{"id":"84b2e882-5d58-4551-a1f5-f2afb709bd23","arxiv_id":"2505.09865","paper_version":2,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":7.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"For circular beta ensembles with beta = 1, 2, 4 and even beta, bulk-scaled correlations and spacing distributions expand in even powers of 1/N, with leading corrections given by a second derivative of the limiting form.","lead":"Large random matrices have universal patterns in how their energy levels repel each other. This paper shows that for several important symmetry classes, the first finite-size correction to those patterns is exactly a simple derivative of the infinite-size pattern.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Prop. 3.2 is false as stated: with E±1 defined as det·Tr in the text, the small-s signs of (3.11) are opposite, so the β=1,4 spacing identities (3.6)/(3.12) are not established.","rationale":"The reader's weakest_assumption focused on the unshown CAS verification and the f±1 characterisation from [14, Prop. 5.10]. My independent check finds a more concrete problem in the same proposition: under the definitions printed in the text, Prop. 3.2 is false at the leading nontrivial order in a small-s expansion. The sign discrepancy is visible directly from (3.9)–(3.10) and the stated definition of E±1, independently of the f±1 computation. The final β=1,4 derivative identities might survive once E±1 is redefined with the correct sign and thinning parameter, so the appropriate verdict remains CONDITIONAL, matching the reader's verdict. I therefore leave the verdict unchanged, but the request for correction should explicitly ask the authors to fix the sign/parameter in the definition of E±1 and to re-verify (3.11) after that fix, in addition to supplying the omitted CAS verification for the f±1 analogue of (2.19).","tokens_in":27503,"tokens_out":47422,"duration_ms":449377,"concrete_test":"Carry out a symbolic small-s expansion of the operator definitions in (3.9)–(3.10) to order s³: compute Tr K±_L, Tr(K±_L)^2, Tr L±_L and Tr(K±_L L±_L) with L=s/2, and compare E±1(s)=−det(I−ξ̂K±_L)Tr((I−ξ̂K±_L)^{-1}ξ̂L±_L) against −(s²/6)(E±0)''. A direct numerical check is also decisive: for s=0.01 and ξ=0.5, evaluate det(I−ξ̂K+_L)Tr((I−ξ̂K+_L)^{-1}ξ̂L+_L) and −(s²/6)(d²/ds²)det(I−ξ̂K+_L) with a high-order Nyström discretisation; the stated identity should fail by a sign in the s³ term, confirming that the printed definition of E±1 needs correction.","verdict_should_be":"UNCHANGED","load_bearing_attack":"Section 3.2 has a sign inconsistency that invalidates Proposition 3.2 as written. Let L=s/2 and ξ̂=2ξ−ξ². From (3.9), E+0(s)=det(I−ξ̂K+_L)=1−ξ̂s+ξ̂π²s³/36+O(s⁵), so −(s²/6)(E+0)''=−ξ̂π²s³/36+O(s⁵). But the object E+1(s) defined immediately before Prop. 3.2 has leading term +ξ Tr L+_L = +ξπ²s³/36, and even if the trace parameter is corrected to ξ̂ the sign is positive, opposite to the RHS. For E−1 the same computation gives E−0=1−ξ̂π²s³/36, so the RHS of (3.11) is +ξ̂π²s³/36 while E−1 has leading term −ξπ²s³/36; again the signs oppose. Thus (3.11) cannot hold under the stated definitions: the definition of E±1 is missing an overall minus sign and should use ξ̂ rather than ξ in the trace, i.e. E±1(s)=−det(I−ξ̂K±_L) Tr((I−ξ̂K±_L)^{-1}ξ̂L±_L). Because Prop. 3.3 derives the β=4 identity from (3.11) via (3.13)–(3.14), both non-unitary spacing identities (3.6) and (3.12) are left without a valid proof as the text stands. The final derivative relations may be salvageable by correcting the sign, but the manuscript contains a false central proposition in its current form.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper investigates the large-N bulk-scaling expansion of circular beta-ensemble observables. For beta = 1, 2 and 4, it claims that n-point correlation functions, spacing-distribution generating functions, and structure functions admit asymptotic expansions in powers of 1/N^2, and that the first correction is, in each case, a second derivative of the limiting form: for beta = 2 this is P_{1,beta=2}^{bulk}(s;xi) = -(1/12) d^2/ds^2(s^2 P_{0,beta=2}^{bulk}(s;xi)) (Prop. 2.1, Eq. (2.8)), with analogous statements for beta = 1 and 4 (Props. 