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When do imprecise copulas avoid sure loss? It depends on the matrix

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load-bearing objection Solid paper with three distinct results; the counterexample in Prop. 4.5 needs its arithmetic made fully transparent. the 1 major comments →

arxiv 2607.06178 v1 pith:5IOKBESY submitted 2026-07-07 math.ST math.PRstat.TH

Discrete imprecise copulas and alternating sign matrices

Toma\v{z} Ko\v{s}ir , Elisa Perrone , Nik Stopar This is my paper
classification math.ST math.PRstat.TH MSC 62H0515B35
keywords alternating sign matricesimprecise copulasquasi-copulasdefect transformationsavoidance of sure losscoherencedense alternating sign matricesdiscrete copulas
verification ladder T0 review T1 audit T2 compute T3 formal T4 reserved

The pith

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

The paper studies discrete quasi-copulas and imprecise copulas of minimal range, which correspond bijectively to alternating sign matrices (ASMs) — matrices with entries in {-1, 0, 1} whose nonzero entries alternate in sign in every row and column. The authors prove three things. First, the family of quasi-copulas corresponding to ASMs is invariant under all six defect transformations (operations that adjust a quasi-copula by its failure to be a proper copula on certain rectangles). This means applying any defect transformation to an ASM-linked quasi-copula always yields another ASM-linked quasi-copula. Second, they construct an explicit 17×17 counterexample showing that imprecise copulas built from ASMs can fail to avoid sure loss, meaning no genuine copula may fit between the lower and upper bounds. Third, they prove that when the underlying ASMs are dense — meaning no zero entries appear between two nonzero entries in any row or column — the resulting imprecise copulas are always coherent, the strongest consistency property. The paper thus draws a sharp line: structural invariance under transformations holds across all ASMs, but probabilistic consistency holds only for the dense subclass.

Core claim

The central discovery is that the combinatorial structure of alternating sign matrices partitions the landscape of imprecise copulas in a specific way: all ASMs are invariant under defect transformations (Theorem 3.3), but only dense ASMs guarantee coherence (Theorem 5.1), while general ASMs can produce imprecise copulas that fail to avoid sure loss (Proposition 4.5). The load-bearing mechanism is the interaction between the alternating-sign pattern of matrix entries and the volume inequalities that define when a copula can be squeezed between two quasi-copulas. For dense ASMs, the block structure decomposes into irreducible pieces (the Brualdi-Schroeder matrices F_n^k) where each piece is a

What carries the argument

The argument proceeds through three mechanisms. (1) Defect transformations: six operations on quasi-copulas defined via directional defect matrices, which measure how far a quasi-copula is from being supermodular on rectangles. The 1-Lipschitz property forces transformed ASM-linked quasi-copulas to remain ASMs. (2) Avoidance of sure loss: characterized by a function L_{(P,Q)} whose sign on a region R determines whether a copula exists between bounds P and Q. The counterexample uses a 17×17 ASM where L evaluates to -1 on a region with 6 positive-multiplicity vertices. (3) Dense ASM structure theorem: dense ASMs decompose into block-diagonal form with irreducible blocks F_n^k, and coherence is

Load-bearing premise

The counterexample to avoidance of sure loss (Proposition 4.5) depends on a single explicit 17×17 alternating sign matrix and a manual count of 6 vertices with positive multiplicity on a shaded region R, yielding L = -1. If the region or its multiplicities were miscounted, the counterexample would fail, and with it the claim that ASM imprecise copulas do not in general avoid sure loss.

What would settle it

Recompute L_{(P0,Q0)}(R) for the explicit 17×17 ASM A in Proposition 4.5. If the region R does not have exactly 6 positive-multiplicity vertices each with defect -1, or if V_{P0}(R) ≠ -7, then L ≠ -1 and the counterexample collapses.

Watch this falsifier — get emailed when new claim-graph text bears on it.

