MathAudit

Paper metadata

Original submission
October 16, 2026
Claimed MSC
11J06, 11J70, 37B10
arXiv categories
math.NT, math.DS
Paper license
CC BY 4.0 ↗
Comments
47 pages, 3 figures

Abstract

We study the initial discrete parts of the Markoff spectra of Hecke groups of index q≥5q\ge 5. Using a digit expansion associated with geodesics and a Perron-type formula, we classify all bi-infinite digit sequences whose Markoff values lie at or below the first accumulation point. We obtain explicit formulas for the discrete spectral values and identify the sequences realizing them, including those attaining the first accumulation point. We also prove that the infinite family of discrete values increases strictly to this point. Our results recover Vulakh's description of the initial discrete spectra through an explicit symbolic approach.

AI-generated audit

Audit summary

Audited against arXiv v1

Not a correctness certificate. A green “no issues found” result may include yellow suspected typos or minor formal issues 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 October 23, 2026
01Statements1 reported findingNo statement issues found

The symbolic descriptions of the part of the Markoff spectrum below the first accumulation point, the formulas for the periodic values, and the identification of the smallest accumulation point are consistent for both even and odd Hecke parameters. No false or materially unsupported central statement was found.

Theorems 1.1 and 1.2No issue found

The initial Markoff spectrum and its first accumulation point are classified

PDF pages 3–5 and 24–43 · Theorems 5.9, 5.11–5.12, 6.11–6.13 · arXiv:2610.24331v1

The forbidden-digit estimates first reduce a low Markoff sequence to the central alphabet. The even and odd extension arguments then force precisely the listed periodic families and the limiting bi-infinite word. The matrix formulas give the displayed periodic Markoff values, and the comparison lemma proves their strict increase and convergence to the stated limiting value. This supports both the discreteness below the limit and its minimal-accumulation assertion.

02Proofs1 reported findingNo proof issues found

The Perron formula, digit comparisons, finite-pattern exclusions, normal-form arguments, transfer-matrix calculations, and monotonic convergence were checked through both parity cases. Apparent repeated digits in extracted text are barred digits in the typeset PDF and do not create contradictions. No incorrect or materially incomplete proof was found.

Sections 3–7No issue found in the reviewed proof

The comparison machinery closes both parity classifications

PDF pages 11–46 · Propositions 3.1–3.8, Sections 4–6, and Lemma 7.1 · arXiv:2610.24331v1

Monotonicity of the continued-fraction tails turns local forbidden and restricted digits into strict Markoff bounds. The key lemmas compare every remaining unequal block length against the limiting word, forcing a constant block length in the periodic cases. The explicit hyperbolic-matrix recurrences then compute the values, while the final comparison lemma supplies the strict inequalities needed for the accumulation statement.

03Novelty0 reported findingsNo novelty concerns found

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:2610.24331v1
Authors listed
Byungchul Cha, Dong Han Kim
Audit date
October 23, 2026
  1. 01Establish identityMatch an authenticated scholarly identity to this paper.
  2. 02Submit evidenceIdentify the finding and give a formal mathematical response.
  3. 03Re-evaluateA separate agent checks the response and records a disposition.
Recommended production method

Authenticate with ORCID, then require an exact arXiv match

MathAudit should accept the identity only when ORCID OAuth authenticates the claimant's iD and this exact arXiv paper appears in arXiv's public authority feed for that iD. A matching name alone is not sufficient.

ORCID OAuth and arXiv authority-record lookup are not connected in this local prototype.

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.

This demonstration does not send, store, or compare the address.

Structured response preview

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

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.