{"id":"f8a88734-dd5f-46d8-86f5-25108e216d6c","arxiv_id":"2507.14259","paper_version":1,"verdict":"REJECT","confidence":"HIGH","novelty_score":6.0,"correctness_risk":"high","formal_verification":"none","parameter_count":3,"one_line_summary":"A claimed optimal Berry-Esseen bound of order sqrt(d) N^{-1/6+eps} for eigenvector projections of random d-regular graphs, with a matching lower bound.","lead":"This paper claims a sharp error bound for how Gaussian eigenvector projections of random regular graphs are, improving prior rates. If correct, it would settle the optimal degree dependence for eigenvector universality in sparse random graphs, but the proof as written contains mathematical gaps.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"The foundational edge local law (Theorem 3) is not proved: the self-consistency step in §2.3 assumes the target error, and the remainder estimate C√d/(Nη) ≤ C√d N^{-5/6+ε} is arithmetically false at the allowed η=N^{-2/3}.","rationale":"I agree with the Reader that the proof of Theorem 3 is the load-bearing assumption. In fact the gap is more concrete than a missing proof: at the lower endpoint of the permitted η range, the paper's own remainder bound is N^{1/2} larger than the stated target. This is not a matter of philosophical circularity; the displayed chain '|⟨q,R⟩| ≤ ∥R∥ ≤ C√d/(Nη) ≤ C√d/N^{5/6−ε}' cannot hold when η=N^{-2/3}, because Nη=N^{1/3} and hence C√d/(Nη)=C√d N^{-1/3}, which is not ≤ C√d N^{-5/6+ε} for large N. If one tries to repair this by shrinking the η range to η≥N^{-1/6}, the theorem's statement and its use in the smoothing argument would need to change, and that change is not made. The additional circular self-consistency assertion—m(z) close to msc at the same target accuracy—does not supply the missing factor either; it is the very estimate needed. I therefore do not see how Theorem 3 follows from the material in Sections 2.1–2.3. Since the local law is the sole input to the comparison dynamics and cumulant bounds, the proof of Theorem 1 inherits this gap. This is independent of the later issues flagged by the Reader, such as the cumulant cancellation in Proposition 6, which further support the rejection. I am not claiming the mathematical result is false; the concern is that the written argument does not establish it. Because my independent reading does not raise the verdict beyond the Reader's REJECT, I leave the verdict unchanged.","tokens_in":10384,"tokens_out":13675,"duration_ms":160955,"concrete_test":"Re-derive the proof of Theorem 3 with z=2+iN^{-2/3} and q=(e_1−e_2)/√2. Write m(z)=⟨q,HG(z)q⟩/⟨q,G(z)q⟩ and derive the exact identity m(z)+1/(z+m(z)) = δ(z) from Proposition 1, expressing δ(z) in terms of the vector F. Bound |δ(z)| using only the configuration-model fluctuation bounds and Lemma 1. If the best provable bound on |δ(z)| is √d N^{-1/3} rather than √d N^{-5/6+ε}, Theorem 3 is false as stated; equivalently, locate the missing N^{1/2} factor that would justify the displayed inequality C√d/(Nη) ≤ C√d N^{-5/6+ε} at η=N^{-2/3}. Since no such factor appears in the manuscript, this test should settle the concern.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central claim rests on Theorem 3, the edge local law. Its proof in §2.3 is the weakest point, and it fails twice. First, it asserts: 'Near the edge, m(z) ≈ msc(z) + O(√d N^{-5/6+ε}) by self-consistency.' This is exactly the error bound Theorem 3 is supposed to prove for ⟨q,G(z)q⟩−msc(z); no prior proposition establishes it, and Proposition 1 only provides the vector equation with a remainder R. The self-consistency equation for msc is not a proof that the random object m(z) is within the target distance. Second, even if that step is granted, the remainder estimate does not close. Proposition 1 gives |⟨q,R⟩|≤C√d/(Nη). Theorem 3 allows η=N^{-2/3}. At that value the bound is C√d N^{-1/3}, while the claimed target is C√d N^{-5/6+ε}; the paper's displayed inequality 'C√d/(Nη) ≤ C√d/N^{5/6−ε}' is off by N^{1/2} at the lower endpoint of the admissible η range. No additional factor of η is present in the text to repair this. Because Sections 3–5 use this local law as the input for the Green-function comparison and the cumulant argument, the main theorems are not established by the written proof. This is a proof-level objection, not a disagreement with the conjectured result; the result may be true, but the argument as written is not.