{"id":"72f5f66c-49d7-4c01-9dd4-9aecfe3c57f2","arxiv_id":"2412.19574","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"New superintegrability formulas are proposed for eigenvalue models built on multivariate Meixner-Pollaczek and Wilson measures, with the Wilson case left partly conjectural.","lead":"This paper finds new examples of matrix models where certain averages can be computed exactly, using multivariate versions of Meixner-Pollaczek and Wilson orthogonal polynomials. It derives formulas for moments of these measures, and shows that the convenient basis of observables differs from the usual Schur basis in this setting.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Wilson moment formula (98) fails its own N=1 single-variable limit: for R=[2] it disagrees with the exact inversion (83)/(84), so the central Wilson superintegrability claim is internally inconsistent as stated.","rationale":"The Wilson model is the advertised headline of the paper. The Meixner–Pollaczek section is comparatively self-contained and may survive; the Wilson section is where the new superintegrability claim must stand. The reader's concern about the unknown denominator α in (89) is real, but the sharper, load-bearing problem is that the final formula (98) is already inconsistent with the exact single-variable case on which the construction is built. Any formula for ⟨Θ_R⟩ must specialize to (83) at N=1, and (98) does not. This is a concrete mathematical counterexample, not merely an incomplete proof. It also explains why the missing α cannot be dismissed as a cosmetic gap: the coefficient that α was meant to fix is known exactly at N=1, and the N=1 output of (98) is wrong. I would adjust the verdict to REJECT for the paper as written, since the central Wilson superintegrability statement fails its own consistency check. I am not claiming the underlying phenomenon is nonexistent; a corrected shift and a proof of the denominator structure could restore a revised claim. But as printed, the central Wilson formula is unverified and contradicted by the paper's own single-variable input.","tokens_in":16232,"tokens_out":24265,"duration_ms":199877,"concrete_test":"Symbolically evaluate both sides of (98) at N=1 with R=[2] and parameters a=b=c=d=1. The left side, by the single-variable inversion (83)/(84), equals ⟨(a−ix)_2(a+ix)_2⟩/⟨1⟩ = (a+b)_2(a+c)_2(a+d)_2/(z)_2 = 54/5. The right side of (98) as printed equals 432/5 under the appendix convention ξ_R(u)=∏(u+j−i), and 2/3 under the ξ_R,∅ reading of (90). If the formula is corrected, repeat the check for R=[3]; the discrepancy persists unless the correction removes the shift uniformly for all partitions.","verdict_should_be":"REJECT","load_bearing_attack":"For N=1, the Wilson Θ-polynomial (85) reduces to θ_r(x|a)=(a−ix)_r(a+ix)_r, and the normalized W_R in (86) has W_0=1. The single-variable inversion (83), which is the input to the whole multivariate construction, fixes the normalized expectation value exactly: ⟨θ_r⟩ = (a+b)_r(a+c)_r(a+d)_r/(z)_r, with z=a+b+c+d. This is the normalized version of (84). Substituting N=1, R=[2] into the claimed formula (98) with the appendix convention ξ_R(u)=∏(u+j−i) gives [(a+b+1)(a+b+2)(a+c+1)(a+c+2)(a+d+1)(a+d+2)]/[z(z+1)], whereas the exact value is [(a+b)(a+b+1)(a+c)(a+c+1)(a+d)(a+d+1)]/[z(z+1)]. For a=b=c=d=1 these are 432/5 and 54/5. The alternative reading of ξ from (90) uses falling products u(u−1)..., giving 2/3 for the same numbers, so no consistent reading of the ambiguous ξ notation matches the N=1 limit. This is not merely the unknown denominator in (89): for Q=∅, N=1, the coefficient C_[2],∅ is fixed by (83), and (98) assigns a different value. A corrected index shift would need to be stated and proved; as written, the headline Wilson formula is contradicted by the paper's own single-variable input.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper develops a framework linking superintegrability of eigenvalue/matrix models to multivariate orthogonal polynomials, using single-variable inversion formulas to construct closed-form expectation values. After reviewing the Hermite and Jacobi cases, it treats two new models: the Meixner–Pollaczek model and the Wilson model. For Meixner–Pollaczek, the author defines determinant-type symmetric functions Θ_R built from Pochhammer symbols and derives the expectation formula (68) from the known single-variable inversion (65) and the multivariate expansion (73). For the Wilson model, the paper defines Θ^W_R and normalized multivariate Wilson polynomials W_R, proposes an expansion with coefficients of the form (89) containing an unknown denominator, and states the superintegrability formula (98). The author explicitly acknowledges that no general formula for this denominator was found. The central claim is that these Θ-functions form a superintegrable basis for the Wilson measure, replacing the usual Schur basis.","tokens_in":16598,"tokens_out":15057,"duration_ms":129747,"significance":"The Meixner–Pollaczek result is a solid and interesting new contribution: it provides an explicit