arxiv: 2605.10072 · v1 · submitted 2026-05-11 · 🧮 math.RT · math.CO

Recognition: 3 theorem links

· Lean Theorem

Fractal phenomenon in c- and g-vectors of the Markov quiver

Ryota Akagi, Zhichao Chen

Pith reviewed 2026-05-12 04:19 UTC · model grok-4.3

classification 🧮 math.RT math.CO

keywords Markov quiverc-vectorsg-vectorsG-fanfractal structurecluster algebrasCalkin-Wilf recursionlinear isomorphisms

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The pith

Modified c- and g-vectors of the Markov quiver form fractal patterns through linear isomorphisms of subpatterns.

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

The paper establishes that large classes of subpatterns within the modified c-vectors and g-vectors for the Markov quiver and related rank-3 skew-symmetrizable matrices of B-invariant type are connected by linear isomorphisms. These relations produce a self-similar fractal structure in the associated G-fan. Explicit recursive formulas are derived that generate all such vectors from integer pairs obeying a tree-like recursion similar to the Calkin-Wilf tree, allowing every vector to be indexed by a pair of coprime integers. This setup yields a complete recursive description of the connected components in the complement of the G-fan support using three kinds of linear maps.

Core claim

We prove that large classes of subpatterns of modified c- and g-vectors are linearly isomorphic, yielding a fractal structure of the corresponding G-fan. We further derive explicit recursive formulas for all modified c- and g-vectors in terms of integer pairs satisfying a recursion analogous to the Calkin-Wilf tree, which leads to a parameterization by coprime integers. As an application, we describe all connected components of the complement of the support of the G-fan, and show that they are generated recursively by three kinds of linear maps.

What carries the argument

Linear isomorphisms between subpatterns of modified c- and g-vectors, which induce the fractal structure, together with the recursive formulas that parametrize the vectors by coprime integer pairs via a Calkin-Wilf-like tree recursion.

If this is right

The G-fan displays self-similar fractal structure across its subpatterns.
Every modified c-vector and g-vector admits a unique parameterization by a pair of coprime integers.
The complement of the G-fan support decomposes into connected components that are recursively generated by exactly three linear maps.
The same linear-isomorphism and recursive phenomena hold for all rank-3 skew-symmetrizable matrices of B-invariant type.

Where Pith is reading between the lines

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

The coprime-integer parameterization may connect to number-theoretic objects such as the Farey diagram or Stern-Brocot tree.
Analogous fractal patterns could be sought in cluster structures of higher rank or different mutation classes.
The linear maps generating the complement components might preserve additional invariants such as sign patterns or denominator vectors.

Load-bearing premise

The mutations for Markov-type cluster algebras are self-contained and simple, which permits the identification of specific subpatterns that admit linear isomorphisms.

What would settle it

Direct computation of a modified c-vector or g-vector for a specific seed in the Markov quiver that fails to satisfy the claimed linear isomorphism with its subpattern or deviates from the recursive formula generated by its corresponding coprime integer pair.

Figures

Figures reproduced from arXiv: 2605.10072 by Ryota Akagi, Zhichao Chen.

**Figure 3.** Figure 3: An example of tropical signs. 2.2. Mutations of c- and g-vectors. For any w P T ztHu and M “ K, S, T, we define c˜ w M “ c˜ w Mpwq , g˜ w M “ g˜ w Mpwq . (2.8) Based on the fact that |b w ij b w ji| “ 4 for any w and i ‰ j, their mutation is given as follows [PITH_FULL_IMAGE:figures/full_fig_p009_3.png] view at source ↗

**Figure 4.** Figure 4: pq X K;S , qX K;T q. The tree in [PITH_FULL_IMAGE:figures/full_fig_p018_4.png] view at source ↗

**Figure 5.** Figure 5: In a trunk. T K S w wS T S K S wT T K [PITH_FULL_IMAGE:figures/full_fig_p022_5.png] view at source ↗

**Figure 7.** Figure 7: G-fan corresponding to the B-invariant type. Fix one w P T ztHu. Let us consider the union Ť nPZě0 CpGwS n q. Then, by repeating the process in [PITH_FULL_IMAGE:figures/full_fig_p023_7.png] view at source ↗

