{"id":"97a3b777-b0a7-4437-96c1-ca05af047123","arxiv_id":"2608.07820","paper_version":1,"verdict":"ACCEPT","confidence":"MODERATE","novelty_score":8.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"Linear spectral statistics of entrywise-transformed spiked Wigner matrices obey a Gaussian CLT under shifted L4 profile admissibility, with explicit centering and covariance.","lead":"This paper proves a central limit theorem for the eigenvalues of spiked random matrices whose entries are passed through a rough, possibly non-differentiable nonlinearity. It shows the fluctuations are Gaussian with an explicit mean and covariance, provided the transform satisfies mild shifted moment conditions.","discovery_kind":"new_method","skeptic_critique":{"model":"deepseek-v4-flash","headline":"No significant objection identified: the proof of Theorem 3.3 is internally coherent, and the flagged W5 condition is restrictive but is exactly what the four-moment replacement and Lindeberg error estimates require.","rationale":"The reader's weakest-assumption identification is accurate: W5 is the least integrable-looking and most restrictive input, and it is genuinely needed twice in the proof. I agree that this is the natural place to focus scrutiny, and I confirm that the paper does not secretly weaken it: Proposition D.12 shows bare L4 is insufficient, and Section 5.1 explicitly proposes a matrix-averaged relaxation as future work rather than claiming it as a theorem. However, I do not see this as a correctness risk for the stated result. A restrictive hypothesis is not a flaw when it is explicit and is used exactly as stated. My own reading of the four proof modules found no hidden gap in the moment matching, the generalized-Wigner reduction, the deterministic response computation, or the outlier completion. The low-degree consistency checks strengthen confidence, and the proof is unusually explicit about imported inputs and their hypotheses. The main residual risk is the usual one for a lengthy analytic paper without machine-checked formalization: a sign or constant error in the contour formulas, or an unverified hypothesis in the imported Li–Xu theorem. That is why my recommended concrete test targets the endpoint formulas and the import; it is a verification step rather than an objection. Given the reader's moderate confidence and my inability to identify a load-bearing defect, the appropriate verdict remains unchanged.","tokens_in":43133,"tokens_out":39012,"duration_ms":362840,"concrete_test":"As a check worth running, independently re-derive Lemma B.3 and Proposition B.6's endpoint formulas (19)–(21) from the cited Li–Xu equations in [31], keeping the resolvent convention G=(A−zI)^{−1} and the zero-diagonal profile S^0_N=N^{−1}(11^T−I). Verify the sign and coefficient of the finite-flat Dyson correction in m_base and the '−Tr T' term in the covariance kernel. If either sign is off, the order-one centering in Theorem 3.3 would shift; if the formulas match, the central centering is confirmed.","verdict_should_be":"UNCHANGED","load_bearing_attack":"I read the proof architecture in good faith and did not find a load-bearing gap. The central claim is conditional on W1–W5 and Assumption 2.3; within those hypotheses every step I checked is consistent. W5 is used precisely twice, as the reader notes: once to obtain a uniformly bounded exact four-moment surrogate (Lemma A.1 and Theorem A.8), and once to control the fifth-order edgewise replacement remainder (Lemma A.2 feeding Lemma C.4). Both uses are explicit and the quantifier order in Lemma A.2 is correct. The bounded-array backend is handled by diagonal balancing plus a positive-diagonal regularization followed by the double limit N→∞ then η↓0 (Lemma B.2); this is a standard convergence-together argument, and the cited Li–Xu theorem is used only for an exactly stochastic, bounded, generalized-Wigner array. The deterministic order-one responses — homogeneous zero-diagonal bias, rank-one Woodbury term, zero-diagonal correction, and quadratic variance-profile term — are derived from Lemmas B.3, B.9 and Proposition B.13, and the low-degree checks in Section 3.8 (φ=1, t, t²) are consistent with exact traces. The global Fourier–Duhamel replacement in Theorem C.6 does not require a common spectral confinement event because the derivative bounds