{"id":"5aebb8a6-d34e-4631-b75f-c5264a5af260","arxiv_id":"2507.19340","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":7.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"For sparse random matrices with q ≥ N^{1/6+δ}, the largest eigenvalue, after a small model-dependent correction, converges to Tracy-Widom at rate almost N^{-1/3} + N^{2/3}/q^4.","lead":"This paper proves an explicit convergence rate for the largest eigenvalue fluctuations of sparse random matrices, including Erdős-Rényi adjacency matrices, toward the universal Tracy-Widom law. The result is a finite-size error bound useful for spectral approximations and network hypothesis tests.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Lemma 6.7's cancellation is only as reliable as unpinned custom code with no certificate; an error in identity generation, equivalence detection, or exact solve would invalidate the GFT and hence Theorem 1.3.","rationale":"The paper's central claim is the quantitative Tracy–Widom law in Theorem 1.3, and the proof path is Theorem 1.3 ← Theorem 1.5 ← Proposition 3.1 ← Lemma 6.7. Lemma 6.7 is the only step that depends on computer-aided symbolic computation, specifically the 13,852×14,246 linear system (6.19) and the 4,288 used identities. The reader's weakest-assumption analysis identified this same tooling assumption: the code at [16] has no pinned commit hash and no machine-checked certificate. I agree with that identification. The concern is not that computer assistance is intrinsically unreliable; rather, the scale and sparsity of the system, combined with the absence of an independent certificate, make this the least secure load-bearing step in the paper. The rest of the proof follows established local-law, Ward-identity, and unmatched-index arguments, and I found no internal inconsistency in those parts. Because the reader already assigned a CONDITIONAL verdict based on this concern, I recommend no change to the verdict. The proposed concrete test—independent exact regeneration of the identities, equivalence classes, and solution—would settle whether the concern actually lands. If the independent computation reproduces the claimed cancellation, the verdict could be upgraded; if not, the central theorem would be unverified.","tokens_in":51448,"tokens_out":3182,"duration_ms":36111,"concrete_test":"Pin the GitHub repository to a specific commit; independently regenerate the 14,246 identities (6.17) from the stated Rules 1–3 using a separate exact-arithmetic implementation, and recompute the basis-term equivalence classes with an independent graph-isomorphism routine. Then verify, with exact rational arithmetic in SageMath, FLINT, or a formal proof checker, that the claimed 4,288-coefficient solution satisfies (6.19) with zero residual and that each used identity is a valid consequence of the resolvent identity and cumulant expansion. If any residual is nonzero or any identity fails, Lemma 6.7 is unsupported; if the independent regeneration reproduces the solution, the conditional concern is resolved.","verdict_should_be":"UNCHANGED","load_bearing_attack":"Theorem 1.3 rests on Theorem 1.5, whose proof reduces to Proposition 3.1 and ultimately to Lemma 6.7. Lemma 6.7 asserts that all leading terms in d/dt E[F(X(t))] cancel up to finitely many non-leading terms O(TF), and the paper's verification of this cancellation is the existence of a rational solution xF to the linear system (6.19) with 13,852 rows and 14,246 columns, of which 4,288 solution entries are nonzero. The identities encoded in (6.17) are generated by the custom code in [16] using Rules 1–3 and an equivalence check for basis terms. The preprint provides no commit hash for [16], no machine-checked certificate that the generated identities are correct consequences of the resolvent identity and cumulant expansions, and no independent formal verification of the rational Gaussian elimination. In particular, if the identity generator misapplies a rule, if the equivalence detection misclassifies two distinct terms as equivalent, or if the sparse exact solver has a pivot error, then the claimed linear combination is not a valid identity; the cancellation in Lemma 6.7 would fail, and the Green function comparison theorem and Theorem 1.3 would lose their support. This is a verification gap rather than a demonstrated mathematical contradiction: the analytic framework around the computer step—second-order cancellations, unmatched-index expansions, Ward identity bounds, and Lemma 5.9—is coherent and standard. The load-bearing assumption is therefore tooling trust, exactly as the reader identified.