{"id":"921201b7-571d-491f-9ae5-74aa930fc43b","arxiv_id":"2510.10370","paper_version":6,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"For root systems A, BC, and D, scaled log-coefficients of an exponential generating function converge if and only if the scaled Dunkl bilinear form does, with limits given by noncrossing-partition sums.","lead":"This paper proves two-way asymptotic equivalences between scaled coefficients of Bessel-type generating functions and scaled limits of Dunkl bilinear forms for root systems of types A, BC, and D. It completes and extends a recent research program connecting such functions to free probability and random-matrix asymptotics.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Remainder bounds in Theorems 6.1/7.1/8.16 are load-bearing but only sketched; the converse inductions need a rigorous or machine-checked verification of the stated degree bounds.","rationale":"The paper's central theorem is an equivalence between scaled log-coefficients and scaled Dunkl bilinear-form evaluations. The forward directions are relatively robust: they only require the leading-order expansion to be correct, and the leading terms are explicitly computed and spot-check correctly. The converse directions are where the remainder bounds become essential: the induction solving for c_λ(N)/(θN)^{ℓ(λ)} treats every non-diagonal term as o_N(1) after the prescribed scaling. If a remainder monomial had one extra power of N or θ, it would not vanish after division by (θN)^{|ν|}N^{ℓ(ν)}, and the triangular matrix inversion used in Theorems 6.10, 7.4, 8.21, and 8.26 would collapse.\n\nThe reader's weakest-assumption analysis identifies exactly this point, and I agree. The manuscript provides a plausible combinatorial argument for Theorem 6.1, but the crucial verification of the degree bounds for R is repeatedly deferred: 'It is not challenging to determine that the remainder term R is a polynomial...' (Theorem 6.1, second proof), 'For more details of a similar argument, see [Yao25, Section 6]' (same proof), and the BC and D analogues say 'we can use the same method' or 'we follow the same framework' without carrying out the casework. The paper's own text says the first proof 'lacks some details.' No machine-checked proof is supplied.\n\nThis is not a detected mathematical error. The diagonal entries and triangularity lemmas (6.9, 7.3, 8.25) are simple and appear correct, and the type-A leading term is independently supported by [Yao25]. The concern is that the proof as written does not fully establish the remainder bound that the converse argument needs. That is precisely the difference between ACCEPT and CONDITIONAL: the result is plausible and probably correct, but a load-bearing estimate is asserted rather than demonstrated in full.\n\nI therefore recommend UNCHANGED: the reader's CONDITIONAL verdict already reflects this gap, and my stress-test does not move it. The proposed computational check would either close the gap or expose a counterexample; if it passes, the remaining work is a written or formalized proof of the same bound.","tokens_in":80929,"tokens_out":18077,"duration_ms":151646,"concrete_test":"Write a short script (Sage, Macaulay2, or Mathematica) that, for each k≤5 and all λ,ν∈Γ[k] (respectively Γ_even[k]) with ℓ(λ)≤ℓ(ν), expands [p_λ,p_ν]_{A_{N-1}(θ)} exactly using the sequence enumeration in (8) with N and θ as formal symbols. Compute R = exact − (claimed main term) and test each monomial: in the type-A/BC/D-even cases, N-degree ≤ k+ℓ(λ)−ℓ(ν)−1 and θ-degree ≤ k−ℓ(ν), and N-degree ≤ θ-degree + ℓ(λ); for the type-D odd [ep_λ,ep_ν], test the same after factoring out ∏_{i=1}^{N-k}(1+2(i−1)θ). Also verify the coefficient of the maximal monomial agrees with the displayed leading coefficient and with Lemmas 6.9, 7.3, and 8.25. If all checks pass for k≤5, repeat for one higher k or compare with an independent derivation; the remaining issue is exposition. If any monomial violates a bound, the corresponding converse theorem is false.