1), the α-dependent term in the finite-N formula (16) for α>r is likely an artifact of that approximation; a CDF that places an atom at x=1 would produce a flat plateau already at finite N.","If the ratio-only behavior extends beyond Gaussian entries — as semicircle-law universality suggests — the same normalized CDF and error formulas could apply to other random-matrix ensembles, making the result a general low-rank approximation tool.","In factorization-machine and black-box optimization settings where the coupling error bounds the regression error, this implies the achievable accuracy is governed by J1/J0 alone, so rank or threshold selection can be guided by this single parameter."],"forward_implications":["Two Gaussian matrices with different absolute scales but the same J1/J0 will show identical normalized eigenvalue CDFs and identical normalized coupling errors; a single parameter captures the spectrum.","The expected coupling error per N J1² saturates at 5 for truncation thresholds α above r, so the entire truncation error is carried by the normalized eigenvalues below r.","For J1 > J0 the normalized spectrum and error become universal, independent of the ratio's value, which simplifies the noise-dominated regime.","The finite-N formulas include explicit 1/N corrections, and the numerical results show convergence from N=100 upward, so the asymptotic predictions are practically usable.","The closed-form expressions allow a practitioner to choose a rank or threshold α that achieves a desired coupling error without performing an eigendecomposition."],"fun_headline_variants":["Spectrum and error hinge on a single ratio","One ratio governs normalized spectra and low-rank error","Min-max normalization: a single ratio decides all","Eigenvalue stats collapse onto a universal curve","Factorization error plateau set by J1/J0 ratio"],"cache_read_input_tokens":2304,"weakest_assumption_plain":"The results rest on the approximate cumulative distribution (4)–(5) taken from the authors' prior work, which spreads the probability mass of the largest normalized eigenvalue uniformly over (r,1) instead of leaving a single atom at 1; if that distribution is inaccurate, the finite-N error formulas fail, although the asymptotic plateau may still hold.","fun_headline_variants_meta":{"raw":{"variants":["Spectrum and error hinge on a single ratio","One ratio governs normalized spectra and low-rank error","Min-max normalization: a single ratio decides all","Eigenvalue stats collapse onto a universal curve","Factorization error plateau set by J1/J0 ratio"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000245,"raw_usage":{"total_tokens":1300,"prompt_tokens":601,"completion_tokens":699,"prompt_tokens_details":{"cached_tokens":256},"prompt_cache_hit_tokens":256,"prompt_cache_miss_tokens":345,"completion_tokens_details":{"reasoning_tokens":627}},"tokens_in":345,"tokens_out":699,"duration_ms":438544,"temperature":1.0,"reasoning_tokens":627,"cache_read_input_tokens":256,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-03T15:49:39.524788+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Measure the empirical cumulative distribution of min-max normalized eigenvalues for a single Gaussian matrix of size N≈100 with J1/J0=0.3, and look near x=1: Eq. (4) predicts a linear rise with slope 1/((1−r)N) throughout (r,1), whereas the true distribution should be flat until a jump of size 1/N at x=1; whichever shape appears decides whether the continuum CDF is reliable.","supporting_citations":[],"review_version":1}