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper proves a quantitative Tracy–Widom law for the largest eigenvalue of sparse random matrices satisfying Assumption 1.1, in the regime q ≥ N^{1/6+δ}. The main result, Theorem 1.3, states that after subtracting the corrected edge 2 + 6κ4/q^2 + χ, the Kolmogorov distance between the rescaled largest eigenvalue distribution and TW1 is bounded by N^ω(N^{-1/3} + N^{2/3}/q^4). The proof follows Schnelli–Xu's long-time Green function comparison strategy: Theorem 1.5 gives a GFT for fine spectral scales, whose proof via Proposition 3.1 and Proposition 4.1 reduces to cancellation among fourth-order cumulant terms and edge-correction terms. A computer-assisted symbolic computation generates identities (Rules 1–3) and solves sparse rational linear systems (110×138 and 13,852×14,246) to verify the cancellations. The paper includes Section 7 on implementation, appendices with proofs of the supporting lemmas, and a GitHub repository with code and identities.","tokens_in":51685,"tokens_out":4569,"duration_ms":42360,"significance":"If correct, Theorem 1.3 is a substantial advance: it gives the first quantitative edge-universality rate for sparse random matrices with q as small as N^{1/6+δ}, recovers the Wigner rate N^{-1/3} from [51] when q ≍ √N, and provides an explicit additional O(N^{2/3}/q^4) error that is natural in view of Remark 1.4. The proof strategy—unmatched-index expansions combined with a computer-generated cancellation between fourth-order cumulant terms and the edge correction—is well matched to the difficulty. The paper is unusually transparent about the computer-assisted component: Section 7 describes the term representation, equivalence checking, and exact rational Gaussian elimination, and the full code and identities are deposited on GitHub [16]. No fitted parameters enter the theorem, and the Wigner baseline [51] is prior peer-reviewed work. These are genuine strengths. The main weakness is the lack of a machine-checked or independently audited certificate for the 13,852×14,246 rational linear system and the equivalence-checking implementation; this is a verifiability concern rather than a demonstrated mathematical error, and it can be addressed within the manuscript's scope.","major_comments":[{"comment":"The load-bearing step in the proof of Lemma 6.7 is the assertion that the rational linear system (6.19), with dimensions 13,852×14,246 and a 4,288-entry solution, exactly expresses the leading terms of d/dt E[F(X(t))] as a linear combination of the identities (6.17). The paper states that the solution is found by a custom C++ sparse Gaussian elimination but does not provide a machine-checkable certificate of the elimination, a proof that the identity-generation rules were implemented without error, or a proof that the equivalence test (Section 7.2) correctly distinguishes all non-equivalent basis terms. The GitHub repository [16] supplies code and data, but the version is unpinned (no commit hash), and the equivalence check for i0 > 1 is a recursive search whose correctness is not formally verified. Since an error in any of these three components—identity generation, equivalence detection, or exact solve—would invalidate Lemma 6.7 and hence Theorem 1.5 and Theorem 1.3, this is a genuine verification gap. Please provide an independent re-verification, a symbolic certificate with a documented verification procedure, or at least a precise description of how a reader can reproduce the full pipeline from the supplied scripts without trusting the same code path that produced the answer.","section":"§6.2, Eq. (6.19)"},{"comment":"The perturbation argument bounding |E[F(X(10 log N))] − E[F(X(∞))]| uses ∥G(10 log N, z(10 log N)) − G(∞, z(∞))∥₂ ≺ 1/(N^{7/2}η²). The stated bound appears to be missing the factor N^{1/2} from the difference of the H-terms in (3.5), and the displayed inequality is not checked against the stated ranges of η (η can be as small as