[
  {
    "result_id": "SH-DC.sheafification-stalks",
    "name": "Sheafification, algebraic structures and the stalk test",
    "statement": "On any topological space, every presheaf of sets has germ-family sheafification preserving stalks and locally represented sections, left adjoint to inclusion of sheaves. Separated presheaves embed sectionwise. Ring and left-module structures are preserved, including associative unital rings. Bijective stalk maps of sheaves are isomorphisms.",
    "conditions": "Arbitrary topological space; empty opens and the zero ring allowed. Elementary sheaf condition, germs and finite algebraic identities only; no abelian category of sheaves assumed.",
    "writing_ai_selfcheck": true,
    "independent_review": false,
    "formalization": "not_formalized",
    "prerequisite_envelope_ids": [],
    "dependency_precision": "Only elementary definitions and germ equivalence; all sheaf-theoretic proof steps supplied locally.",
    "programme_dependency_loci": [],
    "proof": {
      "source": "src/sheaves-of-modules-on-a-ringed-space.md",
      "sha256": "22fcf2e4e2e126fe922ebd68fd81f3dedc581ed2b70cbb60a49250e21da7babe",
      "locus": "Lemma 1.3, full proof",
      "reader": "sheaves-of-modules-on-a-ringed-space.html#sheafification-and-the-stalk-test"
    },
    "licence": "CC0-1.0",
    "proof_status": "complete; writing-AI self-checked"
  },
  {
    "result_id": "SH-DC.modules-abelian",
    "name": "For any topological space X and sheaf of associative unital rings O, left O-module sheaves form an abelian category; stalks are exact and detect exactness.",
    "statement": "For any topological space X and sheaf of associative unital rings O, left O-module sheaves form an abelian category; stalks are exact and detect exactness.",
    "conditions": "Arbitrary topological space and sheaf of unital, possibly noncommutative rings. No Hausdorff, Noetherian or quasi-compact-space hypothesis. The direct-sum sections theorem alone assumes its specified open is quasi-compact.",
    "writing_ai_selfcheck": true,
    "independent_review": false,
    "formalization": "not_formalized",
    "prerequisite_envelope_ids": [
      "SH-DC.sheafification-stalks"
    ],
    "dependency_precision": "Conservative earlier-provider envelope; exact textual uses appear in programme_dependency_loci. Same-lesson preliminary lemmas are inside the full proof locus.",
    "programme_dependency_loci": [
      {
        "provider": "SH-DC.sheafification-stalks",
        "locus": "Lemma 1.3, complete statement and proof in the same lesson",
        "correspondence": "exact sheafification, stalk, separatedness and associative-ring module scope",
        "candidate_path": "course/src/sheaves-of-modules-on-a-ringed-space.md",
        "sha256": "22FCF2E4E2E126FE922EBD68FD81F3DEDC581ED2B70CBB60A49250E21DA7BABE",
        "result_id": "SH-DC.sheafification-stalks",
        "anchor": "sheafification-and-the-stalk-test"
      }
    ],
    "proof": {
      "source": "src/sheaves-of-modules-on-a-ringed-space.md",
      "sha256": "22fcf2e4e2e126fe922ebd68fd81f3dedc581ed2b70cbb60a49250e21da7babe",
      "locus": "Theorem 2.1",
      "reader": "sheaves-of-modules-on-a-ringed-space.html#2-the-abelian-category-of-module-sheaves"
    },
    "licence": "CC0-1.0",
    "proof_status": "complete; writing-AI self-checked"
  },
  {
    "result_id": "SH-DC.filtered-colimits-sums",
    "name": "Module sheaves admit all small limits/colimits; filtered colimits are exact; sections over a quasi-compact open commute with arbitrary direct sums.",
    "statement": "Module sheaves admit all small limits/colimits; filtered colimits are exact; sections over a quasi-compact open commute with arbitrary direct sums.",
    "conditions": "Arbitrary topological space and sheaf of unital, possibly noncommutative rings. No Hausdorff, Noetherian or quasi-compact-space hypothesis. The direct-sum sections theorem alone assumes its specified open is quasi-compact.",
    "writing_ai_selfcheck": true,
    "independent_review": false,
    "formalization": "not_formalized",
    "prerequisite_envelope_ids": [
      "SH-DC.sheafification-stalks"
    ],
    "dependency_precision": "Conservative earlier-provider envelope; exact textual uses appear in programme_dependency_loci. Same-lesson preliminary lemmas are inside the full proof locus.",
    "programme_dependency_loci": [
      {
        "provider": "SH-DC.sheafification-stalks",
        "locus": "Lemma 1.3, complete statement and proof in the same lesson",
        "correspondence": "exact sheafification, stalk, separatedness and associative-ring module scope",
        "candidate_path": "course/src/sheaves-of-modules-on-a-ringed-space.md",
        "sha256": "22FCF2E4E2E126FE922EBD68FD81F3DEDC581ED2B70CBB60A49250E21DA7BABE",
        "result_id": "SH-DC.sheafification-stalks",
        "anchor": "sheafification-and-the-stalk-test"
      }
    ],
    "proof": {
      "source": "src/sheaves-of-modules-on-a-ringed-space.md",
      "sha256": "22fcf2e4e2e126fe922ebd68fd81f3dedc581ed2b70cbb60a49250e21da7babe",
      "locus": "Lemma 1.1 and Theorem 3.1",
      "reader": "sheaves-of-modules-on-a-ringed-space.html#3-limits-colimits-stalks-and-sections-of-sums"
    },
    "licence": "CC0-1.0",
    "proof_status": "complete; writing-AI self-checked"
  },
  {
    "result_id": "SH-DC.pullback-adjunction",
    "name": "For any morphism of spaces with sheaves of unital rings, f* = O_X tensor_{f^{-1}O_Y} f^{-1} is left adjoint to f_*; f^* is right exact, f_* left exact, and inverse image is exact.",
    "statement": "For any morphism of spaces with sheaves of unital rings, f* = O_X tensor_{f^{-1}O_Y} f^{-1} is left adjoint to f_*; f^* is right exact, f_* left exact, and inverse image is exact.",
    "conditions": "Arbitrary topological space and sheaf of unital, possibly noncommutative rings. No Hausdorff, Noetherian or quasi-compact-space hypothesis. The direct-sum sections theorem alone assumes its specified open is quasi-compact.",
    "writing_ai_selfcheck": true,
    "independent_review": false,
    "formalization": "not_formalized",
    "prerequisite_envelope_ids": [
      "SH-DC.sheafification-stalks"
    ],
    "dependency_precision": "Conservative earlier-provider envelope; exact textual uses appear in programme_dependency_loci. Same-lesson preliminary lemmas are inside the full proof locus.",
    "programme_dependency_loci": [
      {
        "provider": "SH-DC.sheafification-stalks",
        "locus": "Lemma 1.3, complete statement and proof in the same lesson",
        "correspondence": "exact sheafification, stalk, separatedness and associative-ring module scope",
        "candidate_path": "course/src/sheaves-of-modules-on-a-ringed-space.md",
        "sha256": "22FCF2E4E2E126FE922EBD68FD81F3DEDC581ED2B70CBB60A49250E21DA7BABE",
        "result_id": "SH-DC.sheafification-stalks",
        "anchor": "sheafification-and-the-stalk-test"
      }
    ],
    "proof": {
      "source": "src/sheaves-of-modules-on-a-ringed-space.md",
      "sha256": "22fcf2e4e2e126fe922ebd68fd81f3dedc581ed2b70cbb60a49250e21da7babe",
      "locus": "Theorems 4.1 and 4.4; Lemma 4.3",
      "reader": "sheaves-of-modules-on-a-ringed-space.html#module-pullback-and-extension-of-scalars"
    },
    "licence": "CC0-1.0",
    "proof_status": "complete; writing-AI self-checked"
  },
  {
    "result_id": "SH-DC.extension-skyscraper",
    "name": "Open extension by zero is exact and left adjoint to restriction; at any point the skyscraper functor is right adjoint to the exact stalk functor.",
    "statement": "Open extension by zero is exact and left adjoint to restriction; at any point the skyscraper functor is right adjoint to the exact stalk functor.",
    "conditions": "Arbitrary topological space and sheaf of unital, possibly noncommutative rings. No Hausdorff, Noetherian or quasi-compact-space hypothesis. The direct-sum sections theorem alone assumes its specified open is quasi-compact.",
    "writing_ai_selfcheck": true,
    "independent_review": false,
    "formalization": "not_formalized",
    "prerequisite_envelope_ids": [
      "SH-DC.sheafification-stalks"
    ],
    "dependency_precision": "Conservative earlier-provider envelope; exact textual uses appear in programme_dependency_loci. Same-lesson preliminary lemmas are inside the full proof locus.",
    "programme_dependency_loci": [
      {
        "provider": "SH-DC.sheafification-stalks",
        "locus": "Lemma 1.3, complete statement and proof in the same lesson",
        "correspondence": "exact sheafification, stalk, separatedness and associative-ring module scope",
        "candidate_path": "course/src/sheaves-of-modules-on-a-ringed-space.md",
        "sha256": "22FCF2E4E2E126FE922EBD68FD81F3DEDC581ED2B70CBB60A49250E21DA7BABE",
        "result_id": "SH-DC.sheafification-stalks",
        "anchor": "sheafification-and-the-stalk-test"
      }
    ],
    "proof": {
      "source": "src/sheaves-of-modules-on-a-ringed-space.md",
      "sha256": "22fcf2e4e2e126fe922ebd68fd81f3dedc581ed2b70cbb60a49250e21da7babe",
      "locus": "Lemmas 5.1 and 5.2",
      "reader": "sheaves-of-modules-on-a-ringed-space.html#5-extension-by-zero-and-skyscrapers"
    },
    "licence": "CC0-1.0",
    "proof_status": "complete; writing-AI self-checked"
  },
  {
    "result_id": "SH-DC.homotopy-localization",
    "name": "The homotopy category and localization at quasi-isomorphisms are triangulated; roofs, representable and cohomology exact sequences, and short-exact-sequence triangles are proved.",
    "statement": "The homotopy category and localization at quasi-isomorphisms are triangulated; roofs, representable and cohomology exact sequences, and short-exact-sequence triangles are proved.",
    "conditions": "Any abelian category; localization in a larger universe before separate original-universe Hom smallness.",
    "writing_ai_selfcheck": true,
    "independent_review": false,
    "formalization": "not_formalized",
    "prerequisite_envelope_ids": [
      "PROGRAMME.D80.diagram-cohomology",
      "SH-DC.filtered-colimits-sums"
    ],
    "dependency_precision": "Conservative earlier-provider envelope; exact textual uses appear in programme_dependency_loci. Same-lesson preliminary lemmas are inside the full proof locus.",
    "programme_dependency_loci": [
      {
        "provider": "D80",
        "locus": "Chapter 2 units 021 and 023; Chapter 3 unit 037",
        "correspondence": "exact_required_scope; stronger arbitrary-ring results specialize to commutative cases"
      },
      {
        "provider": "SH-DC.filtered-colimits-sums",
        "locus": "Lemma1.1 and complete proof, first lesson Section1",
        "candidate_path": "course/src/sheaves-of-modules-on-a-ringed-space.md",
        "source_sha256": "22fcf2e4e2e126fe922ebd68fd81f3dedc581ed2b70cbb60a49250e21da7babe",
        "anchor": "1-coefficients-germs-and-finite-relations",
        "correspondence": "Exact filtered-colimit proof over a fixed ring, specialized to Z and hence abelian groups; not the broader sheaf-colimit theorem.",
        "sha256": "22FCF2E4E2E126FE922EBD68FD81F3DEDC581ED2B70CBB60A49250E21DA7BABE",
        "result_id": "SH-DC.filtered-colimits-sums"
      }
    ],
    "proof": {
      "source": "src/complexes-cones-and-localization.md",
      "sha256": "367394ae4a25dcf64df87302304d8395af84d92765e4e4e45a2333a0071c3862",
      "locus": "Theorems 3.1, 4.2, 5.1; Proposition 5.2; Lemma 5.3 and Lemma 1.2 naturality proof",
      "reader": "complexes-cones-and-localization.html#3-the-triangulated-structure-of-the-homotopy-category"
    },
    "licence": "CC0-1.0",
    "proof_status": "complete; writing-AI self-checked"
  },
  {
    "result_id": "SH-DC.injective-embeddings",
    "name": "Left module sheaves on any space with associative unital coefficient rings have functorial injective embeddings.",
    "statement": "Left module sheaves on any space with associative unital coefficient rings have functorial injective embeddings.",
    "conditions": "No commutativity or separation assumption; product is an injective embedding, not a hull.",
    "writing_ai_selfcheck": true,
    "independent_review": false,
    "formalization": "not_formalized",
    "prerequisite_envelope_ids": [
      "SH-DC.modules-abelian",
      "SH-DC.filtered-colimits-sums",
      "SH-DC.pullback-adjunction",
