What was already known
Cassels and Zong asked whether lattice-packing product inequalities could be strict; D4 is optimal in dimension four. Chiu conjectured that a shortest-vector-maximizing lattice minimizes flat-torus height.
AI involvement
Gargava chose the E8 and Λ9 search directions. ChatGPT 5.5 Pro found the decisive rotation and perturbation; Gargava supplied the mathematical proof, analysis and manuscript.
Human-selected candidate comparisons led to model-discovered retained constructions in both problems.
Importance
The exact eight-dimensional packing answers the Cassels–Zong product question; the nine-dimensional computation challenges the height conjecture.
Limitations
The E8 packing calculation has exact root checks. The Λ9 height comparison uses floating-point cutoffs and lacks interval certification, so the second counterexample remains numerical.
Technical verification
The paper reports exact arithmetic verification of the 240 E8 root projections and ancillary reproduction scripts. The separate Λ9 height computation uses cutoffs and ordinary floating point; the author explicitly says it lacks rigorous interval certification. Neither supports a documented complete check of both claims.
Independent checking
Exact enumeration in anc/cassels/verify_e8_roots.py
- Checker
- Gargava’s ancillary exact verification script
- Scope
- For all 240 E8 roots, rᵀHr=±2, supporting the rotated E8 packing calculation
- Basis
- Reported by the linked checker
Human understanding
The human author explains why the E8 rotation beats the product and how perturbing Λ9 changes height, while the exact model-origin search for the rotation and perturbation is not fully reconstructed as a human derivation.
Correction and publication history
The exact eight-dimensional packing answers the Cassels–Zong product question; the nine-dimensional computation challenges the height conjecture.
Read this published revision