N^{-1+ϵ}+N^{2ϵ}q^{-4}, so N^{7/2}η² may be far larger than N). The subsequent claim |E[F(X(10 log N))] − E[F(X(∞))]| ≤ N^{-1} is therefore not justified as written. This step is load-bearing because it converts the integrated bound from Proposition 3.1 into the full GFT of Theorem 1.5; please write out the complete estimate and verify it for all allowed η.","section":"§3.2, proof of Theorem 1.5"},{"comment":"In the case-2 bound of Lemma 5.9, the manuscript claims that for ph ≥ 1 with an index v distinct from a and b occurring at least twice in the off-diagonal entries, an additional cumulant expansion gains 1/N. The premise is needed to ensure the resulting terms have degree at least 2 after the expansion, but the proof does not verify the degree condition for all subcases (e.g., when the derivative hits different Green function factors in (4.16) and produces terms such as G_{ja}G_{bv}). The bound may be correct, but the argument as written does not check it; a short explicit verification would close the gap.","section":"§5.3, proof of Lemma 5.9, case 2"},{"comment":"The proof of (A.1) extends the averaged local law from Theorem 2.4 to all η ≫ N^{-1} and uses the assertion that y ↦ y Im m_N(E + iy) is strictly increasing in y. That monotonicity is not proved and is used to bound the first term in the extension argument. This is a supporting lemma, but (A.1) enters Lemma 2.6 and hence Theorem 1.3. Please either prove the monotonicity on a high-probability event or replace the argument with a bound that does not rely on it.","section":"Appendix A.1, proof of Lemma 2.6"}],"minor_comments":[{"comment":"The notation ∆ Im is defined in (6.2) but used earlier in the main proof of Proposition 3.1; please reorder so that the definition precedes first use.","section":"§3.2 and §6.2"},{"comment":"The statement says sup_{r > r0}, but the proof works with r ∈ (r0, N^ϵ) and uses rigidity for |r| ≥ N^ϵ; please clarify whether r0 may be negative and whether the supremum should be over r0 < r < ∞ with the small-r behavior covered by rigidity.","section":"§1, Theorem 1.3"},{"comment":"The counts of basis terms and identities (M = 138, L = 110 for the small system; M_F = 14,246, L_F = 13,852 for the large system) would be much easier to verify if the paper included a small table of the distribution of type-0, type-A, and type-AB terms by number of summation indices, together with the counts of identities generated by each rule.","section":"§5.2.3 and §6.2"},{"comment":"The description of the almost-unique identifier AUID is heuristic; while the manuscript states that collisions are checked by exact equivalence, it would help to state the maximum number of summation indices occurring in the computations and why the recursive equivalence check terminates quickly in practice.","section":"§7.2"},{"comment":"The reported runtime of roughly one hour for the large system would be more informative with the exact software versions, hardware, and a description of how the final solution was verified in exact arithmetic (e.g., by re-multiplication and checking zero residues).","section":"§6.2, Remark 6.10"},{"comment":"There are a few typos and undefined notations: '[GOE' appears without definition; 'κ4' in (1.8) is used before Assumption 1.1 states that κ4 = κ4(N); and the line 'd/dt E[m(t,z(t))] = =' in Section 4 has a double equals sign.","section":"Throughout"}],"recommendation":"major_revision","confidential_remarks":"The stress-test concern about the unpinned 13,852×14,246 system is legitimate and is the main reason I recommend major revision rather than minor revision. However, the paper is transparent about the computer-assisted part, and the analytic framework around it—local laws, Ward identities, unmatched-index expansions—is coherent and standard. The central claims are plausible and the result is significant if the computational step can be independently verified. I would be comfortable with acceptance after the authors pin the repository, provide a reproducible verification path for the large linear system, and fix the perturbation bound in Section 3.2. The comparison with [51] is appropriately acknowledged, and the paper's contribution relative to [28,32,42] is clearly stated."