      "SH-DC.extension-skyscraper",
      "SH-DC.homotopy-localization"
    ],
    "dependency_precision": "Conservative earlier-provider envelope; exact textual uses appear in programme_dependency_loci. Same-lesson preliminary lemmas are inside the full proof locus.",
    "programme_dependency_loci": [
      {
        "provider": "sheaves-of-modules-on-a-ringed-space",
        "locus": "Theorems 2.1,3.1; Lemmas 5.1–5.2",
        "correspondence": "exact_required_scope; stronger arbitrary-ring results specialize to commutative cases",
        "candidate_path": "course/src/sheaves-of-modules-on-a-ringed-space.md",
        "sha256": "22FCF2E4E2E126FE922EBD68FD81F3DEDC581ED2B70CBB60A49250E21DA7BABE"
      },
      {
        "provider": "complexes-cones-and-localization",
        "locus": "Lemmas 1.1–1.2,2.4; Theorems 3.1,4.2,5.1; Proposition 5.2",
        "correspondence": "exact_required_scope; stronger arbitrary-ring results specialize to commutative cases",
        "candidate_path": "course/src/complexes-cones-and-localization.md",
        "sha256": "367394AE4A25DCF64DF87302304D8395AF84D92765E4E4E45A2333A0071C3862"
      }
    ],
    "proof": {
      "source": "src/injective-modules-and-bounded-below-derived-functors.md",
      "sha256": "b2dc989efbc8d2d4004c8d24cbc3004a9e81da485cd702bb7d293a7f8af5edba",
      "locus": "Lemmas 2.1-2.2 and Theorem 2.3",
      "reader": "injective-modules-and-bounded-below-derived-functors.html#2-functorial-injective-embeddings"
    },
    "licence": "CC0-1.0",
    "proof_status": "complete; writing-AI self-checked"
  },
  {
    "result_id": "SH-DC.flasque-acyclic",
    "name": "Injective module sheaves are flasque and restrict to injectives; flasque kernels give exact sections; flasque sheaves are acyclic for sections on every open and pushforward.",
    "statement": "Injective module sheaves are flasque and restrict to injectives; flasque kernels give exact sections; flasque sheaves are acyclic for sections on every open and pushforward.",
    "conditions": "Arbitrary topological spaces and unital coefficient rings, without compactness or separation.",
    "writing_ai_selfcheck": true,
    "independent_review": false,
    "formalization": "not_formalized",
    "prerequisite_envelope_ids": [
      "SH-DC.modules-abelian",
      "SH-DC.filtered-colimits-sums",
      "SH-DC.pullback-adjunction",
      "SH-DC.extension-skyscraper",
      "SH-DC.homotopy-localization"
    ],
    "dependency_precision": "Conservative earlier-provider envelope; exact textual uses appear in programme_dependency_loci. Same-lesson preliminary lemmas are inside the full proof locus.",
    "programme_dependency_loci": [
      {
        "provider": "sheaves-of-modules-on-a-ringed-space",
        "locus": "Theorems 2.1,3.1; Lemmas 5.1–5.2",
        "correspondence": "exact_required_scope; stronger arbitrary-ring results specialize to commutative cases",
        "candidate_path": "course/src/sheaves-of-modules-on-a-ringed-space.md",
        "sha256": "22FCF2E4E2E126FE922EBD68FD81F3DEDC581ED2B70CBB60A49250E21DA7BABE"
      },
      {
        "provider": "complexes-cones-and-localization",
        "locus": "Lemmas 1.1–1.2,2.4; Theorems 3.1,4.2,5.1; Proposition 5.2",
        "correspondence": "exact_required_scope; stronger arbitrary-ring results specialize to commutative cases",
        "candidate_path": "course/src/complexes-cones-and-localization.md",
        "sha256": "367394AE4A25DCF64DF87302304D8395AF84D92765E4E4E45A2333A0071C3862"
      }
    ],
    "proof": {
      "source": "src/injective-modules-and-bounded-below-derived-functors.md",
      "sha256": "b2dc989efbc8d2d4004c8d24cbc3004a9e81da485cd702bb7d293a7f8af5edba",
      "locus": "Lemmas 1.1-1.2 and Proposition 1.3",
      "reader": "injective-modules-and-bounded-below-derived-functors.html#1-extending-maps-and-extending-sections"
    },
    "licence": "CC0-1.0",
    "proof_status": "complete; writing-AI self-checked"
  },
  {
    "result_id": "SH-DC.K-injective-comparison",
    "name": "K-injective complexes satisfy Hom_K equals Hom_D into them; bounded-below injective complexes are K-injective; products and cones preserve K-injectivity.",
    "statement": "K-injective complexes satisfy Hom_K equals Hom_D into them; bounded-below injective complexes are K-injective; products and cones preserve K-injectivity.",
    "conditions": "Any abelian category, with existence of indicated degreewise products for the product assertion.",
    "writing_ai_selfcheck": true,
    "independent_review": false,
    "formalization": "not_formalized",
    "prerequisite_envelope_ids": [
      "SH-DC.modules-abelian",
      "SH-DC.filtered-colimits-sums",
      "SH-DC.pullback-adjunction",
      "SH-DC.extension-skyscraper",
      "SH-DC.homotopy-localization"
    ],
    "dependency_precision": "Conservative earlier-provider envelope; exact textual uses appear in programme_dependency_loci. Same-lesson preliminary lemmas are inside the full proof locus.",
    "programme_dependency_loci": [
      {
        "provider": "sheaves-of-modules-on-a-ringed-space",
        "locus": "Theorems 2.1,3.1; Lemmas 5.1–5.2",
        "correspondence": "exact_required_scope; stronger arbitrary-ring results specialize to commutative cases",
        "candidate_path": "course/src/sheaves-of-modules-on-a-ringed-space.md",
        "sha256": "22FCF2E4E2E126FE922EBD68FD81F3DEDC581ED2B70CBB60A49250E21DA7BABE"
      },
      {
        "provider": "complexes-cones-and-localization",
        "locus": "Lemmas 1.1–1.2,2.4; Theorems 3.1,4.2,5.1; Proposition 5.2",
        "correspondence": "exact_required_scope; stronger arbitrary-ring results specialize to commutative cases",
        "candidate_path": "course/src/complexes-cones-and-localization.md",
        "sha256": "367394AE4A25DCF64DF87302304D8395AF84D92765E4E4E45A2333A0071C3862"
      }
    ],
    "proof": {
      "source": "src/injective-modules-and-bounded-below-derived-functors.md",
      "sha256": "b2dc989efbc8d2d4004c8d24cbc3004a9e81da485cd702bb7d293a7f8af5edba",
      "locus": "Lemmas 3.1-3.2, Theorem 3.3 and Corollary 3.4",
      "reader": "injective-modules-and-bounded-below-derived-functors.html#3-complexes-that-admit-comparison-maps"
    },
    "licence": "CC0-1.0",
    "proof_status": "complete; writing-AI self-checked"
  },
  {
    "result_id": "SH-DC.bounded-derived",
    "name": "For A abelian with enough injectives and any additive Phi:A->B with B abelian, RPhi:D+(A)->D+(B) exists and is computed by bounded-below injective resolutions. For left exact Phi its cohomology functors are universal derived functors and bounded-below complexes of Phi-acyclic objects compute RPhi.",
    "statement": "For A abelian with enough injectives and any additive Phi:A->B with B abelian, RPhi:D+(A)->D+(B) exists and is computed by bounded-below injective resolutions. For left exact Phi its cohomology functors are universal derived functors and bounded-below complexes of Phi-acyclic objects compute RPhi.",
    "conditions": "Lower bound for complexes; no filtered-colimit or product exactness assumption on A. Left exactness, or degree-zero comparison as well as positive vanishing, is required in the acyclic-term assertion.",
    "writing_ai_selfcheck": true,
    "independent_review": false,
    "formalization": "not_formalized",
    "prerequisite_envelope_ids": [
      "SH-DC.modules-abelian",
      "SH-DC.filtered-colimits-sums",
      "SH-DC.pullback-adjunction",
      "SH-DC.extension-skyscraper",
      "SH-DC.homotopy-localization"
    ],
    "dependency_precision": "Conservative earlier-provider envelope; exact textual uses appear in programme_dependency_loci. Same-lesson preliminary lemmas are inside the full proof locus.",
    "programme_dependency_loci": [
      {
        "provider": "sheaves-of-modules-on-a-ringed-space",
        "locus": "Theorems 2.1,3.1; Lemmas 5.1–5.2",
        "correspondence": "exact_required_scope; stronger arbitrary-ring results specialize to commutative cases",
        "candidate_path": "course/src/sheaves-of-modules-on-a-ringed-space.md",
        "sha256": "22FCF2E4E2E126FE922EBD68FD81F3DEDC581ED2B70CBB60A49250E21DA7BABE"
      },
      {
        "provider": "complexes-cones-and-localization",
        "locus": "Lemmas 1.1–1.2,2.4; Theorems 3.1,4.2,5.1; Proposition 5.2",
        "correspondence": "exact_required_scope; stronger arbitrary-ring results specialize to commutative cases",
        "candidate_path": "course/src/complexes-cones-and-localization.md",
        "sha256": "367394AE4A25DCF64DF87302304D8395AF84D92765E4E4E45A2333A0071C3862"
      }
    ],
    "proof": {
      "source": "src/injective-modules-and-bounded-below-derived-functors.md",
      "sha256": "b2dc989efbc8d2d4004c8d24cbc3004a9e81da485cd702bb7d293a7f8af5edba",
      "locus": "Theorem 4.1",
      "reader": "injective-modules-and-bounded-below-derived-functors.html#4-right-derived-functors-on-bounded-below-complexes"
    },
    "licence": "CC0-1.0",
    "proof_status": "complete; writing-AI self-checked"
  },
  {
    "result_id": "SH-DC.Grothendieck-injective-embeddings",
    "name": "Every object of a locally small Grothendieck abelian category has a functorial monomorphism into an injective object.",
    "statement": "Every object of a locally small Grothendieck abelian category has a functorial monomorphism into an injective object.",
    "conditions": "AB5 and a generator, within the chosen set universe; no exact-product hypothesis.",
    "writing_ai_selfcheck": true,
    "independent_review": false,
    "formalization": "not_formalized",
    "prerequisite_envelope_ids": [
      "SH-DC.modules-abelian",
      "SH-DC.filtered-colimits-sums",
      "SH-DC.pullback-adjunction",
      "SH-DC.extension-skyscraper",
      "SH-DC.homotopy-localization",
      "SH-DC.injective-embeddings",
      "SH-DC.flasque-acyclic",
      "SH-DC.K-injective-comparison",
      "SH-DC.bounded-derived"
    ],
    "dependency_precision": "Conservative earlier-provider envelope; exact textual uses appear in programme_dependency_loci. Same-lesson preliminary lemmas are inside the full proof locus.",
    "programme_dependency_loci": [
      {
        "provider": "sheaves-of-modules-on-a-ringed-space",
        "locus": "Theorems 2.1,3.1",
        "correspondence": "exact_required_scope; stronger arbitrary-ring results specialize to commutative cases",
        "candidate_path": "course/src/sheaves-of-modules-on-a-ringed-space.md",
        "sha256": "22FCF2E4E2E126FE922EBD68FD81F3DEDC581ED2B70CBB60A49250E21DA7BABE"
      },
      {
        "provider": "complexes-cones-and-localization",
        "locus": "Theorems 3.1,4.2,5.1; Proposition 5.2",
        "correspondence": "exact_required_scope; stronger arbitrary-ring results specialize to commutative cases",
        "candidate_path": "course/src/complexes-cones-and-localization.md",
        "sha256": "367394AE4A25DCF64DF87302304D8395AF84D92765E4E4E45A2333A0071C3862"
      },
      {
        "provider": "injective-modules-and-bounded-below-derived-functors",
        "locus": "Lemmas 3.1–3.2; Theorem 3.3; bounded injective construction",
        "correspondence": "exact_required_scope; stronger arbitrary-ring results specialize to commutative cases",
        "candidate_path": "course/src/injective-modules-and-bounded-below-derived-functors.md",
        "sha256": "B2DC989EFBC8D2D4004C8D24CBC3004A9E81DA485CD702BB7D293A7F8AF5EDBA"
      },
      {
        "provider": "k-injective-resolutions-in-grothendieck-categories",
        "locus": "Proposition 1.1 (generator), complete proof in the same lesson",
        "correspondence": "Generator construction for module sheaves; the abstract Grothendieck-category assertions assume a generator.",
        "candidate_path": "course/src/k-injective-resolutions-in-grothendieck-categories.md",
        "sha256": "75118C6AB567222AEB5E42C60AA5DC206466FFF9EA8A9F193EA1B598DECD760B",
        "anchor": "1-the-grothendieck-hypotheses"
      }
    ],
    "proof": {
      "source": "src/k-injective-resolutions-in-grothendieck-categories.md",
      "sha256": "75118c6ab567222aeb5e42c60aa5dc206466fff9ea8a9f193ea1b598decd760b",
      "locus": "Lemmas 2.1-2.3 and Theorem 2.4",
      "reader": "k-injective-resolutions-in-grothendieck-categories.html#2-sizes-and-functorial-injective-embeddings"
    },
    "licence": "CC0-1.0",
