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REVIEW 3 major objections 4 minor 35 references

Frozen-corner enumeration of Alternating Sign Matrices

T0 review · 3 major / 4 minor · reviewed 2026-08-04 · deepseek-v4-flash

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Pith's one-line read The paper conjectures that the number of n-by-n alternating sign matrices with an s-by-s corner of zeros equals A_n times the determinant of an explicit s-by-s matrix.

desk verdict Honest conjecture paper: the determinant formula is not new (already in [22]) and remains unproved, but the ASM reformulation, reproducible numerical checks, and Tracy-Widom link make it worth refereeing. read the letter →

arxiv 2509.14006 v3 pith:D7AGP3IW submitted 2025-09-17 math.CO cond-mat.stat-mechmath-phmath.MP

Filippo Colomo , Andrei G. Pronko This is my paper · ORCID
classification math.COcond-mat.stat-mechmath-phmath.MP MSC 05A1505A1982B20
keywords alternatingsignmatricesfrozencornerFredholmdeterminantTracy-Widomdistributionsix-vertexmodelmonotonetriangleslimitshapeenumeration
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

The paper proposes an exact formula for B_{n,s}, the number of n x n alternating sign matrices whose top-left corner is an s x s block of zeros. The formula expresses B_{n,s} as A_n times the determinant of an s-by-s matrix M with entries built from hypergeometric functions and refined ASM counts. The authors verify the formula numerically for all n up to 20 and prove it for s=1 through 4 by direct integration, but leave the general case as a conjecture. If true, the determinant form is exactly what one needs to take the large-n limit, and the paper shows it leads to GUE Tracy-Widom fluctuations for the frozen boundary. A rigorous proof would fill a long-standing gap in understanding the fluctuations of large ASMs.

What carries the argument

The central object is the s-by-s matrix M of (1.9)-(1.10), whose entries are double contour integrals with kernel (1-z-w)^{-1} and factors f_i^+(z), f_j^-(w) built from the hypergeometric function g_n(z) that generates refined ASM counts. The matrix converts the multiple-integral representation of B_{n,s}, a constant-term identity from the six-vertex model, into a determinant of size s, small enough for asymptotic analysis via the saddle-point method.

What would settle it

Compute B_{n,s} directly for some n>20, for example n=21, s=5 or s=6, via monotone-triangle enumeration (the paper's code scales) and compare with A_n det(1-M); a mismatch at any (n,s) would disprove Conjecture 1. Alternatively, give an exact symbolic evaluation of the s=5 case and test it against enumeration data.

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Extended reading notes

Core claim

For any n and s up to floor(n/2), the paper conjectures the identity B_{n,s} = A_n det_{1<=i,j<=s}(1-M), where M has entries given by double contour integrals of products of hypergeometric functions, equivalently as an explicit finite sum over refined ASM counts. The determinant is of Fredholm type, so it can be promoted to the Fredholm determinant of an integral operator, and in the appropriate scaling limit its kernel converges to the Airy kernel, yielding the GUE Tracy-Widom distribution for boundary fluctuations.

Load-bearing premise

The formula is assumed to hold for all s based on checks for s=1 through 4 plus consistency constraints, with no proof that the guessed matrix structure persists for larger s.

Editorial extensions

If this is right

  • An exact determinant formula for B_{n,s} valid for every n and s, with entries expressible through known ASM counts.
  • The cumulative distribution of the frozen-boundary intersection with the diagonal converges to the GUE Tracy-Widow distribution under the 2/3-scaling limit (Proposition 4).
  • Tracy-Widom universality extends to ASMs, which are not free-fermionic or determinantal models.
  • The known symmetry and vanishing identities (1.3)-(1.6) are consistent and follow as special cases of the conjecture.
  • The conjecture provides a concrete target for a rigorous proof, for example through integrable-operator or Cauchy-Binet techniques.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • The determinant form hints at a hidden determinantal structure in ASMs that may extend beyond the free-fermion six-vertex point; one could test whether other corner refinements, such as frozen rectangles, admit analogous determinant formulas.
  • The asymptotic machinery could be pushed to describe the full boundary fluctuation field, not just the intersection with the diagonal, yielding a two-parameter Airy-type process.
  • Assuming the conjecture, the saddle-point analysis might provide an independent route to the arctic curve itself, complementing the tangent method and Aggarwal's proof.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

