MathAudit

Paper metadata

Original submission
October 20, 2026
Claimed MSC
11J82
arXiv categories
math.NT

Abstract

We establish degree-dependent irrationality-exponent bounds for logarithms at algebraic arguments and sharper bounds when conjugate logarithms match. A two-torus construction gives an upper bound of three for ratios of logarithms of multiplicatively independent positive rational numbers. We also treat normalized quadratic logarithms and circular logarithms. The rational-logarithm and rational-power exponent-two cases were treated earlier by Liu, Jiang and Zhang; we retain them within the broader framework and its Lean verification. Starting from the interpolation and determinant method in OpenAI's September 24, 2026 preprint on the irrationality exponent of ππ, we track the arithmetic cost of algebraic embeddings and the effects of matching conjugate logarithms. A second, graded Pade construction gives near-optimal measures for powers and bounded-degree polynomial values at sufficiently small logarithmic arguments, with estimates uniform in both coefficient height and rational denominator. We state the branch, independence, parameter-order, and approximation-field hypotheses explicitly. The Lean supplement has reviewed endpoint correspondences for all 62 numbered results: 53 direct checked endpoints or compositions, and nine applications with named classical theorem inputs. Native proofs of those inputs and separate formalizations of every unnumbered assertion are not supplied. Appendix A gives the exact declarations, hypotheses, and verification evidence.

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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
01Statements3 reported findingsNo statement issues found

The degree-dependent single-logarithm bounds, matching-embedding improvements, exponent-three bound for rational logarithm ratios, and the stated small-argument graded approximation theorems are supported with their explicit branch and parameter-order hypotheses.

Theorems 4.1–4.2 and 10.1No issue found

The single-logarithm determinant bounds have the stated field-degree factors

PDF pp. 13–35 · arXiv:2610.28079v1

Weighted interpolation at the moving multiplicative centers supplies a nonzero minor. Clearing denominators and taking the number-field norm gives the arithmetic lower bound, while exact logarithm replacement and the analytic determinant estimate give the upper bound. Retaining the embeddings with a common scalar produces the factor 2d/s2d/s; the rational-power and fully matched cases therefore give exponent two as claimed.

Theorem 11.1No issue found

The two-torus construction yields the exponent-three quotient bound

PDF pp. 36–46 · arXiv:2610.28079v1

Multiplicative independence supplies the qualitative nonvanishing needed for the two-variable minor. The interpolation count, denominator growth, and collision saving balance to a homogeneous linear-form exponent that converts to 2≤μ(log⁡a/log⁡b)≤32\leq\mu(\log a/\log b)\leq3 for positive multiplicatively independent rational a,ba,b.

Theorems 16.1, 17.1, and 18.1No issue found

The graded conclusions retain the necessary smallness and uniformity hypotheses

PDF pp. 76–89 · arXiv:2610.28079v1

The contact-polynomial construction and the stated differential-system rank theorem give adjacent-grade independence. Integral transport controls denominators uniformly in coefficient height, and the analytic remainder is small only after the threshold on the rational argument is imposed. The conclusions therefore match the quantified statements and do not assert exponent two at arbitrary fixed arguments.

02Proofs2 reported findingsNo proof issues found

The central interpolation, blow-up, determinant, norm, analytic-translation, two-torus, and graded-rank arguments are internally consistent. The paper distinguishes its written proofs from the scope of its machine-checked endpoints.

Theorem 5.1 and Sections 6–8No issue found in the reviewed proof

Interpolation and the arithmetic–analytic determinant comparison close the single-logarithm proof

PDF pp. 14–28 · arXiv:2610.28079v1

The blow-up convention is consistent: IOX′=OX′(−E)I\mathcal O_{X'}=\mathcal O_{X'}(-E), −E-E is relatively ample, and A−(1+σ)EA-(1+\sigma)E is nef. Serre vanishing gives the required jet surjection. The norm lower bound and analytic upper bound use the weights in the declared order, and the final parameter inequalities have the correct strict margins.

Sections 11 and 15–18No issue found in the reviewed proof

The quotient and graded constructions preserve their quantifiers

PDF pp. 36–46 and 76–89 · arXiv:2610.28079v1

The two-coordinate interpolation treats the possible collisions before choosing the final parameters, so its constants remain uniform. In the graded applications, the rank input is invoked with its hypotheses explicitly named; coefficient height, denominator, degree, and evaluation-point limits are not interchanged.

03Novelty0 reported findingsNo novelty concerns found

No non-novelty findings.

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Paper
arXiv:2610.28079v1
Authors listed
Gregory V. Chudnovsky
Audit date
October 23, 2026
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