arXiv:2503.16330v2

If a machine did it, it is probably transcendental (even $p$ -adically)

Laura Capuano, Sara Checcoli, Marzio Mula, Lea Terracini

math.NTmath.CO11D8811J7011J8168R15

Paper metadata

Original submission: March 20, 2025
Latest revision: July 13, 2026 · version 2
Claimed MSC: 11D88, 11J70, 11J81, 68R15
arXiv categories: math.NT, math.CO
Paper license: CC BY 4.0 ↗
Comments: 33 pages, accepted for publication on Math. Z

Abstract

Continued fraction expansions provide a well-established bridge between algebraic properties of numbers and combinatorics on words. In this article, we investigate the algebraicity of $p$ -adic numbers whose continued fractions arise from certain classes of words which generalize the classical automatic, periodic and palindromic words. Our main result shows that, under mild conditions on the $p$ -adic continued fraction expansion, such numbers are either algebraic of degree at most 2 or transcendental. This result provides an analogue of results of Bugeaud and Adamczewski-Bugeaud in the real setting and extends previous works that were limited to specific choices of $p$ -adic floor functions and less general classes of words.

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Audit summary

Audited against arXiv v2

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

01Statements3 reported findingsCorrect

The algebraic-degree dichotomy for the two stated classes of $p$ -adic continued fractions and its Ruban, Browkin, and automatic-word corollaries are correct as stated.

Theorems 1.4, 4.1, and 4.3Correct

Quadratic-or-transcendental dichotomy

Pages 3, 9–10, and 22–32 · Theorem 1.4, Theorems 4.1 and 4.3, and Section 6 · arXiv:2503.16330v2

For both combinatorial hypotheses, long repeated or reversed blocks produce simultaneous linear forms in the convergent numerators and denominators. The bounded archimedean and $p$ -adic continuant hypotheses give the height control needed for the Subspace Theorem, while the lower bounds on the partial quotients force the required $p$ -adic smallness. An infinite fixed linear relation then implies eventual periodicity, hence rationality or quadraticity; otherwise the number is transcendental. The separate word configurations in Theorems 4.1 and 4.3 cover the stated alternatives without an omitted case.

Full paper, version 2 ↗

Corollary 1.5Correct

Ruban and Browkin continued fractions

Pages 3–4 and 10 · Corollary 1.5 and its specialization · arXiv:2503.16330v2

For the Ruban and Browkin floor functions, the standard size properties of the partial quotients imply the lower bounds and continuant estimates required by Theorem 1.4. Substitution therefore gives the stated rational, quadratic, or transcendental alternatives; no stronger conclusion than the hypotheses support is asserted.

Corollary 1.6Correct

Automatic two-letter partial-quotient words

Pages 4 and 31–32 · Corollary 1.6 and the final proof · arXiv:2503.16330v2

A non-eventually-periodic automatic word over two allowed partial quotients has the bounded-repetition structure used in Theorem 4.1 or the reversed-prefix structure used in Theorem 4.3. The two fixed digits make all continuant growth conditions uniform. Eventual periodicity gives the rational or quadratic cases, while the nonperiodic case satisfies the transcendence criterion.

02Proofs4 reported findingsCorrect

The continuant estimates and the two Subspace-Theorem arguments are correct and complete. One terminal coefficient list contains a harmless extra symbol, which is a uniquely identifiable typo and does not affect the proof.

Lemmas 3.2–3.6Correct and complete

Continuant identities and height estimates

Pages 6–9 · Section 3 · arXiv:2503.16330v2

The matrix product for $p$ -adic convergents gives the determinant and concatenation identities with the displayed signs. The archimedean bounds follow from the assumed bounded continuants, while the $p$ -adic estimates use the strict size of each partial quotient. These estimates control every coordinate and every local linear form used in Section 6 with constants independent of the repeated-block index.

Lemma 6.1 and proof of Theorem 4.1Correct and complete

Repeated-block Subspace-Theorem argument

Pages 22–27 · Lemma 6.1 and Section 6.1 · arXiv:2503.16330v2

The repeated prefix produces rational approximants whose relevant local products decay faster than the height allowed by the hypotheses. Schlickewei's Subspace Theorem therefore yields a fixed nonzero linear relation along an infinite subsequence. The determinant identities exclude the degenerate coefficient patterns, and the remaining relation forces the continued fraction to be eventually periodic unless the number is transcendental. The rational and quadratic alternatives are retained exactly as stated.

Proof of Theorem 4.3Correct and complete

Palindromic and reversed-block cases

Pages 27–31 · Sections 6.2–6.4 · arXiv:2503.16330v2

The three possible placements of the reversed block lead to the three displayed vectors of convergent data. In each case the continuant symmetry supplies the extra small linear form, and the same height computation meets the Subspace-Theorem exponent. The subsequent coefficient elimination uses nonzero continuant determinants and exhausts all cases, yielding eventual periodicity or transcendence.

Final Subspace-Theorem applicationTypo

An unused fourth coefficient is listed accidentally

Page 25 · final paragraph of the proof using three coordinates · arXiv:2503.16330v2

The proof says that there are $b_1,b_2,b_3,b_4\in\mathbb Z$ not all zero, but the ambient vector has three coordinates and the immediately following relation contains only $b_1,b_2,b_3$ . Delete $b_4$ . The Subspace Theorem supplies exactly the three coefficients used in the displayed equation, so this correction is unique and changes no argument.

Full paper, version 2 ↗

03Novelty0 reported findingsNo non-novelty findings

No non-novelty findings.

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Paper: arXiv:2503.16330v2
Authors listed: Laura Capuano, Sara Checcoli, Marzio Mula, Lea Terracini
Audit date: August 15, 2026

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Source: arXiv v2
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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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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

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If a machine did it, it is probably transcendental (even ppp-adically)

Paper metadata

Abstract

Audit summary

Detailed mathematical audit

Quadratic-or-transcendental dichotomy

Ruban and Browkin continued fractions

Automatic two-letter partial-quotient words

Continuant identities and height estimates

Repeated-block Subspace-Theorem argument

Palindromic and reversed-block cases

An unused fourth coefficient is listed accidentally

Challenge an audit finding

Authenticate with ORCID, then require an exact arXiv match

If a machine did it, it is probably transcendental (even $p$ -adically)