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 , define the theta series and Epstein zeta function by

			(2)
			(3)

where and . For with covolume 1, Luo and Wei (2026) proved that, for every ,

(4)

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

(5)

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) report ChatGPT assistance with preparing the computational verification materials.

For every point lattice spanning , Liu (2026) proved that its normalized second moment satisfies

(6)

with equality iff is obtained from by an orthogonal transformation and positive rescaling. Consequently, is the unique optimal four-dimensional lattice quantizer up to these transformations. The revised computer-assisted proof includes explicit polynomial certificates using nonnegative sums of squares. Liu (2026) reports AI assistance in constructing and checking these certificates.