3 major / 4 minor

Summary. The paper defines B_{n,s}, the number of n×n alternating sign matrices with an s×s all-zero square in a corner, and proposes Conjecture 1: B_{n,s} = A_n det_{1≤i,j≤s}(1-M), where M is given by the double-integral formula (1.9) or the equivalent double-sum formula (1.10). The conjecture is supported by direct enumeration of monotone triangles for all n≤20 and s≤n, which also verifies identities (1.3)–(1.6). The paper also recalls a multiple-integral formula for B_{n,s} (Proposition 2) and derives a conditional Tracy–Widom limit for the frozen-boundary fluctuations (Proposition 4), with the derivation deferred to the authors' earlier work [22]. The central determinant formula is explicitly presented as a conjecture, not a theorem.

Significance. If Conjecture 1 is true, it provides an exact finite-n enumeration for ASMs with a frozen corner of arbitrary size s, and via Proposition 4 it implies GUE Tracy–Widom fluctuations for the frozen boundary, extending this universality beyond determinantal models. The paper is transparent about the conjectural status of its main formula, and it supplies reproducible numerical data: the Appendix A code enumerates monotone triangles up to n=20 and the reported values match the determinant formula. The main weakness is that the determinant ansatz is inferred from the cases s=1,...,4 and is not proved; all later conclusions are conditional on that unproved extrapolation. The paper would be a useful contribution to the combinatorics community if this gap is clearly addressed or if the derivation in [22] is reproduced in sufficient detail.

major comments (3)
  1. [§2.3 and Conjecture 1] The central claim is unproved. Section 2.3 states that the matrix M was guessed after working out s=1,...,4, with the general structure constrained by 'two lemmas' and 'natural recursive properties' deferred to [22]. Because [22] is not reproduced, the reader cannot verify that the structure is forced rather than merely fitted to small cases. The numerical checks in §3 cover n≤20, hence only s≤10, while the conjecture is asserted for all s. Proposition 4 depends entirely on this unproved formula. Please include the two lemmas and the recursive derivation, or give a proof of Conjecture 1, or at least state precisely which assertions in [22] establish the structure for all s.
  2. [§4.2, Proposition 4] Proposition 4 is presented as a numbered result, but its proof is entirely delegated to [22]. The claim that 'a careful analysis shows' the Airy-kernel limit is not supported by any calculation in this paper. As written, the reader cannot tell which part of Proposition 4 is a new contribution and which is a quotation from [22]. If Proposition 4 is meant as a standalone result, provide a proof or detailed sketch; otherwise label it explicitly as a conditional theorem quoted from [22].
  3. [§3.3 and Appendix B] The numerical evidence is finite in both n and s: all n≤20, so s≤10. The step from s=1..4 to arbitrary s is a genuine extrapolation, not a proof. Since Conjecture 1 is the only route to the Tracy–Widom conclusion, the paper should either supply a proof, present additional structural constraints that uniquely determine M for all s, or include checks for larger s (e.g., by high-precision numerical evaluation of the integral representation (2.1)). Without this, the conjecture remains a plausible but weakly supported ansatz.
minor comments (4)
  1. [Abstract] Typos: 'explicitely' should be 'explicitly'; 'in connections with' should be 'in connection with'.
  2. [§1.3, Eq. (1.9)] The contour notation I_{C0} is nonstandard and the description 'containing no other singularity' is ambiguous. The integrand has poles only at 0 from f_i^±, and 1-z-w is nonzero for two small contours around the origin; please clarify, preferably using \oint_{C0}.
  3. [§3.2] The text refers to a frozen square in the 'top-right corner' whereas the earlier definition and Conjecture 1 use the top-left corner. By symmetry this is harmless, but the wording should be made consistent.
  4. [General] The term 'Fredholm type determinant' is applied to an s×s matrix determinant; for finite s this is an ordinary determinant. Consider saying 'determinant of an s×s matrix of Fredholm type' or 'Fredholm determinant of a finite-rank operator' if the limiting operator is being emphasized.

