{"id":"20e5cc5d-6482-4b58-a3cf-32114b8f5c3a","arxiv_id":"1908.00562","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"Arizmendi and Celestino extend Collins-Hasebe-Sakuma spectral formulas to a broad class of matrix-embeddable polynomials in cyclically monotone variables, with applications to limiting eigenvalues of random matrices with discrete spectrum.","lead":"This paper proves that for a special kind of noncommutative independence, the spectrum of many polynomials can be computed from the components' average values, a major simplification. It applies the result to random matrices with discrete spectrum and gives explicit limiting eigenvalue formulas for several example polynomials.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Theorem 12's proof applies Proposition 5 to non-selfadjoint operators, where moment equality does not determine eigenvalues; the stated EV equality is not justified.","rationale":"The reader identified the application of Proposition 5 to non-selfadjoint products as the weakest assumption; I agree. The concern is load-bearing because Theorem 12 is the stated central reduction, and every application in the paper either uses Theorem 12 or inherits the same moment-to-eigenvalue step. The issue is not that the final formulas are known to be false: the CHS formulas and the numerical examples are consistent with the intended selfadjoint setting, and the shift-model test above is likely to confirm the equality even in the non-selfadjoint case. Rather, the proof as written does not establish the theorem in its stated generality, and the definition of EV for non-selfadjoint elements is ambiguous. This warrants the reader's CONDITIONAL verdict: the paper should add selfadjointness hypotheses (or a similarity argument) to Theorem 12 and fix the analogous step in Theorem 7(1). I do not see grounds to reject the paper's intended results, so the verdict is unchanged.","tokens_in":12034,"tokens_out":30265,"duration_ms":309654,"concrete_test":"Test the non-selfadjoint case of Theorem 12 in the exactly solvable CHS model: H = ℓ²(N), A = diagonal trace-class operators, B = alg(1,S) with S the unilateral shift, and τ(S^k)=0. Take k=1, A1 = diag(2^{-i}), B0 = S, B1 = S, and compute the eigenvalues of the truncated n×n matrices (using S with zero boundary, n=100) for X = S A1 S and Y = A1 τ(S²) = 0. If X has any nonzero eigenvalue, Theorem 12 is false. If, as the weighted-shift argument predicts, EV(X) = {0} = EV(Y), the result survives this instance and the flaw is confined to the proof. To settle the proof gap, independently check whether the step from Corollary 11 to Theorem 12 can be replaced by an explicit similarity or selfadjointness argument; if not, Theorem 12 requires a selfadjointness hypothesis.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central reduction of the paper is Theorem 12. Its proof consists of Corollary 11 (a moment equality) followed by 'By applying Proposition 5'. Proposition 5 is CHS Corollary 2.6 and is stated only for selfadjoint trace-class elements; its conclusion is that identical moments imply identical eigenvalue multisets. In Theorem 12 neither X = B0A1B1...AkBk nor Y = A1B'1...Ak(BkB0)' is shown to be selfadjoint, and in general they are not. For non-selfadjoint trace-class operators, the eigenvalue multiset is not determined by the moment sequence: the nilpotent matrix [[0,1],[0,0]] and the zero matrix have the same moments (all zero) but different eigenvalues. Thus Definition 4, which defines EV via any trace-class operator with the same distribution, is ambiguous for non-selfadjoint X, and the equality EV(X)=EV(Y) cannot be inferred from moment equality. The same gap occurs in the proof of Theorem 7(1), where A1B0*B0 is not selfadjoint and the appeal to Proposition 5 is replaced in effect by a similarity (A1B0*B0 is similar to √B A1√B), but this similarity is not stated. Thus the paper's main general claim is, as written, not justified.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper studies cyclically monotone independence, introduced by Collins, Hasebe and Sakuma (CHS), and aims to generalize their spectral formulas for polynomials in cyclically monotone elements. The main technique is a matrix-embedding argument: a product A1B1...AkBk has the same moments as the product with each Bp replaced by its entrywise expectation B'p (Proposition 10), which leads to an eigenvalue reduction theorem (Theorem 12). The paper then gives new proofs of CHS Theorem 7, a result on conjugation invariance (Proposition 14), a theorem on replacing an element b by τ(b) in certain positions (Theorem 17), and several numerical random matrix examples. The intended contribution is a general method for computing spectra of cyclic monotone polynomials.","tokens_in":12260,"tokens_out":17123,"duration_ms":176338,"significance":"If the central reduction were fully justified, this would be a useful and elegant extension of the CHS results: Proposition 10 is a clean moment identity, the random matrix examples are concrete and illustrative, and the paper explicitly shows which information about the B-family enters the limiting spectrum. The paper is clearly written and builds on CHS without introducing ad-hoc parameters. However, the main eigenvalue theorem (Theorem 12) is not proved as stated, because it passes from moment equality to eigenvalue equality for non-selfadjoint operators, where the invoked criterion does not apply. Since this gap is load-bearing for the paper's main general claim, the manuscript needs substantive revision.","major_comments":[{"comment":"The proof of Theorem 12 is not valid. After Corollary 11 the text says 'By applying Proposition 5', but Proposition 5, stated in Section 2.1, applies only to selfadjoint trace-class elements. Neither X = B0A1B1...AkBk nor Y = A1B'1...Ak(BkB0)' is shown to be selfadjoint, and in general they are not. Moreover, Definition 4(3) does not give a well-defined eigenvalue