{
  "id": "the-field-with-one-element",
  "title": "The field with one element",
  "language": "en",
  "lessons": [
    {
      "id": "counting-over-finite-fields-and-the-limit-q-1",
      "title": "Counting over finite fields and the limit q → 1",
      "source": "src/counting-over-finite-fields-and-the-limit-q-1.md",
      "source_sha256": "a9facb1fa030cceff6a9d4f75606ce58b4183ed80599b6d45664f3d35b52066c",
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      "reader_sha256": "2479d55e0415dec1ac2bf795a22f91a993cc85f42dae610c750a9ee1a6062437",
      "licence": "CC0-1.0",
      "authorship_and_checking": "Written by Claude Opus 5.5 (Anthropic), September 2026, revised October 2026. The September text was spot-checked by Claude Opus 5.5 in a separate session; the October revision links the AI Integrated Stacks Project citations and is self-checked by the writing AI. The October revision also corrects points found by GPT-6 Astra (OpenAI), Ultra, in a separate review session. Public domain (CC0)."
    },
    {
      "id": "weil-s-proof-for-curves-and-what-is-missing-over-the-integers",
      "title": "Weil's proof for curves and what is missing over the integers",
      "source": "src/weil-s-proof-for-curves-and-what-is-missing-over-the-integers.md",
      "source_sha256": "e02b3015e87b3fc5a954fab3e3ec42b4d96986f035b2b59eb1e4040aee7cd1e6",
      "reader": "weil-s-proof-for-curves-and-what-is-missing-over-the-integers.html",
      "licence": "CC0-1.0",
      "authorship_and_checking": "Written by Claude Opus 5.5 (Anthropic), September 2026, revised October 2026. The September text was spot-checked by Claude Opus 5.5 in a separate session; the October revision links the AI Integrated Stacks Project citations and is self-checked by the writing AI. The October revision also corrects points found by GPT-6 Astra (OpenAI), Ultra, in a separate review session, and adds the proof of the Hodge index theorem in Section 3.1. Public domain (CC0).",
      "reader_sha256": "fe9fe08db971c94289705e7a59fda57bc9a804a8d9a73956e9ae0f23d1992406"
    },
    {
      "id": "commutative-monoids-and-their-spectra",
      "title": "Commutative monoids and their spectra",
      "source": "src/commutative-monoids-and-their-spectra.md",
      "source_sha256": "c8136ddea8fbf71a99b0454ab0892650f7e4d6a7dedc82795ea1013ad406b82f",
      "reader": "commutative-monoids-and-their-spectra.html",
      "licence": "CC0-1.0",
      "authorship_and_checking": "Written by Claude Opus 5.5 (Anthropic), September 2026, revised October 2026. The September text was spot-checked by Claude Opus 5.5 in a separate session; the October revision links the AI Integrated Stacks Project citations and is self-checked by the writing AI. The October revision also corrects points found by GPT-6 Astra (OpenAI), Ultra, in a separate review session. Public domain (CC0).",
      "reader_sha256": "6306392e082508d401fbf46f85932e46244bae663b31d1dd66f1058bed3c2c19"
    },
    {
      "id": "monoid-schemes",
      "title": "Monoid schemes",
      "source": "src/monoid-schemes.md",
      "source_sha256": "3e72a5016f6da5abdfcabf9bc88d4cb93a4000cb7fdf90716204b26a28da07e1",
      "reader": "monoid-schemes.html",
      "licence": "CC0-1.0",
      "authorship_and_checking": "Written by Claude Opus 5.5 (Anthropic), September 2026, revised October 2026. The September text was spot-checked by Claude Opus 5.5 in a separate session; the October revision links the AI Integrated Stacks Project citations and is self-checked by the writing AI. The October revision also corrects points found by GPT-6 Astra (OpenAI), Ultra, in a separate review session. The proofs of Sumihiro's theorem (Section 6.4) and of (C1) in Facts 6.10, Section 6.6, and Example 8.9, were drafted by GPT-6 Astra (OpenAI) in ChatGPT web, Pro mode, and checked and adapted by Claude Opus 5.5, October 2026. Public domain (CC0).",
      "reader_sha256": "658807e5c9e765675f75f250f71012754a6db4c1662b34367d7877b64a7701b3"
    },
    {
      "id": "torified-varieties-and-the-limits-of-monoid-schemes",
      "title": "Torified varieties and the limits of monoid schemes",
      "source": "src/torified-varieties-and-the-limits-of-monoid-schemes.md",
      "source_sha256": "79decc087bd57cb3fcf2e35a1e816d8e328ed93127c92050a74b6146b0fbd181",
      "reader": "torified-varieties-and-the-limits-of-monoid-schemes.html",
      "reader_sha256": "11cb95d5bccb408a0c75182bc7d7ac62cb1f309e2d7a6b1102489a8fc7e86fde",
      "licence": "CC0-1.0",
      "authorship_and_checking": "Written by Claude Opus 5.5 (Anthropic), September 2026, revised October 2026. The September text was spot-checked by Claude Opus 5.5 in a separate session; the October revision links the AI Integrated Stacks Project citations and is self-checked by the writing AI. The October revision also corrects points found by GPT-6 Astra (OpenAI), Ultra, in a separate review session. Public domain (CC0)."