non-Schur superintegrable basis and a clear derivation from known inversion identities, and it illustrates the general mechanism in a transparent way. The framework itself, connecting multivariate orthogonal polynomials to superintegrability via inversion, is conceptually useful and likely to stimulate further work. However, the Wilson formula (98), which is the paper's headline generalization, is not derived: the expansion coefficients contain an undetermined denominator, the inverse coefficients needed for moments are not given, and the formula contradicts the paper's own N=1 single-variable input. The central Wilson claim therefore remains unsubstantiated, and the paper as a whole cannot be accepted in its present form.","major_comments":[{"comment":"The Wilson moment formula (98) fails its own N=1 single-variable limit. For N=1 and R=[2], the left-hand side of (98) is ∫ w(x) θ_2(x|a) dx. Using the inversion (83) and the moment formula (84), the normalized expectation is (a+b)_2(a+c)_2(a+d)_2/(z)_2, i.e. (a+b)(a+b+1)(a+c)(a+c+1)(a+d)(a+d+1)/(z(z+1)). Substituting N=1, R=[2] into (98) with ξ_R(u)=∏(u+j−i) from (109) gives (a+b+1)(a+b+2)(a+c+1)(a+c+2)(a+d+1)(a+d+2)/(z(z+1)). For a=b=c=d=1 these are 2/5 and 54/5, respectively. Thus (98) contradicts the inversion formula on which the multivariate construction is explicitly based, and the discrepancy is not a normalization issue.","section":"Section 4, Eq. (98)"},{"comment":"The expansion coefficient C_{R,Q} in (89) contains an unknown denominator α^W_{R,Q}, and the author states that no general formula for it was found (text preceding Eq. (92)). Moreover, the inverse coefficients C∨_{R,Q} in (87), which are needed to obtain (98) from orthogonality, are never given. Consequently (98) is an independent conjecture rather than a consequence of the stated expansion. The inconsistency already appears at Q=∅, N=1: using (89) with (92) gives C_{[r],∅} = ξ_{[r]}(1) ξ_{[r]}(1+a+b) ξ_{[r]}(1+a+c) ξ_{[r]}(1+a+d)/(z+1)_r, which for r=1 equals (a+b+1)(a+c+1)(a+d+1)/(z+1), whereas the single-variable inversion (83) fixes this coefficient as (a+b)(a+c)(a+d)/z. The conjectural denominator thus fails to match the input (83) even in the simplest case, reinforcing that the Wilson superintegrability claim is unproved.","section":"Section 4, Eqs. (89)-(96)"}],"minor_comments":[{"comment":"The notation is inconsistent: Eq. (89) uses ξ_{R/Q}, while Eq. (90) defines ξ_{R,Q}. Please clarify whether these denote the same quantity and define the skew content product explicitly.","section":"Eq. (89) and Eq. (90)"},{"comment":"The prefactor in (84), 2π Γ_{a+b}Γ_{a+c}Γ_{b+c}Γ_{b+d}Γ_{c+d}/Γ_z, appears to be missing Γ_{a+d}; compare with the n=0 limit of the normalization (81), which contains all six Gamma factors. Please verify and correct.","section":"Eq. (84)"},{"comment":"The overall sign (-1)^{N(N+7)/4} in the Meixner–Pollaczek moment formula (68) is asserted without derivation; briefly indicate how it follows from the normalization conventions and the inversion formula (65).","section":"Eq. (68)"},{"comment":"The piecewise formula for two-row partitions omits the case q_1 = r_2, and the first line contains a rational expression with denominator 2N+q_1+q_2+z−4; please clarify whether this is intended to be a polynomial and specify the missing boundary case.","section":"Eq. (94)"},{"comment":"The statement that the paper 'solve[s] the moment problem for this generalization' is too strong given the unproved denominator in the Wilson section; rephrase as 'propose' or 'conjecture' until the expansion coefficients are determined.","section":"Introduction, p. 3"}],"recommendation":"major_revision","confidential_remarks":"The Wilson section appears to be at a preliminary stage: the author explicitly admits the denominator in (89) was not found, and the N=1 check shows the stated formula (98) is false as written. The Meixner–Pollaczek section is coherent and may be publishable independently if the Wilson issues are resolved. I see no concerns about citation ethics or novelty disclosure."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Plainly: this paper has one solid new result and one load-bearing error. The Meixner–Pollaczek moment formula (68) is a legitimate step: using inversion identities and multivariate orthogonality to compute expectations of non-Schur observables is a clean idea, and the observation that the superintegrable basis deviates from Schur polynomials is worth recording. The author is also honest that the Wilson denominator structure is unknown.\n\nThe problem is the Wilson section. Formula (98) is the advertised superintegrability statement, but it does not reproduce the single-variable case that the whole construction is built on. With N=1 and R=[2], (98) gives (a+b+1)(a+b+2)(a+c+1)(a+c+2)(a+d+1)(a+d+2)/(z(z+1)), while the inversion (83) fixes the normalized moment as (a+b)(a+b+1)(a+c)(a+c+1)(a+d)(a+d+1)/(z(z+1)). For a=b=c=d=1 these are 432/5 and 54/5. This is not a minor index slip: the central formula is contradicted by the paper's own input. The author admits no general formula for the unknown denominator α^W; the few examples given do not establish the factorization. As written, the Wilson superintegrability claim is a conjecture that fails its simplest consistency check.