Irrationality exponents of logarithmic numbers: Values, quotients, and graded approximation

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 $π$, 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.