in Lemma C.1 are dimension-free, and spectral confinement is used only at the two endpoints. The supercritical outlier contribution is justified through the exterior law and the determinant lemma, with the winding issue in formula (25) correctly avoided by using (24). The weakest point is indeed W5: it is stronger than bare L4 and Proposition D.12 shows L4 alone can fail. However, the paper states this limitation explicitly, gives sufficient criteria (including L^{4+ε} for Gaussian noise), and does not claim W5 is necessary. Assumption 2.3’s weak balance is restrictive but is explicitly identified and discussed as future work. I see no internal inconsistency or unproved assertion that threatens Theorem 3.3 as stated.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper proves a bulk CLT for linear spectral statistics of entrywise-transformed rank-one spiked Wigner matrices with a general microscopic noise law. The entry transform f is only required to satisfy the shifted profile conditions W1–W5 and the spike is delocalized and weakly balanced. At each microscopic shift the authors construct an explicit bounded three-point variable with exactly the same first four centered moments, then apply a generalized-Wigner LSS theorem to the resulting bounded triangular array, compute the order-one deterministic mean as the sum of homogeneous, rank-one, zero-diagonal, and quadratic variance-profile responses, and transfer the CLT to the original rough transform via a global Fourier–Duhamel derivative estimate and a four-moment replacement telescope. The covariance is the standard zero-diagonal real-Wigner covariance with fourth-cumulant parameter κ4^{f,ν}. The paper also treats the supercritical outlier contribution to the full trace and gives a Gaussian-noise corollary for every centered, variance-normalized f in L^{4+ε}(γ), with explicit Hermite-type coefficients.","tokens_in":43429,"tokens_out":32031,"duration_ms":292949,"significance":"If the proof is correct, this is a substantial contribution to low-regularity LSS theory for transformed spiked Wigner matrices. The result removes differentiability assumptions on the entry transform, replaces them with transparent shifted-profile and fourth-tail conditions, and provides explicit formulas for all order-one centering terms and the fluctuation covariance. The three-point moment-matching construction, the unbalanced-profile Dyson computation, and the global Fourier–Duhamel replacement argument are all interesting in their own right. The Gaussian corollary is broad and readily usable in applications, covering every centered and variance-normalized polynomial-growth transform. The proof is internally coherent and, as far as I verified, has no circularity: the coefficients a1, b1, b2, and κ4^{f,ν} are fixed before the CLT is stated, and the imported Li–Xu theorem is applied only to an exactly stochastic, bounded, generalized-Wigner array. The flagged W5 condition is restrictive but is used precisely and explicitly in the two places where it is needed, namely the uniform bounded surrogate construction and the fifth-order replacement remainder estimate.","major_comments":[],"minor_comments":[{"comment":"The inequality on the large-|u| region as printed, |Φ(u)-P(u)| ≤ C ε N^{-2}|u|^4, is not valid: for q=4 the term |Φ^{(4)}(0) u^4/4!| is only bounded by C N^{-2}|u|^4, and for very large |u| the Taylor remainder is controlled by C N^{-5/2}|u|^5. The argument still works if the right-hand side is replaced by C_ε N^{-2}|u|^4 with a constant depending on ε, since ε is fixed before the N→∞ limit and the Lindeberg condition sends the averaged tail to zero. I recommend correcting this display to avoid a false statement in the proof.","section":"Lemma C.4, Eq. (223)"},{"comment":"The notation m(s) for the shifted mean E f(ζ+s) clashes with the Stieltjes transform m(z) used throughout the main text. Since both appear in the same proof chain, I suggest renaming the shifted mean, for instance μ(s) or m_f(s), to prevent confusion.","section":"Appendix A and Section 2.1"},{"comment":"The proof relies on the global generalized-Wigner LSS theorem of Li–Xu [31, Theorem 2.2] without stating its hypotheses or the precise form of the characteristic-function expansion. Given that this theorem is a load-bearing input, the paper should either quote