3.2 and 3.3, Eqs. (3.6) and (3.12)). For even beta, the two-point function is shown to have a 1/N^2 expansion with the leading correction again a second-derivative relation (Prop. 4.3, Eq. (4.16)). The proofs use sigma-Painleve characterizations, explicit Pfaffian kernel forms, Jack-polynomial hypergeometric functions, and Selberg integral identities. The paper also gives explicit differential identities for the spectral form factor and conjectures the general-beta analogue.","tokens_in":27823,"tokens_out":11290,"duration_ms":100150,"significance":"If the main identities hold, the paper establishes a remarkably simple universal structure for finite-size corrections in the bulk of circular beta ensembles: the leading 1/N^2 correction is obtained from the limiting distribution by a single second-derivative operation. Such results are of immediate use in interpreting empirical Riemann-zero spacing data and in guiding further asymptotic analysis. The paper has genuine strengths: explicit functional forms are given for the beta = 1, 2, 4 correlation kernels and structure functions; the even-beta two-point result is proved through a concrete Selberg-integral calculation with no fitted parameters; the general-beta conjectures are clearly labelled; and the differential identities for the structure function are tested against explicit small-tau expansions. The significance is, however, substantially weakened by the fact that one of the central propositions, Prop. 3.2, is false as stated, and the proof of the beta = 2 identity in Prop. 2.1 depends on an unspecified computer-algebra verification.","major_comments":[{"comment":"With E±1 defined in the text as det(I−ξK±_{s/2})Tr((I−ξK±_{s/2})^{-1}ξL±_{s/2}), the identity (3.11) fails already at order s^3. Writing L=s/2 and ξ̂=2ξ−ξ^2, the small-s expansion of the determinant in (3.9) gives E+0(s)=1−ξ̂s+ξ̂π^2s^3/36+O(s^5), so the right-hand side of (3.11) is −ξ̂π^2s^3/36+O(s^5), while E+1(s)=+ξπ^2s^3/36+O(s^5). For E−0 the expansion gives E−0(s)=1−ξ̂π^2s^3/36+O(s^5), so the right-hand side of (3.11) is +ξ̂π^2s^3/36+O(s^5), while E−1(s)=−ξπ^2s^3/36+O(s^5). Thus the signs oppose and the parameter in the trace term is ξ rather than ξ̂. The corrected definition should be E±1(s)=−det(I−ξ̂K±_{s/2})Tr((I−ξ̂K±_{s/2})^{-1}ξ̂L±_{s/2}), which is consistent with the small-s expansions and with (3.11). Since Prop. 3.3 derives the beta=4 identities (3.12) from (3.11) via (3.14), the spacing identities (3.6) and (3.12) are not established as the text stands.","section":"Section 2, proof of Prop. 2.1, Eq. (2.19)"},{"comment":"The key step of the proof of Prop. 2.1 is the claim σ1 = −(1/12)(2tσ0σ0′+t^2σ0′′), which is verified only by the sentence 'With the help of computer algebra, the latter can then be checked upon direct use of (2.14).' No reduced identity in {σ0,σ0′,σ0′′,t} is displayed and no computer-algebra artifact is provided. Because (2.19) is exactly the content of the proposition, the proof is incomplete as submitted. Please provide the explicit algebraic identity, or a verifiable worksheet, and also explain why the boundary condition (2.18) uniquely selects the solution of the second-order equation (2.17).","section":"Section 3.2, proof of Prop. 3.2"},{"comment":"The proof of Prop. 3.2 defers entirely to [14, Prop. 5.10] for the f±0/f±1 characterizations and then states that the resulting identity 'can be established following the procedure used to establish (2.19).' No analogue of the ODE system, the boundary conditions, or the elimination step is written down. In view of the sign error identified above, a deferred proof of this kind is not acceptable; the corrected statement requires a self-contained verification.","section":"Section 3.4"}],"minor_comments":[{"comment":"There is a typo 'invovling' and a