If this is right

  • Defect transformations provide a new tool for studying the combinatorics of ASMs, since they map ASMs to ASMs — one can now ask how rank, Bruhat order position, or other ASM invariants change under these operations.
  • The boundary between coherent and non-coherent ASM imprecise copulas is partially characterized: density suffices for coherence, but the exact conditions for non-dense ASMs remain open.
  • The 17×17 counterexample is constructive and extensible to all larger n, giving a concrete family of imprecise copulas that fail to avoid sure loss.
  • Approximation of full-domain imprecise copulas by discrete ones (announced as forthcoming work) can now leverage the ASM structure to control whether the approximation preserves avoidance of sure loss.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The gap between invariance under defect transformations and coherence suggests there may exist an intermediate combinatorial condition — weaker than density but stronger than the general ASM property — that exactly characterizes avoidance of sure loss for ASM imprecise copulas.
  • Since the counterexample requires n ≥ 17, smaller ASM imprecise copulas might all avoid sure loss; if so, the minimum size for failure would be a combinatorially meaningful quantity.
  • The block-structure argument for dense ASMs may extend to non-dense ASMs with controlled zero patterns, potentially yielding partial coherence results for broader classes.
  • If defect transformations on ASMs interact naturally with the Bruhat order, they could provide a lattice-theoretic route to constructing coherent imprecise copulas from arbitrary ASMs.

Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

1 major / 7 minor

Summary. The paper studies discrete quasi-copulas and imprecise copulas of minimal range, which correspond to alternating sign matrices (ASMs). Three main results are established: (1) the family of minimal-range quasi-copulas is invariant under all six defect transformations (Theorem 3.3); (2) ASM imprecise copulas do not in general avoid sure loss, demonstrated by an explicit 17×17 counterexample (Proposition 4.5); and (3) imprecise copulas corresponding to dense ASMs are always coherent (Theorem 5.1). The proofs are constructive and the arguments for Theorems 3.3 and 5.1 rest on clean general reasoning. The counterexample in Proposition 4.5 involves a finite but unverified computation.

Significance. The paper makes a solid contribution connecting the combinatorial structure of ASMs with probabilistic consistency properties of imprecise copulas. The invariance result (Theorem 3.3) is a clean structural statement proved via the 1-Lipschitz property. Theorem 5.1 provides a positive coherence result for the dense ASM subclass, reducing to known results from [19]. The counterexample in Proposition 4.5 is the most novel contribution, showing that minimal-range imprecise copulas can fail to avoid sure loss. The results are falsifiable and the constructions are explicit.

major comments (1)
  1. Proposition 4.5 (proof, p. 11–12): The computation L_{(P0,Q0)}(R) = -7 + 6 = -1 is the load-bearing calculation for the paper's central counterexample, but none of its four sub-claims is independently verified in the manuscript. Specifically: (1) the region R is described as 'the shaded region in A' but no figure actually shows R overlaid on the 17×17 matrix; (2) the claim that R has exactly 6 vertices with positive multiplicity, each with multiplicity 1, is stated without a table or enumeration; (3) the claim that D^P0_M = -1 at all 6 vertices is not separately verified; (4) V_{P0}(R) = -7 is asserted without an auxiliary computation. Since the entire counterexample fails if any of these is off by one unit (e.g., 7 positive-multiplicity vertices would give L = 0), the authors should provide either a table listing the 6 vertices with their multiplicities and defect values, or a machine-v
minor comments (7)
  1. Theorem 3.3, proof: the argument that consecutive nonzero entries must alternate in sign is sound but could be stated more precisely. The inequality |sum_{r=j}^{k} a_{ir}| <= 1 follows from the partial sum conditions (3), but the step from this to 'distinct sign' for consecutive nonzero entries could be spelled out in one additional sentence.
  2. Proposition 4.5, proof: 'Consequentely' should be 'Consequently'.
  3. Proposition 4.5, proof: 'the pair (P 0, Q0)' — the double space is a typo.
  4. Section 2, p. 4: 'charactarized' should be 'characterized'.
  5. The paper references [20] as 'in preparation, 2026'; this should be updated if possible before final submission.
  6. Figure 1.1 is referenced as showing a schematic summary but the figure content should be verified for legibility in the final version.
  7. In the proof of Theorem 5.1, the reduction to [19, Thm. 4.8] and [19, Thm. 4.10] is stated concisely; a reader without [19] at hand may find it difficult to follow the block-structure argument. A brief restatement of what [19, Thm. 4.4] asserts about the block structure would help.

