{"id":"d3465baf-7e54-4537-8e71-d992a75cc5a6","arxiv_id":"2602.19425","paper_version":2,"verdict":"ACCEPT","confidence":"MODERATE","novelty_score":7.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"For the N=2* U(N) instanton partition function, the absolute convergence radius is exactly 1 for generic parameters with b² = ε₁/ε₂ outside [0,∞), and for b² > 0 it is positive exactly when b² has finite 'exponential type' (a Brjuno-type condition).","lead":"This paper proves exactly when the Nekrasov instanton partition function of the 4d N=2* U(N) gauge theory converges as a sum over Young diagrams. For generic parameters with ε₁/ε₂ outside the non-negative real axis the radius of convergence is exactly 1; for positive real ε₁/ε₂ it is controlled by how well that ratio is approximable by rationals, and shrinks to zero for the most extreme (Liouville-type) ratios.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"No significant objection identified","rationale":"The reader's verdict of ACCEPT with moderate confidence is fair. The central claim is a precise theorem with explicit genericity assumptions, and the proof shows no fatal gap in the parts I scrutinized. The genericity assumption for b² > 0 (excluding real m and real Coulomb differences) is the most delicate point, as the reader noted, but the paper states it clearly and does not overclaim coverage of those cases. The conditional-convergence scope limitation is also explicitly separated in Section 1.2. The minor index typo in Lemma 4.2 is not load-bearing. I therefore see no reason to adjust the verdict.","tokens_in":27760,"tokens_out":45184,"duration_ms":351918,"concrete_test":"Independently re-derive Lemma 4.2 with the indices corrected to a_{n+1} (and q_{n+1} = a_{n+1} q_n + q_{n−1}) and confirm that Proposition 4.6 still yields Csup(x) ≤ 4 + 4Bsup(x). If the corrected proof changes the numerical constant, the lower-bound exponent in (5.26) must be adjusted accordingly, though the qualitative theorem would be unaffected.","verdict_should_be":"UNCHANGED","load_bearing_attack":"I did not find a load-bearing flaw in the central argument. Theorem 1.1 is explicitly conditional on the stated genericity assumptions: m and a_I − a_J must avoid the closure of ε₁Z + ε₂Z. For b² > 0 irrational, this excludes the real line ε₂R, which is a measure-zero but physically common slice; this is a scope limitation the paper acknowledges, not an internal inconsistency. The proof structure is coherent: the b² ∉ [0,∞) case uses box-content counting with decay |λ|^{1/2} log|λ|, the b² > 0 lower bound uses the partially summed logarithmic averages of Proposition 4.6, and the upper bound uses the λ = (p,1^q) family. I cross-checked the summation-by-parts in Section 5 and the off-diagonal factor bounds in Section 6.2; the key inequalities are valid because the numerator factors are bounded away from zero by the genericity condition, not merely the denominators. The only issue I noticed is the index typo in Lemma 4.2 (a_n should be a_{n+1}), but this is readily fixable and does not affect the stated bounds.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper proves Theorem 1.1 about the absolute convergence radius R_abs of the Nekrasov instanton partition function for the 4d N=2* U(N) gauge theory, starting from the explicit sum over colored partitions (2.1), which is taken as the definition and derived from Nekrasov's residue formula in Appendix A. For generic mass and Coulomb differences (avoiding the closure of ε₁Z+ε₂Z), the theorem states: (i) for b²=ε₁/ε₂∈C\\[0,+∞), R_abs=1; (ii) for irrational b²>0, R_abs is governed by the exponential type B_sup(b²), with explicit lower and upper bounds (1.6), so R_abs>0 iff B_sup<∞ and R_abs=0 for super-exponentially well approximated b²; (iii) for rational b²>0 the term-by-term sum is ill-defined due to singular terms. The proof uses box-content counting and a sublinear estimate log|Z_λμ|=O(|λ|^{1/2}log|λ|) in Section 3, a continued-fraction equidistribution analysis and averaged logarithmic bounds in Sections 4–5, and a specific family λ=(p,1^q) for upper bounds in Section 6. The AGT consequences for torus one-point conformal blocks are stated in Section 1.3.","tokens_in":27739,"tokens_out":13133,"duration_ms":104216,"significance":"If correct, this is a substantial advance: it establishes the optimal unit radius for generic parameters, improves the earlier positive-radius bound of [3] for b²∈C\\[0,+∞), and reveals a new Diophantine dichotomy for b²>0. The paper is self-contained and unusually checkable: the central estimates (3.10), (5.23), and (6.19) are derived explicitly, the distinction between absolute and conditional convergence is handled carefully, and the main theorem is stated with its precise genericity assumptions. The scope limitation—for real b² the genericity excludes real masses and real Coulomb differences, a measure-zero but physically common slice—is explicitly acknowledged in Section 1.2 and is not an internal inconsistency. The paper also credits prior related work fairly.","major_comments":[],"minor_comments":[{"comment":"There is an index typo in the proof: in the two tasks, the bounds k∈J2,a_nK and k∈J1,a_n−1K should read a_{n+1}, since q_{n+1}=a_{n+1}q_n+q_{n−1}. The argument is otherwise valid and the typo does not affect the stated bound.","section":"Lemma 4.2"},{"comment":"Same typo: the quotient M is written as M∈J1,a_nK, but it should be M∈J1,a_{n+1}K. This is a local indexing error and does not change the proof.","section":"Proposition 4.6, Eq. (4.18)"},{"comment":"Minor wording: 'developped' should be 'developed'. Also, the notation ϵ₁Z+ϵ₂Z is used both for the lattice and for its closure (e.g., Eq. (2.6) and Eq. (3.12)); an overline or an explicit convention would help avoid ambiguity for readers.","section":"Section 2.3"},{"comment":"The passage from the bound (5.26), R_abs ≥ A e^{-8−8B_sup(b²)}, to the stated lower bound A₁ e^{-8B_sup(b²)/max(1,b²)} in (1.6) uses the b→1/b symmetry. This is valid, but a one-sentence clarification that the constant A₁ absorbs the additional e^{-8} and the transformed prefactor would improve readability.","section":"Theorem 1.1 / Section 5"}],"recommendation":"minor_revision","confidential_remarks":"I agree with the stress-test assessment: no load-bearing flaw was found. The genericity condition is a genuine scope restriction for real b², but it is precisely stated and not a circularity. The paper is within the journal's scope and would be a valuable contribution. The only issues are the indexing typos in Lemma 4.2 and Proposition 4.6 and a few presentation points, which are readily fixable."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Bruno Le Floch's paper proves the expected optimal convergence radius for the N=2* instanton sum in the generic regime, and I think the main theorem is right. The reader's report matches my own checks.\n\nWhat is new: for b^2 outside [0,+∞), the paper upgrades Arnaudo–Bonelli–Tanzini's positive radius to R_abs = 1, which is optimal and consistent with the AGT expectation. For b^2 > 0 it introduces the exponential type B_sup(b^2), proves a genuine dichotomy — positive radius iff B_sup is finite, zero radius for super-exponentially well approximable b^2 — and shows the term-by-term sum is ill-defined for positive rational b^2. That is a real advance, not a repackaging. The proof is self-contained once the residue formula (2.1) is taken as definition, and the combinatorial box-counting arguments in Section 3 and the partial-summation estimate in Section 5 are coherent. I found the same index typo in Lemma 4.2 (a_n should be a_{n+1}); it is harmless.\n\nThe soft spots are the ones the paper itself flags. The genericity condition excludes m and a_I − a_J from the closure of ε1 Z + ε2 Z. For irrational b^2 > 0 with real ε2 this is the real line, so real masses and real Coulomb differences are outside the theorem. That is a large physical slice, and it is not a cosmetic issue: the denominator estimates genuinely break down there. Also, for b^2 > 0 the results concern only absolute convergence; conditional convergence of the physical series is left open, as is the rational-b^2 pole structure (Conjecture C.1). The constants A1, A2 are not explicit, though they are continuous functions of the parameters, so this is a nuisance more than a flaw. Proposition 4.6's proof is compressed; a referee should verify the constants there.