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"The result is real: Theorem 1.3 gives the first explicit finite-N rate for Tracy-Widom fluctuations of the largest eigenvalue of sparse random matrices, O(N^{-1/3}+N^{2/3}/q^4) for q≥N^{1/6+δ}. The paper is also honest about what rests on computation: Lemma 6.7 reduces the leading cancellation to a rational solution of a 13,852 by 14,246 linear system generated by custom code on GitHub.\n\nThe analytic skeleton is coherent. Local laws are applied carefully, the unmatched-index expansion genuinely extends the Wigner treatment, and the shifted edge 2+6κ4/q^2+χ is inherited from earlier work rather than fitted. The Wigner baseline is the authors' own peer-reviewed [51] result, and citing it is appropriate. The paper flags the technical restriction q≥N^{1/6+δ} in Remarks 1.4 and 5.10. No free parameters enter the theorem.\n\nThe soft spot is exactly where the reader and stress test place it: the computer-assisted cancellation. The GitHub repo has no pinned commit hash, no machine-checked certificate for the 4,288 used identities, and no independent verification of the rational Gaussian elimination or the equivalence-checking routine. The equivalence check for i0=1 uses NetworkX's VF2++; for i0>1 it is a recursive mapping heuristic. If the identity generator misapplies a rule, or the equivalence check merges distinct basis terms, the cancellation principle fails and Theorem 1.3 loses its proof. That is a tooling-trust assumption, not a demonstrated contradiction. The surrounding analysis—second-order cancellations, Ward identity bounds, Lemma 5.9—is standard and looks correct. I do not see circularity or fitting.\n\nThe paper deserves a serious referee. Send it to review, but condition the final acceptance on pinning the GitHub artifacts to a commit and on supplying a verifiable certificate: either an independent audit of the solver and identity generator, or a machine-checked proof of the 4,288 identities used. The small 110×138 system is fine with its integer solution; the large system needs the same standard. This is the difference between a convincing computer-assisted proof and a claim.\n\nFor whom: random matrix theorists working on sparse matrices and edge universality; also statisticians who want finite-N bounds for spectral tests, though they will use the theorem rather than audit the code. Bring it to reading group if your group tolerates one session on computational verification of RMT cancellations. I would cite it with the caveat noted.","headline":"First quantitative Tracy-Widom rate for sparse random matrices, with the real risk concentrated in one unpinned computer-assisted linear algebra step.","tokens_in":52267,"tokens_out":3668,"would_cite":true,"duration_ms":35097,"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":"The paper proves that, for sparse random matrices with sparsity parameter $q\\ge N^{1/6+\\delta}$, the shifted largest eigenvalue converges to the Tracy–Widom law with error $N^\\omega(N^{-1/3}+N^{2/3}/q^4)$.","keywords":["Tracy–Widom law","sparse random matrices","Erdős–Rényi graphs","largest eigenvalue","convergence rate","Green function comparison","edge correction","computer-assisted proof"],"falsifier":"Re-run the exact rational Gaussian elimination for the system in equation (6.19) in independent software and verify that the claimed solution, with its 4,288 non-zero entries, has zero residual; a non-zero residual would invalidate Lemma 6.7 and remove the proof of the Green function comparison theorem.","tokens_in":51196,"feed_emoji":"🎲","tokens_out":12491,"duration_ms":110134,"temperature":0.7,"pith_summary":"This paper establishes the first quantitative edge-universality result for sparse random matrices, the model class that contains the normalized adjacency matrix of an Erdős–Rényi graph. The main theorem states that when the mean-degree parameter satisfies $q\\ge N^{1/6+\\delta}$, the largest eigenvalue $N^{2/3}(\\lambda_N-\\widehat L)$ converges in distribution to the Tracy–Widom law $\\mathrm{TW}_1$, uniformly over tail events, with error at most $N^\\omega(N^{-1/3}+N^{2/3}/q^4)$ after subtracting the corrected edge $\\widehat L=2+6\\kappa_4/q^2+\\chi$. This is a rate problem, not just a limit problem: it says how large $N$ must be before the limiting law is a trustworthy approximation. Quantitative