Sharpness of the Cohn-Elkies linear programming bound in dimension two

Theorem 1.1 constructs a real radial Schwartz function on $\mathbb R^2$ with $\widehat f(0)=1$ , $f(0)=2/\sqrt3$ , $\widehat f\ge0$ and $f\le0$ outside the unit disk, so the Cohn-Elkies bound equals the triangular packing density $\pi/(2\sqrt3)$ . Its zeros beyond radius one lie at integral squared radii, which recovers uniqueness of the triangular packing among periodic equality cases. It does not satisfy the stronger zero-set condition of Cohn-Elkies Conjecture 8.1, and it gives no new packing bound (the planar density was already known).

Exact FrontierDelta

Prior state unknown→proved

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Occurred: Sep 23, 2026

Delta type: SOURCE CLAIM

Assumptions: VibeMathed verification: lean-checked. Publication: announcement. AI contribution: ai-discovered. VibeMathed editorial classifications, scores, notes, relations, and dataset structure are CC BY 4.0. Source statements and linked content retain their own rights.

Canonical aliases: Sharpness of the Cohn-Elkies linear programming bound in dimension two · Cohn-Elkies sharpness in the plane

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VibeMathed record: Sharpness of the Cohn-Elkies linear programming bound in dimension two

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No independent mathematician has checked this yet. Checked here: Theorem 1.1 of the manuscript was read against Cohn-Elkies Conjecture 7.3 in dimension two; it gives a real radial Schwartz $f$ with $\widehat f(0)=1$ , $f(0)=2/\sqrt3$ , $\widehat f\ge0$ and $f(x)\le0$ for $|x|\ge1$ , which rescales to the requested witness. The manuscript notes that its witness does not meet the extra zero-set requirement of Cohn-Elkies Conjecture 8.1. Lean: lean/ComparatorChallenges/PlanarPacking.json exists with solution_module OAI.Analysis.PlanarPacking.Main, whose file exists at the pinned commit; this challenge is not in lean/formalization.yaml. The statement PlanarPacking.lean was read here: there exists a Schwartz map on the Euclidean plane, radial, with Fourier transform (kernel $e^{-2\pi i\langle x,\xi\rangle}$ ) equal to 1 at 0, value $2/\sqrt3$ at 0, Fourier transform real and nonnegative everywhere, and $f(x)\le0$ for $\|x\|\ge1$ . This states the headline. Not rebuilt here. Permitted axioms: propext, Quot.sound, Classical.choice.