    "proof_status": "complete; writing-AI self-checked"
  },
  {
    "result_id": "SH-DC.unbounded-K-injective-existence",
    "name": "Every complex in a locally small Grothendieck abelian category has a functorial termwise monic quasi-isomorphism into a K-injective complex of injectives; module sheaves on arbitrary ringed spaces satisfy the hypotheses.",
    "statement": "Every complex in a locally small Grothendieck abelian category has a functorial termwise monic quasi-isomorphism into a K-injective complex of injectives; module sheaves on arbitrary ringed spaces satisfy the hypotheses.",
    "conditions": "AB5 and generator; sheaf coefficients arbitrary associative unital rings, including noncommutative; no separation or exact products.",
    "writing_ai_selfcheck": true,
    "independent_review": false,
    "formalization": "not_formalized",
    "prerequisite_envelope_ids": [
      "SH-DC.modules-abelian",
      "SH-DC.filtered-colimits-sums",
      "SH-DC.pullback-adjunction",
      "SH-DC.extension-skyscraper",
      "SH-DC.homotopy-localization",
      "SH-DC.injective-embeddings",
      "SH-DC.flasque-acyclic",
      "SH-DC.K-injective-comparison",
      "SH-DC.bounded-derived"
    ],
    "dependency_precision": "Conservative earlier-provider envelope; exact textual uses appear in programme_dependency_loci. Same-lesson preliminary lemmas are inside the full proof locus.",
    "programme_dependency_loci": [
      {
        "provider": "sheaves-of-modules-on-a-ringed-space",
        "locus": "Theorems 2.1,3.1",
        "correspondence": "exact_required_scope; stronger arbitrary-ring results specialize to commutative cases",
        "candidate_path": "course/src/sheaves-of-modules-on-a-ringed-space.md",
        "sha256": "22FCF2E4E2E126FE922EBD68FD81F3DEDC581ED2B70CBB60A49250E21DA7BABE"
      },
      {
        "provider": "complexes-cones-and-localization",
        "locus": "Theorems 3.1,4.2,5.1; Proposition 5.2",
        "correspondence": "exact_required_scope; stronger arbitrary-ring results specialize to commutative cases",
        "candidate_path": "course/src/complexes-cones-and-localization.md",
        "sha256": "367394AE4A25DCF64DF87302304D8395AF84D92765E4E4E45A2333A0071C3862"
      },
      {
        "provider": "injective-modules-and-bounded-below-derived-functors",
        "locus": "Lemmas 3.1–3.2; Theorem 3.3; bounded injective construction",
        "correspondence": "exact_required_scope; stronger arbitrary-ring results specialize to commutative cases",
        "candidate_path": "course/src/injective-modules-and-bounded-below-derived-functors.md",
        "sha256": "B2DC989EFBC8D2D4004C8D24CBC3004A9E81DA485CD702BB7D293A7F8AF5EDBA"
      },
      {
        "provider": "k-injective-resolutions-in-grothendieck-categories",
        "locus": "Proposition 1.1 (generator), complete proof in the same lesson",
        "correspondence": "Generator construction for module sheaves; the abstract Grothendieck-category assertions assume a generator.",
        "candidate_path": "course/src/k-injective-resolutions-in-grothendieck-categories.md",
        "sha256": "75118C6AB567222AEB5E42C60AA5DC206466FFF9EA8A9F193EA1B598DECD760B",
        "anchor": "1-the-grothendieck-hypotheses"
      }
    ],
    "proof": {
      "source": "src/k-injective-resolutions-in-grothendieck-categories.md",
      "sha256": "75118c6ab567222aeb5e42c60aa5dc206466fff9ea8a9f193ea1b598decd760b",
      "locus": "Proposition 1.1, Lemmas 3.1-3.2 and Theorem 4.1",
      "reader": "k-injective-resolutions-in-grothendieck-categories.html#4-constructing-the-unbounded-resolution"
    },
    "licence": "CC0-1.0",
    "proof_status": "complete; writing-AI self-checked"
  },
  {
    "result_id": "SH-DC.unbounded-right-derived",
    "name": "An exact F:K(A)->T has an exact right derived RF:D(A)->T when all complexes admit K-injective resolutions. Additive module functors, including pushforward, admit unbounded right derived functors; an additive right adjoint to an exact left adjoint preserves K-injectives.",
    "statement": "An exact F:K(A)->T has an exact right derived RF:D(A)->T when all complexes admit K-injective resolutions. Additive module functors, including pushforward, admit unbounded right derived functors; an additive right adjoint to an exact left adjoint preserves K-injectives.",
    "conditions": "Resolution hypothesis for A; exactness of the left adjoint for the preservation assertion only.",
    "writing_ai_selfcheck": true,
    "independent_review": false,
    "formalization": "not_formalized",
    "prerequisite_envelope_ids": [
      "SH-DC.modules-abelian",
      "SH-DC.filtered-colimits-sums",
      "SH-DC.pullback-adjunction",
      "SH-DC.extension-skyscraper",
      "SH-DC.homotopy-localization",
      "SH-DC.injective-embeddings",
      "SH-DC.flasque-acyclic",
      "SH-DC.K-injective-comparison",
      "SH-DC.bounded-derived"
    ],
    "dependency_precision": "Conservative earlier-provider envelope; exact textual uses appear in programme_dependency_loci. Same-lesson preliminary lemmas are inside the full proof locus.",
    "programme_dependency_loci": [
      {
        "provider": "sheaves-of-modules-on-a-ringed-space",
        "locus": "Theorems 2.1,3.1",
        "correspondence": "exact_required_scope; stronger arbitrary-ring results specialize to commutative cases",
        "candidate_path": "course/src/sheaves-of-modules-on-a-ringed-space.md",
        "sha256": "22FCF2E4E2E126FE922EBD68FD81F3DEDC581ED2B70CBB60A49250E21DA7BABE"
      },
      {
        "provider": "complexes-cones-and-localization",
        "locus": "Theorems 3.1,4.2,5.1; Proposition 5.2",
        "correspondence": "exact_required_scope; stronger arbitrary-ring results specialize to commutative cases",
        "candidate_path": "course/src/complexes-cones-and-localization.md",
        "sha256": "367394AE4A25DCF64DF87302304D8395AF84D92765E4E4E45A2333A0071C3862"
      },
      {
        "provider": "injective-modules-and-bounded-below-derived-functors",
        "locus": "Lemmas 3.1–3.2; Theorem 3.3; bounded injective construction",
        "correspondence": "exact_required_scope; stronger arbitrary-ring results specialize to commutative cases",
        "candidate_path": "course/src/injective-modules-and-bounded-below-derived-functors.md",
        "sha256": "B2DC989EFBC8D2D4004C8D24CBC3004A9E81DA485CD702BB7D293A7F8AF5EDBA"
      },
      {
        "provider": "k-injective-resolutions-in-grothendieck-categories",
        "locus": "Proposition 1.1 (generator), complete proof in the same lesson",
        "correspondence": "Generator construction for module sheaves; the abstract Grothendieck-category assertions assume a generator.",
        "candidate_path": "course/src/k-injective-resolutions-in-grothendieck-categories.md",
        "sha256": "75118C6AB567222AEB5E42C60AA5DC206466FFF9EA8A9F193EA1B598DECD760B",
        "anchor": "1-the-grothendieck-hypotheses"
      }
    ],
    "proof": {
      "source": "src/k-injective-resolutions-in-grothendieck-categories.md",
      "sha256": "75118c6ab567222aeb5e42c60aa5dc206466fff9ea8a9f193ea1b598decd760b",
      "locus": "Theorem 5.1 and Proposition 5.2",
      "reader": "k-injective-resolutions-in-grothendieck-categories.html#5-derived-functors-adjoints-and-products"
    },
    "licence": "CC0-1.0",
    "proof_status": "complete; writing-AI self-checked"
  },
  {
    "result_id": "SH-DC.derived-products",
    "name": "A Grothendieck category has products and coproducts; its derived products are represented by termwise products of K-injective resolutions, derived coproducts by termwise coproducts of arbitrary representatives.",
    "statement": "A Grothendieck category has products and coproducts; its derived products are represented by termwise products of K-injective resolutions, derived coproducts by termwise coproducts of arbitrary representatives.",
    "conditions": "Locally small Grothendieck category; no AB4* requirement.",
    "writing_ai_selfcheck": true,
    "independent_review": false,
    "formalization": "not_formalized",
    "prerequisite_envelope_ids": [
      "SH-DC.modules-abelian",
      "SH-DC.filtered-colimits-sums",
      "SH-DC.pullback-adjunction",
      "SH-DC.extension-skyscraper",
      "SH-DC.homotopy-localization",
      "SH-DC.injective-embeddings",
      "SH-DC.flasque-acyclic",
      "SH-DC.K-injective-comparison",
      "SH-DC.bounded-derived"
    ],
    "dependency_precision": "Conservative earlier-provider envelope; exact textual uses appear in programme_dependency_loci. Same-lesson preliminary lemmas are inside the full proof locus.",
    "programme_dependency_loci": [
      {
        "provider": "sheaves-of-modules-on-a-ringed-space",
        "locus": "Theorems 2.1,3.1",
        "correspondence": "exact_required_scope; stronger arbitrary-ring results specialize to commutative cases",
        "candidate_path": "course/src/sheaves-of-modules-on-a-ringed-space.md",
        "sha256": "22FCF2E4E2E126FE922EBD68FD81F3DEDC581ED2B70CBB60A49250E21DA7BABE"
      },
      {
        "provider": "complexes-cones-and-localization",
        "locus": "Theorems 3.1,4.2,5.1; Proposition 5.2",
        "correspondence": "exact_required_scope; stronger arbitrary-ring results specialize to commutative cases",
        "candidate_path": "course/src/complexes-cones-and-localization.md",
        "sha256": "367394AE4A25DCF64DF87302304D8395AF84D92765E4E4E45A2333A0071C3862"
      },
      {
        "provider": "injective-modules-and-bounded-below-derived-functors",
        "locus": "Lemmas 3.1–3.2; Theorem 3.3; bounded injective construction",
        "correspondence": "exact_required_scope; stronger arbitrary-ring results specialize to commutative cases",
        "candidate_path": "course/src/injective-modules-and-bounded-below-derived-functors.md",
        "sha256": "B2DC989EFBC8D2D4004C8D24CBC3004A9E81DA485CD702BB7D293A7F8AF5EDBA"
      },
      {
        "provider": "k-injective-resolutions-in-grothendieck-categories",
        "locus": "Proposition 1.1 (generator), complete proof in the same lesson",
        "correspondence": "Generator construction for module sheaves; the abstract Grothendieck-category assertions assume a generator.",
        "candidate_path": "course/src/k-injective-resolutions-in-grothendieck-categories.md",
        "sha256": "75118C6AB567222AEB5E42C60AA5DC206466FFF9EA8A9F193EA1B598DECD760B",
        "anchor": "1-the-grothendieck-hypotheses"
      }
    ],
    "proof": {
      "source": "src/k-injective-resolutions-in-grothendieck-categories.md",
      "sha256": "75118c6ab567222aeb5e42c60aa5dc206466fff9ea8a9f193ea1b598decd760b",
      "locus": "Propositions 5.3-5.4",
      "reader": "k-injective-resolutions-in-grothendieck-categories.html#5-derived-functors-adjoints-and-products"
    },
    "licence": "CC0-1.0",
    "proof_status": "complete; writing-AI self-checked"
  },
  {
    "result_id": "SH-DC.flat-module-sheaves",
    "name": "Flatness of module sheaves is stalkwise; sums/filtered colimits and scalar extension preserve flatness; every sheaf has a flat-block epimorphic cover; flat quotients give tensor-exact sequences.",
    "statement": "Flatness of module sheaves is stalkwise; sums/filtered colimits and scalar extension preserve flatness; every sheaf has a flat-block epimorphic cover; flat quotients give tensor-exact sequences.",
    "conditions": "Arbitrary topological space and commutative unital sheaf of rings; no compactness/separation.",
    "writing_ai_selfcheck": true,
    "independent_review": false,
    "formalization": "not_formalized",
    "prerequisite_envelope_ids": [
      "SH-DC.modules-abelian",
      "SH-DC.filtered-colimits-sums",
      "SH-DC.pullback-adjunction",
      "SH-DC.extension-skyscraper",
      "SH-DC.homotopy-localization"
    ],
    "dependency_precision": "Conservative earlier-provider envelope; exact textual uses appear in programme_dependency_loci. Same-lesson preliminary lemmas are inside the full proof locus.",
    "programme_dependency_loci": [
      {
        "provider": "sheaves-of-modules-on-a-ringed-space",
        "locus": "Theorems 2.1,3.1,4.4; Lemmas 5.1–5.2",
        "correspondence": "exact_required_scope; stronger arbitrary-ring results specialize to commutative cases",
        "candidate_path": "course/src/sheaves-of-modules-on-a-ringed-space.md",