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The weakest load-bearing point is the remainder estimate in the leading-order expansions, not the combinatorial leading-term formula itself. Every converse direction in Theorem 1.1 ((b)⇒(a), (d)⇒(c), (h)⇒(f)) is an induction that inverts a triangular matrix MA / MBC / MD after discarding R/((θN)^{|ν|}N^{ℓ(ν)}). If the claimed bounds on R in Theorem 6.1 — x-degree ≤ k+ℓ(λ)−ℓ(ν)−1 and y-degree ≤ k−ℓ(ν) — were off by one, a monomial θ^{k−ℓ(ν)+1}N^{k+ℓ(λ)−ℓ(ν)} would survive after scaling and break the triangularity/invertibility argument. The paper's proof of these bounds is incomplete: the second proof of Theorem 6.1 says 'It is not challenging to determine that the remainder term R is a polynomial...' and defers to [Yao25, Section 6]; Theorems 7.1, 8.10, 8.16, and 8.23 use the same method without carrying out the case analysis. The author explicitly labels the first proof as 'lacks some details.' No formal verification is provided. This is a rigor gap, not a demonstrated falsehood; the leading coefficients and diagonal values spot-check correctly, and the type-A leading term is independently supported by [Yao25].","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper studies the asymptotics of the Dunkl bilinear form [pλ, pν] for type A, BC, and D root systems, with multiplicity parameters varying in N. Its main theorem (Theorem 1.1) asserts equivalences between convergence of scaled logarithmic coefficients cλ(N)/(θN)^{ℓ(λ)} and convergence of scaled Dunkl evaluations [1] D(pν) F_N / ((θN)^{|ν|} N^{ℓ(ν)}), with limits given by explicit products of noncrossing partition sums. Analogous equivalences are proved in the finite-θN regimes (Theorems 6.17, 7.8, 8.26), generalizing [BGCG22], [Xu25], and [Yao25]. The proofs pass through leading-order expansions (Theorems 6.1, 7.1, 8.16, 8.23), triangular matrices MA, MBC, MD, and explicit matrix inversion. Applications include free-convolution corollaries conditional on Conjecture 1.6, uniform convergence results, and coefficient asymptotics for the Bessel functions themselves.","tokens_in":81226,"tokens_out":10672,"duration_ms":74384,"significance":"If the results are fully established, this is a significant contribution: it provides a unified, parameter-free description of coefficient asymptotics for the Bessel functions J^R_a in several asymptotic regimes and for three root systems, with all limiting constants expressed as sums over noncrossing partitions rather than fitted quantities. It generalizes earlier work, answers a question raised in [BGCG22], and treats the type-D odd-degree sector, including the gamma-type denominator ∏_{j=1}^N (1+2(j−1)θ), as a genuinely new feature. The paper is also commendable for its explicit triangularity lemmas and concrete matrix inversions, which make the leading-order formulas independently checkable. The main caveat is the rigor of the remainder estimates, on which the converse directions of the equivalences depend.","major_comments":[{"comment":"The stated bounds on the remainder R are load-bearing for the central claim. Every converse direction in Theorems 6.10, 7.4, 8.21, and 8.26 proceeds by discarding R/((θN)^{|ν|} N^{ℓ(ν)}) (or the analogous scaled denominator), and an off-by-one in the claimed x- or y-degree would leave a surviving monomial and break the triangular-matrix inversion. As written, the proof is not complete: the first proof of Theorem 6.1 is explicitly said to \"lack some details\"; the second proof says \"It is not challenging to determine that the remainder term R is a polynomial...