the relevant theorem explicitly or state its assumptions verbatim so that the verification in Proposition B.6 is checkable by the reader.","section":"Proposition B.6"},{"comment":"The proof of the hybrid-uniform exterior law passes from bounds on expectations of smooth test functions to probability estimates by using smooth approximations of indicators. A sentence explaining the uniformity over the replacement index r in this passage would improve readability, although the argument as written is sound.","section":"Proposition B.19"},{"comment":"The phrase 'derivative-free analytic linear spectral statistics theorem' might be misread as requiring no derivatives of the spectral test function. Since the theorem requires φ analytic, I suggest rewording to make explicit that the derivative-free statement concerns the entry transform f only.","section":"Abstract and Introduction"},{"comment":"For the φ(t)=t^2 check, the statement that 'the rank-one contribution, including any separated outlier, is θ^2' is compressed. It would be clearer to display separately the bulk rank-one response, the zero-diagonal response, and the outlier contribution in the supercritical case, since the equality relies on the cancellation encoded in the contour convention.","section":"Section 3.8"}],"recommendation":"minor_revision","confidential_remarks":"I found no load-bearing gap in the proof architecture. The reliance on the Li–Xu theorem is standard and the cited theorem is used in a regime where the variance profile is exactly stochastic and the entries are bounded, which seems appropriate. The main corrections needed are local: one incorrect constant in Lemma C.4's large-entry bound, a notational clash, and a request for fuller statement of the imported theorem. These do not affect the validity of the central claims, and I expect the paper to be publishable after minor revision."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Short version: this paper is worth taking seriously. It proves a CLT for bulk LSS of entrywise-transformed one-spike Wigner matrices under shifted fourth-moment conditions, with an explicit four-channel centering and the standard covariance. The genuinely new pieces are the exact three-point four-moment completion (Prop A.6) and the global Fourier–Duhamel fifth-derivative transfer (Lemmas C.1–C.4). These are clever and concrete, and they let the paper avoid common truncation and coefficient-stability assumptions. The Gaussian corollary is exactly what it claims: f in L^{4+eps}(gamma), no pointwise differentiability, with explicit Hermite coefficients. Credit where due: the paper states its assumptions cleanly, proves the bounded-array backend carefully, and includes sanity checks for phi=1,t,t^2.\n\nSoft spots. The uniform shifted fourth-moment Lindeberg condition W5 is strong; Proposition D.12 shows bare L4 fails, and the paper says so. That is not a hidden flaw, but it does limit the universality claim. The weak balance condition is also restrictive, though the paper identifies it and sketches the unbalanced extension. The proof leans on Li–Xu and the generalized-Wigner local law; that is standard machinery, but it means the result is not self-contained. The low-degree checks do not substitute for a full verification of the covariance formula, but they are consistent.\n\nNet: the proof architecture holds together under the stated hypotheses. This is a serious paper for people working on transformed spiked models and low-regularity LSS. I would send it to a full referee.","headline":"A careful, genuinely new LSS theorem for rough transforms of spiked Wigner matrices; the proof is coherent and the main caveat is the strength of W5, which the authors own.","tokens_in":44071,"tokens_out":1509,"would_cite":true,"duration_ms":15330,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["60B20","15B52","60F05"],"pacs":[],"model":"deepseek-v4-flash","headline":"Rough entrywise transforms of spiked Wigner matrices still satisfy a Gaussian CLT for linear spectral statistics, with an explicit order-one mean.","keywords":["linear spectral statistics","spiked Wigner matrices","entrywise nonlinear transform","four-moment replacement","shifted profile admissibility","Gaussian fluctuation","generalized Wigner matrices","outlier contribution"],"falsifier":"For Gaussian noise and $\\varphi(t)=t^2$, the theorem predicts $\\mathrm{Var}(\\mathrm{Tr}(M_{N,\\gamma}^f)^2)\\to 4+2\\kappa_f$ and $\\mathbb{E}\\mathrm{Tr}(M_{N,\\gamma}^f)^2=N^{-1}+\\lambda b_2+o(1)$. Compute these quantities exactly for a simple profile-admissible transform, for instance $f(t)=t$, with $\\lambda>0$ and a delocalized spike; if either limit fails for large $N$, the covariance or centering formula is wrong.","tokens_in":42921,"feed_emoji":"🎲","tokens_out":5445,"duration_ms":49440,"temperature":0.7,"pith_summary":"The paper proves a central limit theorem for bulk linear spectral statistics of a Wigner matrix whose entries are first shifted by a rank-one spike and then passed through a possibly nonsmooth function f. The main claim is that, provided the shifted mean, variance, fourth cumulant, and fourth tails of f under the noise law satisfy five 'profile admissibility' conditions, the centered statistics converge jointly to Gaussian with the same zero-diagonal Wigner covariance that appears in the untransformed model, with fourth-cumulant parameter $\\kappa_4^{f,\\nu}$. The deterministic order-one centering is explicit: it separates the homogeneous Wigner bias, a rank-one Woodbury response, a zero-diagonal correction, and a quadratic variance-profile response. A separated supercritical outlier contributes exactly $\\varphi(\\theta+\\theta^{-1})$ to the full-trace centering. A self-contained Gaussian corollary states that any centered, variance-normalized $f\\in L^{4+\\epsilon}(\\gamma)$ satisfies the theorem with no pointwise differentiability of $f$.","feed_headline":"Gaussian CLT survives rough entrywise spiked transforms","feed_subtitle":"Four-moment matching plus a derivative-free proof extends the LSS theorem to nonsmooth transforms of Wigner noise.","key_machinery":"The load-bearing object is the three-point atomic completion (Proposition A.6): for any centered real variable $X$ with variance $v$ and finite fourth moment, there is an explicit three-point law $\\sqrt{v}Q$ whose first four centered moments match $X$ exactly. At each edge shift $s_{ij}$, the paper applies this to $Y_s=f(\\zeta+s)-\\mathbb{E} f(\\zeta+s)$, producing a uniformly bounded surrogate. The second key tool is a global Fourier–Duhamel estimate bounding the fifth derivative of $\\mathrm{Tr}\\tilde{\\varphi}(A+N^{-1/2}uV_e)$ uniformly in $A$, which makes the edgewise replacement error summable once four moments match and the shifted fourth-moment Lindeberg condition W5 holds.","core_discovery":"The central discovery is that an entrywise transform with no pointwise differentiability can be analyzed by matching only the first four centered moments at each microscopic shift. For every shift $s$, the paper constructs an explicit uniformly bounded three-point surrogate with the same mean, variance, and third and fourth centered moments as $f(\\zeta+s)$, which reduces the rough transformed array to a bounded generalized-Wigner array without truncating $f$ or assuming coefficient stability. Applying the generalized-Wigner LSS theorem and a global Fourier–Duhamel fifth-derivative bound, the authors transfer the analytic linear statistic back to the raw transform and identify the limiting covariance as the standard zero-diagonal real-Wigner covariance with fourth-cumulant $\\kappa_4^{f,\\nu}$.","pith_inferences":["Beyond the paper, the same proof structure would likely extend if the uniform shifted-tail condition W5 were relaxed to a matrix-averaged condition along the actual edge shifts, since the bounded surrogate is needed only on the good set of edges.","Beyond the paper, extending the spike to finite rank should replace the scalar Woodbury denominator by an $r\\times r$ determinant and add mixed quadratic variance-profile channels to the centering.","Beyond the paper, the explicit outlier contribution $\\varphi(\\theta+\\theta^{-1})$ suggests that LSS-based detection tests could profitably separate bulk and outlier information in the supercritical regime.","Beyond the paper, replacing analytic test functions by Helffer–Sjöstrand representations could lower the required regularity of $\\varphi$ independently of the roughness of the entry transform."],"forward_implications":["The bulk LSS CLT holds without any derivative of the entry transform when W1–W5 hold; the diagonal-free real-Wigner covariance with $\\kappa_4^{f,\\nu}$ governs the fluctuations.","The order-one mean splits into four explicit channels, so the spike's effect on the mean is exactly computable from $a_1$, $b_2$, $\\beta_2^{f,\\nu}$, and the contour.","In the non-outlier regime $|\\theta_{f,\\nu}|\\le 1$, the full trace equals the bulk statistic with probability tending to one, so the same CLT applies to the full trace.","In the supercritical regime $|\\theta_{f,\\nu}|>1$, a separated outlier contributes exactly $\\varphi(\\theta+\\theta^{-1})$ to the full trace and no fluctuation at this scale.","For Gaussian noise, every centered, variance-normalized $f\\in L^{4+\\epsilon}(\\gamma)$ is profile-admissible, yielding a derivative-free LSS CLT with explicit Hermite coefficients for all polynomial-growth transforms."],"supporting_citations":[{"why":"Supplies the generalized-Wigner LSS theorem used as the stochastic backend for the bounded four-moment surrogate array.","marker":"[31]"},{"why":"Provides the classical homogeneous Wigner LSS CLT whose covariance form the paper extends to the transformed spiked model.","marker":"[33]"},{"why":"Gives the Wigner-type local law used for fixed-contour resolvent estimates and spectral confinement of the surrogate.","marker":"[2]"},{"why":"Supplies the non-identically distributed Bai–Yin spectral-norm bound used to confine the rough and surrogate spectra.","marker":"[37]"},{"why":"Motivates the non-Gaussian score-transform setting in which the paper's shifted-profile assumptions live.","marker":"[13]"},{"why":"Provides the classical largest-eigenvalue bound underlying the triangular-array spectral-norm input.","marker":"[5]"}],"fun_headline_variants":["Four-moment match gives LSS for rough spiked Wigner entries","Derivative-free proof extends LSS to nonsmooth Wigner transforms","Rough entrywise Wigner transforms: CLT without differentiability","Matching four moments handles nonsmooth spiked Wigner entries"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The argument hinges on a uniform tail bound: after centering, the fourth moments of the shifted transformed entries, restricted to values above a large threshold, vanish uniformly over all small shifts as the threshold grows. If this uniformity fails, the bounded surrogate construction and the fifth-order replacement error summation both break down.","fun_headline_variants_meta":{"raw":{"variants":["Four-moment match gives LSS for rough spiked Wigner entries","Derivative-free proof extends LSS to nonsmooth Wigner transforms","Rough entrywise Wigner transforms: CLT without differentiability","Matching four moments handles nonsmooth spiked Wigner entries"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000412,"raw_usage":{"total_tokens":2171,"prompt_tokens":1023,"completion_tokens":1148,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":639,"completion_tokens_details":{"reasoning_tokens":1073}},"tokens_in":639,"tokens_out":1148,"duration_ms":8662,"temperature":1.0,"reasoning_tokens":1073,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-11T04:13:03.774602+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"For Gaussian noise and $\\varphi(t)=t^2$, the theorem predicts $\\mathrm{Var}(\\mathrm{Tr}(M_{N,\\gamma}^f)^2)\\to 4+2\\kappa_f$ and $\\mathbb{E}\\mathrm{Tr}(M_{N,\\gamma}^f)^2=N^{-1}+\\lambda b_2+o(1)$. Compute these quantities exactly for a simple profile-admissible transform, for instance $f(t)=t$, with $\\lambda>0$ and a delocalized spike; if either limit fails for large $N$, the covariance or centering formula is wrong.","supporting_citations":[{"cited_title":"Li and Y","cited_arxiv_id":null,"evidence_quote":"Supplies the generalized-Wigner LSS theorem used as the stochastic backend for the bounded four-moment surrogate array."},{"cited_title":"Lytova and L","cited_arxiv_id":null,"evidence_quote":"Provides the classical homogeneous Wigner LSS CLT whose covariance form the paper extends to the transformed spiked model."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Gives the Wigner-type local law used for fixed-contour resolvent estimates and spectral confinement of the surrogate."},{"cited_title":"O’Rourke, D","cited_arxiv_id":null,"evidence_quote":"Supplies the non-identically distributed Bai–Yin spectral-norm bound used to confine the rough and surrogate spectra."}],"review_version":1}