This is the technical companion to The fifth virial coefficient of hard spheres. It gives the complete proofs and computations: the contact reduction, lens-volume functionals, far-triple measure, curve identifications, and numerical comparisons. Together these arguments establish the representation stated in the companion paper. 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.
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Sungsoo Na (2026) studied this question.
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