The fifth virial coefficient of hard spheres

For hard spheres of equal diameter in three dimensions, the fifth virial coefficient b5 = B5/B24 is an elementary number plus a linear combination, with coefficients in ℚ[π−1], of three integrals over the lens in which two unit balls at unit distance overlap: the integral P(1) of the squared overlap volume, a moment Φ(2) of it over the distance of the balls, and the measure T(1) of ordered triples of lens points at mutual distances at least one. Each of them is a finite linear combination of iterated integrals, of length at most four, of rational one-forms on rational curves and on genus-one curves whose Jacobians are isogenous to the modular curve X0(24), with coefficients in the algebra generated over ℚ̅[π,π − 1] by values of multiple polylogarithms at algebraic arguments; so is the modified star integral of Ree and Hoover. The representation agrees with direct quadrature to 46 digits; the first twelve are b5 = 0.110251372267. MSC2020: 82B21; 52A20, 33C75, 14H52, 65D30. Version 1.0.0, 15 October 2026. Main paper DOI 10.5281/zenodo.23364646; complete proofs and computations DOI 10.5281/zenodo.23364648; supplementary material DOI 10.5281/zenodo.23364650.