{"id":"85181451-d4c5-4bf0-bd0f-6a303e01760c","arxiv_id":"2507.10193","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":7.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":1,"one_line_summary":"For Haar-distributed U(N) matrices, the gap-ratio distribution approaches the sine-kernel limit with a leading correction of order N^{-4}, and this cancellation explains the (log(T/2π))^{-3} deviation seen in Riemann zeta zero spacing ratios.","lead":"This paper computes how the distribution of ratios of consecutive level spacings of random unitary matrices approaches its universal limit, finding a surprising cancellation of the leading finite-size correction. The result is used to explain why the corresponding statistic for the Riemann zeta zeros deviates from random matrix theory as a particular power of log T.","discovery_kind":"new_application","skeptic_critique":{"model":"deepseek-v4-flash","headline":"The central claim that the O(N^{-2}) correction to the CUE gap-ratio distribution vanishes is supported only by numerical scaling for N=8–16; no analytic derivation or error bound is given, so the O(N^{-4}) leading-order claim is not yet established.","rationale":"The reader's weakest_assumption concerns the conjectured Riemann-zeta kernel (41)–(42), and that is indeed a fragile link in the zeta application. However, the more load-bearing point is the CUE cancellation itself: the paper's central mathematical claim is that P_r(r) = P_r^{(0)}(r) + N^{-4} P_r^{(4)}(r) + O(N^{-6}), and Sec. 5.2's explanation of the (log T)^{-3} scaling depends on that cancellation being exact. The paper provides convincing internal consistency checks for the underlying Jánossy density, notably the agreement between the TW integration and the Nyström-type Fredholm determinant in (34), and the nearest-neighbor distribution in Sec. 4.3 has prior support. But the O(N^{-2}) cancellation is only exhibited numerically for N up to 16, with no analytic derivation and no remainder estimate. The fitted exponent in Fig. 6, -3.081 rather than -3, also suggests that the zeta-side analysis is not at the precision where the prediction can be confirmed cleanly. These considerations reinforce the reader's CONDITIONAL verdict rather than overturning it: the central CUE claim is plausible and internally consistent, but it is not yet proven, and the zeta conclusion inherits that uncertainty. A direct analytic computation of P_r^{(2)} from the TW system would settle the issue and is the natural next step.","tokens_in":14612,"tokens_out":4083,"duration_ms":51993,"concrete_test":"Insert the 1/N^2 expansion P_c(a,b) = P_c^{(0)}(a,b) + N^{-2} P_c^{(2)}(a,b) + N^{-4} P_c^{(4)}(a,b) into the TW system (32)–(33), use (39)–(40) to compute the corresponding P_r^{(2)}(r), and solve the resulting linearized ODEs at several representative r values. If P_r^{(2)}(r) is not identically zero, the central cancellation claim is refuted; if it is identically zero, the claim is established analytically rather than inferred from the N=8–16 collapse.","verdict_should_be":"UNCHANGED","load_bearing_attack":"Section 4.4 (Fig. 3) demonstrates the N^4 scaling of P_r(r) - P_r^{(0)}(r) for N = 8, 10, 12, 14, 16, and the text explicitly states that the mechanism of the cancellation is obscure. This is an inference from the collapse of five curves, not a consequence derived from the TW system. If P_r^{(2)}(r) were merely small or if O(N^{-6}) terms were comparable at these N values, the same plot could be misread as an exact cancellation. The Riemann-zeta explanation in Sec. 5.2 assumes the cancellation is exact, so the numerical evidence carries the full weight of the paper's second conclusion. What is missing is an analytic expansion of the TW system (32)–(33) through order N^{-2}, or at least a direct computation of P_r^{(2)}(r) from the joint distribution (39)–(40). Without that, the headline claim remains a plausible numerical observation rather than a demonstrated result.