\n\nWho gets value: anyone working on Nekrasov partition functions, AGT correspondence, or convergence of conformal blocks. It will become a standard reference for this question. I would send it to a serious refereeing process; the referee should check Section 5's summation-by-parts and Section 6.2's upper bound, and possibly ask for more detail on the non-explicit constants. But there is no load-bearing flaw I can identify.","headline":"Solid proof of the optimal convergence radius for N=2* instanton sums, with a new Diophantine dichotomy for b^2>0; worth a careful referee.","tokens_in":28595,"tokens_out":3081,"would_cite":true,"duration_ms":26268,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["05A17","11J70","30B10","81T13"],"pacs":[],"model":"deepseek-v4-flash","headline":"The Nekrasov instanton sum converges with radius 1 for generic parameters, and the paper pins down exactly when the absolute convergence fails.","keywords":["Nekrasov instanton partition function","N=2* gauge theory","absolute convergence","Omega background","gauge/CFT correspondence","conformal blocks","Diophantine approximation","exponential type"],"falsifier":"Take $b^2$ to be a super-exponentially well approximable irrational such as $\\sum_{n \\ge 0} 2^{-2^n}$ (a number with $B_{\\sup}(b^2)=\\infty$) and compute the specific terms $\\lambda=(p,1^q)$ used in Section 6.2 along the sequence of best rational approximants; the values $|Z_{((p,1^q),\\varnothing,\\ldots)}|^{-1/(p+q)}$ should tend to 0, confirming $R_{\\text{abs}}=0$. Conversely, for an irrational $b^2$ with finite exponential type, a numerical evaluation of the $\\liminf$ in (2.2) over partitions of size up to $k$ should approach a positive number consistent with the bounds in (1.6).","tokens_in":27323,"feed_emoji":"🧮","tokens_out":4986,"duration_ms":45727,"temperature":0.7,"texified_at":"2026-08-05T20:57:35.313747+00:00","pith_summary":"This paper proves the absolute convergence radius of the Nekrasov instanton partition function of the 4d N=2* U(N) gauge theory. For generic values of the equivariant parameters, the sum over colored partitions converges exactly when the instanton counting parameter satisfies $|q|<1$, matching the radius expected from the gauge/CFT correspondence. The central new phenomenon is a Diophantine dichotomy for real ratios $b^2=\\varepsilon_1/\\varepsilon_2$: for irrational $b^2$ that is not too well approximated by rationals the radius is positive but may be smaller than 1; for super-exponentially well approximable $b^2$ the absolute sum diverges for every nonzero $q$; and for rational $b^2$ some individual terms carry poles. The proof works by refining the term-by-term bounds, showing that only a small number of boxes in each Young diagram can have a given 'content', which turns previous exponential bounds into subexponential ones.","texify_model":"deepseek-v4-flash","texify_usage":{"total_tokens":7969,"prompt_tokens":949,"completion_tokens":7020,"prompt_tokens_details":{"cached_tokens":0},"prompt_cache_hit_tokens":0,"prompt_cache_miss_tokens":949,"completion_tokens_details":{"reasoning_tokens":6029}},"feed_headline":"Nekrasov instanton sums converge in the unit disk—generically","feed_subtitle":"Exact radius found for the adjoint-matter theory; rational equivariant ratios make the term-by-term sum ill-defined.","key_machinery":"The engine of the proof is a refined bound on each term $Z_{\\vec{Y}}$ in the sum over partitions. Instead of uniformly bounding every factor in the product (2.1), the paper counts how many boxes can share the same 'content' $-\\varepsilon_1(\\mu'_j - i) + \\varepsilon_2(\\lambda_i - j)$, showing that at most $O(|\\lambda|^{1/2})$ boxes have a given value. This converts the previous crude exponential estimates into subexponential growth of $\\log|Z_{\\vec{Y}}|$, which is enough to force the $\\liminf$ in (2.2) to be at least 1. For $b^2>0$, the same box-counting is governed by a Diophantine invariant called the exponential type $B_{\\sup}(b^2)$, a Brjuno-like supremum measuring how well rationals approximate $b^2$; the proofs use an equidistribution lemma for fractiona","core_discovery":"The main result, Theorem 1.1, states that under the genericity assumption that the adjoint mass $m$ and all Coulomb-branch differences $a_I-a_J$ avoid the closure of the lattice $\\varepsilon_1 \\mathbb{Z} + \\varepsilon_2 \\mathbb{Z}$: (i) for $b^2 \\in \\mathbb{C} \\setminus [0,+\\infty)$ the absolute convergence radius is exactly $R_{\\text{abs}}=1$; (ii) for irrational $b^2>0$, $R_{\\text{abs}}$ is controlled by the exponential type $B_{\\sup}(b^2)$, satisfying explicit exponential bounds, with $R_{\\text{abs}}=0$ when $B_{\\sup}(b^2)=\\infty$; (iii) for rational $b^2>0$ the term-by-term sum is ill-defined because some terms are singular. Through the gauge/CFT correspondence this translates into