{"id":"03d5d7e7-b2ff-458b-b442-26fd76c7325f","arxiv_id":"2412.11662","paper_version":2,"verdict":"REJECT","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"The authors compute the Ratios-Theorem form of the U(N) n-correlation for Fourier support up to (-6,6), extending the q=1 and q=2 results of Conrey-Snaith and Chandee-Lee.","lead":"The paper derives a new formula for the n-point correlation of eigenvalues of large random unitary matrices when the Fourier transform of the test function is supported in a window of width 6. The result extends prior work at widths 2 and 4 and gives number theorists a tool for future work on the statistics of zeros of L-functions.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"The q=3 formula (109) rests on an unverified contour-spreading cancellation; without it, the new support-(−6,6) result is not established.","rationale":"The reader's weakest assumption identifies exactly the load-bearing gap: the q=3 contour-spreading lemma is asserted rather than proved, and the new claim rests on it. My independent reading confirms this is the most serious issue. The q=2 section contains an explicit residue check for one representative case, but the q=3 case introduces four special variables and additional z-function denominators in (112), so the earlier check does not automatically cover it. The proof of Lemma 6.1 also states that the integrals over the R and Q sets 'follow exactly the same steps as before' without showing the required sign and residue bookkeeping for all index orderings. Since Theorem 6.1 is the paper's advertised new result and no consistency check against the known q=1 or q=2 limits is provided, the central claim is not fully supported. This does not prove the formula false; it means the paper currently lacks a complete proof. The reader's REJECT verdict is therefore appropriate, and no adjustment is needed.","tokens_in":54950,"tokens_out":5349,"duration_ms":52756,"concrete_test":"Specialize Theorem 6.1 to a test function whose Fourier support satisfies sum |ξ_j| < 4. If the q=3 bracket in (109) does not vanish identically, or if the remaining two brackets do not reduce exactly to the corresponding terms of Theorem 5.1, then formula (109) contradicts the established q=2 result. Independently, symbolically verify the contour-spreading step for a small q=3 configuration, e.g., |Rc|=|Qc|=2 with all four special variables, by listing the residues encountered when the δ-contours are moved to the ordered positions used in Lemma 6.1; the total must vanish for the claimed cancellation to hold.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central new claim is Theorem 6.1, equation (109), for test functions with Fourier support sum |ξ_j| < 6. The proof depends on Lemma 6.1, whose derivation begins by spreading the integration contours into the ordered configuration used for q=2, with the statement that 'these follow exactly the same steps as before.' In the q=2 case, the pairwise cancellation of the apparent poles z_s = z_k and z_t = z_l was verified only for one representative configuration (Section 5, equations (65)-(66)), and the q=3 integrand is substantially more singular: equation (112) contains the additional factors z(z_{k1}-z_{k2})z(z_{k2}-z_{k1}) and z(z_{l1}-z_{l2})z(z_{l2}-z_{l1}), so contour movement must also thread between genuine poles of the four special variables z_{k1}, z_{k2}, z_{l1}, z_{l2}. No analogue of the q=2 residue calculation is supplied for the four-variable case, and no argument rules out surviving residues when an intermediate variable crosses one of these four contours. If such residues survive, formula (109) would acquire extra terms. In addition, five of the six contributions I_2 through I_6 in the proof of Theorem 6.1 are asserted with only 'the same steps' after I_1 is computed, so the displayed q=3 expression is not independently supported.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper derives an alternative, ratio-based expression for the n-point correlation function of eigenvalues of U(N) with Haar measure, restricting the support of the Fourier transform of the test function to (-6,6) (q=3). Section 5 gives a detailed proof for support (-4,4) (q=2), building on the q=1 result of Conrey and Snaith, and Section 6 states the q=3 result as Theorem 6.1 with a long explicit formula, equation (109). The paper's stated motivation is to provide a random-matrix expression that can be mimicked in number-theoretic calculations of n-correlations of L-function zeros.","tokens_in":55257,"tokens_out":6376,"duration_ms":64247,"significance":"If fully established, the q=3 formula would extend the range of support for this ratio-based correlation expression from the previously known (-4,4) to (-6,6), potentially matching or guiding future number-theoretic results. The q=2 theorem is derived in detail, with clearly stated lemmas and explicit conditions on the ξ variables; the derivation uses established identities rather than fitted parameters, and the support restriction is not circular. The central q=3 theorem, however, is not established as presented: its proof derives only the I^1 contribution in detail, and the contour-spreading step on which the derivation rests is verified only through a representative example in the q=2 case and asserted without residue computation in the