missing space in 'denotedet' immediately before the definition of E±0.","section":"Section 3.3"},{"comment":"In Eq. (3.25) the word 'relection' should be 'reflection'.","section":"Section 3.3"},{"comment":"In Remark 2.1.2 there is a stray parenthesis in 'P1,β=2(s;ξ)|)'; the caption of Fig. 1 would benefit from axis labels.","section":"Section 2"},{"comment":"There is a missing space in 'generalk-point' in the first paragraph of Section 4.1.","section":"Section 4.1"},{"comment":"The digamma and harmonic-number asymptotics are cited to Wikipedia articles; the authors may prefer to cite standard handbooks (e.g., Abramowitz and Stegun, already reference [1]).","section":"References"},{"comment":"The factor 1/8 in E1,β=4 is explained in the text, but a one-line derivation of the relative normalization between the beta=1 and beta=4 traces would make the passage easier to follow.","section":"Section 3.3"}],"recommendation":"major_revision","confidential_remarks":"The sign/parameter inconsistency in Prop. 3.2 appears to be a correctable typo rather than a fundamental obstruction, but it is load-bearing: the beta=1,4 spacing identities are not proved in the submitted version. The proof of Prop. 2.1 also needs to be made verifiable by supplying the computer-algebra identity or a code artifact. The paper relies heavily on the authors' own prior work [14,35,76]; this is not circular, but the deferred proofs should be self-contained in the final version. Given the otherwise substantial and explicit results, I recommend major revision rather than rejection."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Bottom line: this is a real contribution, but Section 3.2 has a load-bearing error. The stress-test note lands. As E±0 and E±1 are defined just above Prop 3.2, the claimed identity (3.11) fails already at order s³. For the plus operator, E+0(s)=det(I−ξK+_{s/2})=1−ξs+ξπ²s³/36+O(s⁵), so −(s²/6)(E+0)″=−ξπ²s³/36+O(s⁵), whereas E+1(s)=det(I−ξK+_{s/2})Tr((I−ξK+_{s/2})⁻¹ξL+_{s/2}) has leading term +ξπ²s³/36. The minus operator gives the same sign reversal. So Prop 3.2 is false with the stated definitions, and Prop 3.3, which invokes it, loses its proof. I expect this is fixable — an overall minus sign and consistent ξ̂ in the trace would match the small-s behaviour — but as written the β=1,4 spacing identities (3.6) and (3.12) are unproved.\n\nThe rest of the paper is on much firmer ground. The β=2 result (2.8) is clean and well supported by the explicit small-s expansions (2.4)-(2.5); the Painlevé proof is plausible, with the usual caveat that the CAS verification of (2.19) is not displayed. The even-β two-point expansion and the second-derivative identity (4.16) are genuinely new, and the derivation from the β-dimensional integral is concrete. The structure function results for β=1,4 (Prop 3.4, (3.29)) follow from exact finite-N digamma expressions, so they do not inherit the spacing-defect. The general-β conjectures are labelled as conjectures and Appendix B gives supporting evidence. No fitted parameters, and the self-citations are to characterisations the new identities do not assume, so the circularity burden is low.\n\nMain soft spots, in order: the Section 3.2 sign error; the deferred CAS check and the \"analogous computation\" for β=1; the absence of code or a script to reproduce the CAS verification. None of these are deep — they're the difference between a paper that proves its headline claims and one that asserts them.