Three Irrationality Results for the Dilogarithm: Rhin--Viola and Viola--Zudilin Constructions at $-1/4$, $1/5$, and $-1/3$

Abstract

We prove the irrationality of \[ \Li_2(-1/4),\qquad \Li_2(1/5),\qquad \Li_2(-1/3). \] The three arguments form a natural progression. The first is an endpoint completion of the five-parameter Rhin--Viola method. Rhin and Viola's 2019 treatment already supplies the negative-argument continuation, permutation invariance and factorial divisor; at $z=-4$ we use three bounded shifts, an elementary fixed-contour estimate and a four-term recurrence to remove the remaining nonvanishing problem in the complex-saddle regime. For $1/5$ we pass to the six-parameter Viola--Zudilin family and apply the Rhin--Viola factorial transformations term by term inside two binomial expansions of the same integral. This inherited divisor crosses the arithmetic threshold that the unrefined five-parameter construction does not reach in our computations. The same crossbreed, combined with the negative-argument continuation, proves the result at $-1/3$. All proof-critical finite inequalities are accompanied by exact executable certificates. Supplementary computations propagate $10,000$ exact primitive coefficient pairs for each of $1/5$ and $-1/3$, with complete gcd removal and no failure of the certified smallness or adjacent nonproportionality checks.

arXiv comment28 pages. Ancillary files contain exact verification scripts and certificates for the arithmetic, contour and recurrence arguments, together with supplementary 10,000-term stress tests for Li_2(1/5) and Li_2(-1/3). The numerical stress tests are not used in the proofs