Research log — Exact Cohn–Elkies certificate for planar packing

2026-08-17

• Ben Green, 100 Open Problems, Problem 42 (Dec 2025 update, fetched tonight). Statement: can the Cohn–Elkies scheme prove the optimal circle-packing bound? Comments: sharpness conjectured in dimensions 1, 2, 8, 24; d=1 elementary; d=8 Viazovska; d=24 Cohn–Kumar–Miller–Radchenko–Viazovska. The d=2 case is written as a radial admissible f : R² → R with f ≤ 0 for |x| ≥ 2, ˆf ≥ 0, ˆf(0) > 0, and f(0)/ˆf(0) = √3/6. Related discussion: [274] = Sardari.
• Cohn and Elkies, New upper bounds on sphere packings I, Ann. of Math. 157 (2003), 689–714, arXiv:math/0110009. Fetched tonight (arXiv v3 and the Annals PDF). Theorem 3.1 / 3.2 is the LP scheme. §7 is the Laguerre–Gaussian method with exact rational arithmetic and Sturm. Conjecture 7.3: magic functions exist in dimensions 2, 8, 24. Table 3, n=2: packing and bound both printed 0.28868. Table 4, n=2: 2πr² = 7.25520 and five forced double roots 21.77, 29.02, 50.79, 65.34, 90.19. Coefficients are not printed.
• Cohn, New upper bounds on sphere packings II, Geom. Topol. 6 (2002), 329–353. Theta-series derivation of Theorem 3.1; no extra d=2 function.
• Sardari, Higher Fourier interpolation on the plane, arXiv:2102.08753. Green’s [274]. Interpolation formulas on A2-type nodes; proves CKMRV Conjecture 7.5. Does not produce a CE auxiliary function.
• Viazovska, The sphere packing problem in dimension 8, Ann. of Math. 185 (2017). Magic function in d=8.
• Cohn, Kumar, Miller, Radchenko, Viazovska, The sphere packing problem in dimension 24, Ann. of Math. 185 (2017). Magic function in d=24.
• Cohn, From sphere packing to Fourier interpolation, Bull. Amer. Math. Soc. 61 (2024), 3–22. Survey. d=2 remains the open magic-function case among {1,2,8,24}.
• Mo, Wen, Xia, A new linear programming method in sphere packing, arXiv:2410.04800v2. Periodic trigonometric polynomials on R²/2A2. Their g₂ is not an admissible function on R². They leave the CE-SA function open.
• Li, Dual linear programming bounds for sphere packing, arXiv:2206.09876. Dual bounds show the CE LP is not sharp in dimensions 3–7 and several higher dimensions. d=2 is not on the non-sharpness list.
• Jumagulov, A dual linear programming bound for sphere packing in dimension 36, arXiv:2607.11319 (Jul 2026). Still treats the planar magic function as unknown; dual work is about high-dimensional lower bounds on δ_LP.
• Salmon, Linear programming bounds for fibered sphere packings, arXiv:2607.25254. Quote: “no sharp planar magic function is known.” Conjecture 17 is exactly Green #42.
• Gonçalves, Oliveira e Silva, Steinerberger / related sign-uncertainty papers. Record the CE conjecture A_LP(2) = (4/3)^{1/4}, equivalent to 2πr² = 4π/√3 in Theorem 3.2 normalisation.
• Henry Cohn’s sphere-packing page, https://cohn.mit.edu/sphere-packing/, fetched tonight: no downloadable d=2 magic function or updated exact primal certificate.

No source fetched tonight prints an exact admissible d=2 polynomial (with coefficients) that we failed to meet. The 2003 paper is the published primal record we compare against.

Normalisations used

• Fourier transform: ˆf(t) = ∫ f(x) e^{2πi ⟨x,t⟩} dx (Cohn–Elkies).
• t = 2π |x|². Radial eigenfunctions g_k(x) = L_k(t) exp(−π |x|²), ˆg_k = (−1)^k g_k.
• Theorem 3.2: if f(0)=ˆf(0)>0, f ≤ 0 for |x| ≥ r, ˆf ≥ 0, then center density δ ≤ (r/2)^n. For n=2 this is δ ≤ R/(8π) with R = 2π r².
• Green’s |x| ≥ 2 form is the same bound after a factor-of-two rescaling: δ ≤ f(0)/ˆf(0).