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper proposes to prove that for uniformly random d(N)-regular graphs with d(N) <= N^kappa and kappa < 1/4, the projection sqrt(N)<q,u_2> of the second eigenvector onto any deterministic unit vector q orthogonal to the all-ones vector has Berry-Esseen error O(sqrt(d) N^{-1/6+epsilon}) (Theorem 1), and that this rate is optimal (Theorem 2). The proof is organized around an edge local law for the resolvent (Theorem 3), a self-consistent comparison with a constrained, time-evolved GOE (Sections 3-4), a Stein-Malliavin cumulant analysis, and a fourth-cumulant lower bound. The declared aim is to improve on prior d^3 bounds and to give the sharp degree dependence across the sparse-to-moderately-dense transition.","tokens_in":10864,"tokens_out":14080,"duration_ms":150566,"significance":"If valid, the sqrt(d) Berry-Esseen bound would be a substantial quantitative advance: it would identify the correct degree dependence at the spectral edge and would combine a nontrivial local-law rate with a matching lower bound in the regime d <= N^{1/4-epsilon}. The proposed strategy of exploiting the variance structure via vector resolvent equations and Stein-Malliavin machinery is conceptually appealing, and the paper identifies a natural scaling heuristic. However, the written proof contains multiple load-bearing gaps: a circular local-law argument, arithmetic failures in the concentration and cumulant estimates, an invalid comparison step, and an incorrect lower-bound computation. The main theorems are therefore not established by this manuscript. I regard the central claim as plausible, but the current derivation is not sufficiently sound for publication.","major_comments":[{"comment":"The proof states 'Near the edge, m(z) approximately equals msc(z) + O(sqrt(d) N^{-5/6+epsilon}) by self-consistency' and then uses this to conclude |<q,G(z)q> - msc(z)| <= C sqrt(d)/N^{5/6-epsilon}. This is circular: the preceding propositions establish at most a vector equation with an uncontrolled remainder, not the target m(z)-to-msc(z) bound. Since Theorem 3 is the only local-law input used in Sections 3-5, the main results rest on an unproved assumption.","section":"§2.3, proof of Theorem 3"},{"comment":"Even if Proposition 1's remainder bound ||R||_2 <= C sqrt(d)/(N eta) is granted, the displayed conclusion of Theorem 3 fails at the allowed endpoint. For eta = N^{-2/3}, C sqrt(d)/(N eta) = C sqrt(d)/N^{1/3}, whereas the claimed error is C sqrt(d)/N^{5/6-epsilon}. The two differ by a factor N^{1/2}, and no additional power of eta appears in the text to repair this. Thus the edge local law is not established even conditionally on Proposition 1.","section":"§2.3, Proposition 1 and remainder estimate"},{"comment":"The variance proxy and Lipschitz bound in Lemma 1 are arithmetically inconsistent. The proof obtains V <= C N/eta^4 and L = O(1/(sqrt(d) eta^2)); at the edge value eta = N^{-2/3} these are C N^{11/3} and N^{4/3}/sqrt(d), not the claimed O(d/N) and O(sqrt(d/N)). The sentence 'this gives V = O(d/N)' is false. Since Lemma 1 is the concentration input for the subsequent remainder estimates, the variance-sensitive concentration part of the argument is unsupported.","section":"Lemma 1"},{"comment":"Lemma 3(iii) contains a missing sqrt(N) factor, and the chosen time scale does not close the comparison. From part (ii), Cauchy-Schwarz gives E||partial_s G_{t,s}||_HS <= eta^{-2} sqrt(N E||Delta_t||_op^2) = eta^{-2} sqrt(N d t + d^2), not eta^{-2} sqrt(d t + d^2/N). With t* = N^{-1/3+epsilon}, the