        "sha256": "22FCF2E4E2E126FE922EBD68FD81F3DEDC581ED2B70CBB60A49250E21DA7BABE"
      },
      {
        "provider": "complexes-cones-and-localization",
        "locus": "Lemmas 1.1–1.2,2.4; Proposition 5.2",
        "correspondence": "exact_required_scope; stronger arbitrary-ring results specialize to commutative cases",
        "candidate_path": "course/src/complexes-cones-and-localization.md",
        "sha256": "367394AE4A25DCF64DF87302304D8395AF84D92765E4E4E45A2333A0071C3862"
      }
    ],
    "proof": {
      "source": "src/flat-modules-and-k-flat-resolutions.md",
      "sha256": "7bce2b46408b278a147517161f09ae29a7bc0273b4e271aa2f995966c1aa941d",
      "locus": "Lemmas 1.1-1.3",
      "reader": "flat-modules-and-k-flat-resolutions.html#1-flat-sheaves-and-exact-tensor-sequences"
    },
    "licence": "CC0-1.0",
    "proof_status": "complete; writing-AI self-checked"
  },
  {
    "result_id": "SH-DC.K-flat-existence",
    "name": "Every complex of module sheaves has a termwise-surjective quasi-isomorphism from a K-flat complex of flat terms. There is also a compatible bounded-above resolution system of good upper truncations with split transitions and flat-block terms/quotients.",
    "statement": "Every complex of module sheaves has a termwise-surjective quasi-isomorphism from a K-flat complex of flat terms. There is also a compatible bounded-above resolution system of good upper truncations with split transitions and flat-block terms/quotients.",
    "conditions": "Commutative unital coefficient rings; all complexes unbounded; independent of K-injective existence; direct-sum totalization.",
    "writing_ai_selfcheck": true,
    "independent_review": false,
    "formalization": "not_formalized",
    "prerequisite_envelope_ids": [
      "SH-DC.modules-abelian",
      "SH-DC.filtered-colimits-sums",
      "SH-DC.pullback-adjunction",
      "SH-DC.extension-skyscraper",
      "SH-DC.homotopy-localization"
    ],
    "dependency_precision": "Conservative earlier-provider envelope; exact textual uses appear in programme_dependency_loci. Same-lesson preliminary lemmas are inside the full proof locus.",
    "programme_dependency_loci": [
      {
        "provider": "sheaves-of-modules-on-a-ringed-space",
        "locus": "Theorems 2.1,3.1,4.4; Lemmas 5.1–5.2",
        "correspondence": "exact_required_scope; stronger arbitrary-ring results specialize to commutative cases",
        "candidate_path": "course/src/sheaves-of-modules-on-a-ringed-space.md",
        "sha256": "22FCF2E4E2E126FE922EBD68FD81F3DEDC581ED2B70CBB60A49250E21DA7BABE"
      },
      {
        "provider": "complexes-cones-and-localization",
        "locus": "Lemmas 1.1–1.2,2.4; Proposition 5.2",
        "correspondence": "exact_required_scope; stronger arbitrary-ring results specialize to commutative cases",
        "candidate_path": "course/src/complexes-cones-and-localization.md",
        "sha256": "367394AE4A25DCF64DF87302304D8395AF84D92765E4E4E45A2333A0071C3862"
      }
    ],
    "proof": {
      "source": "src/flat-modules-and-k-flat-resolutions.md",
      "sha256": "7bce2b46408b278a147517161f09ae29a7bc0273b4e271aa2f995966c1aa941d",
      "locus": "Lemmas 2.1-2.4, Theorem 3.1 and Proposition 4.3",
      "reader": "flat-modules-and-k-flat-resolutions.html#3-a-surjective-resolution-by-attaching-cycles"
    },
    "licence": "CC0-1.0",
    "proof_status": "complete; writing-AI self-checked"
  },
  {
    "result_id": "SH-DC.K-flat-calculus",
    "name": "K-flatness is stalkwise, preserved by pullback, direct sums, filtered colimits and tensor products; two out of three in homotopy triangles; tensor comparison holds for quasi-isomorphisms between K-flats against every complex.",
    "statement": "K-flatness is stalkwise, preserved by pullback, direct sums, filtered colimits and tensor products; two out of three in homotopy triangles; tensor comparison holds for quasi-isomorphisms between K-flats against every complex.",
    "conditions": "No arbitrary-colimit closure or equivalence with unbounded termwise flatness; flat quotient terms needed for nonsplit short-exact-sequence closure.",
    "writing_ai_selfcheck": true,
    "independent_review": false,
    "formalization": "not_formalized",
    "prerequisite_envelope_ids": [
      "SH-DC.modules-abelian",
      "SH-DC.filtered-colimits-sums",
      "SH-DC.pullback-adjunction",
      "SH-DC.extension-skyscraper",
      "SH-DC.homotopy-localization"
    ],
    "dependency_precision": "Conservative earlier-provider envelope; exact textual uses appear in programme_dependency_loci. Same-lesson preliminary lemmas are inside the full proof locus.",
    "programme_dependency_loci": [
      {
        "provider": "sheaves-of-modules-on-a-ringed-space",
        "locus": "Theorems 2.1,3.1,4.4; Lemmas 5.1–5.2",
        "correspondence": "exact_required_scope; stronger arbitrary-ring results specialize to commutative cases",
        "candidate_path": "course/src/sheaves-of-modules-on-a-ringed-space.md",
        "sha256": "22FCF2E4E2E126FE922EBD68FD81F3DEDC581ED2B70CBB60A49250E21DA7BABE"
      },
      {
        "provider": "complexes-cones-and-localization",
        "locus": "Lemmas 1.1–1.2,2.4; Proposition 5.2",
        "correspondence": "exact_required_scope; stronger arbitrary-ring results specialize to commutative cases",
        "candidate_path": "course/src/complexes-cones-and-localization.md",
        "sha256": "367394AE4A25DCF64DF87302304D8395AF84D92765E4E4E45A2333A0071C3862"
      }
    ],
    "proof": {
      "source": "src/flat-modules-and-k-flat-resolutions.md",
      "sha256": "7bce2b46408b278a147517161f09ae29a7bc0273b4e271aa2f995966c1aa941d",
      "locus": "Lemmas 2.1-2.4, Corollary 4.4 and Lemma 4.5",
      "reader": "flat-modules-and-k-flat-resolutions.html#2-k-flat-complexes-and-their-closure-properties"
    },
    "licence": "CC0-1.0",
    "proof_status": "complete; writing-AI self-checked"
  },
  {
    "result_id": "SH-DC.derived-tensor",
    "name": "The unbounded derived tensor bifunctor exists, is exact in both variables, is independent of K-flat models and can be computed by resolving either variable; association, Koszul symmetry, unit and open restriction are natural and coherent.",
    "statement": "The unbounded derived tensor bifunctor exists, is exact in both variables, is independent of K-flat models and can be computed by resolving either variable; association, Koszul symmetry, unit and open restriction are natural and coherent.",
    "conditions": "Arbitrary topological space, commutative unital sheaf of rings, direct-sum totalization; no boundedness or finite homological dimension assumption.",
    "writing_ai_selfcheck": true,
    "independent_review": false,
    "formalization": "not_formalized",
    "prerequisite_envelope_ids": [
      "SH-DC.modules-abelian",
      "SH-DC.filtered-colimits-sums",
      "SH-DC.pullback-adjunction",
      "SH-DC.extension-skyscraper",
      "SH-DC.homotopy-localization",
      "SH-DC.flat-module-sheaves",
      "SH-DC.K-flat-existence",
      "SH-DC.K-flat-calculus"
    ],
    "dependency_precision": "Conservative earlier-provider envelope; exact textual uses appear in programme_dependency_loci. Same-lesson preliminary lemmas are inside the full proof locus.",
    "programme_dependency_loci": [
      {
        "provider": "flat-modules-and-k-flat-resolutions",
        "locus": "Lemmas 1.1–1.3,2.1–2.4,3.2,4.1,4.5; Theorem 3.1",
        "correspondence": "exact_required_scope; stronger arbitrary-ring results specialize to commutative cases",
        "candidate_path": "course/src/flat-modules-and-k-flat-resolutions.md",
        "sha256": "7BCE2B46408B278A147517161F09AE29A7BC0273B4E271AA2F995966C1AA941D"
      },
      {
        "provider": "complexes-cones-and-localization",
        "locus": "Theorems 4.2,5.1; Proposition 5.2",
        "correspondence": "exact_required_scope; stronger arbitrary-ring results specialize to commutative cases",
        "candidate_path": "course/src/complexes-cones-and-localization.md",
        "sha256": "367394AE4A25DCF64DF87302304D8395AF84D92765E4E4E45A2333A0071C3862"
      }
    ],
    "proof": {
      "source": "src/derived-tensor-products-and-tor-sheaves.md",
      "sha256": "b78c1ad9864901349e26c6eb3954b046e40b84e35e55ea6b3a71c553e185d292",
      "locus": "Lemmas 1.1 and 2.1, Theorems 1.2 and 2.2",
      "reader": "derived-tensor-products-and-tor-sheaves.html#1-the-two-tensor-comparisons"
    },
    "licence": "CC0-1.0",
    "proof_status": "complete; writing-AI self-checked"
  },
  {
    "result_id": "SH-DC.Tor-flatness",
    "name": "Tor sheaves have stalk Tor comparisons and long exact sequences; a module sheaf is flat iff every first Tor test vanishes iff every positive Tor test vanishes; Tor of two ideal quotients in degree one is intersection modulo product.",
    "statement": "Tor sheaves have stalk Tor comparisons and long exact sequences; a module sheaf is flat iff every first Tor test vanishes iff every positive Tor test vanishes; Tor of two ideal quotients in degree one is intersection modulo product.",
    "conditions": "All test modules are quantified; self Tor alone does not detect flatness; arbitrary ideal quotients have no asserted higher vanishing.",
    "writing_ai_selfcheck": true,
    "independent_review": false,
    "formalization": "not_formalized",
    "prerequisite_envelope_ids": [
      "SH-DC.modules-abelian",
      "SH-DC.filtered-colimits-sums",
      "SH-DC.pullback-adjunction",
      "SH-DC.extension-skyscraper",
      "SH-DC.homotopy-localization",
      "SH-DC.flat-module-sheaves",
      "SH-DC.K-flat-existence",
      "SH-DC.K-flat-calculus"
    ],
    "dependency_precision": "Conservative earlier-provider envelope; exact textual uses appear in programme_dependency_loci. Same-lesson preliminary lemmas are inside the full proof locus.",
    "programme_dependency_loci": [
      {
        "provider": "flat-modules-and-k-flat-resolutions",
        "locus": "Lemmas 1.1–1.3,2.1–2.4,3.2,4.1,4.5; Theorem 3.1",
        "correspondence": "exact_required_scope; stronger arbitrary-ring results specialize to commutative cases",
        "candidate_path": "course/src/flat-modules-and-k-flat-resolutions.md",
        "sha256": "7BCE2B46408B278A147517161F09AE29A7BC0273B4E271AA2F995966C1AA941D"
      },
      {
        "provider": "complexes-cones-and-localization",
        "locus": "Theorems 4.2,5.1; Proposition 5.2",
        "correspondence": "exact_required_scope; stronger arbitrary-ring results specialize to commutative cases",
        "candidate_path": "course/src/complexes-cones-and-localization.md",
        "sha256": "367394AE4A25DCF64DF87302304D8395AF84D92765E4E4E45A2333A0071C3862"
      }
    ],
    "proof": {
      "source": "src/derived-tensor-products-and-tor-sheaves.md",
      "sha256": "b78c1ad9864901349e26c6eb3954b046e40b84e35e55ea6b3a71c553e185d292",
      "locus": "Propositions 3.1-3.2, Theorem 3.3 and Lemma 3.4",
      "reader": "derived-tensor-products-and-tor-sheaves.html#3-tor-sheaves-and-the-flatness-criterion"
    },
    "licence": "CC0-1.0",
    "proof_status": "complete; writing-AI self-checked"
  },
  {
    "result_id": "SH-DC.holomorphic-coordinate-germs",
    "name": "Coordinate divisibility and successive quotients of holomorphic germs",
    "statement": "Coordinates are injective on holomorphic germs; evaluation has the coordinate-ideal kernel; R2/(x)=O_C,0 and R2/(x,y)=C; nonzero coordinates are units.",
    "conditions": "Holomorphic germs in one or two complex variables with the explicit continuous, coordinatewise holomorphic definition.",
    "writing_ai_selfcheck": true,
    "independent_review": false,
    "formalization": "not_formalized",
    "prerequisite_envelope_ids": [
      "PROGRAMME.CA.holomorphic-Taylor"
    ],
    "programme_dependency_loci": [
      {
        "provider": "PROGRAMME.CA.holomorphic-Taylor",
        "candidate_path": "course/src/prerequisites/holomorphic-power-series.md",
        "locus": "Definition 1.1, polydisc Cauchy Theorem 1.2, Taylor Theorem 2.1 and the power-series implication of Proposition 2.2.",
        "sha256": "3be99e15592a58e053c211500e47a5457d205ab1ae2738b472d319cf33166b16",
        "anchor": "power-series-and-regularity",
        "correspondence": "Exact Taylor, coefficient uniqueness, coefficient estimates and convergent-series holomorphy inputs."