\" and defers to [Yao25, Section 6]; Theorems 7.1 and 8.16 make similar assertions without carrying out the case analysis. Since the leading-order coefficients and diagonal values spot-check correctly, I am not claiming the formula is false; rather, the equivalence theorems are not yet established without a complete proof of these rem","section":"Section 6.1, Theorems 6.1, 7.1, 8.16, 8.23"},{"comment":"The type-D odd-sector argument is more delicate than the type-A case because the leading term contains the N-dependent factor ∏_{i=1}^{N-k}(1+2(i−1)θ), and the applications divide by the full product ∏_{j=1}^{N}(1+2(j−1)θ). The proof of Lemma 8.20 also uses a signed cancellation between contributions corresponding to ∂1 and −2θ∂1, and Lemma 8.24 uses an analogous cancellation involving (1+2Nθ)^{-1}. These cancellations are stated in words rather than proved by an explicit bijection or generating-function identity. Since Lemma 8.20 and Lemma 8.24 are load-bearing for part (C) of Theorem 1.1, the paper should give a fully detailed proof of these cancellations and of the remainder bounds after division by the gamma-type product.","section":"Section 8.3, Theorems 8.16 and 8.23; Corollaries 8.22 and 8.26"}],"minor_comments":[{"comment":"The notation [∏_{l=1}^{ℓ(ν)} x^{ν_l}] is used in the leading-order formula and in the definition of MA, but it is not defined. It appears to denote coefficient extraction; please define it explicitly in Section 2.","section":"Section 6.1, Theorem 6.1"},{"comment":"The variables x and y in the remainder bounds R(N,θ) are not explicitly identified as N and θ. State this identification in each theorem statement to make the degree bounds unambiguous.","section":"Theorems 6.1, 7.1, 8.16"},{"comment":"The lemma calls MA upper-triangular, but this depends on the ordering of Γ[k] (e.g., by decreasing ℓ(λ)). Make the ordering explicit so that the triangularity claim is unambiguous.","section":"Lemma 6.9"},{"comment":"Statements such as \"Theorem 6.1 with A_{N-1}(θ) replaced by D_N(θ) and θ replaced by 2θ is true\" are not self-contained. Please restate the full theorem or state precisely which substitutions are used in the proof.","section":"Theorems 8.10 and 8.12"},{"comment":"The proof invokes [Yao25, Theorem 1.6] with a Bessel generating function that may not exist in the measure-theoretic setting. Since the argument is formal, please explain why the proof of that theorem applies in the formal power series setting without the existence assumption.","section":"Corollary 6.12"}],"recommendation":"major_revision","confidential_remarks":"To the editor: the main risk is not novelty or the leading-order combinatorics, but the rigor of the remainder estimates on which all converse implications rest. The companion paper [Yao25] is cited for the key type-A bound; if the journal permits reliance on a companion paper, it would be prudent to verify that the companion states the exact off-by-one-free bound needed here. I did not find evidence of improper self-citation; the reliance is substantive and explained."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Short version: this is a real step forward. The type D results (even and odd degree sectors), the converse directions for A and BC in the |θN|→∞ regime, and the |θ0N|→∞ BC regime are genuinely new. Theorem 1.1, if correct, resolves the open question in the appendix of [BGCG22] and upgrades the one-directional results of [Yao25] to equivalences. The asymptotic constants are derived as sums over noncrossing partitions and matrix inverses, with no parameter fitting, and the paper is unusually honest about its own gaps.\n\nThe spot-checks I ran support the main structure: the upper-triangularity of MA, MD, MBC, the diagonal evaluations, and the c=−1 / negative-integer exclusions are consistent. The type A leading term has independent support from [Yao25].\n\nThe soft spots are where the stress-test points. Every converse induction in Theorems 6.10, 7.4, 8.21, and 8.26 deletes a remainder term R/((θN)^{|ν|}N^{ℓ(ν)}) using the degree bounds in Theorem 6.1 (and its analogues). Those bounds — x-degree ≤ k+ℓ(λ)−ℓ(ν)−1, y-degree ≤ k−ℓ(ν) — are load-bearing, and the proof is 'It is not challenging to determine...' with details deferred to [Yao25, Section 6] or left implicit for BC and D. An off-by-one in N or θ would collapse the triangularity and invertibility arguments. That is a genuine rigor gap, not a cosmetic one. Also, the invertibility of W^{D;odd} in Lemma 8.25 is asserted as 'straightforward' and used in Theorem 8.26(d)⇒(b) without proof. Lemma 10.6(B) is admittedly incomplete and supports the type BC one-variable reduction, leaning on [BF97]/[BR25]. These are all addressable in revision, and the leading coefficients check out, so I don't suspect falsehood. But they need to be fixed before the paper can be relied on.