What was already known
The Cohn–Triantafillou modular-form approach had certified dual lower bounds through dimension 32; the best known dimension-36 packing had center density 2^18/3^10.
AI involvement
Jumagulov set the target and explains the certificate. Claude Fable 5 and Opus 4.8 helped construct the exact certificate and verification tools; GPT-5.6 Codex supplied a separate cross-check.
Claude helped construct the retained dual certificate itself, while the paper and problem framing remain human-led; Codex’s separate role was checking.
Importance
The certificate rules out this LP method certifying optimality of that known packing.
Limitations
It does not determine the densest dimension-36 packing or prove strict nonsharpness of the LP bound.
Technical verification
The paper reports exact rational feasibility checks through coefficient 800 and certified tail positivity using outward interval bounds, with an independent Arb recheck. The author reran the proof pipeline; the computations were not reproduced here.
Formal checking
Exact rational coefficient and vertex checks plus certified interval tail constants and Arb re-verification
- Checker
- Jumagulov’s exact certificate pipeline, with separate Arb recheck
- Scope
- Theorem 1.1 dual-feasible dimension-36 point and B>146.1036 bound
- Basis
- Reported by the linked checker
Checking a formal statement does not establish that it matches the original problem.
Human understanding
Jumagulov explains the dual-certificate construction, constraint-generation obstacle and lift-aware tail bound needed to close eventual positivity.
Correction and publication history
The certificate rules out this LP method certifying optimality of that known packing.
Read this published revision