      }
    ],
    "proof_status": "complete_local_proof_with_readable_earlier_analytic_prerequisites",
    "proof": {
      "source": "src/derived-tensor-products-and-tor-sheaves.md",
      "sha256": "b78c1ad9864901349e26c6eb3954b046e40b84e35e55ea6b3a71c553e185d292",
      "locus": "Section 4, Lemma (holomorphic coordinate germs), full proof",
      "reader": "derived-tensor-products-and-tor-sheaves.html#holomorphic-coordinate-germs"
    },
    "licence": "CC0-1.0 original programme proof; course component terms remain as stated in LICENCE.md"
  },
  {
    "result_id": "SH-DC.derived-pull-push-adjunction",
    "name": "Unbounded derived pullback and pushforward exist for arbitrary commutative ringed-space morphisms, are adjoint, compose canonically, and pullback coherently preserves derived tensor.",
    "statement": "Unbounded derived pullback and pushforward exist for arbitrary commutative ringed-space morphisms, are adjoint, compose canonically, and pullback coherently preserves derived tensor.",
    "conditions": "No flatness of f required for adjunction or composition; nonflat pushforward is not asserted to preserve K-injectives.",
    "writing_ai_selfcheck": true,
    "independent_review": false,
    "formalization": "not_formalized",
    "prerequisite_envelope_ids": [
      "SH-DC.modules-abelian",
      "SH-DC.filtered-colimits-sums",
      "SH-DC.pullback-adjunction",
      "SH-DC.extension-skyscraper",
      "SH-DC.homotopy-localization",
      "SH-DC.injective-embeddings",
      "SH-DC.flasque-acyclic",
      "SH-DC.K-injective-comparison",
      "SH-DC.bounded-derived",
      "SH-DC.Grothendieck-injective-embeddings",
      "SH-DC.unbounded-K-injective-existence",
      "SH-DC.unbounded-right-derived",
      "SH-DC.derived-products",
      "SH-DC.flat-module-sheaves",
      "SH-DC.K-flat-existence",
      "SH-DC.K-flat-calculus",
      "SH-DC.derived-tensor",
      "SH-DC.Tor-flatness"
    ],
    "dependency_precision": "Conservative earlier-provider envelope; exact textual uses appear in programme_dependency_loci. Same-lesson preliminary lemmas are inside the full proof locus.",
    "programme_dependency_loci": [
      {
        "provider": "derived-tensor-products-and-tor-sheaves",
        "locus": "Lemmas 1.1,2.1; Theorems 1.2,2.2",
        "correspondence": "exact_required_scope; stronger arbitrary-ring results specialize to commutative cases",
        "candidate_path": "course/src/derived-tensor-products-and-tor-sheaves.md",
        "sha256": "B78C1AD9864901349E26C6EB3954B046E40B84E35E55EA6B3A71C553E185D292"
      },
      {
        "provider": "k-injective-resolutions-in-grothendieck-categories",
        "locus": "Theorems 4.1,5.1; Proposition 5.2",
        "correspondence": "exact_required_scope; stronger arbitrary-ring results specialize to commutative cases",
        "candidate_path": "course/src/k-injective-resolutions-in-grothendieck-categories.md",
        "sha256": "75118C6AB567222AEB5E42C60AA5DC206466FFF9EA8A9F193EA1B598DECD760B"
      },
      {
        "provider": "sheaves-of-modules-on-a-ringed-space",
        "locus": "Theorem 4.4; Lemma 5.1",
        "correspondence": "exact_required_scope; stronger arbitrary-ring results specialize to commutative cases",
        "candidate_path": "course/src/sheaves-of-modules-on-a-ringed-space.md",
        "sha256": "22FCF2E4E2E126FE922EBD68FD81F3DEDC581ED2B70CBB60A49250E21DA7BABE"
      },
      {
        "provider": "injective-modules-and-bounded-below-derived-functors",
        "locus": "Theorems 3.3,4.1",
        "correspondence": "exact_required_scope; stronger arbitrary-ring results specialize to commutative cases",
        "candidate_path": "course/src/injective-modules-and-bounded-below-derived-functors.md",
        "sha256": "B2DC989EFBC8D2D4004C8D24CBC3004A9E81DA485CD702BB7D293A7F8AF5EDBA"
      },
      {
        "provider": "complexes-cones-and-localization",
        "locus": "Theorems 4.2, 5.1; Proposition 5.2",
        "correspondence": "exact_required_scope; stronger arbitrary-ring results specialize to commutative cases",
        "use_scope": "Exact textual uses in the complete lesson; conservative prerequisite envelope remains distinct.",
        "candidate_path": "course/src/complexes-cones-and-localization.md",
        "sha256": "367394AE4A25DCF64DF87302304D8395AF84D92765E4E4E45A2333A0071C3862"
      },
      {
        "provider": "flat-modules-and-k-flat-resolutions",
        "correspondence": "exact_required_scope",
        "locus": "Lemmas 2.2, 2.4; Corollary 4.4",
        "use_scope": "Exact textual uses in the complete lesson; conservative prerequisite envelope remains distinct.",
        "candidate_path": "course/src/flat-modules-and-k-flat-resolutions.md",
        "sha256": "7BCE2B46408B278A147517161F09AE29A7BC0273B4E271AA2F995966C1AA941D"
      }
    ],
    "proof": {
      "source": "src/derived-pullback-and-pushforward.md",
      "sha256": "fece9386a64e3298cd133f9ed65d4d74aac82b9a8b266fb5ccc6363c8c063ab2",
      "locus": "Theorem 1.1, Proposition 1.2, Lemma 2.1, Theorem 2.2 and Proposition 3.1",
      "reader": "derived-pullback-and-pushforward.html#1-derived-pullback-on-k-flat-models"
    },
    "licence": "CC0-1.0",
    "proof_status": "complete; writing-AI self-checked"
  },
  {
    "result_id": "SH-DC.local-cohomology-comparisons",
    "name": "Cohomology on opens and local derived direct images admit their full restriction and sheafification descriptions; module and underlying-abelian unbounded cohomology/pushforward agree.",
    "statement": "Cohomology on opens and local derived direct images admit their full restriction and sheafification descriptions; module and underlying-abelian unbounded cohomology/pushforward agree.",
    "conditions": "Exact forgetting alone does not preserve K-injectivity; coefficient comparison uses the derived identity-space adjunction and works with integer torsion.",
    "writing_ai_selfcheck": true,
    "independent_review": false,
    "formalization": "not_formalized",
    "prerequisite_envelope_ids": [
      "SH-DC.modules-abelian",
      "SH-DC.filtered-colimits-sums",
      "SH-DC.pullback-adjunction",
      "SH-DC.extension-skyscraper",
      "SH-DC.homotopy-localization",
      "SH-DC.injective-embeddings",
      "SH-DC.flasque-acyclic",
      "SH-DC.K-injective-comparison",
      "SH-DC.bounded-derived",
      "SH-DC.Grothendieck-injective-embeddings",
      "SH-DC.unbounded-K-injective-existence",
      "SH-DC.unbounded-right-derived",
      "SH-DC.derived-products",
      "SH-DC.flat-module-sheaves",
      "SH-DC.K-flat-existence",
      "SH-DC.K-flat-calculus",
      "SH-DC.derived-tensor",
      "SH-DC.Tor-flatness"
    ],
    "dependency_precision": "Conservative earlier-provider envelope; exact textual uses appear in programme_dependency_loci. Same-lesson preliminary lemmas are inside the full proof locus.",
    "programme_dependency_loci": [
      {
        "provider": "derived-tensor-products-and-tor-sheaves",
        "locus": "Lemmas 1.1,2.1; Theorems 1.2,2.2",
        "correspondence": "exact_required_scope; stronger arbitrary-ring results specialize to commutative cases",
        "candidate_path": "course/src/derived-tensor-products-and-tor-sheaves.md",
        "sha256": "B78C1AD9864901349E26C6EB3954B046E40B84E35E55EA6B3A71C553E185D292"
      },
      {
        "provider": "k-injective-resolutions-in-grothendieck-categories",
        "locus": "Theorems 4.1,5.1; Proposition 5.2",
        "correspondence": "exact_required_scope; stronger arbitrary-ring results specialize to commutative cases",
        "candidate_path": "course/src/k-injective-resolutions-in-grothendieck-categories.md",
        "sha256": "75118C6AB567222AEB5E42C60AA5DC206466FFF9EA8A9F193EA1B598DECD760B"
      },
      {
        "provider": "sheaves-of-modules-on-a-ringed-space",
        "locus": "Theorem 4.4; Lemma 5.1",
        "correspondence": "exact_required_scope; stronger arbitrary-ring results specialize to commutative cases",
        "candidate_path": "course/src/sheaves-of-modules-on-a-ringed-space.md",
        "sha256": "22FCF2E4E2E126FE922EBD68FD81F3DEDC581ED2B70CBB60A49250E21DA7BABE"
      },
      {
        "provider": "injective-modules-and-bounded-below-derived-functors",
        "locus": "Theorems 3.3,4.1",
        "correspondence": "exact_required_scope; stronger arbitrary-ring results specialize to commutative cases",
        "candidate_path": "course/src/injective-modules-and-bounded-below-derived-functors.md",
        "sha256": "B2DC989EFBC8D2D4004C8D24CBC3004A9E81DA485CD702BB7D293A7F8AF5EDBA"
      },
      {
        "provider": "complexes-cones-and-localization",
        "locus": "Theorems 4.2, 5.1; Proposition 5.2",
        "correspondence": "exact_required_scope; stronger arbitrary-ring results specialize to commutative cases",
        "use_scope": "Exact textual uses in the complete lesson; conservative prerequisite envelope remains distinct.",
        "candidate_path": "course/src/complexes-cones-and-localization.md",
        "sha256": "367394AE4A25DCF64DF87302304D8395AF84D92765E4E4E45A2333A0071C3862"
      },
      {
        "provider": "flat-modules-and-k-flat-resolutions",
        "correspondence": "exact_required_scope",
        "locus": "Lemmas 2.2, 2.4; Corollary 4.4",
        "use_scope": "Exact textual uses in the complete lesson; conservative prerequisite envelope remains distinct.",
        "candidate_path": "course/src/flat-modules-and-k-flat-resolutions.md",
        "sha256": "7BCE2B46408B278A147517161F09AE29A7BC0273B4E271AA2F995966C1AA941D"
      }
    ],
    "proof": {
      "source": "src/derived-pullback-and-pushforward.md",
      "sha256": "fece9386a64e3298cd133f9ed65d4d74aac82b9a8b266fb5ccc6363c8c063ab2",
      "locus": "Proposition 3.2 and Theorem 4.1",
      "reader": "derived-pullback-and-pushforward.html#3-composition-leray-and-the-local-descriptions"
    },
    "licence": "CC0-1.0",
    "proof_status": "complete; writing-AI self-checked"
  },
  {
    "result_id": "SH-DC.Leray-spectral-sequence",
    "name": "For a module sheaf M, E2^(p,q)=H^p(Y,R^q f_*M) converges to H^(p+q)(X,M), with a finite filtration in each total degree and differential (r,1-r).",
    "statement": "For a module sheaf M, E2^(p,q)=H^p(Y,R^q f_*M) converges to H^(p+q)(X,M), with a finite filtration in each total degree and differential (r,1-r).",
    "conditions": "First-quadrant sequence for sheaves in degree zero; no asserted unconditional convergence for arbitrary two-sided unbounded inputs.",
    "writing_ai_selfcheck": true,
    "independent_review": false,
    "formalization": "not_formalized",
    "prerequisite_envelope_ids": [
      "SH-DC.modules-abelian",
      "SH-DC.filtered-colimits-sums",
      "SH-DC.pullback-adjunction",
      "SH-DC.extension-skyscraper",
      "SH-DC.homotopy-localization",
      "SH-DC.injective-embeddings",
      "SH-DC.flasque-acyclic",
      "SH-DC.K-injective-comparison",
      "SH-DC.bounded-derived",
      "SH-DC.Grothendieck-injective-embeddings",
      "SH-DC.unbounded-K-injective-existence",
      "SH-DC.unbounded-right-derived",
      "SH-DC.derived-products",
      "SH-DC.flat-module-sheaves",
      "SH-DC.K-flat-existence",
      "SH-DC.K-flat-calculus",
      "SH-DC.derived-tensor",
      "SH-DC.Tor-flatness"
    ],
    "dependency_precision": "Conservative earlier-provider envelope; exact textual uses appear in programme_dependency_loci. Same-lesson preliminary lemmas are inside the full proof locus.",
    "programme_dependency_loci": [
      {
        "provider": "derived-tensor-products-and-tor-sheaves",
        "locus": "Lemmas 1.1,2.1; Theorems 1.2,2.2",
        "correspondence": "exact_required_scope; stronger arbitrary-ring results specialize to commutative cases",