\n\nReaders who care about finite-N corrections in circular ensembles, or the Riemann-zero interpretation, will get value from the β=2 and even-β parts now and the β=1,4 parts after repair. This paper deserves a serious referee, but the referee's first job is to make the authors fix Prop 3.2. I'd send it out, and in my own work I'd cite the β=2 and even-β results once the sign issue is acknowledged.","headline":"Genuinely new finite-size results for circular β ensembles, but Prop 3.2 is false as stated due to a sign error; the β=1,4 spacing proofs need rework, while β=2 and even-β parts look solid.","tokens_in":28409,"tokens_out":11689,"would_cite":true,"duration_ms":102341,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["15B52","60B20","34M55","33C70"],"pacs":[],"model":"deepseek-v4-flash","headline":"The leading finite-size correction to bulk eigenvalue statistics in circular beta ensembles is a second derivative of the limiting form, with constants -1/12, -1/6 and -1/24 for beta = 2, 1 and 4.","keywords":["circular beta ensembles","bulk scaling","finite-size corrections","spacing distributions","sigma-Painleve","structure function","Riemann zeros","random matrix theory"],"falsifier":"Compute the two sides of (2.9) independently: for a grid of $s\\in(0,3)$ and $\\xi=1$, evaluate $E_0(s;\\xi)=\\det(I-\\xi K_s)$ numerically with a high-accuracy Fredholm determinant routine, form $-\\frac{s^2}{12}E_0''$, and compare with $E_1$ from the trace formula (2.12) using the kernel (2.11); any discrepancy beyond the numerical tolerance rules out Proposition 2.1. A direct combinatorial check is to simulate CUE eigenangles at several $N$, form $(2\\pi/N)^2 P_N(2\\pi s/N;\\xi)$, subtract the known limit, multiply by $N^2$, and check that the curves converge to the right-hand side of (2.8).","tokens_in":27248,"feed_emoji":"📐","tokens_out":10972,"duration_ms":97074,"temperature":0.7,"pith_summary":"This paper asks whether the finite-size corrections seen in the bulk of circular $\\beta$ ensembles have any structure beyond being small. It proves that they do: for $\\beta$ = 1, 2 and 4, and for even $\\beta$ in the two-point case, the bulk-scaled correlation functions and spacing-distribution generating functions expand in powers of 1/$N^{2}$, and the leading correction is a fixed second derivative of the infinite-N limiting statistic. Explicitly, the coefficient of 1/$N^{2}$ is -1/(6 $\\beta$) times the second derivative of $s^{2}$ times the limit for the relevant statistic. This matters because the correction is then determined with no free parameters once the limiting distribution is known, turning large-N eigenvalue data, including empirically measured spacing distributions of the Riemann zeros at large height, into a quantitative test of the random-matrix prediction.","feed_headline":"The 1/N^2 correction is a second derivative of the bulk limit","feed_subtitle":"For circular beta ensembles, the N^{-2} term is fixed by a second derivative of the infinite-N statistic.","key_machinery":"The argument runs through three mechanisms. For $\\beta=2$, the leading spacing generating function is the Fredholm determinant $\\det(I-\\xi K_s)$ of the sine kernel, whose $\\sigma$-Painleve V tau-function representation supplies a differential equation; the correction $E_1$ is a trace formula involving the kernel $L_\\infty(x,y)=(\\pi(x-y)/6)\\sin(\\pi(x-y))$, and the paper verifies the derivative relation using a second-order linear ODE for the correction function $\\sigma_1$ together with computer algebra. For $\\beta=1,4$, Pfaffian kernels and classical-group identities reduce the generating functions to combinations of determinants of $K^\\pm_s = K_s \\pm$ reflected kernels, whose $\\sigma$-Painleve III-prime characterisations yield the same derivative identities. For even $\\beta$, the $n$-point correlation is written as a Jack-polynomial hypergeometric function, and the two-point case as a $\\beta$-dimensional Selberg integral; integration by parts on the integral extracts the $1/N^2$ coefficient and identifies it with the second derivative.","core_discovery":"The central discovery is a derivative identity tying the first nontrivial finite-size term to the bulk limit. For the circular unitary ensemble, Proposition 2.1 states $P^{\\mathrm{bulk}}_{1,\\beta=2}(s;\\xi) = -\\frac{1}{12}\\frac{d^2}{ds^2}\\left(s^2 P^{\\mathrm{bulk}}_{0,\\beta=2}(s;\\xi)\\right)$, equivalently $E^{\\mathrm{bulk}}_{1,\\beta=2}(s;\\xi) = -\\frac{s^2}{12}\\frac{d^2}{ds^2}E^{\\mathrm{bulk}}_{0,\\beta=2}(s;\\xi)$. The