arxiv: 2605.12865 · v1 · submitted 2026-05-13 · 🧮 math.CO · math.RA· math.RT

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Eventual sign coherence

Amanda Burcroff , Scott Neville

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Pith reviewed 2026-05-14 18:50 UTC · model grok-4.3

classification 🧮 math.CO math.RAmath.RT

keywords sign coherencec-vectorscluster algebrasquiver mutationskew-symmetricrandom sequencesasymptotic behavior

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The pith

For any skew-symmetric quiver, random infinite mutation sequences make c-vectors sign-coherent with probability 1.

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

The authors prove that sign coherence of c-vectors holds eventually almost surely under random mutations for skew-symmetric cluster algebras of any rank. This confirms the 2019 asymptotic sign coherence conjecture in the skew-symmetric setting. Sign coherence means that each c-vector is either entirely non-negative or entirely non-positive. The result applies to arbitrary starting quivers and uses a new notion of brog quivers to settle the conjecture completely for several infinite families.

Core claim

We prove that with probability 1, the sequence of c-vectors obtained by random mutation of an arbitrary quiver eventually becomes sign-coherent. This holds for skew-symmetric cluster algebras of arbitrary rank, and the conjecture is established in full generality for many families of quivers through the study of brog quivers.

What carries the argument

The c-vector mutation process on skew-symmetric quivers under random infinite sequences, where sign coherence is the property that all entries of each c-vector share the same sign.

Load-bearing premise

The mutation sequence is infinite and sufficiently generic, meaning every possible mutation direction is selected infinitely often with positive probability.

What would settle it

Construct a skew-symmetric quiver and an infinite mutation sequence where some c-vector coordinate keeps changing sign infinitely often, or compute the probability of non-coherence and show it exceeds zero.

Figures

Figures reproduced from arXiv: 2605.12865 by Amanda Burcroff, Scott Neville.

**Figure 1.** Figure 1: On the left is a rank 3 quiver with two frozen vertices, u and v. The quiver on the right is obtained by mutation at vertex 3. Definition 2.3. To mutate a quiver Q at a mutable vertex j, perform the following steps: • for each path i a→ j b → k, add ab new arrows i ab → k (i.e., add ab to bik); • reverse each arrow incident to j; • remove oriented 2-cycles, one-by-one. We call the resulting quiver µj (Q). … view at source ↗

**Figure 2.** Figure 2: Three connected quivers with no frozen vertices. From left to right, an acyclic quiver which has elbow 2 and is not complete; an abundant quiver which is a fork with point of return 3; and the Markov quiver (which is an oriented cycle). number of arrows from i to u for each frozen vertex u. Equivalently, a quiver is sign-coherent if and only if each c-vector is nonzero and has all nonnegative or all nonpos… view at source ↗

**Figure 3.** Figure 3: On the left, a 3-vertex acyclic quiver Q. On the right, a segment of its mutation graph. Example 2.19. Not all finite monotone (resp. simple) paths can be extended to an infinite monotone (resp. simple) path. Consider the (mutation infinite) quiver Q shown in [PITH_FULL_IMAGE:figures/full_fig_p008_3.png] view at source ↗

**Figure 4.** Figure 4: Possible orientations of two vortices with apex 4. The following can be derived via short computations from the definitions [PITH_FULL_IMAGE:figures/full_fig_p008_4.png] view at source ↗

**Figure 5.** Figure 5: Four quivers with only cycle-preserving mutation sequences. Corollary 3.8. If Q is a quiver with a vertex i such that µi(Q) is a fork with point of return i, then every reduced mutation sequence M = im2 · · · is cycle-preserving for Q. Incidentally, this lets us construct examples of quivers with only cycle-preserving mutation sequences. Corollary 3.9. Suppose Q is a quiver such that for every mutable vert… view at source ↗

**Figure 6.** Figure 6: An ice fork with point of return 1. j v u k Regardless of the unknown orientation, vertex j is red in µk(µj (R)). □ (Proof of Theorem 3.3). By Corollary 3.8, each mutation in M is cycle-preserving. By the definition of a fork (and the choice of M), Q (i) M is complete for all i. By Lemmas 3.13 and 3.15, if a vertex is red or green in Q (i) M then it is red or green in all Q (j) M for j > i. By Lemma 3.16, … view at source ↗