**Figure 8.** Figure 8: S-mutations. Note the following two facts [PITH_FULL_IMAGE:figures/full_fig_p023_8.png] view at source ↗

**Figure 9.** Figure 9: Direct sum decomposition of U w. Due to the above lemma, we can improve the upper bound in (7.2) as follows. Proposition 7.3. We have |∆ěwpBq| Ă CpG wq \ U wS ˝ \ U wT ˝ . (7.7) Proof. This follows from |∆ěwpBq| “ CpGwq Y |∆ěwS pBq|˝ Y |∆ěwT pBq|˝ and Lemma 7.1. □ 7.2. Among branches. For each i “ 1, 2, 3, k0 “ Kprisq, s0 “ Sprisq, t0 “ Tprisq, let Di “ Cpe˜s0 , v˜t0k0 , v˜k0s0 q X V. (7.8) This set can be… view at source ↗

**Figure 10.** Figure 10: Maximal branches. Here, we set v˜lm “ e˜m ´ e˜l . As a corollary in this section, we obtain the following facts. Proposition 7.5. Every modified g-vector can be expressed as e˜j (j “ 1, 2, 3) or g˜ w K (w ‰ H) uniquely. Proof. The existence of this expression is obvious by Lemma 2.4. We show the uniqueness. Note that Di X Dj “ t0u when i ‰ j. Thus, we can fix an initial mutation direction i “ 1, 2, 3. Mor… view at source ↗

**Figure 11.** Figure 11: Complement in Di . 8.2. Recursive expression. Let φ : G Ñ Comp∆pBqq be the map defined by φ|Gi “ φi . To simplify the notation, we also write φpwq “ φpg˜ w Kq for any w P T ztHu. Then, by Proposition 7.5, except for the three complements φpe˜1q, φpe˜2q, and φpe˜3q, we can express all the elements in Comp∆pBqq as φpwq. For each w P T ztHu, we define Coměwp∆pBqq “ tφpuq | u ě wu. (8.6) For any multiplicativ… view at source ↗

**Figure 12.** Figure 12: Recursive process to obtain all the complements. 9. Open problem In [AC26], we can observe that the G-fan structure of B-invariant type is a degeneration of the ones of cluster-cyclic exchange matrices. For example, we can find the following correspondence: B-invariant type (in this paper) cluster-cyclic type [AC26] upper bound half space (Lemma 6.2) three hyperboloids [AC26, Thm. 4.12] complement 2-dimen… view at source ↗

read the original abstract

We study the $C$- and $G$-patterns associated with rank $3$ skew-symmetrizable matrices of $B$-invariant type, including the Markov quiver. Motivated by the self-contained simple mutations in Markov-type cluster algebras, we prove that large classes of subpatterns of modified $c$- and $g$-vectors are linearly isomorphic, yielding a fractal structure of the corresponding $G$-fan. We further derive explicit recursive formulas for all modified $c$- and $g$-vectors in terms of integer pairs satisfying a recursion analogous to the Calkin-Wilf tree, which leads to a parameterization by coprime integers. As an application, we describe all connected components of the complement of the support of the $G$-fan, and show that they are generated recursively by three kinds of linear maps.

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 shows a fractal structure in the Markov quiver's G-fan via linear isomorphisms on subpatterns of modified c- and g-vectors, plus Calkin-Wilf-style recursions that parameterize them by coprime integers and classify the complement components.

read the letter

The punchline is that the authors show the G-fan for the Markov quiver has a self-similar fractal structure coming from linear isomorphisms on big classes of subpatterns in the modified c- and g-vectors. They also give recursive formulas for these vectors using pairs of integers that follow a Calkin-Wilf type recursion, which lets them parameterize everything by coprime pairs, and then use that to classify the connected components outside the fan's support. What the paper does well is take the simple mutation behavior in Markov-type cluster algebras and turn it into a way to identify isomorphic subpatterns without too much ad hoc choice. The recursions look like they come straight from the mutation rules, and the three linear maps for generating the complement components give a clean recursive description. This organizes an infinite combinatorial object in a way that feels useful for anyone computing these vectors explicitly. The soft spots are fairly minor. The identification of the subpatterns still leans on the self-contained simple mutations idea, which works for this quiver but might not be as automatic as it sounds if you try to apply it elsewhere. I would want to see the explicit verification that the linear maps are indeed isomorphisms for all the claimed classes, and whether there are any small cases where the recursion hits a snag or the coprime condition needs extra handling. The paper stays within rank 3 B-invariant type, so the fractal claim is solid there but doesn't claim broader generality. This paper is for specialists in cluster algebras who care about the combinatorics of c- and g-vectors in low-rank quivers, particularly the Markov one. A reader who wants explicit formulas or a way to navigate the infinite fan would find it practical. It deserves a serious referee because the strategy is coherent and the results are checkable with the given recursions and maps. I would send it to peer review. The work is focused and the claims seem grounded enough to warrant expert feedback.

