Lattice packing of B₂⁴ × B₂⁴
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Must the best lattice packing density of a Cartesian product of Euclidean balls equal the product of their separate optimal lattice densities? The concrete test case is B₂⁴×B₂⁴ in dimension eight.
Why it matters and what counts as progress
Why it matters: A strict improvement would show that a coupled lattice can outperform independent product packings.
What counts as progress: An exact product-body lattice whose density exceeds the D4×D4 benchmark, or a theorem identifying when equality must hold.
How the problem developed
- 13 Jul 2026 · Exact counterexample [1]
Nihar P. Gargava Gave an exact counterexample in dimension eight to the Cartesian-product equality for critical determinants and lattice-packing densities associated with product bodies.
AI contribution
On two counterexamples in the geometry of numbers
Gargava gives an exact rotated E8 lattice packing of B₂⁴ × B₂⁴ with density π⁴(1+1/√5)⁴/1024, about 9.67% above the product D4 packing. A separate determinant-preserving perturbation of the nine-dimensional densest lattice Λ9 has lower computed flat-torus height.
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