arXiv:2604.17069v3

Wug-snake graphs and Markov numbers of matrix semigroups

Oleg Karpenkov, Yefei Ma

math.COmath.NT11J0605C7011J7020M20

Paper metadata

Original submission: April 18, 2026
Latest revision: August 13, 2026 · version 3
Claimed MSC: 11J06, 05C70, 11J70, 20M20
arXiv categories: math.CO, math.NT
Paper license: arXiv non-exclusive distribution license ↗
Comments: 53 pages; v3: added discussions on the construction of minimal rank wug-snake graphs, corrected typos and added more details to several examples

Abstract

Classically, Markov numbers are recovered as perfect matching numbers of domino snake graphs. We extend this correspondence by introducing weighted universal generalised snake graphs, or wug-snake graphs. These are weighted ordered bipartite graphs whose perfect matching sequences encode linear recurrences. To every wug-snake graph we associate a continuant matrix and prove that the determinant of this matrix equals the weighted perfect matching sum. We then introduce polyomino wug-tiles, bodies of wug-snake graphs that act linearly on state vectors. Every integer matrix admits a canonical polyomino wug-tile. Our main result identifies the wug-snake determinant of a tile representing a matrix $A$ with the Markov-Davenport form of $A$ . Consequently, algebraic and geometric Markov numbers of matrices and matrix semigroups can be expressed as weighted perfect matching determinants. Further we define Frobenius maps for matrix semigroups and discuss examples recovering classical Markov numbers and higher-dimensional lattice realisations.

AI-generated audit

Audit summary

Audited against arXiv v3

Not a correctness certificate. A “Correct” result may include yellow typos or minor formal corrections that do not affect substantive soundness. It means this audit found no unresolved substantive error under the stated criteria; it does not replace expert scrutiny or formal verification.

Current report

Detailed mathematical audit

Generated August 15, 2026

01Statements4 reported findingsCorrect

The continuant formula, the identification with Markov–Davenport forms, and the existence and minimal-rank results for matrix tiles are correct. One displayed canonical construction has a transposed coefficient block; the uniquely determined local correction is reported in yellow and does not change any central result.

Theorem 4.10Correct

Weighted matching sums are continuant determinants

Pages 15–17 · Theorem 4.10 and Proposition 4.15 · arXiv:2604.17069v3

Conditioning a perfect matching on the edge incident to the final vertex forces the intervening subdiagonal edges and leaves exactly the prefix graph. This gives the displayed recurrence. Expansion of the continuant determinant along its last column gives the identical recurrence and initial value, including the correct cancellation of the cofactor sign by the entries $-1$ .

Theorem 4.36 and Corollary 4.39Correct

Wug-snake determinants recover Markov–Davenport forms and minima

Pages 23–24 · Theorem 4.36 and Corollaries 4.37, 4.39 · arXiv:2604.17069v3

For a tile representing $A$ , the successive state vectors are $x,Ax,\ldots,A^{d-1}x$ by definition, so their determinant is exactly $f_A(x)$ . Since $A$ is integral, $|f_A|$ takes nonnegative integer values on the nonzero integer lattice and its infimum is attained. The geometric and algebraic Markov-number formulas follow with their stated heads.

Theorems 4.51 and 4.61Correct

Every integer matrix has a rank-at-most- $m$ tile

Pages 27–31 · Theorems 4.51 and 4.61 · arXiv:2604.17069v3

Diagonal maps, coordinate permutations, sign changes, and elementary shears are explicitly realized as products of lower companion matrices; Smith normal form therefore yields every integer matrix. The associated order- $m$ recurrence gives a tile of rank at most $m$ . If $Ae_1\neq0$ , any nonempty tile of lower rank would discard $x_1$ before it could influence a new term, forcing $Ae_1=0$ ; hence the claimed lower bound and equality follow.