same relation holds with constants $-1/6$ and $-1/24$ for $\\beta=1$ and $\\beta=4$, and for even $\\beta$ the two-point correlation obeys $\\rho^{\\mathrm{bulk}}_{(2),1,\\beta}(x,0) = -\\frac{1}{6\\beta}\\frac{d^2}{dx^2}\\left(x^2 \\rho^{\\mathrm{bulk}}_{(2),0,\\beta}(x,0)\\right)$. For $\\beta=1,4$ the spectral form factor also has a $1/N^2$ expansion, with first and second corrections given by differential operators of the form $\\tau^2\\frac{d^2}{d\\tau^2}$ and $\\tau^4\\frac{d^4}{d\\tau^4}+8\\tau^3\\frac{d^3}{d\\tau^3}+12\\tau^2\\frac{d^2}{d\\tau^2}$ applied to the limiting form.","pith_inferences":["An implication the authors leave implicit: if the pattern holds to all orders, each finite-$N$ bulk statistic is completely determined by its $N=\\infty$ limit through a tower of even-order derivative operators; the recurrence in Appendix A is the natural seed for that expansion.","The thinning parameter $\\xi$ enters the correction only through the limiting function $P_0(s;\\xi)$. One can therefore test (2.8) cheaply by Monte Carlo: for $\\xi\\in(0,1)$, the difference $N^2(P_N-P_0)$ should collapse onto the same second-derivative curve for every $\\xi$.","The conjectured $\\beta$-independence of (4.22) is testable for non-even $\\beta$ such as $\\beta=3$ using the Hessenberg unitary matrix construction of the circular $\\beta$ ensemble; no Pfaffian or determinantal structure is needed to sample it.","The form-factor identities (3.29) resemble a degenerate diffusion in $\\tau$ acting on the limit; combined with the small-$\\tau$ expansions (3.30)-(3.32), they predict exact coefficients at $N^{-2}$ and $N^{-4}$ that a direct Fourier transform of finite-$N$ data could verify."],"forward_implications":["For $\\beta=2$, the derivative identity makes the $1/N^2$ correction to the Riemann-zero spacing distribution and to its thinned version a known function of the limiting distribution, so empirical large-height zero data can be compared with the prediction with no fitted parameters.","All $n$-point correlation functions in the bulk for $\\beta=1,2,4$ expand in powers of $1/N^2$ only; odd inverse powers of $N$ do not appear at bulk scaling.","The spacing-distribution generating functions and gap probabilities inherit the same $1/N^2$ expansion, so the correction $E_1$ is explicitly $-\\frac{s^2}{6\\beta}E_0''$ in the cases proved.","For $\\beta=1,4$, the spectral form factor has a $1/N^2$ expansion whose first two corrections are differential operators applied to the limiting form, with constants $c_1=-1/6$, $c_4=-1/24$, $d_1=7/360$, and $d_4=7/5760$.","For even $\\beta$, the two-point correlation has a $1/N^2$ expansion with leading correction given by (4.16), and the paper conjectures, with supporting evidence in Appendix B, that the same derivative identity holds for the spacing generating function for all $\\beta>0$."],"supporting_citations":[{"why":"Supplies the second-order linear ODE and boundary condition for the correction function $\\sigma_1$ on which the proof of Proposition 2.1 rests.","marker":"[35]"},{"why":"Provides the operator expression for $E_1$ and the sigma-Painleve III-prime characterisations of $f^\\pm_0$ and $f^\\pm_1$ used for $\\beta=1$ and $\\beta=4$.","marker":"[14]"},{"why":"Gives the sigma-Painleve V tau-function representation of the sine-kernel Fredholm determinant that underlies the $\\beta=2$ proof.","marker":"[50]"},{"why":"Supplies the standard sine-kernel and Pfaffian-kernel forms for the circular ensembles and their limiting bulk statistics.","marker":"[59]"},{"why":"Provides the book-level framework for correlation functions, Fredholm determinants, Jack-polynomial hypergeometric functions, and classical-group identities used throughout.","marker":"[29]"},{"why":"Gives the Jack-polynomial hypergeometric evaluation of even-$\\beta$ $n$-point correlation functions used in Proposition 4.1.","marker":"[26]"},{"why":"Gives the $\\beta$-dimensional Selberg integral form of the even-$\\beta$ two-point correlation