Paper 2026/2827

γ₁₁: Contact Geometry — the macadamias 0001, 0014, 0059 and 0015

Abstract

The root paper of this series reduces the value $\gamma_{11}=4/\sqrt[11]{972}$ of the Hermite constant to a statement about chains of lattices $L\supset M\supset N$ of ranks 11, 10 and 9 whose rank-9 member lies, in the sense of Voronoi, over one of 4 487 813 perfect forms; the $\gamma_{11}$ paper excludes all of them but ten, the macadamias. This paper settles four macadamias, the forms 0001:010488327, 0014:001749549, 0059:016902802 and 0015:030291998, by one method: the geometry of the contacts of the form. A section frame of short dual vectors forces the first gluing of $M$ over $N$ to lie near one of finitely many coherent centres; at such a centre the rank-10 metric is compared with an explicit reference lattice, whose Delaunay cells and contact identities give a uniform bound on its covering radius; and a primitive-extension inequality turns that bound into $\det L\ge 972$. The proof is computer-assisted: every finite check it uses is stated as a Computational result with the records of the certificate archive that certify it, and all comparisons are between explicit rational numbers.

Note: Data on https://zenodo.org/records/23979316