\n\nThere are also smaller issues: a missing right-hand side in (97), half-integer phases in (71)–(73), and the ξ notation in (90) conflicts with the appendix convention. These typos would be trivial to fix, but they add noise.\n\nWhat survives is the Meixner–Pollaczek part. The derivation is mostly coherent, and the N=2 checks seem to support it, though I would want the sign conventions cleaned up. The paper is clearly exploratory, and the author says so. But the headline Wilson result should not appear as a theorem.\n\nWho should read this: people working on superintegrability of matrix models and on multivariate orthogonal polynomials. The MP section is worth reading; the Wilson section is a cautionary example. I would not cite the Wilson formula until it is fixed. A serious referee could usefully push the author to either derive the denominator or downgrade (98) to a conjecture with explicit verification. I would send it to peer review, not desk reject, because the MP result and the non-Schur phenomenon are real and the Wilson gap is concretely diagnosable.","headline":"Fresh MP moments, but the Wilson formula (98) fails its own N=1 limit, so the headline claim is not just unproved—it's wrong as written.","tokens_in":17100,"tokens_out":16572,"would_cite":false,"duration_ms":122773,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["33C45","33C50","05E05","15B52"],"pacs":[],"model":"deepseek-v4-flash","headline":"Pochhammer-built observables replace Schur polynomials and give exact moments in two new matrix models.","keywords":["superintegrability","matrix models","eigenvalue models","multivariate orthogonal polynomials","Meixner-Pollaczek polynomials","Wilson polynomials","moments","Schur polynomials"],"falsifier":"Compute the left side of the Wilson expectation formula (98) directly for a partition not covered by the low-lying examples, for instance $R=[4,3,2]$ and $Q=[2,1]$ in the expansion of $\\Theta^W_R$, and compare with the proposed content-product answer; any disagreement would refute the claim.","tokens_in":16004,"feed_emoji":"🔢","tokens_out":14699,"duration_ms":121032,"temperature":0.7,"pith_summary":"This paper argues that two eigenvalue models built from the Wilson family of hypergeometric orthogonal polynomials are superintegrable in a new sense: the observables with closed-form expectation values are not Schur polynomials but determinant-type functions built from Pochhammer symbols, the rising factorials $(x)_n=x(x+1)\\cdots(x+n-1)$. For the Meixner-Pollaczek measure, the expectation of $\\Theta_R(x|\\lambda)$ is given by an explicit product of Schur specializations, and an analogous formula holds for the Wilson measure. The point is that superintegrability is not tied to characters of $GL(N)$: it can survive with observables that depend on the number of eigenvalues and on the parameters of the measure. If correct, this solves the multivariable moment problem for these measures and provides a new class of exactly solvable matrix models.","feed_headline":"Forget Schur: Pochhammer determinants close Wilson matrix models","feed_subtitle":"Expectation values of Pochhammer-built observables are explicit, solving the moment problem for two new measures.","key_machinery":"The load-bearing object is the determinant-type symmetric function $\\Theta_R(x|a)=\\det_{i,j}(\\theta_{R_j+N-j}(x_i|a))/\\Delta(x)$, where $\\Delta(x)=\\prod_{i<j}(x_i-x_j)$ is the Vandermonde determinant, $\\theta_n(x|a)=(a-ix)_n(a+ix)_n$ for Wilson and $\\theta_n(x|\\lambda)=(\\sqrt{-1}x+\\lambda)_n$ for Meixner-Pollaczek, and the multivariate orthogonal polynomials are defined by the same determinant-over-Vandermonde ratio. The mechanism is the inversion formula: the single-variable expansion of $\\theta_n$ in the corresponding orthogonal polynomials is lifted to partitions by replacing Pochhammer symbols with products over boxes of the Young diagram of quantities like $j-i$ (the content), and orthogonality of the multivariate polynomials converts the inverted expansion into the moment formula. The hypergeometric structure of the Wilson family is what makes the coefficients factor into Pochhammer and content products; the nonstandard denominator in the single-variable Wilson inversion is exactly the place where a general formula is missing.","core_discovery":"The central claim is that for the multivariate Meixner-Pollaczek measure, the functions $\\Theta_R(x|\\lambda) = \\det_{i,j}((\\sqrt{-1}x_i+\\lambda)_{R_j+N-j})/\\Delta(x)$ have expectation values given by a closed product of Schur function specializations, equation (68), and that the analogous statement for the Wilson measure is the content-product formula (98). These equalities are derived from inversion formulas that expand the $\\Theta$-functions in multivariate Meixner-Pollaczek and Wilson polynomials and then use orthogonality; they are