{
  "id": "metric-markov-cotype-of-l1",
  "title": "Metric Markov cotype of ℓ₁",
  "description": "OpenAI's answer (2026) to a question of Mendel and Naor: the space ℓ₁ has metric Markov cotype two. The course compares geometric and uniform averages of time along stationary Markov chains, writes finite subsets of ℓ₁ through weighted cuts, proves a flatness inequality for the cubic 3r²−2r³, and builds the required points from a killed random walk; they turn out to be expected medians of three stopped-walk endpoints.",
  "units": [
    {
      "id": "markov-chains-and-metric-markov-cotype",
      "title": "Markov chains and metric Markov cotype",
      "reader": "markov-chains-and-metric-markov-cotype.html",
      "source": "src/markov-chains-and-metric-markov-cotype.md",
      "source_sha256": "496480c9cf67f9f300e0b1f660ef9ca885532799aaa765c08cbc9e9356cede75",
      "status": "author_self_checked"
    },
    {
      "id": "cuts-and-a-flat-cubic",
      "title": "Cuts and a flat cubic",
      "reader": "cuts-and-a-flat-cubic.html",
      "source": "src/cuts-and-a-flat-cubic.md",
      "source_sha256": "34dd0e3027b8a35b6e165d44ba39cb1daf65ca415ffcd9bbff119cff750b3796",
      "status": "author_self_checked"
    },
    {
      "id": "a-stopped-walk-and-the-cotype-of-l1",
      "title": "A stopped walk and the cotype of ℓ₁",
      "reader": "a-stopped-walk-and-the-cotype-of-l1.html",
      "source": "src/a-stopped-walk-and-the-cotype-of-l1.md",
      "source_sha256": "eaa4c1ac47ff8b200ddf742e0c2bd53e87ec31db66159b2517e5b4c1a7bcf279",
      "status": "author_self_checked"
    }
  ],
  "prerequisites": [],
  "status": "author_self_checked",
  "proof_providers": "providers.json",
  "provenance": "provenance.json",
  "licence": "CC0-1.0"
}