{
  "id": "dimension-reduction-in-lp",
  "title": "Dimension reduction in L_p",
  "description": "OpenAI's theorem (2026) that n points of L_p, 1 < p < infinity, p != 2, embed into l_p with any fixed distortion above 1 using n^o(1) coordinates, while exact embeddings need about n^2/16; through weighted moments, the localization of electrical flows, and Poisson sums.",
  "units": [
    {
      "id": "finite-subsets-of-lp",
      "title": "Finite subsets of L_p",
      "reader": "finite-subsets-of-lp.html",
      "source": "src/finite-subsets-of-lp.md",
      "source_sha256": "a9d9a0e27e229fbc942ecf9612a3d54f316cb8dfe808235574e13ccb063eb929",
      "status": "author_self_checked"
    },
    {
      "id": "weighted-moments-and-sampling",
      "title": "Weighted moments and sampling",
      "reader": "weighted-moments-and-sampling.html",
      "source": "src/weighted-moments-and-sampling.md",
      "source_sha256": "92f69c6518fcade92c9649ca76af3a1ae01f861644cafd0b1200df0061ac2f7b",
      "status": "author_self_checked"
    },
    {
      "id": "electrical-flows-and-a-weighted-projection",
      "title": "Electrical flows and a weighted projection",
      "reader": "electrical-flows-and-a-weighted-projection.html",
      "source": "src/electrical-flows-and-a-weighted-projection.md",
      "source_sha256": "0421aecd4f7e47d4befa19ab22173d6344dff5092f591d0c34d7313e295fda71",
      "status": "author_self_checked"
    },
    {
      "id": "signed-poisson-sums-below-two",
      "title": "Signed Poisson sums and exponents below two",
      "reader": "signed-poisson-sums-below-two.html",
      "source": "src/signed-poisson-sums-below-two.md",
      "source_sha256": "5c9e807f86c2481dc69fe99ea2adaa29be4cd2a819d4b6a1a7ae58ab03950a93",
      "status": "author_self_checked"
    },
    {
      "id": "truncation-ramps-above-two",
      "title": "Truncation ramps and exponents above two",
      "reader": "truncation-ramps-above-two.html",
      "source": "src/truncation-ramps-above-two.md",
      "source_sha256": "aa38f2387db7e36d042abb67f90c30a10319e944e5f57fb26ab6b277822f5f20",
      "status": "author_self_checked"
    }
  ],
  "prerequisites": [],
  "status": "author_self_checked",
  "proof_providers": "providers.json",
  "provenance": "provenance.json",
  "licence": "CC0-1.0"
}