Paper 2026/2849

γ₁₁: Unequal Triangles and Adapted References — the macadamia 0050

Abstract

This paper settles the ninth of the ten macadamias of the series on the Hermite constant in dimension 11: the perfect form $P$ = 0050:024468269 of rank 9, of determinant 8192/9 at minimum 4. The root paper reduces $\gamma_{11}$ to chains $L\supset M\supset N$ of lattices of ranks 11, 10, 9, each with minimum at least 4; we prove that every admissible chain whose rank-9 member is admissibly marked by $P$ has $\det L\ge 972$. No argument at rank 10 can do this, because an actual lattice of rank 10 with determinant $3328/3<4\cdot 972^{9/11}$ lies over $P$. The proof is computer-assisted, with explicit finite checks. Its decisive step is elementary: the three additive dual vectors of a section covector and the first normal usually have different lengths, and keeping the three lengths in the circle obstruction, instead of their maximum, gives a determinant cap that sharpens every subsequent contact and spectral bound. Adapted reference forms turn the contact conditions into exact rational tests, a common slack is retained throughout, and four computational results reduce the problem to a five-parameter root, exhausted by a finite cover of 6 993 terminal regions replayed in exact arithmetic.

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