        "candidate_path": "course/src/derived-tensor-products-and-tor-sheaves.md",
        "sha256": "B78C1AD9864901349E26C6EB3954B046E40B84E35E55EA6B3A71C553E185D292"
      },
      {
        "provider": "k-injective-resolutions-in-grothendieck-categories",
        "locus": "Theorems 4.1,5.1; Proposition 5.2",
        "correspondence": "exact_required_scope; stronger arbitrary-ring results specialize to commutative cases",
        "candidate_path": "course/src/k-injective-resolutions-in-grothendieck-categories.md",
        "sha256": "75118C6AB567222AEB5E42C60AA5DC206466FFF9EA8A9F193EA1B598DECD760B"
      },
      {
        "provider": "sheaves-of-modules-on-a-ringed-space",
        "locus": "Theorem 4.4; Lemma 5.1",
        "correspondence": "exact_required_scope; stronger arbitrary-ring results specialize to commutative cases",
        "candidate_path": "course/src/sheaves-of-modules-on-a-ringed-space.md",
        "sha256": "22FCF2E4E2E126FE922EBD68FD81F3DEDC581ED2B70CBB60A49250E21DA7BABE"
      },
      {
        "provider": "injective-modules-and-bounded-below-derived-functors",
        "locus": "Theorems 3.3,4.1",
        "correspondence": "exact_required_scope; stronger arbitrary-ring results specialize to commutative cases",
        "candidate_path": "course/src/injective-modules-and-bounded-below-derived-functors.md",
        "sha256": "B2DC989EFBC8D2D4004C8D24CBC3004A9E81DA485CD702BB7D293A7F8AF5EDBA"
      },
      {
        "provider": "complexes-cones-and-localization",
        "locus": "Theorems 4.2, 5.1; Proposition 5.2",
        "correspondence": "exact_required_scope; stronger arbitrary-ring results specialize to commutative cases",
        "use_scope": "Exact textual uses in the complete lesson; conservative prerequisite envelope remains distinct.",
        "candidate_path": "course/src/complexes-cones-and-localization.md",
        "sha256": "367394AE4A25DCF64DF87302304D8395AF84D92765E4E4E45A2333A0071C3862"
      },
      {
        "provider": "flat-modules-and-k-flat-resolutions",
        "correspondence": "exact_required_scope",
        "locus": "Lemmas 2.2, 2.4; Corollary 4.4",
        "use_scope": "Exact textual uses in the complete lesson; conservative prerequisite envelope remains distinct.",
        "candidate_path": "course/src/flat-modules-and-k-flat-resolutions.md",
        "sha256": "7BCE2B46408B278A147517161F09AE29A7BC0273B4E271AA2F995966C1AA941D"
      }
    ],
    "proof": {
      "source": "src/derived-pullback-and-pushforward.md",
      "sha256": "fece9386a64e3298cd133f9ed65d4d74aac82b9a8b266fb5ccc6363c8c063ab2",
      "locus": "Proposition 3.1 and Exercise 3 solution",
      "reader": "derived-pullback-and-pushforward.html#6-exercises-with-checked-solutions"
    },
    "licence": "CC0-1.0",
    "proof_status": "complete; writing-AI self-checked"
  },
  {
    "result_id": "SH-DC.internal-derived-Hom",
    "name": "Internal derived Hom is well defined and exact in both variables on unbounded complexes; it commutes with open restriction, satisfies internal and map tensor-Hom adjunctions, and its local hypercohomology equals all shifted local derived-map groups.",
    "statement": "Internal derived Hom is well defined and exact in both variables on unbounded complexes; it commutes with open restriction, satisfies internal and map tensor-Hom adjunctions, and its local hypercohomology equals all shifted local derived-map groups.",
    "conditions": "Arbitrary commutative ringed space; Hom uses products, tensor uses sums. No unrestricted stalk-Hom comparison or exactness of sheaf products.",
    "writing_ai_selfcheck": true,
    "independent_review": false,
    "formalization": "not_formalized",
    "prerequisite_envelope_ids": [
      "SH-DC.modules-abelian",
      "SH-DC.filtered-colimits-sums",
      "SH-DC.pullback-adjunction",
      "SH-DC.extension-skyscraper",
      "SH-DC.homotopy-localization",
      "SH-DC.injective-embeddings",
      "SH-DC.flasque-acyclic",
      "SH-DC.K-injective-comparison",
      "SH-DC.bounded-derived",
      "SH-DC.Grothendieck-injective-embeddings",
      "SH-DC.unbounded-K-injective-existence",
      "SH-DC.unbounded-right-derived",
      "SH-DC.derived-products",
      "SH-DC.flat-module-sheaves",
      "SH-DC.K-flat-existence",
      "SH-DC.K-flat-calculus",
      "SH-DC.derived-tensor",
      "SH-DC.Tor-flatness",
      "SH-DC.derived-pull-push-adjunction",
      "SH-DC.local-cohomology-comparisons",
      "SH-DC.Leray-spectral-sequence"
    ],
    "dependency_precision": "Conservative earlier-provider envelope; exact textual uses appear in programme_dependency_loci. Same-lesson preliminary lemmas are inside the full proof locus.",
    "programme_dependency_loci": [
      {
        "provider": "derived-pullback-and-pushforward",
        "locus": "Proposition 3.2; Theorem 4.1",
        "correspondence": "exact_required_scope; stronger arbitrary-ring results specialize to commutative cases",
        "candidate_path": "course/src/derived-pullback-and-pushforward.md",
        "sha256": "FECE9386A64E3298CD133F9ED65D4D74AAC82B9A8B266FB5CCC6363C8C063AB2"
      },
      {
        "provider": "derived-tensor-products-and-tor-sheaves",
        "locus": "Lemmas 1.1,2.1; Theorem 2.2",
        "correspondence": "exact_required_scope; stronger arbitrary-ring results specialize to commutative cases",
        "candidate_path": "course/src/derived-tensor-products-and-tor-sheaves.md",
        "sha256": "B78C1AD9864901349E26C6EB3954B046E40B84E35E55EA6B3A71C553E185D292"
      },
      {
        "provider": "k-injective-resolutions-in-grothendieck-categories",
        "locus": "Theorems 4.1,5.1",
        "correspondence": "exact_required_scope; stronger arbitrary-ring results specialize to commutative cases",
        "candidate_path": "course/src/k-injective-resolutions-in-grothendieck-categories.md",
        "sha256": "75118C6AB567222AEB5E42C60AA5DC206466FFF9EA8A9F193EA1B598DECD760B"
      },
      {
        "provider": "injective-modules-and-bounded-below-derived-functors",
        "locus": "Lemmas 2.1, 3.1; Theorem 3.3",
        "correspondence": "exact_required_scope; stronger arbitrary-ring results specialize to commutative cases",
        "use_scope": "Exact textual uses in the complete lesson; conservative prerequisite envelope remains distinct.",
        "candidate_path": "course/src/injective-modules-and-bounded-below-derived-functors.md",
        "sha256": "B2DC989EFBC8D2D4004C8D24CBC3004A9E81DA485CD702BB7D293A7F8AF5EDBA"
      },
      {
        "provider": "complexes-cones-and-localization",
        "locus": "Theorems 3.1,4.2",
        "correspondence": "exact_required_scope; stronger arbitrary-ring results specialize to commutative cases",
        "candidate_path": "course/src/complexes-cones-and-localization.md",
        "sha256": "367394AE4A25DCF64DF87302304D8395AF84D92765E4E4E45A2333A0071C3862"
      },
      {
        "provider": "sheaves-of-modules-on-a-ringed-space",
        "correspondence": "exact_required_scope",
        "locus": "Lemma 5.1",
        "use_scope": "Exact textual uses in the complete lesson; conservative prerequisite envelope remains distinct.",
        "candidate_path": "course/src/sheaves-of-modules-on-a-ringed-space.md",
        "sha256": "22FCF2E4E2E126FE922EBD68FD81F3DEDC581ED2B70CBB60A49250E21DA7BABE"
      }
    ],
    "proof": {
      "source": "src/internal-derived-hom-and-ext-sheaves.md",
      "sha256": "b5b5e565b53e99059d5992f69dd576fa7d9fb771366c844afea3d09f1aa8c825",
      "locus": "Lemmas 1.1 and 2.1-2.3; Theorems 2.2 and 3.1; Section 3",
      "reader": "internal-derived-hom-and-ext-sheaves.html#2-constructing-internal-derived-hom"
    },
    "licence": "CC0-1.0",
    "proof_status": "complete; writing-AI self-checked"
  },
  {
    "result_id": "SH-DC.composition-dinaturality",
    "name": "Derived composition is associative and unital, natural in its outside variables and dinatural in the middle; evaluation, tensor-Hom and dual maps are canonical, with their exact variances.",
    "statement": "Derived composition is associative and unital, natural in its outside variables and dinatural in the middle; evaluation, tensor-Hom and dual maps are canonical, with their exact variances.",
    "conditions": "Arbitrary unbounded complexes over a commutative ringed space; the dual map is not asserted to be an isomorphism without perfectness.",
    "writing_ai_selfcheck": true,
    "independent_review": false,
    "formalization": "not_formalized",
    "prerequisite_envelope_ids": [
      "SH-DC.modules-abelian",
      "SH-DC.filtered-colimits-sums",
      "SH-DC.pullback-adjunction",
      "SH-DC.extension-skyscraper",
      "SH-DC.homotopy-localization",
      "SH-DC.injective-embeddings",
      "SH-DC.flasque-acyclic",
      "SH-DC.K-injective-comparison",
      "SH-DC.bounded-derived",
      "SH-DC.Grothendieck-injective-embeddings",
      "SH-DC.unbounded-K-injective-existence",
      "SH-DC.unbounded-right-derived",
      "SH-DC.derived-products",
      "SH-DC.flat-module-sheaves",
      "SH-DC.K-flat-existence",
      "SH-DC.K-flat-calculus",
      "SH-DC.derived-tensor",
      "SH-DC.Tor-flatness",
      "SH-DC.derived-pull-push-adjunction",
      "SH-DC.local-cohomology-comparisons",
      "SH-DC.Leray-spectral-sequence"
    ],
    "dependency_precision": "Conservative earlier-provider envelope; exact textual uses appear in programme_dependency_loci. Same-lesson preliminary lemmas are inside the full proof locus.",
    "programme_dependency_loci": [
      {
        "provider": "derived-pullback-and-pushforward",
        "locus": "Proposition 3.2; Theorem 4.1",
        "correspondence": "exact_required_scope; stronger arbitrary-ring results specialize to commutative cases",
        "candidate_path": "course/src/derived-pullback-and-pushforward.md",
        "sha256": "FECE9386A64E3298CD133F9ED65D4D74AAC82B9A8B266FB5CCC6363C8C063AB2"
      },
      {
        "provider": "derived-tensor-products-and-tor-sheaves",
        "locus": "Lemmas 1.1,2.1; Theorem 2.2",
        "correspondence": "exact_required_scope; stronger arbitrary-ring results specialize to commutative cases",
        "candidate_path": "course/src/derived-tensor-products-and-tor-sheaves.md",
        "sha256": "B78C1AD9864901349E26C6EB3954B046E40B84E35E55EA6B3A71C553E185D292"
      },
      {
        "provider": "k-injective-resolutions-in-grothendieck-categories",
        "locus": "Theorems 4.1,5.1",
        "correspondence": "exact_required_scope; stronger arbitrary-ring results specialize to commutative cases",
        "candidate_path": "course/src/k-injective-resolutions-in-grothendieck-categories.md",
        "sha256": "75118C6AB567222AEB5E42C60AA5DC206466FFF9EA8A9F193EA1B598DECD760B"
      },
      {
        "provider": "injective-modules-and-bounded-below-derived-functors",
        "locus": "Lemmas 2.1, 3.1; Theorem 3.3",
        "correspondence": "exact_required_scope; stronger arbitrary-ring results specialize to commutative cases",
        "use_scope": "Exact textual uses in the complete lesson; conservative prerequisite envelope remains distinct.",
        "candidate_path": "course/src/injective-modules-and-bounded-below-derived-functors.md",
        "sha256": "B2DC989EFBC8D2D4004C8D24CBC3004A9E81DA485CD702BB7D293A7F8AF5EDBA"
      },
      {
        "provider": "complexes-cones-and-localization",
        "locus": "Theorems 3.1,4.2",
        "correspondence": "exact_required_scope; stronger arbitrary-ring results specialize to commutative cases",
        "candidate_path": "course/src/complexes-cones-and-localization.md",
        "sha256": "367394AE4A25DCF64DF87302304D8395AF84D92765E4E4E45A2333A0071C3862"
      },
