The 24-cell conjecture in the all-contact case and near D4

The 24-cell conjecture states that every Voronoi cell of a packing of unit balls in R⁴ has volume at least 8, the volume of the regular 24-cell. It implies that no packing of equal balls in R⁴ is denser than the lattice packing D₄. We prove the conjecture for two classes of cells; in each class, equality holds only for the D₄ root configuration. The first consists of the cells whose ball touches every ball centred at distance less than 2√2 from its centre, and its proof uses a certificate at the second level of the Lasserre hierarchy. The second consists of the cells whose 24 nearest centres lie in an explicit neighbourhood of the D₄ configuration, whatever the other centres are. For them the volume is at least 8 plus a fixed multiple of the total outward displacement of the 24 centres, which we prove by a quantitative rigidity estimate and a second-order expansion of the volume about D₄. Our proofs for both classes are computer-assisted and partly formalised in Lean.

Version 3 has a rewritten abstract and introduction. The introduction now gives the background with references and ends with a short subsection relating the results to the work of Bhattacharjee et al. Section 7 opens with an outline of the proof. The verification is collected in Section 8 and Table 3, while the Lean names of the formalised results and hypotheses have moved to the supplement, which has a new version. Theorems and proofs are as in version 2.