    },
    {
      "id": "varieties-over-the-field-with-one-element-after-soule-and-connes-consani",
      "title": "Varieties over the field with one element after Soulé and Connes–Consani",
      "source": "src/varieties-over-the-field-with-one-element-after-soule-and-connes-consani.md",
      "source_sha256": "1e55f6fa9af4e8eb4b1c4126bf5f5208e4dfdec0fadb3231e2c1f6663105f965",
      "reader": "varieties-over-the-field-with-one-element-after-soule-and-connes-consani.html",
      "reader_sha256": "086369dea85605c2997b2a1c00be49045003bbea6ae57e4d8098b79e92d0d499",
      "licence": "CC0-1.0",
      "authorship_and_checking": "Written by Claude Opus 5.5 (Anthropic), September 2026, revised October 2026. The September text was spot-checked by Claude Opus 5.5 in a separate session; the October revision links the AI Integrated Stacks Project citations and is self-checked by the writing AI. The October revision also corrects points found by GPT-6 Astra (OpenAI), Ultra, in a separate review session. Proposition 5.6, Example 5.7 and Theorem 5.8 were drafted by GPT-6 Astra (OpenAI) in ChatGPT web, Pro mode, and checked and adapted by Claude Opus 5.5, October 2026. Public domain (CC0)."
    },
    {
      "id": "schemes-relative-to-a-symmetric-monoidal-category",
      "title": "Schemes relative to a symmetric monoidal category",
      "source": "src/schemes-relative-to-a-symmetric-monoidal-category.md",
      "source_sha256": "e45b645d84f259f3910ada0fb97b6bee06ba471ce0e842f8e1e624a63e89c420",
      "reader": "schemes-relative-to-a-symmetric-monoidal-category.html",
      "licence": "CC0-1.0",
      "authorship_and_checking": "Written by Claude Opus 5.5 (Anthropic), September 2026, revised October 2026. The September text was spot-checked by Claude Opus 5.5 in a separate session; the October revision links the AI Integrated Stacks Project citations and is self-checked by the writing AI. Public domain (CC0).",
      "reader_sha256": "ae63336bff03c8cc27f9ad673159a05220c6fcaa6a79c7f8a9d92b1704ce9a43"
    },
    {
      "id": "lambda-rings-and-descent-to-the-field-with-one-element",
      "title": "Λ-rings and descent to the field with one element",
      "source": "src/lambda-rings-and-descent-to-the-field-with-one-element.md",
      "source_sha256": "e3927fecccc666e8203e9f7188bf5a031cf6af9f3c8600f595a0b12e888474d5",
      "reader": "lambda-rings-and-descent-to-the-field-with-one-element.html",
      "licence": "CC0-1.0",
      "authorship_and_checking": "Written by Claude Opus 5.5 (Anthropic), September 2026, revised October 2026. The September text was spot-checked by Claude Opus 5.5 in a separate session; the October revision links the AI Integrated Stacks Project citations and is self-checked by the writing AI. The October revision also corrects points found by GPT-6 Astra (OpenAI), Ultra, in a separate review session. Public domain (CC0).",
      "reader_sha256": "d8c7aab36fdb595d9b06438153c4dd37c55ada8b66bcf25135cc92610a5beaba"
    },
    {
      "id": "generalized-rings",
      "title": "Generalized rings",
      "source": "src/generalized-rings.md",
      "source_sha256": "43809fa60fa8a790895d0b613c6e8aefebb0b1d9143091b9d595da3ae04d1fd0",
      "reader": "generalized-rings.html",
      "licence": "CC0-1.0",
      "authorship_and_checking": "Written by Claude Opus 5.5 (Anthropic), September 2026, revised October 2026. The September text was spot-checked by Claude Opus 5.5 in a separate session; the October revision links the AI Integrated Stacks Project citations and is self-checked by the writing AI. Public domain (CC0).",
      "reader_sha256": "b958d843489196d01a85046ee1a896513f8ab1be1d81a2134b710e8a57a2bd8c"
    },
    {
      "id": "blueprints-and-blue-schemes",
      "title": "Blueprints and blue schemes",
      "source": "src/blueprints-and-blue-schemes.md",
      "source_sha256": "bc40051542b18cc321510c756fee00b94720bd81270d4826d6e0354561c44add",
      "reader": "blueprints-and-blue-schemes.html",
      "licence": "CC0-1.0",
      "authorship_and_checking": "Written by Claude Opus 5.5 (Anthropic), September 2026, revised October 2026. The September text was spot-checked by Claude Opus 5.5 in a separate session; the October revision links the AI Integrated Stacks Project citations and is self-checked by the writing AI. The October revision also corrects points found by GPT-6 Astra (OpenAI), Ultra, in a separate review session. Public domain (CC0).",
      "reader_sha256": "44e854c7d79d7b27d59ffc341e384f0284f3cc07031bc0140921cda11285fea1"
    },
    {
      "id": "characteristic-one-and-hyperrings",
      "title": "Characteristic one and hyperrings",
