arXiv:2602.14142v3

Convergence and combinatorics of the Reverse algorithm

Hiroaki Ito, Niels Langeveld, Jörg Thuswaldner

math.DS37B1037A2568R1511J70

Paper metadata

Original submission: February 15, 2026
Latest revision: July 9, 2026 · version 3
Claimed MSC: 37B10, 37A25, 68R15, 11J70
arXiv categories: math.DS
Paper license: CC BY 4.0 ↗
Comments: Implemented more suggestions of the referee

Abstract

We study the Reverse algorithm, a multidimensional continued fraction algorithm, which is not unimodular. We show that the Reverse algorithm is ergodic and, by proving that its second Lyapunov exponent is negative, that it is a.e. exponentially convergent. In addition to that, we attach substitutions to this algorithm and study the $S$ -adic languages generated by sequences of these substitutions. The negativity of the second Lyapunov exponent implies that almost all of these languages are balanced. By a thorough study of the combinatorics of the substitutions, we are even able to obtain a concrete generic family of balanced languages that is characterized in terms of a simple condition on the underlying sequence of substitutions.

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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 16, 2026

01Statements3 reported findingsContains unsupported statements

The ergodicity theorem and the factor-balancedness criterion are supported. The negativity and explicit upper bound for the second Lyapunov exponent are not able to be verified because the decisive $4^{12}$ -cylinder computation is reported without the calculation or an auditable error certificate.

Theorem 1.3Correct

Ergodicity of the Reverse algorithm

Pages 5 and 8–10 · Theorem 1.3 and Section 3 · arXiv:2602.14142v3

The induced transformation whose branches end in symbol $4$ is full. Recurrence together with the zero measure of the Rauzy gasket gives positive matrix blocks almost everywhere and hence almost-everywhere topological convergence. Lemma 3.2 controls the row-sum ratio after a final $M_4$ , yielding the uniform Rényi constant $2^3$ . The Rényi criterion therefore proves ergodicity of the induced map, and the standard inducing argument transfers it to $f_R$ with the previously established invariant probability measure.

Theorem 1.4Not able to verify

The numerical Lyapunov bound is not independently verifiable from the supplied material

Pages 5 and 12–13 · Theorem 1.4, equation (4.6), and Remark 4.5 · arXiv:2602.14142v3

The analytic reduction is explicit: a negative value of $(1/12)\int_\Delta\log\|D^{(12)}(x)\|_1\,d\mu(x)$ implies the result. The sign then depends on the claimed finite estimates $L_1(12)<0.024002$ and $L_2(12)<-0.044610$ , obtained by maximizing over nearly $4^{12}$ cylinders. The paper states that each summand was evaluated to 18-digit precision with error below $10^{-18}$ , but gives neither code nor interval endpoints, rounding rules, exact cylinder data, or per-operation bounds from which that assertion can be checked; the arXiv source bundle contains only the TeX and bibliography. The margin is large compared with the claimed aggregate error, but the decisive numerical values themselves cannot be reconstructed or certified from the available evidence.

Theorem 1.5Correct

Positive density of the contracting block implies factor balancedness

Pages 5 and 13–18 · Theorem 1.5 and Section 5 · arXiv:2602.14142v3

The substitution estimates first control how letter-balance constants change under every generator. Occurrences of $(\sigma_1\sigma_2\sigma_3)^9$ provide seven consecutive contracting triples, while the intervening blocks have a uniform operator bound. Positive occurrence density makes the resulting geometric series converge, so all shifted languages are letter balanced. Grouping each run of $\sigma_4$ with the following $\sigma_i$ produces proper substitutions without changing the language, and the cited proper-sequence theorem then gives factor balancedness.

02Proofs5 reported findingsContains incorrect or incomplete proofs

The analytic Lyapunov reduction is correct, but its sole numerical input is not accompanied by enough data to verify the asserted upper bounds. The balancedness proof also omits the separated-occurrence selection required by its factorization, although a verified greedy-thinning repair is available.