\n\nThe probabilistic applications are conditional on the open Conjecture 1.6; the corollaries are fine as conditional statements, but they should be labeled that way in any final version.\n\nAudience: researchers in asymptotic moments of Bessel generating functions, Dunkl operators, and free or rectangular free convolution. The paper deserves a serious referee and probably a 'revise' decision. I would push the author for a fully detailed remainder analysis (or a machine-checked version) before accepting.","headline":"The type D and BC equivalences are new and structurally coherent, but the converse directions rest on remainder estimates that are sketched rather than proved; worth refereeing, not yet a finished paper.","tokens_in":81795,"tokens_out":3006,"would_cite":true,"duration_ms":32496,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["33C52","05A18","41A60","46L54"],"pacs":[],"model":"deepseek-v4-flash","headline":"Scaled Bessel coefficients are equivalent to Dunkl bilinear-form limits","keywords":["Bessel functions","Dunkl operators","Dunkl bilinear form","noncrossing partitions","root systems A_N, BC_N, D_N","asymptotic expansions","free convolution","power-sum symmetric functions"],"falsifier":"Choose N=2, θ_N=N, and evaluate [p_(2),p_(1,1)]_{A_{N-1}(θ)} directly from the definition of the Dunkl operators. Compare the exact value with the main term in Theorem 6.1: if the difference, divided by (θ_N N)^2 N^2, does not tend to zero as N→∞, the theorem's remainder bound is false. More generally, any explicit pair (λ,ν) and sequence θ_N for which the claimed degree bounds on the remainder fail would falsify the equivalence.","tokens_in":1438,"feed_emoji":"🧮","tokens_out":2055,"duration_ms":70953,"temperature":0.7,"pith_summary":"This paper sets out to prove that, in the large-N limit, the asymptotic coefficients of Bessel functions attached to the type A, BC, and D root systems are governed by a single equivalence: scaling the log-coefficients of an exponential formal power series by powers of θN or θ₀N yields the same information as scaling evaluations of the Dunkl bilinear form under power-sum differential operators. The explicit limiting formulas are weighted noncrossing-partition products, with type BC adding a (1+c)^{o(π)} factor and type D carrying an extra product ∏(1+2(j−1)θ) in its odd-degree sector. A sympathetic reader would care because these equivalences convert the difficult task of estimating Bessel coefficients into checkable moment asymptotics, and they yield free-convolution and rectangular-free-convolution limits for the associated measures. The paper also proves analogues in the θN→c regime, generalizing earlier statements and adding new type D results.","feed_headline":"Asymptotic Bessel coefficients boil down to noncrossing partitions","feed_subtitle":"Scaling limits of Bessel coefficients match Dunkl bilinear-form limits for root systems A, BC, and D.","key_machinery":"The central object is the Dunkl bilinear form [f,g]=[1]D(f)g, evaluated on power-sum symmetric functions p_λ. Its leading-order behavior is encoded in triangular infinite matrices whose entries are sums over noncrossing partitions, weighted by block sizes, powers of 2, and (for BC) factors (1+c)^{o(π)}. The proof controls all subleading terms with remainder-degree bounds, so that division by the appropriate powers of θN and N leaves only the leading triangular system, whose invertibility drives the equivalence.","core_discovery":"The central claim is Theorem 1.1: for exponential formal power series F_N = exp(Σ c_λ(N)p_λ), the scaled log-coefficients c_λ(N)/(θN)^{ℓ(λ)} converge if and only if the scaled Dunkl bilinear-form evaluations [1]∏ᵢ(Σ_jD_j)^{νᵢ}F_N / ((θN)^{|ν|}N^{ℓ(ν)}) converge. The common limit is the product over blocks of noncrossing partitions of |B|c(|B|), with type BC contributing (1+c)^{o(π)} and type D splitting into even and odd sectors; the odd