      "source": "src/characteristic-one-and-hyperrings.md",
      "source_sha256": "4eed1556f50fa6d116e812052ef83df24d0a11c2d5f81a2665819da10d290ea5",
      "reader": "characteristic-one-and-hyperrings.html",
      "licence": "CC0-1.0",
      "authorship_and_checking": "Written by Claude Opus 5.5 (Anthropic), September 2026, revised October 2026. The September text was spot-checked by Claude Opus 5.5 in a separate session; the October revision is self-checked by the writing AI and corrects a point found by GPT-6 Astra (OpenAI), Ultra, in a separate review session. The proofs in Sections 6.5 and 6.6 were drafted by GPT-6 Astra (OpenAI) in ChatGPT web, Pro mode, and checked and adapted by Claude Opus 5.5, October 2026. Public domain (CC0).",
      "reader_sha256": "06a315d66a3f6767e9b25aa85d696be23f42a24cfe5a10491343b57ad8bb9e2e"
    },
    {
      "id": "gamma-sets-and-algebras-over-the-sphere",
      "title": "Γ-sets and algebras over the sphere",
      "source": "src/gamma-sets-and-algebras-over-the-sphere.md",
      "source_sha256": "df25a6605c08d5040bcacd185936a2a51970c50376a76e6a17a3684f4a2d38ae",
      "reader": "gamma-sets-and-algebras-over-the-sphere.html",
      "licence": "CC0-1.0",
      "authorship_and_checking": "Written by Claude Opus 5.5 (Anthropic), September 2026, revised October 2026. The September text was spot-checked by Claude Opus 5.5 in a separate session; the October revision links the AI Integrated Stacks Project citations and is self-checked by the writing AI. The October revision also corrects points found by GPT-6 Astra (OpenAI), Ultra, in a separate review session. Public domain (CC0).",
      "reader_sha256": "3d892f5b7acb965e57612a4a00b475436b6d837be4f608d43130761161c9de64"
    },
    {
      "id": "the-arithmetic-site",
      "title": "The arithmetic site",
      "source": "src/the-arithmetic-site.md",
      "source_sha256": "ba8be043a396be43fb6cf88b7750266424795bc90c396bf183bfbc4c4162dff1",
      "reader": "the-arithmetic-site.html",
      "licence": "CC0-1.0",
      "authorship_and_checking": "Written by Claude Opus 5.5 (Anthropic), September 2026, revised October 2026. The September text was spot-checked by Claude Opus 5.5 in a separate session; the October revisions (AI Integrated Stacks Project citations; the general topos statements located in a later lesson) are self-checked by the writing AI. Public domain (CC0).",
      "reader_sha256": "1f1117debbe3119d32226dfe22c243cf3c4171514d3993b4da2970da8c501731"
    },
    {
      "id": "the-scaling-site",
      "title": "The scaling site",
      "source": "src/the-scaling-site.md",
      "source_sha256": "9dc2264887e0a55d412960afd386a9c428a9ea12bcec3deae63c2639736d92a8",
      "reader": "the-scaling-site.html",
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      "licence": "CC0-1.0",
      "authorship_and_checking": "Written by Claude Opus 5.5 (Anthropic), September 2026, revised October 2026. The September text was spot-checked by Claude Opus 5.5 in a separate session; the October revisions (AI Integrated Stacks Project citations; the points of a topos of sheaves, Lebesgue's covering theorem, three elementary facts and one implication of Weil's criterion proved) are self-checked by the writing AI. The October revision also corrects points found by GPT-6 Astra (OpenAI), Ultra, in a separate review session. Public domain (CC0)."
    },
    {
      "id": "the-projective-line-over-f-1-and-the-abc-conjecture",
      "title": "The projective line over F_1 and the ABC conjecture",
      "source": "src/the-projective-line-over-f-1-and-the-abc-conjecture.md",
      "source_sha256": "348c3eac098203824e38b8869228d93b5cfdb80bfdc79fadca6a5acfa2fbfc15",
      "reader": "the-projective-line-over-f-1-and-the-abc-conjecture.html",
      "licence": "CC0-1.0",
      "authorship_and_checking": "Written by Claude Opus 5.5 (Anthropic), September 2026, revised October 2026. The September text was spot-checked by Claude Opus 5.5 in a separate session; the October revision links the AI Integrated Stacks Project citations and is self-checked by the writing AI. The October revision also corrects points found by GPT-6 Astra (OpenAI), Ultra, in a separate review session. Public domain (CC0).",
      "reader_sha256": "6a867822eab908d2b1a75504926a385122605993b20ea98817d605487c4d1573"
    }
  ],
  "published": false,
  "reading_url": "https://kokunoyumeto.github.io/open-math-courses/courses/the-field-with-one-element/",
  "licence": "CC0-1.0",
  "availability": "private_preview",
  "download_scope": "All fifteen course lessons and editable sources; linked proofs in other courses and external references are not bundled."
}