{
  "id": "universal-optimality-of-the-triangular-lattice",
  "title": "Universal optimality of the triangular lattice",
  "description": "A complete proof that the triangular lattice is universally optimal in the plane, following OpenAI's proof (2026): among all point configurations of density one it has the least energy for every completely monotone interaction of the squared distance. The linear programming bound for configurations of density one; completely monotone interactions as positive mixtures of exponentials; the reduction to one normalized statement about functions of one variable; the construction of these functions from a sine product, cardinal interpolation, Gaussian damping and the Fourier transform; the solution of the interpolation system; and the proof of the two sign conditions. Three steps are finite sets of inequalities between explicitly defined numbers, established by computations with rigorous error bounds; the programs are printed in the lessons. A last lesson derives that the triangular lattice has the least periodic energy for the interactions |x|^-s, 0 < s < 2, and -log|x| among all periodic configurations.",
  "units": [
    {
      "id": "energy-density-and-the-linear-programming-bound",
      "title": "Energy, density and the linear programming bound",
      "reader": "energy-density-and-the-linear-programming-bound.html",
      "source": "src/energy-density-and-the-linear-programming-bound.md",
      "source_sha256": "2c5c4e02feca0569b563806a99070fd7a844302fa56f0e1ac5b8453f69a8a1d2",
      "status": "author_self_checked"
    },
    {
      "id": "completely-monotone-interactions",
      "title": "Completely monotone interactions",
      "reader": "completely-monotone-interactions.html",
      "source": "src/completely-monotone-interactions.md",
      "source_sha256": "44a56f7439f691f2f714ecae150d654d4d117479c44ad18a755ea3f27d587532",
      "status": "author_self_checked"
    },
    {
      "id": "shells-and-gaussian-duality",
      "title": "Shells and Gaussian duality",
      "reader": "shells-and-gaussian-duality.html",
      "source": "src/shells-and-gaussian-duality.md",
      "source_sha256": "fe4777d2375453e9ec988cc36498e3e60beaba39e9e7e23b69b34e63a202f567",
      "status": "author_self_checked"
    },
    {
      "id": "the-sine-product-and-cardinal-functions",
      "title": "The sine product and cardinal functions",
      "reader": "the-sine-product-and-cardinal-functions.html",
      "source": "src/the-sine-product-and-cardinal-functions.md",
      "source_sha256": "dae9e24089c7b61b313ae98aac32daf005e0a8e513dc3d41ab360636dc9d6413",
      "status": "author_self_checked"
    },
    {
      "id": "gaussian-damping-and-fourier-pairs",
      "title": "Gaussian damping and Fourier pairs",
      "reader": "gaussian-damping-and-fourier-pairs.html",
      "source": "src/gaussian-damping-and-fourier-pairs.md",
      "source_sha256": "e0b5df7630eda81b7c465dd66cf2fff2b2348d977108bfab86a74fab9fe65cde",
      "status": "author_self_checked"
    },
    {
      "id": "the-transformed-function-at-distant-nodes",
      "title": "The transformed function at distant nodes",
      "reader": "the-transformed-function-at-distant-nodes.html",
      "source": "src/the-transformed-function-at-distant-nodes.md",
      "source_sha256": "29bfa01db039a70eed5ab3af19a9936a25afbc6b939c083ca1b325bc93088f11",
      "status": "author_self_checked"
    },
    {
      "id": "a-positive-quadrature-rule",
      "title": "A positive quadrature rule",
      "reader": "a-positive-quadrature-rule.html",
      "source": "src/a-positive-quadrature-rule.md",
      "source_sha256": "fea9cdd2b4b6f8741b07b7129d7a18dc9c5078d979f5d06040d8be0bf3f82f94",
      "status": "author_self_checked"
    },
    {
      "id": "the-reference-lists",
      "title": "The reference lists",
      "reader": "the-reference-lists.html",
      "source": "src/the-reference-lists.md",
      "source_sha256": "1421c58759fc5720e48eb0200c484697678256a38e85e4ffd8430c937ea5ccca",
      "status": "author_self_checked"
    },
    {
      "id": "exact-interpolation",
      "title": "Exact interpolation",
      "reader": "exact-interpolation.html",
      "source": "src/exact-interpolation.md",
      "source_sha256": "3d3d7d5cb4fc31cbff90810bd3e3e0b89f5d9f91b85b24a8a254e9ccf3c67fe4",
      "status": "author_self_checked"
    },
    {
      "id": "quotients-on-the-half-gaps",
      "title": "Quotients on the half-gaps",
      "reader": "quotients-on-the-half-gaps.html",
      "source": "src/quotients-on-the-half-gaps.md",
      "source_sha256": "2ae6ac24a789a1d78041d98c7e32b481faafa3c31d531bc58daa3a00ed0930c0",
      "status": "author_self_checked"
    },
    {
      "id": "positivity-on-the-half-gaps",
      "title": "Positivity on the half-gaps",
      "reader": "positivity-on-the-half-gaps.html",
      "source": "src/positivity-on-the-half-gaps.md",
      "source_sha256": "aaf7b88e4e2ecd106b7339781ec89d21389a50d816e0d718d34205a222ab7694",
      "status": "author_self_checked"
    },
    {
      "id": "the-tail-and-the-theorem",
      "title": "The tail and the theorem",
      "reader": "the-tail-and-the-theorem.html",
      "source": "src/the-tail-and-the-theorem.md",
      "source_sha256": "e66b874094604db365a165df18c1be01789865c98eb7be23efc919c9550a82bb",
      "status": "author_self_checked"
    },
    {
      "id": "periodic-riesz-and-logarithmic-energies",
      "title": "Periodic Riesz and logarithmic energies",
      "reader": "periodic-riesz-and-logarithmic-energies.html",
      "source": "src/periodic-riesz-and-logarithmic-energies.md",
      "source_sha256": "15d34603b409619d93c0f6bc5643a2120eec29c7238480c9513c08e9bbe3bc09",
      "status": "author_self_checked"
    }
  ],
  "prerequisites": [
    "core course Measure and Integration (D10): Lebesgue integration in the plane, Fubini's theorem, dominated convergence",
    "course Harmonic analysis on locally compact abelian groups: Fourier inversion for integrable functions with integrable transform, self-duality of Lebesgue measure, the Poisson summation formula on a Euclidean lattice",
    "core course Real Analysis II (C20): the Weierstrass approximation theorem",
    "course Harmonic analysis on locally compact abelian groups, prerequisite lesson Finite Radon representation on locally compact spaces: positive functionals on C(K) are integrals",
    "course Harmonic analysis on locally compact abelian groups, prerequisite lesson Real-variable calculations for the Fourier examples: the Fourier transform of the Gaussian in one variable",
    "core course Complex Analysis (C50): power series, Taylor expansion of entire functions, Cauchy's estimate for Taylor coefficients",
    "the lessons prove every other fact they use"
  ],
  "status": "author_self_checked",
  "proof_providers": "providers.json",
  "provenance": "provenance.json",
  "licence": "CC0-1.0"
}