used to prove Proposition 4.3.","marker":"[28]"},{"why":"Provides the exact finite-$N$ structure functions and small-$\\tau$ expansions used to prove Proposition 3.4 and to identify the differential relations (3.29).","marker":"[76]"}],"fun_headline_variants":["1/N^2 bulk correction is a second derivative","Finite-size term: second derivative of bulk","Derivative identity ties finite-N term to bulk","For circular beta, N^{-2} term is second derivative"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The load-bearing premise is that the $\\sigma$-Painleve V and $\\sigma$-Painleve III-prime characterisations of the leading gap probability, together with the linear ODE for the correction term $\\sigma_1$ (or $f^\\pm_1$) quoted from the literature, are valid on the full parameter range used; if that characterisation is restricted or the computer-algebra verification of (2.19) is incomplete, Proposition 2.1 fails and Proposition 3.2 inherits the same fragility.","fun_headline_variants_meta":{"raw":{"variants":["1/N^2 bulk correction is a second derivative","Finite-size term: second derivative of bulk","Derivative identity ties finite-N term to bulk","For circular beta, N^{-2} term is second derivative"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000639,"raw_usage":{"total_tokens":3050,"prompt_tokens":1158,"completion_tokens":1892,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":774,"completion_tokens_details":{"reasoning_tokens":1838}},"tokens_in":774,"tokens_out":1892,"duration_ms":14024,"temperature":1.0,"reasoning_tokens":1838,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-15T21:22:51.581324+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Compute the two sides of (2.9) independently: for a grid of $s\\in(0,3)$ and $\\xi=1$, evaluate $E_0(s;\\xi)=\\det(I-\\xi K_s)$ numerically with a high-accuracy Fredholm determinant routine, form $-\\frac{s^2}{12}E_0''$, and compare with $E_1$ from the trace formula (2.12) using the kernel (2.11); any discrepancy beyond the numerical tolerance rules out Proposition 2.1. A direct combinatorial check is to simulate CUE eigenangles at several $N$, form $(2\\pi/N)^2 P_N(2\\pi s/N;\\xi)$, subtract the known limit, multiply by $N^2$, and check that the curves converge to the right-hand side of (2.8).","supporting_citations":[{"cited_title":"Forrester and A","cited_arxiv_id":null,"evidence_quote":"Supplies the second-order linear ODE and boundary condition for the correction function $\\sigma_1$ on which the proof of Proposition 2.1 rests."},{"cited_title":"Bornemann, P.J","cited_arxiv_id":null,"evidence_quote":"Provides the operator expression for $E_1$ and the sigma-Painleve III-prime characterisations of $f^\\pm_0$ and $f^\\pm_1$ used for $\\beta=1$ and $\\beta=4$."},{"cited_title":"Jimbo, T","cited_arxiv_id":null,"evidence_quote":"Gives the sigma-Painleve V tau-function representation of the sine-kernel Fredholm determinant that underlies the $\\beta=2$ proof."},{"cited_title":"Mehta,Random matrices, 3rd ed., Elsevier, San Diego, 2004","cited_arxiv_id":null,"evidence_quote":"Supplies the standard sine-kernel and Pfaffian-kernel forms for the circular ensembles and their limiting bulk statistics."},{"cited_title":"Forrester,Selberg correlation integrals and the1/r2 quantum many body system, Nucl","cited_arxiv_id":null,"evidence_quote":"Gives the Jack-polynomial hypergeometric evaluation of even-$\\beta$ $n$-point correlation functions used in Proposition 4.1."},{"cited_title":"Forrester,Addendum to Selberg correlation integrals and the1/r2 quantum many body system, Nucl","cited_arxiv_id":null,"evidence_quote":"Gives the $\\beta$-dimensional Selberg integral form of the even-$\\beta$ two-point correlation used to prove Proposition 4.3."},{"cited_title":"Witte and P.J","cited_arxiv_id":null,"evidence_quote":"Provides the exact finite-$N$ structure functions and small-$\\tau$ expansions used to prove Proposition 3.4 and to identify the differential relations (3.29)."}],"review_version":1}