| # Triangular-lattice optimality, long-range Riesz and Coulomb energies, and spherical logarithmic energy |
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| The following describes the scope of the Lean formalization related to the following accompanying paper(s): |
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| - [An atomic certificate for triangular-lattice universal optimality](../../preprints/An-atomic-certificate-for-triangular-lattice-universal-optimality-September-26-2026/paper.pdf) |
| - [A sharp Fourier certificate for planar circle packing](../../preprints/A-sharp-Fourier-certificate-for-planar-circle-packing-September-23-2026/paper.pdf) |
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| ## Scope |
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| The formalization proves that the density-one triangular lattice minimizes lower energy per particle among all locally finite planar configurations of centered-disk density one, for every nonnegative completely monotone function of squared distance. Infinite energies are allowed in the comparison. |
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| It also constructs sharp radial Schwartz minorants for every Gaussian potential: each minorant lies below the Gaussian, has nonnegative real Fourier transform, agrees with the Gaussian at nonzero triangular-lattice points, and vanishes on nonzero dual-lattice points after Fourier transformation. For the stated parameter range, the formalization includes the explicit atomic interpolation construction. |
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| The planar Cohn–Elkies sharpness conjecture asks whether the two-point Fourier method attains the optimal circle-packing density. The formalization constructs a radial Schwartz function $f$ on $\mathbb R^2$ with $\widehat f(0)=1$, $f(0)=2/\sqrt3$, $\widehat f$ real and nonnegative everywhere, and $f(x)\le0$ whenever $\|x\|\ge1$. These are the sharp Fourier certificate conditions yielding density $\pi/(2\sqrt3)$. The separate uniqueness statement for periodic equality cases is outside this selected theorem. |
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| ## Comparator links |
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| | Result | Comparator statement | |
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| | Sharp Gaussian minorants from an atomic certificate | [AtomicGaussian.lean](../ComparatorChallenges/AtomicGaussian.lean) | |
| | Universal energy minimality of the triangular lattice | [TriangularEnergy.lean](../ComparatorChallenges/TriangularEnergy.lean) | |
| | Gaussian Fourier minorants with their construction | [TriangularGaussian.lean](../ComparatorChallenges/TriangularGaussian.lean) | |
| | Sharp Fourier certificate for planar circle packing | [PlanarPacking.lean](../ComparatorChallenges/PlanarPacking.lean) | |
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