{
  "id": "infinite-matroid-packing-and-covering",
  "title": "Packing and covering in infinite matroids",
  "description": "OpenAI's theorem (2026) that the infinite matroid packing/covering conjecture is false in ZFC; through density ideals, an ultrafilter with an ordinal rank, local bases defined by probes, and a self-dual uniform matroid placed along a double ray.",
  "units": [
    {
      "id": "infinite-matroids-and-the-conjecture",
      "title": "Infinite matroids and the packing/covering conjecture",
      "reader": "infinite-matroids-and-the-conjecture.html",
      "source": "src/infinite-matroids-and-the-conjecture.md",
      "source_sha256": "d11dc9fdac545c79e9582294a26b4243a532080ccaa50b4d484b2756ca528ddb",
      "status": "author_self_checked"
    },
    {
      "id": "ideals-ultrafilters-and-an-ordinal-rank",
      "title": "Ideals, ultrafilters and an ordinal rank",
      "reader": "ideals-ultrafilters-and-an-ordinal-rank.html",
      "source": "src/ideals-ultrafilters-and-an-ordinal-rank.md",
      "source_sha256": "94b3df24ece6007c3985c5fccdd7a24d1ffe334fe72b2c7d2fb2ffd7d19def6f",
      "status": "author_self_checked"
    },
    {
      "id": "probes-and-local-bases",
      "title": "Probes and local bases",
      "reader": "probes-and-local-bases.html",
      "source": "src/probes-and-local-bases.md",
      "source_sha256": "41af6a317c4f59742bff5df85f074e3fb09d8d5b4db3c10c634346e0dbc3046f",
      "status": "author_self_checked"
    },
    {
      "id": "assembling-a-self-dual-matroid",
      "title": "Assembling a self-dual matroid",
      "reader": "assembling-a-self-dual-matroid.html",
      "source": "src/assembling-a-self-dual-matroid.md",
      "source_sha256": "152ebee8b177c87bacf1333563ebc715987d1e980cab3322a7fe141c04e63f44",
      "status": "author_self_checked"
    },
    {
      "id": "no-independent-covering",
      "title": "No independent covering",
      "reader": "no-independent-covering.html",
      "source": "src/no-independent-covering.md",
      "source_sha256": "3da263d3869d11b5f4fe98805476adde2047e5c70edb2732c1961a8c2509382d",
      "status": "author_self_checked"
    }
  ],
  "prerequisites": [
    "core course Mathematical Logic, Set Theory, and Computability (C80): Button, Set Theory: An Open Introduction (Open Logic Project), Proposition 10.2, Theorem 11.4, Lemma 14.2, Corollary 14.10, Theorem 15.11, Proposition 15.12, Theorem 5.16, Theorem 16.6"
  ],
  "status": "author_self_checked",
  "proof_providers": "providers.json",
  "provenance": "provenance.json",
  "licence": "CC0-1.0",
  "contributions": [
    {
      "contributor": "GPT-6 Astra (OpenAI)",
      "date": "2026-10-10",
      "licence": "CC0-1.0",
      "scope": "Lesson 1 Lemma 2.2 through Corollary 2.8 and Proposition 4.3; lesson 5 intersection-consequence binding; lesson 4 Proposition 4.2(b) containment-direction correction.",
      "review": "Author self-check; no independent review or formal verification",
      "original_author": "Claude Opus 5.5 (Anthropic)"
    },
    {
      "contributor": "GPT-6 Astra (OpenAI)",
      "date": "2026-10-10",
      "licence": "CC0-1.0",
      "scope": "Lesson 1 Lemmas 4.4-4.5, Proposition 4.6 and Theorem 4.7; exact lesson 5 consequence.",
      "review": "Author self-check; not independent mathematical review or formal verification."
    }
  ]
}