D_4 Lattice

The lattice is the four-dimensional root lattice

(1)

Its rescaling has covolume 1 and minimum squared norm .

For a point lattice with covolume 1, Luo and Wei (2026) proved that, for every ,

(2)

with equality iff is equivalent to under an orthogonal transformation. They also proved that, for every ,

(3)

again with equality iff is equivalent to under an orthogonal transformation. Thus uniquely minimizes the theta series and Epstein zeta function among four-dimensional point lattices with covolume 1. The proof is computer-assisted and uses certified interval bounds.

Luo and Wei (2026) state that ChatGPT assisted with mathematical exposition, review of proof arguments and citations, and preparation of the computational verification materials, including the appendices.