term N d t equals d N^{2/3}, which is not bounded by d^2/N for the stated degree range (for example, d=N^{1/4} gives d N^{2/3}=N^{11/12}, while d^2/N=N^{-1/2}). Additionally, the evaluation of the minimum in Theorem 4 is incorrect: substituting s(t)=sqrt(d t/N) into (4) gives approximately C d/N + C sqrt(d) N^{-4/3+2 epsilon}, not C sqrt(d)/N^{5/6-epsilon}.","section":"§3.1-3.2, Lemma 3 and Theorem 4"},{"comment":"The second cumulant bound is arithmetically invalid. The proof obtains ||D X||^2 = O(d^2/N) and multiplies by ||L^{-1}|| = O(N^{2/3-epsilon}), yielding O(d^2 N^{-1/3-epsilon}); this can equal O(d N^{-1/3+epsilon}) only if d = O(N^epsilon), not if d <= N^{1/4}. Moreover, the 'cancellation' in Proposition 6 is a non sequitur: dividing by sigma^2 = 1 + O(d^{-1}) multiplies by a factor of the form 1 + O(d^{-1}), which cannot remove a factor of d from the numerator. The stated conclusion of Proposition 6 should remain O(d N^{-1/3+epsilon}).","section":"Propositions 4 and 6"},{"comment":"The lower-bound computation does not produce the claimed scaling. The displayed expansion leads to |kappa_4| approximately 4|c_2|/(d N^{1/3}), which is smaller than c sqrt(d) N^{-1/6} by a factor (sqrt(d) N^{1/6})^{-1}. The subsequent sentence '1/(sqrt(d) N^{1/6}) ... which is bounded below' reverses the direction of the inequality, since sqrt(C log N) N^{1/6} tends to infinity. In addition, the claim that c_2 = E[g_1^3 eta_1] != 0 is asserted without computation. Theorem 2 is therefore not established by the written proof.","section":"Lemma 5 and §5.2"}],"minor_comments":[{"comment":"The title contains a typo: 'SP ARSE' should read 'SPARSE'.","section":"Title"},{"comment":"The subsection titled 'Proof of Theorem 1' actually gives a heuristic derivation of Proposition 1; Proposition 1 itself is not formally proved, and the local GOE coupling used there is not rigorously specified.","section":"§2.2"},{"comment":"The stochastic differential equation for X_i^{(q)} is introduced without derivation, without a precise definition of the Brownian motions B_{ij}, and without a justification of the error term E_i(t); these objects need to be defined for the argument to be checkable.","section":"§5.1, Step 2"},{"comment":"The 'quantitative Berry-Esseen lower bound' cited to [2] and [5] is not a standard result in those references and is not stated or proved in the paper; if it is a new lemma, it must be proved.","section":"§5.2 and references [2], [5]"},{"comment":"The bound ||L^{-1}|| = O(N^{2/3-epsilon}) on the complement of constants is cited to [4] and [6], but those references concern different operators and settings; the paper should state precisely which theorem applies to the switching operator on regular graphs and why.","section":"References [4], [6]"},{"comment":"The arithmetic in equations (4)-(5) is incorrect, although the resulting bound C d/N is actually at least as strong as C sqrt(d)/N^{5/6-epsilon} for d <= N^{1/3}; this should be corrected locally.","section":"§3.2, Theorem 4"}],"recommendation":"reject","confidential_remarks":"The manuscript depends on the author's own unpublished preprint [12] for the fixed-degree case and invokes it without stating the precise theorem or proof needed; the editor may wish to verify that this does not create a circularity in the literature. There is also a pattern of citing standard references for nonstandard statements (the Berry-Esseen lower bound in §5.2 and the operator norm bound on L^{-1} in Proposition 4); I would recommend checking these citations carefully if a revised version is submitted."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"First, the thesis is worth taking seriously. The theorems are new: previous work had d^3 or fixed degree bounds, and the sqrt(d) rate is the natural variance heuristic. The paper is clearly organized and transparent about relying on the author's fixed-degree preprint. But the proof as written fails at the load-bearing step, and the failures are not cosmetic.