multiset for non-selfadjoint X: the moment sequence of a trace-class operator does not determine the multiplicity of the eigenvalue 0. For example, the zero matrix and a nilpotent Jordan block of larger size have identical moment sequences but different eigenvalue multisets. Thus the equality EV(X) = EV(Y) does not follow from the moment equality of Corollary 11. This is a load-bearing gap because Theorem 12 is the central general reduction and is invoked in the proof of Theorem 7(1).","section":"Section 3.1 (Theorem 12)"},{"comment":"The proof of Theorem 7(1) contains the same error at the step 'due to Corollary 5, B0A1B0* and A1B0*B0 must have the same eigenvalues'. Here A1B0*B0 is not selfadjoint in general, so Proposition 5 cannot be applied to it. Even if Theorem 12 were repaired, this particular step needs a separate argument: one would have to compare moments of the selfadjoint element B0A1B0* directly with those of the selfadjoint element √B diag(a1,...,ak)√B, or use a similarity argument after replacing B0*B0 by its expectation. As written, part 1 of Theorem 7 is not proven.","section":"Section 3.1 (proof of Theorem 7(1))"}],"minor_comments":[{"comment":"In the first lines of the proof, the text writes 'ω(((i(ab+ba))^m)' where the polynomial should be i(ab−ba); this typo appears twice.","section":"Section 3.1 (proof of Theorem 7(4))"},{"comment":"The reference 'Corollary 5' should be 'Proposition 5', and the notation 'AB' is introduced without definition; the intended meaning is likely 'A1B' with A1 = diag(a1,...,ak) and B = (τ(b_i* b_j)).","section":"Section 3.1 (proof of Theorem 7(1))"},{"comment":"The sentence 'Since B is positive definite' should say 'positive semidefinite', since the matrix (τ(b_i* b_j)) is only positive semidefinite in general.","section":"Section 3.1 (proof of Theorem 7(1))"},{"comment":"The statement uses both ℓ and k for the number of b-variables, and the proof only treats the case k = ℓ; the indexing should be made consistent.","section":"Section 3.3 (Theorem 17)"},{"comment":"The displayed limiting matrix has off-diagonal entries 1, but for independent centered GUE matrices B and C one expects trn(CB) and trn(BC) to converge to 0; the example needs a correction or a clarification of the normalization and correlation of B and C.","section":"Section 4 (Example 21)"},{"comment":"In part 1, the moment set is described with inconsistent indices (m, n, and ϵ_n); the notation should be made uniform.","section":"Section 2.1 (Definition 4)"}],"recommendation":"major_revision","confidential_remarks":"The gap is localized to the moment-to-eigenvalue passage for non-selfadjoint operators, and the paper's core idea (Proposition 10 plus the matrix embedding) is sound. I recommend sending the manuscript back for major revision rather than rejecting it, since the authors can likely repair the statements by restricting EV comparisons to selfadjoint elements or by defining EV up to zero eigenvalue multiplicity, and by adding the missing argument in the proof of Theorem 7(1)."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Colleague,\n\nThe thing to know: this paper has a genuinely useful trick—write a cyclic polynomial as the (1,1) corner of a block product, then replace each B block by its entrywise expectation. That replacement is clean and correct for moments (Prop 10), and it does extend the CHS formulas beyond degree 2 and 3. Prop 13 and Theorem 17 are nice consequences. If the only question were whether the moment machinery works, I'd be positive.\n\nThe soft spot is the step from moments to eigenvalues. Theorem 12 passes from the moment equality in Corollary 11 to EV(B0A1...AkBk) = EV(A1B'1...Ak(BkB0)') by invoking Proposition 5, which is CHS Corollary 2.6. That proposition is stated for selfadjoint trace-class elements, and the operators in Theorem 12 are not selfadjoint in general. Moment equality does not determine the eigenvalue multiset for non-selfadjoint operators—the standard counterexample is 0 vs a nilpotent Jordan block if you only match power moments; with *-moments the example fails, but the paper's own definition doesn't give you the spectral result for non-selfadjoint elements. So as written, Theorem 12 is under-justified. It is probably fixable in the cases that matter (the block products that arise in the applications are often selfadjoint after a cyclic permutation, or their spectral behavior follows from the polar decomposition), but the general statement needs an extra argument or a selfadjointness hypothesis.\n\nThere is also a concrete algebra slip in the proof of Theorem 7(4). The displayed B0 = [[i, -ib],[0,0]] with B1 = [[b,0],[1,0]] gives a 2x2 matrix whose eigenvalues are ± i√(τ(b^2)-τ(b)^2), not ±√(...), and the moment calculation as printed loses a factor of i^m. The final formula happens to be the known CHS result, but this proof needs reworking, not just a typo fix. The proof of part (1) also implicitly uses that A1B0*B0 has the same eigenvalues as √B A1√B via a similarity; that's true but not stated.\n\nThe numerics are illustrative rather than demonstrative—no error bars, just a few eigenvalue plots and a three-row moment table. They support the formulas, but they wouldn't carry weight if the proofs had a real hole.\n\nNet: worth a serious referee. The core Proposition 10 is solid, and the extensions are useful. But Theorem 12 should be corrected, the proof of Theorem 7(4) fixed, and the non-selfadjoint gap closed. I'd send it to review.","headline":"Good matrix-corner trick for cyclic monotone independence, but the main eigenvalue theorem overreaches by applying a selfadjoint-only result to non-selfadjoint operators.","tokens_in":12810,"tokens_out":7616,"would_cite":true,"duration_ms":73884,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":null,"created_at":"2026-08-14T15:49:30.853552+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":null,"supporting_citations":[],"review_version":1}