Circularity Check

0 steps flagged · score 2.0 of 10

No significant circularity; the central claim is an openly stated conjecture verified against independent monotone-triangle enumeration.

full rationale

The paper's central object, Conjecture 1, is explicitly presented as a conjecture, not as a derived theorem. Section 2.3 states that the determinant form was guessed from the cases s=1,...,4, with the general structure 'constrained by two lemmas ... and by some natural recursive properties, see [22]'. This is a self-citation to the authors' prior work, and the cited material is not reproduced, so the paper does not provide a complete derivation of the matrix M. However, this is not circularity: the conjecture is not used as an input to prove itself, and no fitted parameter is renamed as a prediction. The numerical checks in Section 3 compare the conjectured determinant formula with an independent monotone-triangle enumeration for all n≤20, and the paper explicitly notes that the identity has gone through all checks. Proposition 4 is conditional on Conjecture 1 and clearly states 'If Conjecture 1 is true'. The derivation of the multiple-integral representation (Proposition 2) is attributed to the six-vertex model literature and can be checked numerically. The main weakness is the lack of proof for the guessed determinant structure, not a circular reduction of the conclusion to its premises. The self-citation to [22] is load-bearing for the ansatz's motivation but not for any claimed proof, since no proof is claimed. Therefore the appropriate circularity score is low, reflecting the presence of self-citation without circular dependence.

Assumptions & free parameters 0 free parameters · 5 assumptions · 0 invented entities

No free parameters are fitted and no new entities are introduced. The central claim rests on proved ASM enumeration theorems, the monotone-triangle bijection, an imported exact integral representation from the six-vertex model, and an unproved structural guess for the determinant form. The numerical checks are independent but do not remove the unproved step.

assumptions (5)
  • standard math Zeilberger's ASM enumeration formula (1.1) and refined ASM enumeration formula (1.2)
    Used to define A_n and A_{n,r}, to normalize g_n(z), and to express the matrix M in terms of known enumerations.
  • domain assumption Bijection between ASMs and monotone triangles with the frozen-corner condition translated to (3.1)
    Section 3 relies on the standard ASM-to-monotone-triangle bijection for the direct enumeration; no proof is repeated here.
  • domain assumption Proposition 2, the exact multiple-integral representation for B_{n,s}
    The paper takes the identity (2.1) from the authors' earlier six-vertex model work via integrability and the Quantum Inverse Scattering Method; no self-contained derivation appears in this preprint.
  • ad hoc to paper The inferred s by s determinant structure of M in (1.9) holds for all s
    Section 2.3 states that the determinant form was obtained for s=1..4 and then guessed in general; this is exactly the unproved content of Conjecture 1.
  • domain assumption Saddle-point analysis leading to the Airy kernel and Tracy-Widom distribution in Proposition 4
    Proposition 4 is stated with details deferred to the authors' previous paper [22]; it is conditional on Conjecture 1.

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Cite this review

Pith. "Pith review of Frozen-corner enumeration of Alternating Sign Matrices." pith.science (2026). https://pith.science/paper/D7AGP3IW

@misc{pith2026250914006,
  author       = {Pith},
  title        = {Pith review of: Frozen-corner enumeration of Alternating Sign Matrices},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/D7AGP3IW}},
  note         = {Machine review of arXiv:2509.14006}
}
abstract

An Alternating Sign Matrix (ASM) is a square matrix with entries in $\{0,1,-1\}$, and such that: $i)$ in each row and columns, nonzero entries alternate in sign; $ii)$ for any given row or column, entries sum up to $1$. We define the frozen-square enumeration as the enumeration of $n\times n$ ASMs under the refinement of having, located in a corner, an $s\times s$ square of entries that are all zeroes. We state a conjectural formula for such enumeration, in terms of the determinant of some $s\times s$ matrix whose entries are given explicitly. We provide numerical support in favour of our conjecture. We also illustrate the relevance of the conjectured formula in connection with the limit shape observed in large ASMs, its fluctuations, and the Tracy--Widom distribution.

Discussion (0). Continue with ORCID to comment.

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