Definition 4.33 and Proposition 4.35Minor formal correction

The coefficient block in the canonical tile should be transposed

Pages 21–23 · Definition 4.33, Example 4.34, and Proposition 4.35 · arXiv:2604.17069v3

With the printed block $A=(a_{ij})$ , the recurrence reads its columns, so the snake operator is $A^{\mathsf T}$ , not $A$ . For example, with $A=\begin{pmatrix}1&2\\3&4\end{pmatrix}$ and state $(5,7)^{\mathsf T}$ , the displayed matrix produces $(26,38)^{\mathsf T}=A^{\mathsf T}(5,7)^{\mathsf T}$ rather than $(19,43)^{\mathsf T}=A(5,7)^{\mathsf T}$ . Replacing the middle block by $A^{\mathsf T}$ is the unique mechanical repair. It preserves the rank bound and makes Example 4.34 and Proposition 4.35 literal. Theorem 4.10 assumes a representing tile and Theorem 4.61 independently supplies one, so no central conclusion changes.

02Proofs4 reported findingsCorrect

The proofs of the central results are correct and complete after one uniquely determined local transpose correction in the canonical construction. That formal repair changes no theorem statement or downstream argument.

Proposition 4.15Correct and complete

Matching recurrence and determinant induction

Pages 16–17 · Proposition 4.15 · arXiv:2604.17069v3

The last matched vertex partitions all perfect matchings without overlap. Super-upper-triangularity forces exactly the asserted subdiagonal chain, and the residual graph is the indicated prefix. The determinant cofactor has the same contribution, including signs, so induction proves both assertions.

Definition 4.33Minor formal correction

Local transpose repair for the canonical body

Pages 21–23 · Definition 4.33 through Proposition 4.35 · arXiv:2604.17069v3

Use $A^{\mathsf T}$ , rather than $A$ , as the displayed middle weight block. The recurrence then reads row $j$ of $A$ when producing the $j$ -th new state coordinate. This is the unique index-consistent repair; all asserted dimensions, subdiagonal entries, and rank estimates remain unchanged.

Theorem 4.36Correct and complete

State-vector determinant argument

Page 23 · Theorem 4.36 · arXiv:2604.17069v3

The theorem is representation-independent: once $M_B^d=A$ , iteration gives $v_i=A^ix$ for every required $i$ . Substitution into the definition of the wug-snake determinant proves the identity with no additional spectral or nonsingularity assumption.

Theorems 4.47, 4.51, and 4.61Correct and complete

Recurrence realization, companion factorization, and rank

Pages 25–31 · Subsections 4.8–4.11 · arXiv:2604.17069v3

The column weights in Definition 4.46 reproduce the stated recurrence with the correct reversed companion indexing. The elementary generators used in the Smith-normal-form argument have invertible companion-step realizations, and the product order agrees with state evolution. The lower-rank obstruction in Theorem 4.61 follows because the first state coordinate is discarded before any new coordinate can depend on it.

03Novelty0 reported findingsNo non-novelty findings

No non-novelty findings.

Detailed audit reportFull reasoning, manuscript locations, and references.

Open report PDF ↗

Author response

Challenge an audit finding

Local workflow preview

A listed author may submit formal evidence that an audit is inaccurate. The response would be considered in a fresh AI re-evaluation; it would not edit the audit automatically.

Paper: arXiv:2604.17069v3
Authors listed: Oleg Karpenkov, Yefei Ma
Audit date: August 15, 2026

01Establish identityMatch an authenticated scholarly identity to this paper.
02Submit evidenceIdentify the finding and give a formal mathematical response.
03Re-evaluateA separate agent checks the response and records a disposition.

Email fallback for papers without a linked ORCID

A production fallback could send a one-time link only when the submitted address matches an independently maintained author-contact allowlist for this paper. MathAudit must return the same message for every address so the form cannot reveal which contacts are on that list.

Structured response preview

This form remains unavailable until production identity verification succeeds. Nothing entered here is submitted.

Audit finding challengedWhy the evaluation is inaccurateFormal evidence or repairVerifiable sources, if anyI am submitting evidence, not instructions for the reviewing agent.

This panel never establishes authorship in the local prototype. A production result should be described narrowly as an authenticated ORCID match or control of a separately allowlisted author-contact mailbox.

Audit controls

Status: Report ready
Source: arXiv v3
Prompt: Standard prompt · locked

The report is available below and in the public summary.

