{
  "id": "three-marginal-coulomb-transport",
  "title": "Three-marginal Coulomb transport",
  "description": "OpenAI's counterexample (2026) to the Monge ansatz: for the Coulomb cost of three particles in space, and every Riesz cost in dimension at least two, a smooth density for which the optimal coupling exists but is given by no pair of maps, although maps come arbitrarily close. A fourth lesson restates the result with infinitesimals: an internal pair of maps at an infinitely fine resolution exceeds the minimum by a positive infinitesimal, and its standard part is the optimal coupling.",
  "units": [
    {
      "id": "couplings-and-supporting-potentials",
      "title": "Couplings and supporting potentials",
      "reader": "couplings-and-supporting-potentials.html",
      "source": "src/couplings-and-supporting-potentials.md",
      "source_sha256": "b211ff8cd7c795e4ca9c105ceaebf81f50523ec82c143c82a9c28733930d3754",
      "status": "author_self_checked"
    },
    {
      "id": "two-branches-with-a-shared-potential",
      "title": "Two branches with a shared potential",
      "reader": "two-branches-with-a-shared-potential.html",
      "source": "src/two-branches-with-a-shared-potential.md",
      "source_sha256": "9dd764aafe46d739f570086278147a6720b7e7b043c5e62f9ae9832fbe4a2380",
      "status": "author_self_checked"
    },
    {
      "id": "monge-plans-approach-the-minimum",
      "title": "Monge plans approach the minimum",
      "reader": "monge-plans-approach-the-minimum.html",
      "source": "src/monge-plans-approach-the-minimum.md",
      "source_sha256": "6039f967da869325ff8fe8ddeb4e540dc2819d067263258bd19fb69824e865dc",
      "status": "author_self_checked"
    },
    {
      "id": "monge-plans-at-infinitesimal-resolution",
      "title": "Monge plans at infinitesimal resolution",
      "reader": "monge-plans-at-infinitesimal-resolution.html",
      "source": "src/monge-plans-at-infinitesimal-resolution.md",
      "source_sha256": "fe1f72d91da6f872284cca9445c3daf1418c171c0dc2e962f9926d658e3d87cc",
      "status": "author_self_checked"
    }
  ],
  "prerequisites": [],
  "status": "author_self_checked",
  "proof_providers": "providers.json",
  "provenance": "provenance.json",
  "licence": "CC0-1.0"
}