Exact Cohn–Elkies certificate for planar circle packing

• Slug: cohn-elkies
• List: P27
• Solver: SuperGrok CLI grok-4.6 --reasoning-effort xhigh
• Status: open
• Area: Sphere packing / Fourier analysis
• Sources: Green 100 #42; Cohn–Elkies Annals 2003
• Started: 2026-08-17

Statement

It is unknown whether the Cohn–Elkies linear-programming scheme itself can attain the sharp hexagonal circle-packing density in dimension 2: equivalently, whether an exact admissible auxiliary function with the required optimal ratio exists.

Tonight

An exact admissible function (algebraic coefficients, independently checkable Fourier sign conditions) that meets or strictly improves a published Cohn–Elkies ratio, or a documented obstruction. Numerical LP output without exact certificates is an incomplete search. Fetch Green #42 and Cohn–Elkies before searching.

Outcome (2026-08-17)

Exact Laguerre–Gaussian certificate in compute/certs/ce_d2_m5.json, replay compute/verify.py.

R = 3627599/500000 = 7.255198 gives center density ≤ R/(8π) ≈ 0.28867515620, ratio 1.00000007485 against hexagonal √3/6. Meets Cohn–Elkies 2003 Table 3 (0.28868) and is strictly below the printed Table 4 value 2πr² = 7.25520. Not a magic function; Green #42 remains open.