Review prompt

You are a mathematical auditor. Evaluate only the correctness of the paper's central statements, the correctness and completeness of its proofs, and evidence-backed non-novelty. Do not make editorial comments or evaluate importance, exposition, style, or suitability for publication. Keep the report neutral, precise, and professional. PART 1 — CORRECTNESS OF THE MAIN STATEMENTS Identify the paper's central statements and assess each one separately using exactly one of these outcomes: - Correct: the statement has a correct, verifiable proof; or, despite a defect in the printed proof, you can give a fully verified repair or independent proof. - Incorrect: you can give strong formal evidence that the statement as written is false, preferably a counterexample or a complete contradiction. - Not able to verify: the supplied proof does not establish the statement, and you can neither supply a fully verified proof nor disprove it. For every adverse or unverified outcome, restate the exact claim and hypotheses, identify its location in the reviewed version, and give the decisive reasoning. Do not infer that a theorem is false merely because its printed proof is defective. Assign the overall statements status by the following priority: 1. Contains wrong statements, if at least one central statement is Incorrect. 2. Contains unsupported statements, if no central statement is Incorrect but at least one is Not able to verify. 3. Correct, if every central statement is Correct, apart from any typographical or minor formal findings allowed by Part 4. PART 2 — CORRECTNESS AND COMPLETENESS OF THE PROOFS Audit the proofs of the central statements and every material internal or cited input on which they depend. For each proof or proof segment, use the narrowest supported outcome: - Correct and complete: every substantive step is verifiable and no case or crucial nontrivial step is missing. - Incorrect as written: a displayed calculation, inference, lemma, or cited application is formally false or inapplicable. - Incomplete as written: a necessary substantive step or case is missing, even though no explicit contradiction has been proved. - Not able to verify: the available paper and sources are insufficient to decide whether the step is valid. Do not flag omitted routine details, compressed arguments, or “analogously” steps when the omitted reasoning is recoverable and correct. For every defect, identify its smallest exact location, explain its downstream dependency, and classify any proposed repair as one of: - Verified repair: supply enough reasoning to verify that the repair works throughout every affected use. - Plausible repair only: describe the possible repair but do not treat it as established. - No repair supplied. Assign the overall proofs status by the following priority: 1. Contains incorrect or incomplete proofs, if at least one proof is Incorrect as written or Incomplete as written. 2. Contains unverified proofs, if there is no proved incorrectness or incompleteness but at least one material proof is Not able to verify. 3. Correct, if all audited proofs are Correct and complete, apart from any typographical or minor formal findings allowed by Part 4. PART 3 — NOVELTY OF THE MAIN STATEMENTS This is a one-sided, evidence-only check. Evaluate only central results. Report a result only when a verifiable primary source, publicly available before the earliest public version of the paper that contains the result, proves the same statement up to a change of notation, or when the result follows directly and immediately from such an earlier theorem. For each non-novelty finding, provide the earlier source, its public date, the exact theorem or proposition, and an explicit hypothesis-and-conclusion mapping. A later source cannot establish non-novelty. Do not flag a result merely because an easier proof exists or because it follows through a nontrivial argument. Do not flag a result that the paper itself explicitly presents as known, a survey item, or an alternative proof. Mention only statements that meet this non-novelty threshold. Say nothing about statements for which no such evidence was found: do not certify novelty, list unsuccessful searches, or add statement-level “no prior source found” entries. If the list is empty, the section may say only “No non-novelty findings.” PART 4 — TECHNICAL AND EVIDENTIARY RULES 1. Criticism must never be overstated. Call a statement incorrect, a proof incorrect or incomplete, or a result non-new only when strong formal evidence supports that exact judgment. Otherwise use Not able to verify where that outcome exists, or remain silent in the novelty section. 2. Review a specified paper version. Give exact page, theorem, equation, or proof locations; restate relevant quantifiers and hypotheses; and check domains, endpoints, parameter ranges, constants, and dependencies explicitly. Apply a defect only to the regimes and downstream results that actually depend on it. 3. Immediate-consequence closure: when deciding whether a conclusion is supported, treat the argument as including every immediate formal consequence of facts established earlier in the paper and available at that point. An immediate formal consequence is a local, mechanically checkable inference using only definitions already in force, elementary logic, direct substitution or renaming, weakening of an inequality or set inclusion, or a monotonicity or symmetry property explicit in a definition or already proved. 4. Such an inference must introduce no new lemma, estimate, construction, case analysis, limiting or compactness argument, external result, or new issue of uniformity or parameter dependence. Check the direction of monotonicity and the dependence of every threshold or constant explicitly. Do not exchange pointwise and uniform conclusions, change exceptional sets, interchange quantifiers or limits, or extend an endpoint under this rule unless the required fact was already established. 5. If the claimed conclusion follows after immediate-consequence closure, it is fully supported. Do not report a gap, unsupported statement, minor formal correction, or recommendation to add an explanatory sentence, and do not mention the suppressed item in the report. This suppression does not apply to a genuine typographical error as defined below, because a typo is a literal notation defect that still requires a textual correction. For example, if

