The fifth virial coefficient of hard disks: complete proofs and computations

This paper gives the complete proofs and computations behind the companion article (S. Na, The fifth virial coefficient of hard disks, Zenodo, DOI 10.5281/zenodo.23097001), which proves a formula for the fifth virial coefficient of the hard-disk fluid in terms of one two-dimensional lens integral P1 and relates P1 to Eichler integrals at a CM point. The proof of the one-constant formula is completed through four-step planar walks, a pairing lemma for closed unit pentagons, ladders of disk and circle steps with rational Stokes and Hankel certificates, and an exact primitive on the chambers of the surface of closed unit pentagons. We prove the change of variables from P1 to a singular K3 surface, the Picard–Fuchs system of its twist family, and the decomposition of P1 into two Eichler moments. The contact identity and the identity among the polar-pair diagrams are proved for unit balls in ℝd; in the plane the latter is the linear relation of Ree and Hoover (1964) among the five-point star integrals, and in ℝd its correction term is one of their modified star integrals. For hard spheres the two identities give B5 to twelve digits. We describe the numerical methods and list a computational receipt for every computation. MSC2020: 82B21 (primary); 11F67, 14J28, 33C10, 60G50, 65D30, 11Y60 (secondary) Files: the paper (PDF, 117 pages) and its flattened LaTeX source (ZIP). Companion to the paper The fifth virial coefficient of hard disks (DOI 10.5281/zenodo.23097001). Scripts, certificates and data: DOI 10.5281/zenodo.23097007.