      {
        "provider": "sheaves-of-modules-on-a-ringed-space",
        "correspondence": "exact_required_scope",
        "locus": "Lemma 5.1",
        "use_scope": "Exact textual uses in the complete lesson; conservative prerequisite envelope remains distinct.",
        "candidate_path": "course/src/sheaves-of-modules-on-a-ringed-space.md",
        "sha256": "22FCF2E4E2E126FE922EBD68FD81F3DEDC581ED2B70CBB60A49250E21DA7BABE"
      }
    ],
    "proof": {
      "source": "src/internal-derived-hom-and-ext-sheaves.md",
      "sha256": "b5b5e565b53e99059d5992f69dd576fa7d9fb771366c844afea3d09f1aa8c825",
      "locus": "Propositions 4.1-4.2 and Exercise 3",
      "reader": "internal-derived-hom-and-ext-sheaves.html#4-evaluation-composition-and-their-variances"
    },
    "licence": "CC0-1.0",
    "proof_status": "complete; writing-AI self-checked"
  },
  {
    "result_id": "SH-DC.internal-Hom-examples",
    "name": "RHom(j!O_U,F)=Rj*(F|U); analytic-curve residue self Hom has cohomology in degrees 0,1; Hom-stalk comparison can fail injectivity; dual-number residue canonical dual map fails even in degree zero.",
    "statement": "RHom(j!O_U,F)=Rj*(F|U); analytic-curve residue self Hom has cohomology in degrees 0,1; Hom-stalk comparison can fail injectivity; dual-number residue canonical dual map fails even in degree zero.",
    "conditions": "Open extension on any commutative ringed space; analytic example over holomorphic functions on C; explicit convergent-sequence and dual-number counterexamples have their stated coefficients.",
    "writing_ai_selfcheck": true,
    "independent_review": false,
    "formalization": "not_formalized",
    "prerequisite_envelope_ids": [
      "SH-DC.modules-abelian",
      "SH-DC.filtered-colimits-sums",
      "SH-DC.pullback-adjunction",
      "SH-DC.extension-skyscraper",
      "SH-DC.homotopy-localization",
      "SH-DC.injective-embeddings",
      "SH-DC.flasque-acyclic",
      "SH-DC.K-injective-comparison",
      "SH-DC.bounded-derived",
      "SH-DC.Grothendieck-injective-embeddings",
      "SH-DC.unbounded-K-injective-existence",
      "SH-DC.unbounded-right-derived",
      "SH-DC.derived-products",
      "SH-DC.flat-module-sheaves",
      "SH-DC.K-flat-existence",
      "SH-DC.K-flat-calculus",
      "SH-DC.derived-tensor",
      "SH-DC.Tor-flatness",
      "SH-DC.derived-pull-push-adjunction",
      "SH-DC.local-cohomology-comparisons",
      "SH-DC.Leray-spectral-sequence",
      "SH-DC.holomorphic-coordinate-germs"
    ],
    "dependency_precision": "Conservative earlier-provider envelope; exact textual uses appear in programme_dependency_loci. Same-lesson preliminary lemmas are inside the full proof locus.",
    "programme_dependency_loci": [
      {
        "provider": "derived-pullback-and-pushforward",
        "locus": "Proposition 3.2; Theorem 4.1",
        "correspondence": "exact_required_scope; stronger arbitrary-ring results specialize to commutative cases",
        "candidate_path": "course/src/derived-pullback-and-pushforward.md",
        "sha256": "FECE9386A64E3298CD133F9ED65D4D74AAC82B9A8B266FB5CCC6363C8C063AB2"
      },
      {
        "provider": "derived-tensor-products-and-tor-sheaves",
        "locus": "Lemmas 1.1,2.1; Theorem 2.2",
        "correspondence": "exact_required_scope; stronger arbitrary-ring results specialize to commutative cases",
        "candidate_path": "course/src/derived-tensor-products-and-tor-sheaves.md",
        "sha256": "B78C1AD9864901349E26C6EB3954B046E40B84E35E55EA6B3A71C553E185D292"
      },
      {
        "provider": "k-injective-resolutions-in-grothendieck-categories",
        "locus": "Theorems 4.1,5.1",
        "correspondence": "exact_required_scope; stronger arbitrary-ring results specialize to commutative cases",
        "candidate_path": "course/src/k-injective-resolutions-in-grothendieck-categories.md",
        "sha256": "75118C6AB567222AEB5E42C60AA5DC206466FFF9EA8A9F193EA1B598DECD760B"
      },
      {
        "provider": "injective-modules-and-bounded-below-derived-functors",
        "locus": "Lemmas 2.1, 3.1; Theorem 3.3",
        "correspondence": "exact_required_scope; stronger arbitrary-ring results specialize to commutative cases",
        "use_scope": "Exact textual uses in the complete lesson; conservative prerequisite envelope remains distinct.",
        "candidate_path": "course/src/injective-modules-and-bounded-below-derived-functors.md",
        "sha256": "B2DC989EFBC8D2D4004C8D24CBC3004A9E81DA485CD702BB7D293A7F8AF5EDBA"
      },
      {
        "provider": "complexes-cones-and-localization",
        "locus": "Theorems 3.1,4.2",
        "correspondence": "exact_required_scope; stronger arbitrary-ring results specialize to commutative cases",
        "candidate_path": "course/src/complexes-cones-and-localization.md",
        "sha256": "367394AE4A25DCF64DF87302304D8395AF84D92765E4E4E45A2333A0071C3862"
      },
      {
        "provider": "sheaves-of-modules-on-a-ringed-space",
        "correspondence": "exact_required_scope",
        "locus": "Lemma 5.1",
        "use_scope": "Exact textual uses in the complete lesson; conservative prerequisite envelope remains distinct.",
        "candidate_path": "course/src/sheaves-of-modules-on-a-ringed-space.md",
        "sha256": "22FCF2E4E2E126FE922EBD68FD81F3DEDC581ED2B70CBB60A49250E21DA7BABE"
      },
      {
        "provider": "SH-DC.holomorphic-coordinate-germs",
        "candidate_path": "course/src/derived-tensor-products-and-tor-sheaves.md",
        "locus": "Section 4, Lemma (holomorphic coordinate germs), full proof",
        "anchor": "holomorphic-coordinate-germs",
        "sha256": "B78C1AD9864901349E26C6EB3954B046E40B84E35E55EA6B3A71C553E185D292",
        "correspondence": "Holomorphic coordinate germ calculation, exact curve/surface scope."
      }
    ],
    "proof": {
      "source": "src/internal-derived-hom-and-ext-sheaves.md",
      "sha256": "b5b5e565b53e99059d5992f69dd576fa7d9fb771366c844afea3d09f1aa8c825",
      "locus": "Section 5 and Exercise 5",
      "reader": "internal-derived-hom-and-ext-sheaves.html#5-worked-examples"
    },
    "licence": "CC0-1.0",
    "proof_status": "complete; writing-AI self-checked"
  },
  {
    "result_id": "SH-DC.localization-support",
    "name": "Unbounded closed support has derived adjunction to exact closed direct image, local-to-global composition, natural sheaf/global localization triangles and supported module-Ab comparisons; locally closed pairs have the corresponding natural triangle.",
    "statement": "Unbounded closed support has derived adjunction to exact closed direct image, local-to-global composition, natural sheaf/global localization triangles and supported module-Ab comparisons; locally closed pairs have the corresponding natural triangle.",
    "conditions": "Any sheaf of unital rings; unchanged closed-set coefficients; K-injective resolutions with injective terms; negative cone projection gives positive lift boundary.",
    "writing_ai_selfcheck": true,
    "independent_review": false,
    "formalization": "not_formalized",
    "prerequisite_envelope_ids": [
      "SH-DC.modules-abelian",
      "SH-DC.filtered-colimits-sums",
      "SH-DC.pullback-adjunction",
      "SH-DC.extension-skyscraper",
      "SH-DC.homotopy-localization",
      "SH-DC.injective-embeddings",
      "SH-DC.flasque-acyclic",
      "SH-DC.K-injective-comparison",
      "SH-DC.bounded-derived",
      "SH-DC.Grothendieck-injective-embeddings",
      "SH-DC.unbounded-K-injective-existence",
      "SH-DC.unbounded-right-derived",
      "SH-DC.derived-products",
      "SH-DC.derived-tensor",
      "SH-DC.Tor-flatness",
      "SH-DC.derived-pull-push-adjunction",
      "SH-DC.local-cohomology-comparisons",
      "SH-DC.Leray-spectral-sequence",
      "SH-DC.internal-derived-Hom",
      "SH-DC.composition-dinaturality",
      "SH-DC.internal-Hom-examples",
      "SH-DC.K-flat-calculus"
    ],
    "dependency_precision": "Conservative earlier-provider envelope; exact textual uses appear in programme_dependency_loci. Same-lesson preliminary lemmas are inside the full proof locus.",
    "programme_dependency_loci": [
      {
        "provider": "sheaves-of-modules-on-a-ringed-space",
        "locus": "Theorems 2.1,4.4",
        "correspondence": "exact_required_scope; stronger arbitrary-ring results specialize to commutative cases",
        "candidate_path": "course/src/sheaves-of-modules-on-a-ringed-space.md",
        "sha256": "22FCF2E4E2E126FE922EBD68FD81F3DEDC581ED2B70CBB60A49250E21DA7BABE"
      },
      {
        "provider": "injective-modules-and-bounded-below-derived-functors",
        "locus": "Lemma 1.1; Proposition 1.3; Theorem 4.1(4)",
        "correspondence": "exact_required_scope; stronger arbitrary-ring results specialize to commutative cases",
        "candidate_path": "course/src/injective-modules-and-bounded-below-derived-functors.md",
        "sha256": "B2DC989EFBC8D2D4004C8D24CBC3004A9E81DA485CD702BB7D293A7F8AF5EDBA"
      },
      {
        "provider": "k-injective-resolutions-in-grothendieck-categories",
        "locus": "Theorems 4.1,5.1",
        "correspondence": "exact_required_scope; stronger arbitrary-ring results specialize to commutative cases",
        "candidate_path": "course/src/k-injective-resolutions-in-grothendieck-categories.md",
        "sha256": "75118C6AB567222AEB5E42C60AA5DC206466FFF9EA8A9F193EA1B598DECD760B"
      },
      {
        "provider": "derived-pullback-and-pushforward",
        "locus": "Theorem 1.1; Proposition 3.2; Theorem 4.1",
        "correspondence": "exact_required_scope; stronger arbitrary-ring results specialize to commutative cases",
        "use_scope": "Exact textual uses in the complete lesson; conservative prerequisite envelope remains distinct.",
        "candidate_path": "course/src/derived-pullback-and-pushforward.md",
        "sha256": "FECE9386A64E3298CD133F9ED65D4D74AAC82B9A8B266FB5CCC6363C8C063AB2"
      },
      {
        "provider": "complexes-cones-and-localization",
        "locus": "Theorem 3.1; Proposition 5.2",
        "correspondence": "exact_required_scope; stronger arbitrary-ring results specialize to commutative cases",
        "candidate_path": "course/src/complexes-cones-and-localization.md",
        "sha256": "367394AE4A25DCF64DF87302304D8395AF84D92765E4E4E45A2333A0071C3862"
      },
      {
        "provider": "derived-tensor-products-and-tor-sheaves",
        "locus": "Lemma 1.1; Theorem 2.2 for cup products",
        "correspondence": "exact_required_scope; stronger arbitrary-ring results specialize to commutative cases",
        "use_scope": "Exact textual uses in the complete lesson; conservative prerequisite envelope remains distinct.",
        "candidate_path": "course/src/derived-tensor-products-and-tor-sheaves.md",
        "sha256": "B78C1AD9864901349E26C6EB3954B046E40B84E35E55EA6B3A71C553E185D292"
      },
      {
        "provider": "internal-derived-hom-and-ext-sheaves",
        "locus": "Theorem 3.1 for cup products",
        "correspondence": "exact_required_scope; stronger arbitrary-ring results specialize to commutative cases",
        "candidate_path": "course/src/internal-derived-hom-and-ext-sheaves.md",
        "sha256": "B5B5E565B53E99059D5992F69DD576FA7D9FB771366C844AFEA3D09F1AA8C825"
      },
      {
        "provider": "flat-modules-and-k-flat-resolutions",
        "correspondence": "exact_required_scope",
        "locus": "Lemma 2.2",
        "use_scope": "Exact textual uses in the complete lesson; conservative prerequisite envelope remains distinct.",
        "candidate_path": "course/src/flat-modules-and-k-flat-resolutions.md",
        "sha256": "7BCE2B46408B278A147517161F09AE29A7BC0273B4E271AA2F995966C1AA941D"