Section 3Correct and complete

Inducing and Rényi-condition proof

Pages 8–10 · Lemmas 3.1–3.3 and Proposition 3.4 · arXiv:2602.14142v3

For almost every coding, recurrence may be applied after an occurrence of $4$ , giving a repeated positive block and shrinking cylinders. Appending $M_4$ replaces the three row sums by their pairwise sums. The strict triangle inequality of Lemma 3.2 then bounds the largest-to-smallest new row-sum ratio by $2$ , and the cubic Jacobian formula gives a branch-independent bound below $8$ . These facts supply all four hypotheses of the cited ergodicity criterion on the full-measure induced system.

Equation (4.6)Not able to verify

The computer-assisted sign estimate lacks a checkable certificate

Pages 12–13 · definitions of $L_1,L_2$ , equation (4.6), and Remark 4.5 · arXiv:2602.14142v3

The formulas for $L_1$ and $L_2$ do give upper bounds for the two integral pieces: positive cylinder maxima are paired with upper density bounds and negative maxima with lower density bounds. What remains is the finite assertion $L_1(12)+L_2(12)<-0.020608.$ Eighteen-digit working precision by itself does not establish a $10^{-18}$ error for each term, and no executable calculation or interval certificate is supplied. Repair classification: No repair supplied. Publishing the cylinder enumeration together with directed-rounding bounds for every maximum, volume, density factor, logarithm, and final sum would make this step auditable.

Lemma 5.9Typo

The accumulated balance constant has its indices reversed

Page 17 · first paragraph of the proof of Lemma 5.9 · arXiv:2602.14142v3

After applying $n-m$ substitutions, Proposition 5.2 gives the constant $\frac{44}{7}C+4(n-m),$ not the printed $\frac{44}{7}C+4(m-n)$ . The next displayed estimate already uses $C+(n-m)$ , and iterating the preceding proposition forces the positive sign. Correcting the order of the indices is unique and leaves every subsequent bound unchanged.

Proposition 5.10Incomplete as written

The selected block occurrences must be nonoverlapping

Pages 17–18 · definition of $(\ell_k)$ in the proof of Proposition 5.10 · arXiv:2602.14142v3

The factorization using intervals $[\ell_{k-1}+21,\ell_k)$ requires $\ell_k\geq\ell_{k-1}+21$ , whereas the manuscript chooses only a strictly increasing occurrence sequence. This leaves overlapping intervals and does not justify the displayed factorization. Repair classification: Verified repair. Greedily retain the first eligible occurrence at least $21$ positions after the preceding retained occurrence. Thinning by a fixed separation preserves positive lower density up to a fixed factor, so $g_k=o(k)$ and the contraction series remain valid.

Section 5Correct and complete

Projection, contraction, and proper-blocking arguments

Pages 13–18 · Lemmas 5.1–5.9 and proof of Theorem 1.5 · arXiv:2602.14142v3

The letter-count identities verify the balance loss for each substitution. The extremal-ray computation gives the stated projection norm, the block $B_{\sigma_1}B_{\sigma_2}B_{\sigma_3}$ contracts the relevant hyperplane by $5/7$ , and the intervening products have uniform norm at most $10$ . After the local occurrence-selection correction above, seven contractions per selected block yield $10(5/7)^7<1$ . The final proper blocking satisfies the hypotheses of the cited factor-balance theorem.

03Novelty0 reported findingsNo non-novelty findings

No non-novelty findings.

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Paper: arXiv:2602.14142v3
Authors listed: Hiroaki Ito, Niels Langeveld, Jörg Thuswaldner
Audit date: August 16, 2026

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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. 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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. 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P(delta)

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P

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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

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Convergence and combinatorics of the Reverse algorithm

Paper metadata

Abstract

Audit summary

Detailed mathematical audit

Ergodicity of the Reverse algorithm

The numerical Lyapunov bound is not independently verifiable from the supplied material

Positive density of the contracting block implies factor balancedness

Inducing and Rényi-condition proof

The computer-assisted sign estimate lacks a checkable certificate

The accumulated balance constant has its indices reversed

The selected block occurrences must be nonoverlapping

Projection, contraction, and proper-blocking arguments

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