sector is normalized by ∏_{j=1}^N(1+2(j−1)θ). Equivalent statements are proved for the θN→c regime, with non-degeneracy conditions such as c≠−1 for BC and c not a negative integer for the converse directions.","pith_inferences":["The paper's leading-order matrices are triangular; an implication it does not spell out is that the same triangular structure should organize finite-N corrections, so one could test the remainder bounds by computing exact small-example bilinear forms and checking the stated x- and y-degree decay.","At c=−1 for type BC, the converse direction is deliberately excluded, suggesting a genuine loss of information at that parameter value; a natural extension would classify what extra data would restore equivalence there.","The odd-degree type D prefactor ∏(1+2(j−1)θ) hints at an interpretation of that sector as moments of a signed measure with a gamma-type density; checking the sign pattern for small N would be a direct test."],"forward_implications":["If the theorem is right, computing the asymptotic coefficients of J^A, J^{BC}, and J^D reduces to computing noncrossing-partition sums: for each ν the limit is ∏ᵢ Σ_{π∈NC(νᵢ)} ∏_{B∈π} |B|c(|B|).","Assuming the conjectural integral representation for products of Bessel functions, the paper's corollaries give weak convergence of the associated measures to the free convolution (type A), the rectangular free convolution (type BC), and a symmetrized free convolution (type D).","In the θN→c regime the same machinery works under explicit non-degeneracy conditions, so the paper covers both |θN|→∞ and θN finite in one framework.","For fixed-degree coefficients, the paper obtains asymptotic inverses of the Dunkl-bilinear-form matrices and, under moment bounds, uniform convergence of Bessel functions on compact sets.","The type D odd-degree sector is genuinely non-polynomial: its leading terms contain ∏_{j=1}^{N-k}(1+2(j−1)θ), so coefficient asymptotics there involve gamma-function-type factors."],"fun_headline_variants":["Noncrossing partitions govern Bessel coefficient limits","Bessel coefficient scaling matches Dunkl bilinear limits","Root systems A, D, BC: Bessel asymptotics unified","Asymptotic Bessel coefficients: a partition limit law","Bessel generating coefficients: scaling to noncrossing partitions"],"cache_read_input_tokens":82944,"weakest_assumption_plain":"The load-bearing premise is the leading-order expansion with remainder bounds: after removing the main term, every remainder monomial is lower order in x and y by at least one power, so it vanishes after division by (θN)^{|ν|}N^{ℓ(ν)}. Every converse direction is an induction that erases those remainders, so an off-by-one power of N or θ would collapse the equivalence.","fun_headline_variants_meta":{"raw":{"variants":["Noncrossing partitions govern Bessel coefficient limits","Bessel coefficient scaling matches Dunkl bilinear limits","Root systems A, D, BC: Bessel asymptotics unified","Asymptotic Bessel coefficients: a partition limit law","Bessel generating coefficients: scaling to noncrossing partitions"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000784,"raw_usage":{"total_tokens":3304,"prompt_tokens":756,"completion_tokens":2548,"prompt_tokens_details":{"cached_tokens":256},"prompt_cache_hit_tokens":256,"prompt_cache_miss_tokens":500,"completion_tokens_details":{"reasoning_tokens":2468}},"tokens_in":500,"tokens_out":2548,"duration_ms":16122,"temperature":1.0,"reasoning_tokens":2468,"cache_read_input_tokens":256,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-04T10:20:31.417916+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Choose N=2, θ_N=N, and evaluate [p_(2),p_(1,1)]_{A_{N-1}(θ)} directly from the definition of the Dunkl operators. Compare the exact value with the main term in Theorem 6.1: if the difference, divided by (θ_N N)^2 N^2, does not tend to zero as N→∞, the theorem's remainder bound is false. More generally, any explicit pair (λ,ν) and sequence θ_N for which the claimed degree bounds on the remainder fail would falsify the equivalence.","supporting_citations":[],"review_version":1}