statements of this kind are what turn edge universality into a tool for finite-$N$ inference, such as hypothesis tests based on the largest eigenvalue.","feed_headline":"Sparse random matrices converge to Tracy–Widom with explicit rate","feed_subtitle":"The largest eigenvalue of Erdős–Rényi graphs follows Tracy–Widom after a corrected edge, at a provable rate.","key_machinery":"The load-bearing mechanism is a Green function comparison theorem for fine spectral scales, built on averaged products of Green function entries $G_{xy}(z)$, with $z=\\widehat L_t+x+i\\eta$. Terms are encoded as rational combinations of such products up to permutation of summation indices, and the comparison is driven by three operations: the cumulant expansion formula for the interpolation flow between the sparse matrix and a Gaussian Wigner matrix; resolvent identities that generate algebraic relations among the leading terms; and iterative expansion of unmatched indices, which turns odd-order terms into negligible ones. The fourth-order cancellation is verified by computer algebra: 4,288 identities, obtained from the resolvent identities, are assembled into a 13,852-by-14,246 rational linear system whose exact solution expresses the leading terms as a combination of the identities, proving that only non-leading terms remain.","core_discovery":"On the paper's own terms, the central discovery is Theorem 1.3: under Assumption 1.1 with $q\\ge N^{1/6+\\delta}$, for every fixed $r_0$ and small $\\omega>0$, the bound $\\sup_{r>r_0}|P(N^{2/3}(\\lambda_N-2-6\\kappa_4/q^2-\\chi)\\le r)-\\mathrm{TW}_1(r)|\\le N^\\omega(N^{-1/3}+N^{2/3}/q^4)$ holds for all large $N$. The proof flows from a long-time Green function comparison theorem (Theorem 1.5) that compares the sparse matrix with a Gaussian Wigner matrix at spectral scales $\\eta\\gg N^{-1}+q^{-4}$ and time $t\\asymp\\log N$; the comparison error is the same $N^{-1/3}+N^{2/3}/q^4$ rate. What makes the comparison possible is a cancellation principle: all leading fourth-order cumulant terms in the time derivative are shown to cancel against the terms generated by the edge correction, leaving only non-leading terms that can be bounded by the local law and the Ward identity.","pith_inferences":["The threshold $q\\ge N^{1/6+\\delta}$ and the constraint $\\eta\\ge N^{2\\epsilon}q^{-4}$ come from truncating the cumulant expansion at sixth order; the authors themselves indicate that higher-order edge corrections should extend the result down to $q\\ge N^\\delta$, but that extension is not proved here.","The large sparse linear system has only about 0.07% non-zero entries, which suggests the cancellation may have a compact structural explanation; finding an explicit combinatorial identity for the fourth-order terms would remove the reliance on computer verification.","The same identity-generation pipeline, resolvent identities plus cumulant expansions encoded as a rational linear system, looks transferable to other high-order edge statistics such as joint fluctuations of the largest eigenvalues, where explicit cancellation formulas are not known."],"forward_implications":["For Erdős–Rényi adjacency matrices with $p\\gg N^{-2/3}$, the largest eigenvalue after the corrected edge has Tracy–Widom fluctuations at the stated rate, giving the first quantitative edge law for this sparse model class.","In the dense limit $q\\asymp\\sqrt{N}$, the bound reduces to $N^\\omega N^{-1/3}$, matching the sharpest known convergence rate for Wigner matrices up to the $N^\\omega$ factor.","The edge shift $2+6\\kappa_4/q^2+\\chi$ is part of the statement, so centering at the unshifted semicircle edge $2$ would leave a systematic $q^{-2}$ bias in any finite-$N$ comparison.","The Green function comparison at spectral scales below the eigenvalue spacing is a reusable tool for other fine-scale edge statistics of sparse matrices, such as counting eigenvalues in $N^{-2/3}$ windows.","Quantitative Kolmogorov-type bounds provide finite-$N$ error control for p-values in tests based on the largest eigenvalue, the application to community detection in stochastic block models mentioned in the introduction."],"supporting_citations":[{"why":"Supplies the Green function comparison method and the rigidity estimates that anchor the proof.","marker":"[22]"},{"why":"Provides