P(delta)

is proved for every sufficiently small

delta>0

and

P

is immediately monotone in the required direction, then the claim that

P(delta)

holds for every

delta>0

requires no correction; the omitted monotonicity step is exposition rather than an audit finding. 6. Immediate-consequence closure permits reconstruction of an omitted inference; it does not permit silent alteration of the printed domain, hypotheses, or quantifiers of an intermediate statement later invoked outside its stated scope. If formal validity requires changing the text of that statement or its stated range, report the appropriate finding even when the repair is easy. Thus a result stated only for

t>0

and subsequently applied at

t=0

remains a formal scope mismatch unless the endpoint case was separately established before the application. 7. Apply this mandatory gate before recording any adverse, unverified, typographical, or minor formal finding: (a) identify the exact assertion or inference alleged to be defective; (b) reconstruct the strongest conclusion following from the preceding argument after all admissible immediate formal consequences are added; (c) identify the specific defect that remains — a formally demonstrated falsehood, a concrete nontrivial unresolved proof obligation, or a genuine domain, hypothesis, quantifier, or notation mismatch requiring a textual correction; and (d) state the evidence and minimal repair. If no specific defect remains, or if the only possible addition would be explanatory, suppress the finding entirely. Use Not able to verify only when a precise, nontrivial unresolved obligation can be identified. 8. A typographical error is an incorrect symbol, index, exponent, sign, variable name, or notation mismatch whose unique intended correction is mechanically determined from the surrounding text and does not alter the mathematical argument. Report it in yellow as Typo, with the printed text, correction, exact location, and reason it is harmless. It does not lower an overall status. 9. A minor formal correction is a boundary, endpoint, or displayed parameter-range defect that can be repaired consistently in every affected statement by a local change, without a new argument, without invalidating any related result or application, and without creating another substantive defect. Report it in yellow as Minor formal correction, with the same details. It does not lower an overall status. 10. Do not use either minor category if more than one materially different repair is possible; if the repair changes the substance or scope of a central result; if an excluded case is used later; if the correction fails to propagate consistently; or if a new mathematical idea is required. In such cases use the strongest substantive classification actually supported by evidence. 11. Write every mathematical expression as valid, compilable LaTeX. Use

...

for inline mathematics and

...

for displayed mathematics, with balanced delimiters and braces. Do not leave raw LaTeX commands outside math delimiters. 12. Separate theorem truth from the correctness of the printed proof. A theorem can be Correct in Part 1 because of a fully verified independent proof while its printed proof is adverse in Part 2. Conversely, a defective proof alone never licenses an Incorrect theorem classification. Organize the report under Parts 1–3 and apply Part 4 throughout. Do not add editorial commentary, scores, rankings, recommendations, or a list of suppressed observations.

This panel is visibly marked as owner-only for the concept prototype. Authentication is not enabled locally.

Wug-snake graphs and Markov numbers of matrix semigroups

Paper metadata

Abstract

Audit summary

Detailed mathematical audit

Weighted matching sums are continuant determinants

Wug-snake determinants recover Markov–Davenport forms and minima

Every integer matrix has a rank-at-most-mmm tile

The coefficient block in the canonical tile should be transposed

Matching recurrence and determinant induction

Local transpose repair for the canonical body

State-vector determinant argument

Recurrence realization, companion factorization, and rank

Challenge an audit finding

Authenticate with ORCID, then require an exact arXiv match

Every integer matrix has a rank-at-most- $m$ tile