Uniform cubic bounds for irrational values of the dilogarithm

Abstract

We study explicit denominator conditions ensuring the irrationality of the dilogarithm at positive and negative rational points. We prove that $\operatorname{Li}_2(p/q)$ is irrational whenever $q>(4e^4/27)p^3$, and obtain an explicit, slightly smaller threshold for $\operatorname{Li}_2(-p/q)$. The positive argument specializes the Rhin--Viola integral family and uses a uniform estimate for a restricted prime divisor. The negative argument combines the same arithmetic construction with three adjacent integral moments and an explicit maximization. We also give finite-numerator refinements by interpolating certified partition bounds and by introducing a comparison measure with a quadratic tail. In particular, $\operatorname{Li}_2(5/q)$ is irrational for every integer $q\ge418$. The proof separates the established integral and determinant constructions from the parameter estimates and finite certificates used here.