the Wigner-matrix convergence rate and the proof structure that this paper adapts to the sparse setting.","marker":"[51]"},{"why":"Introduces the edge correction with $\\kappa_4$ and $\\chi$ and proves the transition in edge fluctuations that the quantitative law sharpens.","marker":"[28]"},{"why":"Supplies the refined spectral edge, averaged local law, and eigenvalue rigidity used for smaller $q$.","marker":"[32]"},{"why":"Gives higher-order edge corrections and the local-law framework near the edge used to justify the cancellation principle.","marker":"[42]"},{"why":"Establishes the Tracy–Widom limit for sparse matrices with $q\\ge N^{1/6+\\delta}$, the regime assumed here.","marker":"[44]"},{"why":"Provides the entrywise local semicircle law for sparse random matrices that controls off-diagonal Green function entries.","marker":"[19]"},{"why":"Supplies the computer-generated identities and exact rational solution of the linear system used to verify the leading-term cancellation.","marker":"[16]"}],"fun_headline_variants":["Sparse matrices hit Tracy–Widom with proven rate","Erdős–Rényi graphs: edge law with explicit TW rate","Tracy–Widom for sparse matrices: cancellation gives sharp rate","Sparse spectra: largest eigenvalue follows TW with explicit error","Sparse matrices: rate to Tracy–Widom pinned down"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The proof rests on the correctness of a large computer-generated algebraic database, namely the rational solution of the 13,852-by-14,246 linear system and the 4,288 identities used to cancel the leading terms, and no machine-checked certificate or independent formal verification of that computation is supplied.","fun_headline_variants_meta":{"raw":{"variants":["Sparse matrices hit Tracy–Widom with proven rate","Erdős–Rényi graphs: edge law with explicit TW rate","Tracy–Widom for sparse matrices: cancellation gives sharp rate","Sparse spectra: largest eigenvalue follows TW with explicit error","Sparse matrices: rate to Tracy–Widom pinned down"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.001102,"raw_usage":{"total_tokens":4583,"prompt_tokens":920,"completion_tokens":3663,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":536,"completion_tokens_details":{"reasoning_tokens":3572}},"tokens_in":536,"tokens_out":3663,"duration_ms":27650,"temperature":1.0,"reasoning_tokens":3572,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-15T17:55:20.572895+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Re-run the exact rational Gaussian elimination for the system in equation (6.19) in independent software and verify that the claimed solution, with its 4,288 non-zero entries, has zero residual; a non-zero residual would invalidate Lemma 6.7 and remove the proof of the Green function comparison theorem.","supporting_citations":[{"cited_title":"Rigidity of eigenvalues of generalized Wigner matrices","cited_arxiv_id":null,"evidence_quote":"Supplies the Green function comparison method and the rigidity estimates that anchor the proof."},{"cited_title":"Convergence rate to the Tracy-Widom laws for the largest eigenvalue of Wigner matrices","cited_arxiv_id":null,"evidence_quote":"Provides the Wigner-matrix convergence rate and the proof structure that this paper adapts to the sparse setting."},{"cited_title":"Transition from Tracy–Widom to Gaussian fluctuations of extremal eigenvalues of sparse Erdős–Rényi graphs","cited_arxiv_id":null,"evidence_quote":"Introduces the edge correction with $\\kappa_4$ and $\\chi$ and proves the transition in edge fluctuations that the quantitative law sharpens."},{"cited_title":"Local law and Tracy-Widom limit for sparse random matrices","cited_arxiv_id":null,"evidence_quote":"Establishes the Tracy–Widom limit for sparse matrices with $q\\ge N^{1/6+\\delta}$, the regime assumed here."},{"cited_title":"Spectral statistics of Erdős-Rényi graphs I: Local semicircle law","cited_arxiv_id":null,"evidence_quote":"Provides the entrywise local semicircle law for sparse random matrices that controls off-diagonal Green function entries."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Supplies the computer-generated identities and exact rational solution of the linear system used to verify the leading-term cancellation."}],"review_version":2}