not obtained by evaluating the $\\Theta$-functions at special points, so the superintegrable basis is genuinely different from Schur functions.","pith_inferences":["A natural extension the paper does not take is the $q$-deformation: the same inversion-formula route should produce dual superintegrable models whose answers are sums over partitions, analogous to gauge-theory partition functions.","If the missing Wilson denominator has no closed form for generic partitions, the simple determinant-ratio basis may need to be replaced by a deformed basis; this is an editorial conjecture, not a claim of the paper.","The same determinant-ratio construction could be applied to other polynomials in the Askey scheme of hypergeometric orthogonal polynomials, which would test the paper's suggestion that hypergeometricity is the mechanism behind the closed formulas.","The $N$-dependence and parameter-dependence of the superintegrable basis suggest a connection to the infinite-dimensional algebra structures mentioned in the introduction, but establishing that connection is left for future work."],"forward_implications":["The Meixner-Pollaczek model is superintegrable with a basis of non-homogeneous symmetric functions that depend on $N$ and on the parameters of the measure, without any deformation of characters.","The Wilson model provides the most general Hermitian eigenvalue model in this family, with all previously known superintegrable Hermitian models recovered in suitable limits.","The moment problem for the multivariate Meixner-Pollaczek and Wilson measures is solved in the sense that expectation values of the full $\\Theta$-basis are explicit.","The multivariate Wilson polynomials introduced here, together with their orthogonality and expansion formulas, become available for further study even though the general expansion denominator remains conjectural.","The new combinatorial structures in the answers give concrete data for any future classification of superintegrable bases."],"supporting_citations":[{"why":"Supplies the single-variable inversion formulas and moments for classical orthogonal polynomials that are lifted to the multivariable setting.","marker":"[19]"},{"why":"Provides the multivariate Meixner-Pollaczek polynomials and their orthogonality and difference equation used in the MP model.","marker":"[25]"},{"why":"Gives the determinant-over-Vandermonde definition of multivariate orthogonal polynomials and recursive methods for their expansion coefficients.","marker":"[16]"}],"fun_headline_variants":["Pochhammer determinants close Wilson matrix models","Beyond Schur: Wilson polynomials unlock explicit moments","Explicit moments for Wilson and Meixner-Pollaczek measures","New superintegrability: Wilson matrix models without Schur","Pochhammer basis solves Wilson moment problem"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The load-bearing premise is that the Wilson expansion coefficients follow the factored pattern proposed in the paper, a pattern that is verified only for low-lying partitions; if that pattern gives the wrong value for a generic partition, the final Wilson moment formula collapses.","fun_headline_variants_meta":{"raw":{"variants":["Pochhammer determinants close Wilson matrix models","Beyond Schur: Wilson polynomials unlock explicit moments","Explicit moments for Wilson and Meixner-Pollaczek measures","New superintegrability: Wilson matrix models without Schur","Pochhammer basis solves Wilson moment problem"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000993,"raw_usage":{"total_tokens":4137,"prompt_tokens":804,"completion_tokens":3333,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":420,"completion_tokens_details":{"reasoning_tokens":3257}},"tokens_in":420,"tokens_out":3333,"duration_ms":22341,"temperature":1.0,"reasoning_tokens":3257,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-11T00:11:32.545104+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Compute the left side of the Wilson expectation formula (98) directly for a partition not covered by the low-lying examples, for instance $R=[4,3,2]$ and $Q=[2,1]$ in the expansion of $\\Theta^W_R$, and compare with the proposed content-product answer; any disagreement would refute the claim.","supporting_citations":[{"cited_title":"On moments of classical orthogonal polynomials,","cited_arxiv_id":null,"evidence_quote":"Supplies the single-variable inversion formulas and moments for classical orthogonal polynomials that are lifted to the multivariable setting."},{"cited_title":"Hermitian Symmetric Spaces of Tube Type and Multivariate Meixner-Pollaczek Polynomials","cited_arxiv_id":"0812.1292","evidence_quote":"Provides the multivariate Meixner-Pollaczek polynomials and their orthogonality and difference equation used in the MP model."},{"cited_title":"Mops: Multivariate orthogonal polynomials (symbolically),","cited_arxiv_id":null,"evidence_quote":"Gives the determinant-over-Vandermonde definition of multivariate orthogonal polynomials and recursive methods for their expansion coefficients."}],"review_version":1}