Local minimality of flat-torus height at Λ9
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The product-ball lattice-packing question asks whether a lattice packing of B₂⁴×B₂⁴ can exceed the product D4 construction. A distinct question asks whether the densest nine-dimensional lattice Λ9 minimizes flat-torus height under determinant-preserving perturbations.
Why it matters and what counts as progress
Why it matters: These are separate optimization targets: exact packing density in eight dimensions and a spectral height comparison in nine dimensions.
What counts as progress: An exact improvement of the product-ball lattice density, or a certified height comparison for a determinant-preserving Λ9 perturbation; these are distinct objectives.
How the problem developed
- 1997 · Partial result [1]
Patrick Chiu Proved that the height of volume-one flat tori attains a minimum in the moduli space.
- 2000 · Conjecture context [2]
Kok Seng Chua Recorded Sarnak’s conjecture that a global height minimizer is given by a lattice with longest shortest vector, equivalently a densest lattice packing.
- 13 Jul 2026 · Numerical counterexample [3]
Nihar Gargava Reported a numerical counterexample in dimension nine to the proposed global-minimum principle, using Λ₉; the comparison is numerical within floating-point accuracy.
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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