\n\nThe edge local law, Theorem 3, is the foundation. Its proof in §2.3 asserts \"m(z) ≈ msc(z) + O(√d N^{-5/6+ε}) by self-consistency.\" That is exactly the bound the theorem must prove; no earlier proposition supplies it. Then the remainder estimate is arithmetically wrong. Proposition 1 gives |⟨q,R⟩| ≤ C√d/(Nη). At the admissible endpoint η = N^{-2/3}, this is C√d N^{-1/3}, not C√d N^{-5/6+ε}. The missing N^{1/2} is not recoverable, because the line \"≤ C√d/N^{5/6-ε}\" just changes the exponent without a new factor. Sections 3–5 all consume this theorem, so the main eigenvector statements do not follow.\n\nThe problems are systematic. Lemma 1's variance proxy is internally inconsistent: the computation yields V = C/(Nη^4), and with η = N^{-2/3} that is C N^{5/3}, not O(d/N). The claimed Lipschitz constant L = O(1/(√d η^2)) is also far from O(√(d/N)) at the edge. Proposition 6's cumulant cancellation is algebraically false: dividing κ2 = O(d N^{-1/3+ε}) by σ^2 = 1+O(d^{-1}) cannot remove the d factor; the bound stays O(d N^{-1/3+ε}), which ruins the Berry-Esseen rate. Lemma 5's lower bound also does not work: the expansion gives κ4 = O(1/(d N^{1/3})), which is much smaller than the claimed √d N^{-1/6}; choosing q supported on d coordinates only makes it worse. Theorem 4's optimization has a similar substitution error, and the stated minimum is not what the formula yields.\n\nCredit where due: the paper identifies a plausible rate and a reasonable overall architecture—variance-sensitive concentration, vector resolvent, Stein-type cumulant control. But the architecture is not implemented. The local law, the variance proxy, the cumulant cancellation, and the lower bound all need real work, not editing.\n\nWho should read this? Someone tracking the sparse-to-dense transition might skim it for the conjecture and the discussion around d≈N^{1/4}. It should not be sent to peer review in this form. A desk reject with an invitation to resubmit once the local law is genuinely proved, with correct arithmetic, is the right call.","headline":"The new sqrt(d) theorem statements are plausible but the proof's local law is circular and arithmetically wrong; desk reject.","tokens_in":11284,"tokens_out":4831,"would_cite":false,"duration_ms":54110,"reading_group":"no","serious_thinker":"no","would_accept_peer_review":false},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["05C80","60B20","60F05"],"pacs":[],"model":"deepseek-v4-flash","headline":"Random regular graph eigenvector projections are Gaussian to error O(sqrt(d) N^{-1/6+eps}), and this rate is optimal.","keywords":["random regular graphs","eigenvector universality","Berry-Esseen bound","edge local law","resolvent concentration","sparse random matrices","degree dependence","fourth cumulant"],"falsifier":"Simulate random $d$-regular graphs with $d=N^{0.2}$ and $N$ around $10^5$, choosing $q$ with $d$ equal nonzero coordinates, and measure the sup-norm distance between the empirical distribution of $\\sqrt{N}\\langle q,u_2\\rangle$ and $\\Phi$. If the distance grows faster than $\\sqrt{d}\\,N^{-1/6+\\varepsilon}$, or if the fourth cumulant of the overlap does not stay bounded below by a constant times $\\sqrt{d}\\,N^{-1/6}$, the claimed sharp rate is wrong.","tokens_in":10158,"feed_emoji":"🎲","tokens_out":13515,"duration_ms":131288,"temperature":0.7,"pith_summary":"This paper studies the second eigenvector $u_2$ of the normalized adjacency matrix of a uniformly random $d(N)$-regular graph on $N$ vertices, with degree growing no faster than $N^\\kappa$, $\\kappa<1/4$. It claims that for every deterministic unit vector $q$ orthogonal to the all-ones vector, the projection $\\sqrt{N}\\langle q,u_2\\rangle$ is approximately Gaussian, with a Berry-Esseen error at most $C_\\kappa \\sqrt{d}\\,N^{-1/6+\\varepsilon}$. Previous bounds carried a factor $d^3$, so this would place the sparse-to-moderately-dense regime in the same $N^{-1/6}$ universality class as fixed degree, with only a square-root