The fifth virial coefficient of hard disks

We determine the fifth virial coefficient of the hard-disk fluid. We prove that B5/B24 = 140/81 − 101√3/(9π) + 1169/(18π2) − 71√3/(2π3) + 4P1/π3, where P1 is a two-dimensional integral of squared triple-overlap areas of unit disks, and that B5/B24 is an explicit expression in two Eichler moments of a weight-four meromorphic modular form for Γ0(6)+ at the CM point (3 + i√15)/12, with coefficients built from algebraic numbers, a product of four values of Γ at fifteenths, Cl2(π/3), log 3 and π. Equivalently, B5 is one thirtieth of the complete-graph Mayer integral plus a polynomial in π with coefficients in ℚ(√3). Evaluated independently to 1000 digits, the two formulas agree to 7.5 × 10−1015. The most precise published value has eight digits, and a direct evaluation of the ten Mayer diagrams by two quadrature methods agrees with the formulas to 44 digits. The proof of the first formula rests on a pairing lemma for closed equilateral pentagons and on an exact-form identity on the chambers of the surface of such pentagons; parts of both proofs are computer-assisted, through certificates checked in exact arithmetic. The constant P1 is a period of a singular K3 surface with complex multiplication by ℚ(√−15). MSC2020: 82B21 (primary); 11F67, 14J28, 33C10, 11Y60 (secondary) Files: the paper (PDF, 25 pages) and its flattened LaTeX source (ZIP). Companion paper with the complete proofs and computations: DOI 10.5281/zenodo.23097003. Scripts, certificates and data: DOI 10.5281/zenodo.23097007.