convergence of torus one-point conformal blocks of the Virasoro and $W_N$ algebras for central charges in $\\mathbb{C} \\setminus [25,+\\infty)$ in the Virasoro case, w","pith_inferences":["If cancellations are as strong as the contour-integral representation suggests, the conditional convergence radius of the physical series may still be 1 even when the absolute radius is smaller, including for the b^2>0 cases; the paper explicitly leaves this open.","The same box-content counting technique should extend to 5d instanton partition functions and to related SQCD theories, potentially producing a similar Diophantine dichotomy there; the paper notes the technique appears to extend straightforwardly to the 5d setting.","The genericity assumption excludes masses and Coulomb differences lying on the real line ε2R, a common physical convention; a testable question is whether summing in carefully chosen groups removes the singular rational-b^2 terms and restores convergence for those parameter values.","The resemblance to convergence of basic hypergeometric series and of recent Virasoro fusion-kernel series suggests that exponential type may be the controlling invariant for a wider family of conformal-block expansions."],"forward_implications":["For b^2∈C\\[0,+∞), the instanton partition function is absolutely summable in the unit disk, so any rearrangement of the sum is legitimate; this supports constructions that resum subsets of terms, such as Higgsing limits.","The gauge/CFT correspondence implies that one-point torus conformal blocks of Virasoro and W_N converge in |q|<1 for generic external and internal parameters when the central charge lies outside [25,+∞) in the Virasoro case.","For irrational b^2>0 with finite exponential type there is a positive absolute convergence radius, so the absolute Nekrasov sum is meaningful for small coupling, with the radius explicitly bounded by Diophantine properties of b^2.","For super-exponentially well approximable b^2>0 the absolute sum diverges for every nonzero q, so any convergence of the physical series would have to come from systematic cancellations between terms.","For rational b^2>0 the term-by-term sum is ill-defined, confirming that rational ratios require special handling, such as grouping terms or cancelling poles."],"fun_headline_variants":["Exact radius for Nekrasov instanton sums","Nekrasov sums: radius 1 for non-real ε1/ε2","Rational ε1/ε2 makes Nekrasov sums ill-defined","Conformal blocks converge for central charges outside [25,∞)"],"cache_read_input_tokens":2304,"weakest_assumption_plain":"The proof requires the adjoint mass and all Coulomb-branch differences to avoid the closure of the lattice $\\varepsilon_1 \\mathbb{Z} + \\varepsilon_2 \\mathbb{Z}$; when $\\varepsilon_2$ is real and $b^2$ is irrational this closure is the entire real line, so the genericity condition excludes all real masses and real Coulomb differences, and if a denominator can approach zero the lower-bound estimates break down.","fun_headline_variants_meta":{"raw":{"variants":["Exact radius for Nekrasov instanton sums","Nekrasov sums: radius 1 for non-real ε1/ε2","Rational ε1/ε2 makes Nekrasov sums ill-defined","Conformal blocks converge for central charges outside [25,∞)"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000402,"raw_usage":{"total_tokens":2009,"prompt_tokens":898,"completion_tokens":1111,"prompt_tokens_details":{"cached_tokens":256},"prompt_cache_hit_tokens":256,"prompt_cache_miss_tokens":642,"completion_tokens_details":{"reasoning_tokens":1042}},"tokens_in":642,"tokens_out":1111,"duration_ms":9646,"temperature":1.0,"reasoning_tokens":1042,"cache_read_input_tokens":256,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-02T21:43:14.536090+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Take $b^2$ to be a super-exponentially well approximable irrational such as $\\sum_{n \\ge 0} 2^{-2^n}$ (a number with $B_{\\sup}(b^2)=\\infty$) and compute the specific terms $\\lambda=(p,1^q)$ used in Section 6.2 along the sequence of best rational approximants; the values $|Z_{((p,1^q),\\varnothing,\\ldots)}|^{-1/(p+q)}$ should tend to 0, confirming $R_{\\text{abs}}=0$. Conversely, for an irrational $b^2$ with finite exponential type, a numerical evaluation of the $\\liminf$ in (2.2) over partitions of size up to $k$ should approach a positive number consistent with the bounds in (1.6).","supporting_citations":[],"review_version":1}