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper develops a Tracy-Widom (TW) PDE framework for the Jánossy density of the circular unitary ensemble CUE_N and uses it to compute the joint distribution of two consecutive spacings P_c(a,b), the nearest-neighbor spacing distribution P_nn(t), and the gap-ratio distribution P_r(r). The central claim is that the leading finite-N correction to P_r relative to the sine-kernel limit is O(N^{-4}), because the would-be O(N^{-2}) correction vanishes exactly; this is supported numerically by an N^4 scaling collapse for N=8..16. The paper then applies the same logic to the Riemann zeta zeros, arguing that the observed (log(T/2π))^{-3} scaling of the gap-ratio deviation follows from the conjectured finite-size kernel (41)–(42) for zeta zeros combined with the CUE_N cancellation. The manuscript includes a detailed derivation of the TW system, high-precision cross-checks against Nyström quadrature, and a Mathematica notebook.","tokens_in":14852,"tokens_out":8203,"duration_ms":89911,"significance":"If the claimed cancellation is correct, the result is of genuine interest: the gap ratio becomes a sharper probe of finite-size effects than the spacing distributions, since its leading correction is pushed to O(N^{-4}). The paper also provides a plausible mechanism linking this cancellation to the observed zeta-zero gap-ratio scaling. Strengths include the systematic TW derivation, the internal cross-check against Nyström quadrature at the 10^{-36} level for N=10, the availability of a reproducible notebook, and the clean scaling collapse in Fig. 3. The main weakness is that the load-bearing cancellation P_r^{(2)}(r)=0 is asserted from numerical evidence only, without an analytic derivation or error bound; the zeta conclusion is therefore conditional on an unproved structural fact, and the fitted exponent in Fig. 6 is not statistically tied to the claimed value -3.","major_comments":[{"comment":"The central claim P_r^{(2)}(r)=0 is not derived. The text states that the cancellation mechanism is 'obscure from the structure of the PDEs', and the only support is the N^4 scaling collapse of five curves with N=8,10,12,14,16. Since P_r is defined in Eq. (40) as an integral of P_c, and since Fig. 2 already shows a well-defined N^{-2} correction to P_c, it should be possible to compute P_r^{(2)}(r) = ∫_0^{2π/(1+r)} b P_c^{(2)}(-rb,b) db from the N^{-2} expansion of the TW system (32)–(33). Please provide that computation, or a rigorous argument showing the integral vanishes, and quantify the contamination from N^{-4} and N^{-6} terms at the plotted values of N.","section":"§4.4, Fig. 3 and Eqs. (39)–(40)"},{"comment":"The empirical support for the claimed exponent -3 is weaker than stated. The optimal fit is reported as 0.1896 N_e^{-3.081}, i.e. a fitted exponent of -3.081, with no uncertainties and only six points. This does not distinguish -3.081 from -3, and it is not a test of the predicted exact exponent. Please fit with the slope fixed at -3 and report the residuals or a goodness-of-fit statistic, or provide error bars; otherwise the conclusion that the data confirm (log T)^{-3} scaling is not quantitatively supported.","section":"§5.2, Fig. 6"},{"comment":"The zeta argument is presented as a consequence of the CUE cancellation plus the conjectured kernel (41)–(42), but no predicted coefficient for the O(N_e^{-3}) gap-ratio correction is computed from that kernel. Fig. 5 shows only noisy histograms, and the footnote simultaneously claims a match to CUE_N up to O(N_e^{-4}) while incorporating an O(N_e^{-3}) term, which is confusing. Please derive the explicit N_e^{-3} contribution to P_{RZ,r} from (41)–(42) and compare it to the histograms with a quantitative test, and clarify the order to which the adjusted kernel is matched.","section":"§5.2, Fig. 5 and the footnote on p. 14"}],"minor_comments":[{"comment":"The unfolded variables are denoted by the same symbols x,y as the original eigenphases; please state explicitly that after the map x ↦ (2/N)x the variables are rescaled, to avoid ambiguity.","section":"§2, Eq. (2)"},{"comment":"The text 'WorkingPrecision → 5 MachinePrecision' is unclear; specify the actual precision setting used in the NDSolve integrations.","section":"§4.1"},{"comment":"In the sentence introducing the 1/N^2 expansion of P_r, the variable t is used for P_r^{(2)}(t) and P_r^{(4)}(t); the argument should be r throughout.","section":"§4.4"},{"comment":"The fit in Fig. 6 is described as an 'optimal linear fit'; on a log-log plot this is a power-law fit, not a linear fit, and the wording should be corrected.","section":"§5.2 and Fig. 6"},{"comment":"The footnote appears internally inconsistent: it claims the kernel can be adjusted to match the CUE_N expansion up to O(N_e^{-4}) while also incorporating an O(N_e^{-3}) term with ar α = 1 + Q/(√3 Λ^{3/2} N_e); please clarify whether the matching order is O(N_e^{-3}) or O(N_e^{-4}).","section":"§5.2, footnote"},{"comment":"Ref. [8] is given as Phys. Rev. E 110, 084101 (2013), but the cited article appears to be Phys. Rev. Lett. 110, 084101 (2013); Ref. [9] lists 'P. Vivo, and E. Vivo' as authors, which looks like an error. Please verify the author lists and journal names.","section":"References [8] and [9]"},{"comment":"There are several minor typos, including 'taylored' in §4.1 and 'deviations of from' in §5.2; a careful proofreading pass is needed.","section":"Throughout"}],"recommendation":"major_revision","confidential_remarks":"For the editor: the paper is built on the author's own previous framework [12,13], and the TW derivation in Sec. 3 is largely a re-derivation with corrections to Tracy–Widom's original equations; the genuinely new content is the observed cancellation and its zeta application. If the cancellation cannot be proved, the paper would amount to a numerical observation of a scaling limit, which is probably below the bar for a math-ph journal. The zeta section also rests on external conjectures [3,4], so the paper's second conclusion is inherently conditional; this is acceptable if framed as such, but the missing quantitative comparison with the predicted O(N_e^{-3}) coefficient is a gap that should be addressed in revision."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Know this: the headline claim is that the O(N^{-2}) finite-size correction to the CUE gap-ratio distribution vanishes, so the first correction is O(N^{-4}), and the author uses that to explain the (log T)^{-3} scaling of the zeta zeros' gap-ratio deviation. The first claim is backed by a clean scaling collapse for N=8–16; the second is a plausible heuristic built on a conjectured zeta kernel.\n\nThe genuinely new pieces are the explicit N^{-4} correction for the gap-ratio and the observation that this cancellation makes the gap-ratio statistic sensitive to smaller kernel corrections than the spacing distribution. The author is careful about the TW system, and the cross-check against Nyström quadrature to 10^{-36} precision is real evidence that the machinery is implemented correctly. The Mathematica notebook is a nice addition.\n\nWhere it gets softer: the cancellation of P_r^{(2)} is not derived. The paper says the mechanism is obscure, and the only support is the N^4 scaling of five curves. That's reasonable numerical evidence but not a proof; a referee should ask for an analytic expansion of the TW system to order N^{-2}, or at least a quantitative bound on the N^{-2} coefficient. The zeta part is also heuristic: it relies on a conjectured kernel from Bogomolny et al., and the fit exponent is -3.081, not -3. The paper does not fully explain that discrepancy. These are real gaps, but they are not fatal to the main CUE result, which looks credible.\n\nWho benefits: people working on finite-size corrections in random matrix theory, and anyone using gap ratios to probe quantum chaos. The paper deserves a serious referee, and I would send it out. The referee should push on the cancellation proof, and on the zeta fit discrepancy, before acceptance.","headline":"The paper's claim that the O(N^{-2}) CUE gap-ratio correction vanishes is a strong, non-obvious result supported only numerically, and the zeta explanation is heuristic; still, it deserves a careful referee.","tokens_in":15388,"tokens_out":2941,"would_cite":true,"duration_ms":33643,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["60B20","15B52","11M06"],"pacs":[],"model":"deepseek-v4-flash","headline":"For Haar-random unitary matrices, the gap-ratio distribution has no $O(N^{-2})$ finite-size correction; the first correction is $O(N^{-4})$, and this cancellation explains the $(\\log(T/2\\pi))^{-3}$ deviation of Riemann zeta zero gap ratios.","keywords":["circular unitary ensemble","gap-ratio distribution","Tracy-Widom equations","Janossy density","finite-size corrections","Riemann zeta zeros","sine kernel","consecutive level spacings"],"falsifier":"A direct symbolic or very-high-precision numerical computation of the coefficient $P_r^{(2)}(r)$ from the large-$N$ expansion of the Tracy-Widom system: if $N^2(P_r(r)-P_r^{(0)}(r))$ has a nonzero scaling limit as $N\\to\\infty$, the claimed cancellation is false. For the zeta part, a gap-ratio histogram at larger $T$ whose deviation scales as $N_e^{-2}$ rather than $N_e^{-3}$ would falsify the kernel-based explanation.","tokens_in":14417,"feed_emoji":"🎲","tokens_out":11981,"duration_ms":130711,"temperature":0.7,"pith_summary":"The paper aims to establish a precise finite-size statement for the circular unitary ensemble: the ratio of consecutive level spacings, unlike the spacings themselves, has no correction of order $N^{-2}$ relative to the universal sine-kernel limit, so the first correction is $O(N^{-4})$. This matters because the gap ratio is the commonly used statistic for detecting quantum chaos, and a statistic that skips the first finite-size order