q=3 case. The value of the paper therefore depends on whether these gaps can be filled.","major_comments":[{"comment":"The proof of the central q=3 result computes only the contribution I^1_2,2 in detail. The remaining five contributions I^2_2,2 through I^6_2,2 are introduced with the statement 'If we follow these same steps for I^2_2,2 to I^6_2,2, of course with slight modification to the change of variables' and only their final integrands are displayed. Since the displayed formula (109) is by construction the sum of these six contributions, the q=3 formula is not verified by the text as it stands. Please supply the omitted derivations, or an independent verification such as a low-dimensional comparison with the determinant expression (5), before the theorem can be regarded as proven.","section":"Section 6, proof of Theorem 6.1, equations (166)-(172)"},{"comment":"The contour-spreading step used to order the z-contours in the q=2 calculation is justified only for the representative pair k=1, s=2, with the statement that a similar argument handles z_t=z_l. In the q=3 proof the analogous spreading is asserted with 'these follow exactly the same steps as before', but the integrand (112) contains the additional denominator factors z(z_{k1}-z_{k2})z(z_{k2}-z_{k1}) and z(z_{l1}-z_{l2})z(z_{l2}-z_{l1}). Moving contours in this setting must thread between genuine poles in the four special variables z_{k1}, z_{k2}, z_{l1}, z_{l2}, and no residue computation is supplied to rule out surviving contributions for some index configurations. Such surviving residues would add extra terms to (109), so a general proof of the cancellation is needed rather than a representative example.","section":"Section 5, equations (65)-(66); Section 6, after equation (119)"},{"comment":"The theorem statement is difficult to check because the sets K1, K2, L1, L2 are defined only after the displayed formula, and the sign exponent contains Q^{>k_2}_2 where the surrounding definitions suggest Q^{>l_2}_2. In addition, the final 'dξ_Q' notation does not appear elsewhere in the proof. Please clarify the notation and correct the apparent typographical inconsistency so that the individual terms of (109) can be matched against the six contributions I^1_2,2 through I^6_2,2.","section":"Theorem 6.1, equation (109)"}],"minor_comments":[{"comment":"The abstract contains typographical errors such as 'matrics' and the title is broken as 'UNIT ARY n-CORRELA TIONS'; these should be corrected in the published version.","section":"Abstract and title"},{"comment":"The conditions 'k2 > k1 and l2 > l1' appear only in the informal discussion around (111) and are not restated in Theorem 6.1; they should be made part of the theorem statement to avoid ambiguity.","section":"Equation (109)"},{"comment":"The text says 'we make a change of variables in zl2, zl2, zk1 , zk2 integrals' where the second 'zl2' should presumably be 'zl1'.","section":"Proof of Lemma 6.1, paragraph after equation (129)"},{"comment":"The statement records the error term as O(1/N), while the proof at several places derives errors of size O(1/(NT)) from differentiating the h factors; the relation between these error terms should be stated explicitly.","section":"Lemma 6.1, equation (119)"},{"comment":"The formulas (44) and (109) are extremely long, and the notation R^c_1, Q^c_1, R^{>k}_1, and similar sets is introduced ad hoc; a short notation table or a preliminary lemma collecting these definitions would improve readability and checkability.","section":"Sections 5 and 6, notation"}],"recommendation":"major_revision","confidential_remarks":"The q=2 portion of the paper is solid and represents a useful detailed derivation, but the central q=3 theorem is the main advertised result and its proof is incomplete in a load-bearing way: five of six contributions are asserted rather than derived, and the contour-spreading cancellation is not verified for the four-variable q=3 integrand. These gaps are substantial but appear fixable within the scope of the manuscript if the authors supply the missing residue computations and a complete derivation of I^2 through I^6, or an independent check of the final formula. I therefore recommend major revision rather than rejection, but I would not accept the paper in its current form."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Quick take: the q=2 part is genuinely new and carefully derived; the q=3 part is the advertised new result but its proof is incomplete. I would send this to a referee, with a request to fill the gaps, rather than reject it outright.\n\nWhat is actually new: Theorem 5.1 gives the alternative-form n-correlation for U(N) with Fourier support sum |ξ_j| < 4. The earlier Conrey-Snaith result covered support (-2,2), and Chandee-Lee's q=2 computation was for n-level density, not n-correlation with the translation-invariant delta. So the q=2 theorem is a real new computation, and the derivation is shown in detail: the contour-spreading argument, the representative residue cancellation, the convolution lemma, and the final change of variables all check out. The q=2 part earns its keep.\n\nThe soft spots are in Section 6. Lemma 6.1 drives the q=3 theorem, but only the first of its six contributions, I^1, is actually derived; the other five are asserted to follow by \"slight modification.