发表机构
Wuhan University; Wuhan Institute for Math & AI, Wuhan University; Peking University（武汉大学; 武汉大学数学与人工智能研究院; 北京大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文通过分离权重行列式论证，证明任意正有理数 r≠1 的对数的无理性指数 μ(log r)=2，并给出完整参数选择与估计，附带几何推广及有限度界。
AI 中文摘要
我们为正有理数 r ≠ 1 的每个值提出了一种直接的分离权重行列式论证，以证明 μ(log r) = 2。我们首先证明该论证中使用的移动中心插值定理。然后固定任意 r = a/b，并给出该论证的所有参数选择、分母估计、行平移和解析行列式界。有理线性组合和有理仿射变换作为推论随之而来。几何推广和显式有限度界收集在附录中。本文仍为待审核的研究草稿；此次重组不构成对其数学结论的独立认证。
英文摘要
We study rational approximation to logarithms of positive rational numbers and give a separated-weight determinant argument for $μ(\log r)=2$ whenever $r\in\mathbb Q_{>0}\setminus\{1\}$. The geometric input is an interpolation theorem for logarithmic jets at points with distinct multiplicative coordinates. Its successive weight thresholds are independent of the interpolation centres; the eventual degree threshold may depend on them. A local intersection-length bound and a curve inequality lead to surjectivity through a blow-up and Serre vanishing. For a fixed rational argument, this produces a nonzero rational determinant. Clearing denominators gives an arithmetic lower bound, while an exact row translation and repeated Taylor indices yield a contradictory analytic upper bound. The consequences include nonzero rational linear combinations of such logarithms, rational affine changes, and logarithms of positive numbers with a rational positive integer power. We also give a bounded-fibre interpolation extension, an explicit bound for the final analytic degree, and a counterexample to interpolation under the volume condition alone.
Comments21 pages. Revised manuscript and abstract, with expanded historical context, explanations of the parameter choices, and a discussion of the connection with the argument for pi through the exponential map
