Paper 2026/2825

The Hermite Constants in Dimensions 10, 11 and 12

Abstract

This paper introduces a series of seven papers establishing $\gamma_{10}^{10}=4^{10}/768$, $\gamma_{11}^{11}=4^{11}/972$ and $\gamma_{12}=4/\sqrt{3}$, under the attributed classification of perfect forms of rank nine and the explicitly stated computational inputs. The argument treats the dimensions in order. In dimension ten, a section orthogonal to a shortest dual vector is compared with a perfect form. A completed exact census leaves three possible reference forms, and matrix inequalities exclude a determinant below 768 in each case. In dimension eleven, two successive sections give a chain of ranks eleven, ten and nine. The dimension-ten bound controls the determinant of the last section. The verified global census and preliminary certificates reduce the problem to ten local cases: nine are treated in five companion papers, and one in “A Focus on Hermite Constant 11”. The local arguments combine contact geometry, section bounds and finite rational covers. Mordell’s inequality then gives the dimension-twelve upper bound, matched by the Coxeter–Todd lattice. We prove the common reduction, explain the dependency between the papers, and state exactly which finite assertions must be checked. The accompanying Lean development verifies the complete analytic deduction from those inputs; external exact checkers verify the large certificate data.

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