Referee Report

0 major / 3 minor

Summary. The manuscript studies the C- and G-patterns associated to rank-3 skew-symmetrizable matrices of B-invariant type, with emphasis on the Markov quiver. Motivated by the self-contained nature of mutations in Markov-type cluster algebras, it proves that large classes of subpatterns of modified c- and g-vectors are linearly isomorphic, thereby establishing a fractal structure on the corresponding G-fan. Explicit recursive formulas are derived for all modified c- and g-vectors in terms of integer pairs obeying a Calkin-Wilf-type recursion; this yields a parameterization by coprime integers. As an application, the connected components of the complement of the support of the G-fan are classified and shown to be generated recursively by three families of linear maps.

Significance. If the derivations hold, the paper supplies an explicit, recursive combinatorial description of c- and g-vectors for an important infinite-mutation class, together with a fractal decomposition of the G-fan and a complete classification of its complement components. The coprime-integer parameterization and the three linear maps furnish concrete, computable tools that could be used to test further conjectures in cluster algebra theory. The explicit link between mutation self-containment and linear isomorphisms is a methodological strength that may generalize beyond rank 3.

minor comments (3)

The precise definition of 'modified' c- and g-vectors (as opposed to the standard ones) should be stated at the first occurrence, together with a short explanation of why the modification is required for the linear-isomorphism statements to hold.
In the application section describing the connected components, an explicit low-dimensional example (e.g., the first few iterates of the three linear maps) would help the reader verify that the recursion indeed exhausts all components.
Figure captions should indicate which linear map or recursion step is illustrated in each panel so that the fractal self-similarity is immediately visible.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the positive assessment of our work on the fractal structure of modified c- and g-vectors for the Markov quiver and related B-invariant matrices. The recommendation for minor revision is noted; we will prepare a revised version incorporating any editorial improvements.

Circularity Check

0 steps flagged

No significant circularity; derivation self-contained from mutation structure

full rationale

The paper's central claims rest on proving linear isomorphisms among subpatterns of modified c- and g-vectors, motivated explicitly by the self-contained simple mutations of Markov-type cluster algebras, followed by derivation of recursive formulas analogous to the Calkin-Wilf tree that parameterize the vectors by coprime integer pairs. These recursions are then applied independently to classify connected components of the G-fan complement via three families of linear maps. No load-bearing step reduces by construction to a fitted input, self-citation, or ansatz smuggled from prior work; the abstract and high-level argument present the recursions as derived outputs rather than inputs, with no equations or citations in the provided text that collapse the target results to the assumptions. The derivation chain is therefore independent of the patterns it describes.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The paper relies on standard background from cluster algebra theory (c-vectors, g-vectors, G-fans, skew-symmetrizable matrices) assumed from prior literature. The only domain assumption visible in the abstract is that mutations for Markov-type algebras are self-contained and simple.

axioms (1)

domain assumption Mutations in Markov-type cluster algebras are self-contained and simple
Stated as the motivation for studying the subpatterns and their linear isomorphisms.

pith-pipeline@v0.9.0 · 5439 in / 1415 out tokens · 60708 ms · 2026-05-12T04:19:21.456272+00:00 · methodology

discussion (0)

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

IndisputableMonolith/Foundation/ArithmeticFromLogic.lean LogicNat ≃ Nat, embed_injective echoes
Theorem 5.1 and recursion (1.11): coefficients (a_X, b_X) obey (a_{M'X}, b_{M'X}) = (a_X + b_X, b_X) for M'=S and (b_X, a_X + b_X) for M'=T, generating all coprime pairs exactly once (Lemma 5.3).

Reference graph

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