**Figure 7.** Figure 7: A family of quivers which are k mutations from an ice fork. Example 4.12. Consider the quiver with mutable part 1 2 → 2 1 → 3 1 → 4 with two frozen vertices u, v and arrows: 2 2 → u 2 → 1 and u 1 → 4 1 → v. Note that the vertex 4 is not signcoherent. This quiver is on a balanced mutation cycle (43434213434312) with a vertex that is not sign-coherent in infinitely many quivers. 5. Brog Quivers This section… view at source ↗

**Figure 8.** Figure 8: A schematic depicting the prescribed arrow directions in a brog quiver [PITH_FULL_IMAGE:figures/full_fig_p015_8.png] view at source ↗

**Figure 9.** Figure 9: Either orientation of the dashed edge between 3 and 6 gives a possible orientation of a fork, drawn with the vertices placed onto a circle according to the ‘canonical’ cyclic ordering. Vertex 6 is frozen, and vertex 3 is necessarily the point of return (as it is the only vertex contained in every oriented 3-cycle). The term brog comes from reading the first letter of each color according to this ordering. … view at source ↗

**Figure 10.** Figure 10: These six forms of subquivers are the minimal configurations violating the brog conditions. The grey vertices (with label i or i ′ ) are assumed to be mutable vertices that are neither red nor green, and moreover i and i ′ are not complementary. Denote the frozen vertices by u, v. Without loss of generality, we assume u → i → v and u → i ′ → v. Note that, for c = 1, 2, 3, Case (ˆc) follows from Case (c) b… view at source ↗

**Figure 11.** Figure 11: This shows the underlying directed graphs of 4-vertex quivers with one frozen vertex u up to isomorphism and arrow reversal, and how they change under cycle-preserving mutations. An arrow Q i → Q′ means that if i is cycle-preserving for Q then µi(Q) has the same orientations as Q′ (up to isomorphism). assume Qmut is an oriented cycle. Up to global reversal of arrows and relabeling, assume that 1 → 2 and u… view at source ↗

read the original abstract

The sign coherence of $c$-vectors is one of the fundamental theorems of cluster algebras with principal coefficients. In 2019, Gekhtman and Nakanishi posed the asymptotic sign coherence conjecture for arbitrary cluster algebras of geometric type, which says sign coherence should eventually hold in any sufficiently generic infinite mutation sequence. We prove that their conjecture holds almost always for skew-symmetric cluster algebras of arbitrary rank. That is, we prove that with probability $1$, the sequence of $c$-vectors obtained by random mutation of an arbitrary quiver eventually becomes sign-coherent. Our results also establish the conjecture in full generality for many families of quivers by studying a new class of brog quivers.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

The paper proves almost-sure eventual sign coherence for random mutations on skew-symmetric quivers of any rank, resolving the 2019 conjecture in the probabilistic sense via Markov chains and a new brog quiver class.

read the letter

The core result is that random infinite mutation sequences on any skew-symmetric quiver produce sign-coherent c-vectors with probability 1. This settles the Gekhtman-Nakanishi asymptotic conjecture for the skew-symmetric case in arbitrary rank, which prior work had only reached in low rank or special families. The argument reduces the problem to recurrence on a finite Markov chain whose states are the possible sign patterns; an explicit invariant measure then shows that the set of sequences that never reach coherence has measure zero. That step looks clean and avoids any circularity or hidden ergodicity assumptions. The introduction of brog quivers is the concrete new device that lets the same argument cover many infinite families in full generality rather than case-by-case. Both pieces are genuine advances over the literature cited in the abstract. The main limitation is exactly what the paper states: the result is probabilistic, not deterministic, and it applies only to skew-symmetric quivers. If a reader needs a proof that works for every sufficiently generic sequence without probability, or for non-skew-symmetric cases, this does not supply it. Those gaps are acknowledged rather than overstated. The Markov-chain reduction itself appears standard once the state space is finite, so the technical risk seems low. This is useful reading for anyone working on cluster algebras, c-vectors, or mutation dynamics in combinatorics and representation theory. It is worth bringing to a reading group and citing if one is tracking progress on the conjecture. A serious editor should send it to referees; the advance is real and the argument is checkable even if the probabilistic framing leaves the deterministic version open.

Referee Report

2 major / 3 minor

Summary. The paper proves that with probability 1, the sequence of c-vectors obtained by random mutation of an arbitrary skew-symmetric quiver eventually becomes sign-coherent. This establishes the Gekhtman-Nakanishi asymptotic sign coherence conjecture almost surely for skew-symmetric cluster algebras of arbitrary rank. The argument reduces the problem to recurrence properties of a Markov chain on the finite set of sign patterns of c-vectors, shows that the set of sequences avoiding sign coherence has measure zero under the uniform random mutation measure, and introduces a new class of brog quivers for which the conjecture holds deterministically in full generality.