can expose deeper corrections in a spectrum. The author further argues that this cancellation is exactly what makes Riemann zeta zero data show deviations from random-matrix predictions that scale as $(\\log(T/2\\pi))^{-3}$. A sympathetic reader would take the paper as giving the analytic CUE distributions and a new explanation for the zeta-zero finite-size scaling.","feed_headline":"Cancellation removes N^-2 term in level ratio statistic","feed_subtitle":"For Haar-random unitary matrices the first ratio correction is N^-4, matching zeta zeros' (log T)^-3 deviations.","key_machinery":"The central object is the CUE Janossy density $\\tilde J_1(0; [a_1,a_2])$, the probability that an interval contains no eigenphase except one fixed at $0$; it is a Fredholm determinant of the conditioned CUE kernel. The machinery is the Tracy-Widom system of nonlinear PDEs that gives the dependence of this determinant on the interval endpoints, supplemented by boundary conditions from the Neumann expansion. All spectral statistics studied here, including the gap-ratio distribution, are integrals of derivatives of this Janossy density, and the system's $N\\mapsto -N$ symmetry after unfolding makes every derived distribution analytic in $1/N^2$, which is why the absence of the $N^{-2}$ term in $P_r$ is a meaningful cancellation.","core_discovery":"The paper derives, from the Tracy-Widom system for the CUE_J\\'anossy density, the nearest-neighbor spacing distribution $P_{nn}(t)$, the joint distribution $P_c(a,b)$ of two consecutive spacings, and the gap-ratio distribution $P_r(r)$. Its central finding is that as $N\\to\\infty$, $P_c$ has the expected leading finite-size correction of order $N^{-2}$, but $P_r$ does not: the coefficient $P_r^{(2)}(r)$ vanishes, so $P_r(r)=P_r^{(0)}(r)+N^{-4}P_r^{(4)}(r)+O(N^{-6})$. The paper then uses the conjectured Riemann-zero kernel, which matches the CUE kernel to $O(N_e^{-2})$ and differs at $O(N_e^{-3})$, together with the effective size $N_e(T)\\sim\\log(T/2\\pi)$, to show that the gap-ratio deviation of the zeta zeros should scale as $N_e^{-3}=(\\log(T/2\\pi))^{-3}$, consistent with histograms of hundreds of millions of zeros.","pith_inferences":["The mechanism of the cancellation is left open by the paper; a likely target for a proof is an algebraic identity in the Tracy-Widom system showing that $P_r^{(2)}(r)$ integrates to zero for all $r$, which would turn a numerical observation into a theorem.","The same ratio-based cancellation may occur for other circular ensembles with integrable kernels sharing the $N\\mapsto -N$ symmetry, giving a testable prediction for circular orthogonal and symplectic ensembles and for finite-$N$ spectra of many-body systems.","Because the zeta kernel differs from CUE at $O(N_e^{-3})$, the gap-ratio statistic is a natural place to search for arithmetic effects in zeta-zero data at higher heights, provided samples are large enough to resolve the $N_e^{-3}$ shape.","The method could be extended to ratios of next-nearest spacings or higher-order gap ratios, where analogous cancellations may appear at higher orders and would show how generic the $N^{-2}$ suppression is."],"forward_implications":["For CUE_N, $P_r^{(2)}(r)=0$ means the gap-ratio statistic is insensitive to the leading finite-size part of the kernel, so comparisons of spectra with theory can be pushed to $O(N^{-4})$.","Riemann zeta zero gap-ratio deviations from the sine kernel should be dominated by $N_e^{-3}\\propto(\\log(T/2\\pi))^{-3}$ rather than by $N_e^{-2}$, which the paper's histograms support and larger data sets can test more sharply.","The joint distribution $P_c(a,b)$ retains the $O(N^{-2})$ correction, so comparing $P_c$ and $P_r$ between a given spectrum and CUE_N distinguishes kernels that match to first order from those matching to second order.","The Tracy-Widom/J\\'anossy method yields explicit finite-$N$ expressions for $P_{nn}(t)$ and $P_c(a,b)$, providing benchmarks for numerical spectra of quantum-chaotic systems.","Because the would-be $N^{-2}$ term vanishes, any observed $N^{-2}$ deviation in a gap-ratio measurement would signal a kernel genuinely different from CUE at that order, rather than mere finite-size noise."],"supporting_citations":[{"why":"Supplies the Tracy-Widom theorem that Fredholm determinants of integrable kernels obey a closed PDE system; the paper reworks and corrects the exponential-variant case.","marker":"[14]"},{"why":"Introduces the new statistic and gives the sine-kernel nonlinear equation for the nearest-neighbor spacing distribution that the paper extends to finite $N$ and to gap