dependence on the degree. The paper also states a matching lower bound, asserting that no smaller degree-dependent prefactor is possible.","feed_headline":"Square-root degree bound is optimal for edge eigenvectors","feed_subtitle":"New Berry-Esseen bound for d-regular graphs cuts the d^3 rate to sqrt(d) and proves no faster rate is possible.","key_machinery":"The argument rests on a sharp edge local law for the resolvent $G(z)=(\\tilde H-z)^{-1}$: for $z$ in the edge window, $\\langle q,G(z)q\\rangle$ is within $O(\\sqrt{d}\\,N^{-5/6+\\varepsilon})$ of the Stieltjes transform $m_{\\rm sc}(z)=(-z+\\sqrt{z^2-4})/2$ of the semicircle law. The mechanisms are refined martingale concentration for the configuration model, a vector-outlier resolvent equation $v=-q/(z+m(z))+R$ with a small remainder, and a variance-normalization step in the cumulant computation that cancels the apparent factor of $d$. The interpolation parameter $s(t)=\\sqrt{dt/N}$ balances the graph-structure error against the error from a Gaussian (GOE) evolution, producing the claimed $O(\\sqrt d\\,N^{-5/6+\\varepsilon})$ bound.","core_discovery":"The core claim is Theorem 1: for $3\\le d(N)\\le N^\\kappa$ with $\\kappa<1/4$, any deterministic unit $q\\perp e$ satisfies $\\sup_x |\\mathbb{P}(\\sqrt{N}\\langle q,u_2\\rangle \\le x)-\\Phi(x)|\\le C_\\kappa\\sqrt{d}\\,N^{-1/6+\\varepsilon}$. Theorem 2 asserts the matching lower bound $\\sup_{q\\perp e,\\|q\\|=1}\\sup_x |\\cdots|\\ge c\\sqrt{d}\\,N^{-1/6}$ once $d\\ge C\\log N$. Together the two theorems identify $\\sqrt{d}\\,N^{-1/6}$ as the sharp scaling of edge eigenvector universality, meaning the $N^{-1/6}$ fluctuation rate of the edge survives degree growth and the degree enters only through its square root.","pith_inferences":["Because the concentration lemma is stated for general resolvent entries, the same variance-sensitive argument would likely improve degree dependence for bulk eigenvector fluctuations, not just the edge second eigenvector.","The lower-bound construction singles out vectors $q$ supported on $d$ equal-size coordinates; a direct numerical check of the fourth cumulant for such $q$ should show the predicted $\\approx c/(d N^{1/3})$ scaling and would test the mechanism independently of the full theorem.","The paper conjectures a new universality class near $d=N^{1/2+o(1)}$ with error $O(d N^{-2/3})$; if correct, simulations across that degree range should exhibit a crossover from $\\sqrt{d}N^{-1/6}$ to $dN^{-2/3}$ scaling."],"forward_implications":["The edge local law gives uniform resolvent concentration in the window $|E-2|\\le N^{-2/3+\\varepsilon}$, so other edge statistics of random regular graphs inherit the same $\\sqrt{d}$ error.","The previous $d^3$ dependence for eigenvector projections is replaced by $\\sqrt{d}$, extending quantitative eigenvector universality to degrees as large as $N^{1/4}$.","The matching lower bound implies the $\\sqrt{d}$ prefactor is unavoidable for any Gaussian approximation at this scale, not an artifact of the proof."],"supporting_citations":[{"why":"Prior fixed-degree quantitative eigenvector universality result that this paper extends to growing degree.","marker":"[12]"},{"why":"Sparse-matrix eigenvector statistics result giving the d^3 Berry-Esseen bound that this paper improves.","marker":"[1]"},{"why":"Dynamical random-matrix approach whose martingale concentration framework is adapted for the local law.","marker":"[6]"},{"why":"Vector-based resolvent technique used to avoid iterative bootstrap in the local-law proof.","marker":"[9]"},{"why":"The normal-approximation equation used to control higher-order fluctuations of the overlap.","marker":"[13]"},{"why":"Spectral gap of the switching chain used to bound the inverse operator in the cumulant computation.","marker":"[4]"},{"why":"Local Gaussian-ensemble approximation used to compare graph and GOE spectral statistics near the edge.","marker":"[8]"},{"why":"Fourth-cumulant Berry-Esseen lower bound used to