{
  "id": "circulant-hadamard-matrices",
  "title": "Circulant Hadamard matrices",
  "description": "OpenAI's proof (2026) of the circulant Hadamard conjecture, usually attributed to Ryser: real circulant Hadamard matrices exist only in orders 1 and 4. The course works in the group ring of the cyclic group: it proves the local structure of cyclotomic integers near a prime and Turyn's restriction to orders 4u² with u odd, shows that alternating products of character values are roots of unity of odd order, and derives a contradiction modulo a prime above 2; even-length Barker sequences follow.",
  "units": [
    {
      "id": "circulant-hadamard-matrices-and-group-rings",
      "title": "Circulant Hadamard matrices and group rings",
      "reader": "circulant-hadamard-matrices-and-group-rings.html",
      "source": "src/circulant-hadamard-matrices-and-group-rings.md",
      "source_sha256": "21be2bbc046612cb53ea67680891d03d40aac33a4740f707e5a9ccef6196d544",
      "status": "author_self_checked"
    },
    {
      "id": "local-rings-of-cyclotomic-integers",
      "title": "Local rings of cyclotomic integers",
      "reader": "local-rings-of-cyclotomic-integers.html",
      "source": "src/local-rings-of-cyclotomic-integers.md",
      "source_sha256": "cfcb4e7fbfeb0f7e07d4c015fbafa738294d560fdbc2833dc4800b919e40a02c",
      "status": "author_self_checked"
    },
    {
      "id": "alternating-character-products",
      "title": "Alternating character products",
      "reader": "alternating-character-products.html",
      "source": "src/alternating-character-products.md",
      "source_sha256": "5fab9f5e4742124eb4f27a1057c6db5b374600677fa19b5bc5b3de0097519133",
      "status": "author_self_checked"
    },
    {
      "id": "the-circulant-hadamard-conjecture",
      "title": "The circulant Hadamard conjecture",
      "reader": "the-circulant-hadamard-conjecture.html",
      "source": "src/the-circulant-hadamard-conjecture.md",
      "source_sha256": "3a950cc1eabe7c78c5aee832e938d7b0eb2989fd336128a1decdf274e5b9057b",
      "status": "author_self_checked"
    }
  ],
  "prerequisites": [
    "core course Abstract Algebra II (rings, ideals, quotients, localization, Chinese remainder theorem)",
    "core course Linear Algebra (diagonalization of commuting matrices of finite order)"
  ],
  "status": "author_self_checked",
  "proof_providers": "providers.json",
  "provenance": "provenance.json",
  "licence": "CC0-1.0"
}