Simulated Author's Rebuttal

1 responses · 0 unresolved

The referee requests detailed verification of the four sub-claims underlying the computation L_{(P0,Q0)}(R) = -7 + 6 = -1 in Proposition 4.5, which is the paper's central counterexample. The request is entirely reasonable: the computation is load-bearing, the margin is tight (a single additional positive-multiplicity vertex would change the conclusion), and the manuscript currently provides no figure, table, or auxiliary computation to support the sub-claims.

read point-by-point responses

  1. Referee: Proposition 4.5 (proof, p. 11–12): The computation L_{(P0,Q0)}(R) = -7 + 6 = -1 is the load-bearing calculation for the paper's central counterexample, but none of its four sub-claims is independently verified in the manuscript. Specifically: (1) the region R is described as 'the shaded region in A' but no figure actually shows R overlaid on the 17×17 matrix; (2) the claim that R has exactly 6 vertices with positive multiplicity, each with multiplicity 1, is stated without a table or enumeration; (3) the claim that D^P0_M = -1 at all 6 vertices is not separately verified; (4) V_{P0}(R) = -7 is asserted without an auxiliary computation. Since the entire counterexample fails if any of these is off by one unit (e.g., 7 positive-multiplicity vertices would give L = 0), the authors should provide either a table listing the 6 vertices with their multiplicities and defect values, or a machine-v

    Authors: The referee is correct on all four points. The computation in Proposition 4.5 is the most consequential calculation in the paper, and as presented it is under-documented. We will address each sub-claim in the revised manuscript as follows: (1) We will add a figure showing the region R overlaid on the 17×17 matrix A, with the 39 constituent squares clearly marked. (2) We will add a table listing all vertices of R with nonzero multiplicity, their multiplicities, and their coordinates. (3) For each of the 6 vertices with positive multiplicity, the table will include the value of D^{P0}_M at that vertex, along with a brief indication of which rectangle achieves the minimum. (4) We will add an auxiliary computation showing how V_{P0}(R) = -7 is obtained, by summing the entries of A over the 39 squares comprising R and accounting for the negative entries. We agree that without these details the reader cannot independently verify the counterexample, and we will include them in full. revision: no

Circularity Check

0 steps flagged

No significant circularity found; self-citations are to parameter-free structural theorems, not to fitted or self-definitional inputs

full rationale

The paper's three main results are examined for circularity. (1) Theorem 3.3 (invariance of ASM quasi-copulas under defect transformations) is proved directly from the definitions of defect matrices, the 1-Lipschitz property, and the ABM conditions (3). The only external citation is [8, Thm. 4.3] by Dibala et al. (non-overlapping authors), confirming transformations yield quasi-copulas. No circularity. (2) Proposition 4.5 (ASM imprecise copulas need not avoid sure loss) uses [28, Prop. 16] (Omladič and Stopar, one current author) as a characterization tool: the function L_{(P,Q)} and its sign criterion. This is a parameter-free theorem with stated assumptions that do not include the target result. The counterexample is an explicit 17×17 ASM with a finite, checkable computation L = -7 + 6 = -1. The self-citation provides a method, not a fitted input. The computation is fragile (unverified vertex/multiplicity count) but that is a correctness risk, not circularity. (3) Theorem 5.1 (coherence of dense ASM imprecise copulas) cites [19, Thm. 4.8] and [19, Thm. 4.10] (Košir and Perrone, two current authors). These are structural results about dense ASMs (block decomposition, main defect computation) that do not assume coherence of the imprecise copula. The proof chains: block structure → main defect has compatible block structure → each irreducible block is coherent → patchwork preserves coherence. The self-cited theorems are inputs about matrix structure, not about the target conclusion. No step reduces to its own output by construction. The self-citations are normal academic practice providing independent structural lemmas, not load-bearing circular dependencies.