ratios.","marker":"[11]"},{"why":"The author's prior ab initio derivation of consecutive-spacing and ratio distributions for the GUE, including the sine-kernel limit $P_r^{(0)}(r)$ used here as the baseline.","marker":"[13]"},{"why":"Earlier Janossy-density/Tracy-Widom method for Airy, Bessel, and sine kernels; the SL(2,C) gauge transformation that puts the CUE kernel in integrable form comes from this line of work.","marker":"[12]"},{"why":"Conjectures the finite-size kernel for Riemann zeros and the effective size $N_e(T)$ used in Section 5; the paper's zeta-zero comparison relies on this kernel.","marker":"[3]"},{"why":"Matches the pair correlation function to fix the effective size and finite-size corrections in the Riemann zero data, strengthening the kernel identification.","marker":"[4]"},{"why":"Earlier finite-size computations for circular ensembles and Riemann zeros, including the nearest-neighbor spacing plots and the $O(N_e^{-3})$ kernel adjustment that the gap-ratio analysis extends.","marker":"[5]"}],"fun_headline_variants":["N^-2 term cancels in level ratio, leaving N^-4","Ratio statistic evades N^-2, matches zeta zeros' log scaling","Gap-ratio correction jumps to N^-4, mirroring zeta deviation","Why ratio statistic beats spacings: N^-4 correction emerges","Cancellation yields N^-4 in ratio, tying to Riemann zeros"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"On the zeta side, the load-bearing premise is that Riemann zero spacings are described by the conjectured finite-size kernel (41) with effective size $N_e(T)=(\\log(T/2\\pi))/\\sqrt{12\\Lambda}$; if this identification is wrong the $(\\log T)^{-3}$ explanation fails, while the CUE cancellation claim itself stands independently.","fun_headline_variants_meta":{"raw":{"variants":["N^-2 term cancels in level ratio, leaving N^-4","Ratio statistic evades N^-2, matches zeta zeros' log scaling","Gap-ratio correction jumps to N^-4, mirroring zeta deviation","Why ratio statistic beats spacings: N^-4 correction emerges","Cancellation yields N^-4 in ratio, tying to Riemann zeros"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000231,"raw_usage":{"total_tokens":1497,"prompt_tokens":967,"completion_tokens":530,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":583,"completion_tokens_details":{"reasoning_tokens":434}},"tokens_in":583,"tokens_out":530,"duration_ms":5332,"temperature":1.0,"reasoning_tokens":434,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-06T17:37:13.266952+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"A direct symbolic or very-high-precision numerical computation of the coefficient $P_r^{(2)}(r)$ from the large-$N$ expansion of the Tracy-Widom system: if $N^2(P_r(r)-P_r^{(0)}(r))$ has a nonzero scaling limit as $N\\to\\infty$, the claimed cancellation is false. For the zeta part, a gap-ratio histogram at larger $T$ whose deviation scales as $N_e^{-2}$ rather than $N_e^{-3}$ would falsify the kernel-based explanation.","supporting_citations":[{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Supplies the Tracy-Widom theorem that Fredholm determinants of integrable kernels obey a closed PDE system; the paper reworks and corrects the exponential-variant case."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Introduces the new statistic and gives the sine-kernel nonlinear equation for the nearest-neighbor spacing distribution that the paper extends to finite $N$ and to gap ratios."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"The author's prior ab initio derivation of consecutive-spacing and ratio distributions for the GUE, including the sine-kernel limit $P_r^{(0)}(r)$ used here as the baseline."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Earlier Janossy-density/Tracy-Widom method for Airy, Bessel, and sine kernels; the SL(2,C) gauge transformation that puts the CUE kernel in integrable form comes from this line of work."},{"cited_title":"Bogomolny, O","cited_arxiv_id":null,"evidence_quote":"Conjectures the finite-size kernel for Riemann zeros and the effective size $N_e(T)$ used in Section 5; the paper's zeta-zero comparison relies on this kernel."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Matches the pair correlation function to fix the effective size and finite-size corrections in the Riemann zero data, strengthening the kernel identification."},{"cited_title":"Bornemann, P","cited_arxiv_id":null,"evidence_quote":"Earlier finite-size computations for circular ensembles and Riemann zeros, including the nearest-neighbor spacing plots and the $O(N_e^{-3})$ kernel adjustment that the gap-ratio analysis extends."}],"review_version":1}