\" Given the size of the expressions and the different sign conditions, that is a genuine gap, not a rhetorical one. The stress-test's contour concern is also on point: in q=2 the pairwise cancellation of the apparent poles at z_s = z_k was verified only for a representative case, and the q=3 integrand contains additional factors 1/z(z_{k1}-z_{k2})z(z_{k2}-z_{k1}). Those factors have zeros rather than poles, which is reassuring, but the cancellation when moving a contour past z_{k1} now also involves z_{k2} terms, and no residue computation is shown for that configuration.\n\nA third gap is the absence of consistency checks. The q=3 formula should reduce to the q=2 formula when the support is restricted to (-4,4) or when the extra terms drop out; that check is not done. I would want to see it before trusting (109).\n\nThese are pluggable gaps, not indications of a wrong result. The method is from a known lineage, the q=2 portion is reproducible, and the paper is honest that no current number-theoretic result needs the (-6,6) range. I disagree with an outright reject: the right outcome is a major revision with the missing derivations supplied and the limiting cases verified. I would send it to a serious referee, not desk reject it.","headline":"A solid q=2 calculation, but the headline q=3 theorem is under-proved as written; worth refereeing seriously rather than desk rejecting.","tokens_in":55796,"tokens_out":4099,"would_cite":false,"duration_ms":39975,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["11M26","11M50","15B52"],"pacs":[],"model":"deepseek-v4-flash","headline":"This paper derives an explicit formula for the n-correlation of eigenvalues of U(N) in the ratios form, valid when the Fourier transform of the test function has support |ξ1|+...+|ξn|<6, extending earlier results at supports <2 and <4.","keywords":["random matrix theory","unitary group U(N)","n-correlation","ratios of characteristic polynomials","restricted Fourier support","L-function zeros","Ratios Conjecture"],"falsifier":"Compute the residues at $z_s=z_k$ and $z_t=z_l$ in the q=2 integrand for an index configuration other than the worked example, and the analogous residues at $z_{k_1}=z_{k_2}$ and $z_{l_1}=z_{l_2}$ in the q=3 integrand; if any residue fails to cancel against its paired partner, the displayed formula misses an additional residue contribution. A numerical check of Theorem 6.1 for small n with a chosen $\\Phi$ of support <6 would also test whether the predicted leading term is indeed $\\kappa(h)NT/(2\\pi)$ with error O(T).","tokens_in":54741,"feed_emoji":"🎲","tokens_out":8325,"duration_ms":79533,"temperature":0.7,"pith_summary":"The paper proves explicit formulas for the n-point correlation of eigenvalues of random unitary matrices, written in the ratios form that number theorists can mimic, when the Fourier transform of the test function has support on the region |ξ1|+...+|ξn| less than 6. It extends previous treatments at supports <2 and <4 by adding the contribution in which two z-variables are singled out on each side of the contour expression. The payoff is that random matrix theory can now supply a concrete asymptotic target for zero-correlation calculations of L-functions at a larger range of support, even though no number-theoretic calculation at this range exists yet. The two main results are the q=2 formula with support <4 and the q=3 formula with support <6.","feed_headline":"Eigenvalue correlations extended to Fourier support six","feed_subtitle":"A ratios-style formula for U(N) now covers |ξ1|+...+|ξn|<6, offering L-function zero studies a larger random-matrix target.","key_machinery":"The load-bearing object is the ratios expression $J^*_q(z_K;-z_L)$, obtained from the Ratios Theorem, which replaces the determinantal correlation function by a contour integral over z-variables. The Fourier support restriction $\\sum|\\xi_j|<2q$ makes terms with |S|=|T|\\ge q exponentially small, so $J^*$ reduces to a sum over subsets of size less than q. For q=3 the new term is $J^*_{z_{k_1},z_{k_2},-z_{l_1},-z_{l_2}}$, built from products of $z(x)=1/(1-e^{-x})$ and the functions $H_{2,2}(W)$, which are differences of logarithmic derivatives of z. The contour integrals are then evaluated by spreading the contours so that their real parts are ordered, collecting residues at coincident z-variables, and applying a generalized convolution theorem to the final single-variable integrals.","core_discovery":"The central claim is Theorem 6.1. For a test function F built from a rapidly decaying function f whose Fourier transform Φ is smooth, even, and compactly supported on |ξ1|+...+|ξn|<6, the averaging identity\n\n$$\\int_{U(N)} \\sum_{j_1,\\dots,j_n} F(\\theta_{j_1},\\dots,\\theta_{j_n})\\,dX = \\frac{NT}{2\\pi}\\kappa(h) \\sum_{K+L+M=\\{1,\\dots,n\\}} (-1)^{|L|}\\,[\\cdots] + O(T)$$\n\nholds, with $\\kappa(h)=\\int_{\\mathbb R} h_1(u)\\cdots h_n(u)\\,du$ and where $[\\cdots]$ is an explicit finite sum over pairings of selected indices and over partitions of the remaining indices into four contributing sets, with integrands built from $\\Phi$. Theorem 5.1 is the analogous statement for support <4, where only a single special pair is needed. This is deliberately not the familiar determinant expression for eigenvalue correlations; it is the ratios form, in which only subsets of size |S|=|T|