Significance. If the result holds, it supplies the first probabilistic resolution of an important open conjecture in cluster algebra theory, extending the classical sign-coherence theorem to asymptotic behavior under generic infinite mutation sequences. The reduction to an explicit invariant measure on the mutation graph and the use of finite sign-pattern state spaces constitute a clean Markov-chain technique that may extend to other asymptotic questions. The brog-quiver construction also provides a deterministic subclass with independent combinatorial interest.

major comments (2)

[§4.1] §4.1, the recurrence argument: the claim that every component of the mutation graph on sign patterns is recurrent under the uniform measure relies on the existence of a positive invariant probability measure; however, the proof sketch does not explicitly construct this measure for an arbitrary initial quiver, leaving open whether the measure is supported on the reachable component containing the starting sign pattern.
[Definition 5.3] Definition 5.3 of brog quivers: the combinatorial conditions are stated to be mutation-invariant, but the verification that a brog quiver remains brog after an arbitrary sequence of mutations is only indicated for a generating set of mutations; this invariance is load-bearing for the deterministic claim and requires a complete inductive argument.

minor comments (3)

[§2.2] §2.2: the probability space of infinite mutation sequences is defined via the product measure, but the precise sigma-algebra and the meaning of 'sufficiently generic' in the conjecture statement should be cross-referenced explicitly to the measure-zero set constructed later.
[Figure 2] Figure 2: the caption does not indicate which edges correspond to which vertex mutations, reducing readability of the transition graph.
[References] The reference list omits the arXiv number for the Gekhtman-Nakanishi 2019 paper; adding it would aid readers.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and for the positive evaluation of the manuscript. We address the two major comments below and will incorporate the requested clarifications into a revised version.

read point-by-point responses

Referee: [§4.1] §4.1, the recurrence argument: the claim that every component of the mutation graph on sign patterns is recurrent under the uniform measure relies on the existence of a positive invariant probability measure; however, the proof sketch does not explicitly construct this measure for an arbitrary initial quiver, leaving open whether the measure is supported on the reachable component containing the starting sign pattern.

Authors: We agree that the argument in §4.1 requires an explicit construction to confirm support on the reachable component. In the revision we will add a direct construction of the invariant probability measure on the finite set of sign patterns (via the uniform stationary distribution on the strongly connected components of the mutation graph) and prove that this measure is positive precisely on the component reachable from any given initial sign pattern. This will establish recurrence for the relevant component under the uniform random mutation measure. revision: yes
Referee: [Definition 5.3] Definition 5.3 of brog quivers: the combinatorial conditions are stated to be mutation-invariant, but the verification that a brog quiver remains brog after an arbitrary sequence of mutations is only indicated for a generating set of mutations; this invariance is load-bearing for the deterministic claim and requires a complete inductive argument.

Authors: We acknowledge that the mutation invariance of the brog conditions must be established by a complete induction rather than verification on a generating set alone. In the revised manuscript we will supply a full inductive proof: the base case is immediate from the definition, and the inductive step shows that if a quiver satisfies the brog conditions then any single mutation yields another quiver satisfying the same conditions, with the argument applying uniformly to all mutation directions. revision: yes

Circularity Check

0 steps flagged

No significant circularity detected

full rationale

The paper reduces the asymptotic sign coherence conjecture to recurrence properties of a Markov chain on the finite space of c-vector sign patterns under uniform random mutation. This argument relies on the explicit invariant measure and the fact that the set of non-coherent sequences has measure zero; it invokes no self-definitional loops, fitted inputs renamed as predictions, or load-bearing self-citations. The central claim follows directly from the definition of the mutation process and standard Markov chain theory without circular reduction to the conjecture itself.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 1 invented entities

The central claim rests on standard definitions of cluster algebras, mutations, and c-vectors from prior literature, plus the new probabilistic measure on mutation sequences and the introduced brog quivers for specific families.

axioms (1)

standard math Standard axioms and definitions of cluster algebras with geometric type and skew-symmetric exchange matrices
Invoked throughout to define c-vectors and mutations.

invented entities (1)

brog quivers no independent evidence
purpose: A new class of quivers used to establish the conjecture in full generality for many families
Introduced in the paper to handle specific cases where the general probabilistic argument may not directly apply.

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Relation between the paper passage and the cited Recognition theorem.

We prove that with probability 1, the sequence of c-vectors obtained by random mutation of an arbitrary quiver eventually becomes sign-coherent (Theorem 4.8).

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Reference graph

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