prove the matching lower bound.","marker":"[5]"},{"why":"Configuration-model edge-exposure process underlying the martingale concentration lemma.","marker":"[3]"}],"fun_headline_variants":["Optimal sqrt(d) bound for edge eigenvector universality","Square-root degree rate proven optimal for edge eigenvectors","Sharp sqrt(d) error rate for sparse random regular graphs","Best possible N^{-1/6} sqrt(d) bound for eigenvector universality","Optimal degree scaling for edge eigenvector Gaussianity"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"Near the spectral edge, the proof's self-consistency step assumes that the deterministic approximation to the resolvent is already accurate to the error $O(\\sqrt{d}\\,N^{-5/6+\\varepsilon})$ that the local law is supposed to prove; if that accuracy is not established independently, the bound does not follow.","fun_headline_variants_meta":{"raw":{"variants":["Optimal sqrt(d) bound for edge eigenvector universality","Square-root degree rate proven optimal for edge eigenvectors","Sharp sqrt(d) error rate for sparse random regular graphs","Best possible N^{-1/6} sqrt(d) bound for eigenvector universality","Optimal degree scaling for edge eigenvector Gaussianity"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000448,"raw_usage":{"total_tokens":2242,"prompt_tokens":910,"completion_tokens":1332,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":526,"completion_tokens_details":{"reasoning_tokens":1247}},"tokens_in":526,"tokens_out":1332,"duration_ms":11845,"temperature":1.0,"reasoning_tokens":1247,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-06T16:15:08.843878+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Simulate random $d$-regular graphs with $d=N^{0.2}$ and $N$ around $10^5$, choosing $q$ with $d$ equal nonzero coordinates, and measure the sup-norm distance between the empirical distribution of $\\sqrt{N}\\langle q,u_2\\rangle$ and $\\Phi$. If the distance grows faster than $\\sqrt{d}\\,N^{-1/6+\\varepsilon}$, or if the fourth cumulant of the overlap does not stay bounded below by a constant times $\\sqrt{d}\\,N^{-1/6}$, the claimed sharp rate is wrong.","supporting_citations":[{"cited_title":"Quantitative Edge Eigenvector Universality for Random Regular Graphs: Berry-Esseen Bounds with Explicit Constants","cited_arxiv_id":"2507.12502","evidence_quote":"Prior fixed-degree quantitative eigenvector universality result that this paper extends to growing degree."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Sparse-matrix eigenvector statistics result giving the d^3 Berry-Esseen bound that this paper improves."},{"cited_title":"and Yau, H.-T.: A dynamical approach to random matrix theory","cited_arxiv_id":null,"evidence_quote":"Dynamical random-matrix approach whose martingale concentration framework is adapted for the local law."},{"cited_title":"A., Bourgade, P.: Extreme gaps between eigenvalues of random matrices","cited_arxiv_id":null,"evidence_quote":"Vector-based resolvent technique used to avoid iterative bootstrap in the local-law proof."},{"cited_title":"In Proceedings of the Sixth Berkeley Symposium on Math- ematical Statistics and Probability , Vol","cited_arxiv_id":null,"evidence_quote":"The normal-approximation equation used to control higher-order fluctuations of the overlap."},{"cited_title":"M., and Richthammer, T.: Proof of Aldous’ spectral gap conjecture","cited_arxiv_id":null,"evidence_quote":"Spectral gap of the switching chain used to bound the inverse operator in the cumulant computation."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Local Gaussian-ensemble approximation used to compare graph and GOE spectral statistics near the edge."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Fourth-cumulant Berry-Esseen lower bound used to prove the matching lower bound."},{"cited_title":"European J","cited_arxiv_id":null,"evidence_quote":"Configuration-model edge-exposure process underlying the martingale concentration lemma."}],"review_version":1}