Axiom & Free-Parameter Ledger

0 free parameters · 7 axioms · 0 invented entities

The paper introduces no new mathematical entities or free parameters. All objects (discrete quasi-copulas, ASMs, defect transformations, imprecise copulas) are from prior literature. The axioms are all standard mathematical results from cited references, several co-authored by the present authors, but each is a parameter-free theorem with independent proofs.

axioms (7)
  • standard math Discrete quasi-copulas on uniform grids correspond bijectively to points of the ASM polytope (Striker 2009, [41, Thm. 2.1])
    Used in Section 2 to establish the matrix representation of quasi-copulas.
  • standard math The six defect transformations on quasi-copulas yield quasi-copulas (Dibala et al. [8, Thm. 4.3])
    Invoked in the proof of Theorem 3.3 to ensure transformed objects remain quasi-copulas before showing they are ASMs.
  • standard math A pair (P, Q) of quasi-copulas is an imprecise copula iff P_M ≤ Q and P ≤ Q_O (Dibala et al. [8, Thm. 5.2])
    Used in Section 4 and in the proof of Theorem 5.1 to verify imprecise copula conditions.
  • standard math Characterization of avoidance of sure loss via the function L_{(P,Q)} (Omladič and Stopar [28, Prop. 16])
    Used in the proof of Proposition 4.5 to verify the counterexample does not avoid sure loss.
  • standard math Structure theorem for dense ASMs: irreducible dense ASMs are F^k_n, and general dense ASMs have block structure (Košir and Perrone [19, Thm. 4.4])
    Used in the proof of Theorem 5.1 to reduce to irreducible blocks.
  • standard math For irreducible dense ASM F^k_n, the main defect transformation yields F^{k-1}_n (Košir and Perrone [19, Thm. 4.8])
    Used in the proof of Theorem 5.1 to identify the transformed quasi-copula.
  • standard math Quasi-copulas corresponding to irreducible dense ASMs satisfy P = min_{C in C(P, P_M)} C (Košir and Perrone [19, Thm. 4.10])
    Used in the proof of Theorem 5.1 to establish coherence for irreducible blocks.

pith-pipeline@v1.1.0-glm · 21780 in / 3236 out tokens · 413431 ms · 2026-07-08T14:04:37.500739+00:00 · methodology

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read the original abstract

In this paper, we study discrete quasi-copulas and discrete imprecise copulas of minimal range, which naturally correspond to alternating sign matrices. We show that this family is invariant under all defect transformations on quasi-copulas and give a constructive proof demonstrating that discrete imprecise copulas of minimal range do not, in general, avoid sure loss. In contrast, we show that discrete imprecise copulas of minimal range that correspond to dense alternating sign matrices are always coherent, and hence avoid sure loss.

Figures

Figures reproduced from arXiv: 2607.06178 by Elisa Perrone, Nik Stopar, Toma\v{z} Ko\v{s}ir.

Figure 1.1
Figure 1.1. Figure 1.1: Schematic summary of the main paper results. invariant. In Section 4, we introduce discrete imprecise copulas and give an example of a discrete imprecise copula of minimal range that does not avoid sure loss. In Section 5, we show that discrete imprecise copulas that correspond to dense ASMs are always coherent. In the last section, we provide some concluding remarks. 2. Preliminaries In this section, we… view at source ↗
Figure 3.1
Figure 3.1. Figure 3.1: A rectangle in an 8 × 8 matrix corresponding to the grid L 2 8 . coordinates (2, 1), (2, 5), (5, 5) and (5, 1) on the grid. Each of these is marked by a small x on the picture. The Q-volume of R is equal to the sum of all entries of A(Q) with indices in (2, 5] × (1, 5]. These entries represent masses in each of 1 × 1 squares of R. Now, for each discrete quasi-copula Q ∈ Qn we define four directional defe… view at source ↗

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