      }
    ],
    "proof": {
      "source": "src/sections-with-support-and-the-localization-triangle.md",
      "sha256": "d5e0f4079d0a4a0dedd36622b944e75d6072b3781976b6c8328271997ab1282a",
      "locus": "Proposition 1.1, Theorem 2.1, Proposition 2.2 and Proposition 3.1",
      "reader": "sections-with-support-and-the-localization-triangle.html#1-closed-supports-and-their-adjunction"
    },
    "licence": "CC0-1.0",
    "proof_status": "complete; writing-AI self-checked"
  },
  {
    "result_id": "SH-DC.Euclidean-point-support",
    "name": "R Gamma_{0}(R^n,Z)=Z[-n], including n=0; ambient local support sheaf is the point skyscraper in degree n. In dimension one delta(a,b)=b-a for negative/positive rays.",
    "statement": "R Gamma_{0}(R^n,Z)=Z[-n], including n=0; ambient local support sheaf is the point skyscraper in degree n. In dimension one delta(a,b)=b-a for negative/positive rays.",
    "conditions": "Euclidean topology and constant integer coefficients; singular comparison proved for all open Euclidean sets and arbitrary abelian coefficient groups.",
    "writing_ai_selfcheck": true,
    "independent_review": false,
    "formalization": "not_formalized",
    "prerequisite_envelope_ids": [
      "SH-DC.modules-abelian",
      "SH-DC.filtered-colimits-sums",
      "SH-DC.pullback-adjunction",
      "SH-DC.extension-skyscraper",
      "SH-DC.homotopy-localization",
      "SH-DC.injective-embeddings",
      "SH-DC.flasque-acyclic",
      "SH-DC.K-injective-comparison",
      "SH-DC.bounded-derived",
      "SH-DC.Grothendieck-injective-embeddings",
      "SH-DC.unbounded-K-injective-existence",
      "SH-DC.unbounded-right-derived",
      "SH-DC.derived-products",
      "SH-DC.derived-tensor",
      "SH-DC.Tor-flatness",
      "SH-DC.derived-pull-push-adjunction",
      "SH-DC.local-cohomology-comparisons",
      "SH-DC.Leray-spectral-sequence",
      "SH-DC.internal-derived-Hom",
      "SH-DC.composition-dinaturality",
      "SH-DC.internal-Hom-examples",
      "SH-DC.K-flat-calculus"
    ],
    "dependency_precision": "Conservative earlier-provider envelope; exact textual uses appear in programme_dependency_loci. Same-lesson preliminary lemmas are inside the full proof locus.",
    "programme_dependency_loci": [
      {
        "provider": "sheaves-of-modules-on-a-ringed-space",
        "locus": "Theorems 2.1,4.4",
        "correspondence": "exact_required_scope; stronger arbitrary-ring results specialize to commutative cases",
        "candidate_path": "course/src/sheaves-of-modules-on-a-ringed-space.md",
        "sha256": "22FCF2E4E2E126FE922EBD68FD81F3DEDC581ED2B70CBB60A49250E21DA7BABE"
      },
      {
        "provider": "injective-modules-and-bounded-below-derived-functors",
        "locus": "Lemma 1.1; Proposition 1.3; Theorem 4.1(4)",
        "correspondence": "exact_required_scope; stronger arbitrary-ring results specialize to commutative cases",
        "candidate_path": "course/src/injective-modules-and-bounded-below-derived-functors.md",
        "sha256": "B2DC989EFBC8D2D4004C8D24CBC3004A9E81DA485CD702BB7D293A7F8AF5EDBA"
      },
      {
        "provider": "k-injective-resolutions-in-grothendieck-categories",
        "locus": "Theorems 4.1,5.1",
        "correspondence": "exact_required_scope; stronger arbitrary-ring results specialize to commutative cases",
        "candidate_path": "course/src/k-injective-resolutions-in-grothendieck-categories.md",
        "sha256": "75118C6AB567222AEB5E42C60AA5DC206466FFF9EA8A9F193EA1B598DECD760B"
      },
      {
        "provider": "derived-pullback-and-pushforward",
        "locus": "Theorem 1.1; Proposition 3.2; Theorem 4.1",
        "correspondence": "exact_required_scope; stronger arbitrary-ring results specialize to commutative cases",
        "use_scope": "Exact textual uses in the complete lesson; conservative prerequisite envelope remains distinct.",
        "candidate_path": "course/src/derived-pullback-and-pushforward.md",
        "sha256": "FECE9386A64E3298CD133F9ED65D4D74AAC82B9A8B266FB5CCC6363C8C063AB2"
      },
      {
        "provider": "complexes-cones-and-localization",
        "locus": "Theorem 3.1; Proposition 5.2",
        "correspondence": "exact_required_scope; stronger arbitrary-ring results specialize to commutative cases",
        "candidate_path": "course/src/complexes-cones-and-localization.md",
        "sha256": "367394AE4A25DCF64DF87302304D8395AF84D92765E4E4E45A2333A0071C3862"
      },
      {
        "provider": "derived-tensor-products-and-tor-sheaves",
        "locus": "Lemma 1.1; Theorem 2.2 for cup products",
        "correspondence": "exact_required_scope; stronger arbitrary-ring results specialize to commutative cases",
        "use_scope": "Exact textual uses in the complete lesson; conservative prerequisite envelope remains distinct.",
        "candidate_path": "course/src/derived-tensor-products-and-tor-sheaves.md",
        "sha256": "B78C1AD9864901349E26C6EB3954B046E40B84E35E55EA6B3A71C553E185D292"
      },
      {
        "provider": "internal-derived-hom-and-ext-sheaves",
        "locus": "Theorem 3.1 for cup products",
        "correspondence": "exact_required_scope; stronger arbitrary-ring results specialize to commutative cases",
        "candidate_path": "course/src/internal-derived-hom-and-ext-sheaves.md",
        "sha256": "B5B5E565B53E99059D5992F69DD576FA7D9FB771366C844AFEA3D09F1AA8C825"
      },
      {
        "provider": "flat-modules-and-k-flat-resolutions",
        "correspondence": "exact_required_scope",
        "locus": "Lemma 2.2",
        "use_scope": "Exact textual uses in the complete lesson; conservative prerequisite envelope remains distinct.",
        "candidate_path": "course/src/flat-modules-and-k-flat-resolutions.md",
        "sha256": "7BCE2B46408B278A147517161F09AE29A7BC0273B4E271AA2F995966C1AA941D"
      }
    ],
    "proof": {
      "source": "src/sections-with-support-and-the-localization-triangle.md",
      "sha256": "d5e0f4079d0a4a0dedd36622b944e75d6072b3781976b6c8328271997ab1282a",
      "locus": "Lemma 4.1, Proposition 4.2 and Exercise 3",
      "reader": "sections-with-support-and-the-localization-triangle.html#4-integer-coefficients-near-a-euclidean-point"
    },
    "licence": "CC0-1.0",
    "proof_status": "complete; writing-AI self-checked"
  },
  {
    "result_id": "SH-DC.support-cup-pullback",
    "name": "Support cup product commutes with forgetting supports and association; pullback has a natural compositional map to inverse-image supports which commutes with forgetting supports.",
    "statement": "Support cup product commutes with forgetting supports and association; pullback has a natural compositional map to inverse-image supports which commutes with forgetting supports.",
    "conditions": "Commutative coefficient sheaves; arbitrary unbounded inputs and arbitrary ringed-space morphisms; no general invertibility assertion.",
    "writing_ai_selfcheck": true,
    "independent_review": false,
    "formalization": "not_formalized",
    "prerequisite_envelope_ids": [
      "SH-DC.modules-abelian",
      "SH-DC.filtered-colimits-sums",
      "SH-DC.pullback-adjunction",
      "SH-DC.extension-skyscraper",
      "SH-DC.homotopy-localization",
      "SH-DC.injective-embeddings",
      "SH-DC.flasque-acyclic",
      "SH-DC.K-injective-comparison",
      "SH-DC.bounded-derived",
      "SH-DC.Grothendieck-injective-embeddings",
      "SH-DC.unbounded-K-injective-existence",
      "SH-DC.unbounded-right-derived",
      "SH-DC.derived-products",
      "SH-DC.derived-tensor",
      "SH-DC.Tor-flatness",
      "SH-DC.derived-pull-push-adjunction",
      "SH-DC.local-cohomology-comparisons",
      "SH-DC.Leray-spectral-sequence",
      "SH-DC.internal-derived-Hom",
      "SH-DC.composition-dinaturality",
      "SH-DC.internal-Hom-examples",
      "SH-DC.K-flat-calculus"
    ],
    "dependency_precision": "Conservative earlier-provider envelope; exact textual uses appear in programme_dependency_loci. Same-lesson preliminary lemmas are inside the full proof locus.",
    "programme_dependency_loci": [
      {
        "provider": "sheaves-of-modules-on-a-ringed-space",
        "locus": "Theorems 2.1,4.4",
        "correspondence": "exact_required_scope; stronger arbitrary-ring results specialize to commutative cases",
        "candidate_path": "course/src/sheaves-of-modules-on-a-ringed-space.md",
        "sha256": "22FCF2E4E2E126FE922EBD68FD81F3DEDC581ED2B70CBB60A49250E21DA7BABE"
      },
      {
        "provider": "injective-modules-and-bounded-below-derived-functors",
        "locus": "Lemma 1.1; Proposition 1.3; Theorem 4.1(4)",
        "correspondence": "exact_required_scope; stronger arbitrary-ring results specialize to commutative cases",
        "candidate_path": "course/src/injective-modules-and-bounded-below-derived-functors.md",
        "sha256": "B2DC989EFBC8D2D4004C8D24CBC3004A9E81DA485CD702BB7D293A7F8AF5EDBA"
      },
      {
        "provider": "k-injective-resolutions-in-grothendieck-categories",
        "locus": "Theorems 4.1,5.1",
        "correspondence": "exact_required_scope; stronger arbitrary-ring results specialize to commutative cases",
        "candidate_path": "course/src/k-injective-resolutions-in-grothendieck-categories.md",
        "sha256": "75118C6AB567222AEB5E42C60AA5DC206466FFF9EA8A9F193EA1B598DECD760B"
      },
      {
        "provider": "derived-pullback-and-pushforward",
        "locus": "Theorem 1.1; Proposition 3.2; Theorem 4.1",
        "correspondence": "exact_required_scope; stronger arbitrary-ring results specialize to commutative cases",
        "use_scope": "Exact textual uses in the complete lesson; conservative prerequisite envelope remains distinct.",
        "candidate_path": "course/src/derived-pullback-and-pushforward.md",
        "sha256": "FECE9386A64E3298CD133F9ED65D4D74AAC82B9A8B266FB5CCC6363C8C063AB2"
      },
      {
        "provider": "complexes-cones-and-localization",
        "locus": "Theorem 3.1; Proposition 5.2",
        "correspondence": "exact_required_scope; stronger arbitrary-ring results specialize to commutative cases",
        "candidate_path": "course/src/complexes-cones-and-localization.md",
        "sha256": "367394AE4A25DCF64DF87302304D8395AF84D92765E4E4E45A2333A0071C3862"
      },
      {
        "provider": "derived-tensor-products-and-tor-sheaves",
        "locus": "Lemma 1.1; Theorem 2.2 for cup products",
        "correspondence": "exact_required_scope; stronger arbitrary-ring results specialize to commutative cases",
        "use_scope": "Exact textual uses in the complete lesson; conservative prerequisite envelope remains distinct.",
        "candidate_path": "course/src/derived-tensor-products-and-tor-sheaves.md",
        "sha256": "B78C1AD9864901349E26C6EB3954B046E40B84E35E55EA6B3A71C553E185D292"
      },
      {
        "provider": "internal-derived-hom-and-ext-sheaves",
        "locus": "Theorem 3.1 for cup products",
        "correspondence": "exact_required_scope; stronger arbitrary-ring results specialize to commutative cases",
        "candidate_path": "course/src/internal-derived-hom-and-ext-sheaves.md",
        "sha256": "B5B5E565B53E99059D5992F69DD576FA7D9FB771366C844AFEA3D09F1AA8C825"
      },
      {
        "provider": "flat-modules-and-k-flat-resolutions",
        "correspondence": "exact_required_scope",
        "locus": "Lemma 2.2",
        "use_scope": "Exact textual uses in the complete lesson; conservative prerequisite envelope remains distinct.",
        "candidate_path": "course/src/flat-modules-and-k-flat-resolutions.md",
        "sha256": "7BCE2B46408B278A147517161F09AE29A7BC0273B4E271AA2F995966C1AA941D"
      }
    ],
    "proof": {
      "source": "src/sections-with-support-and-the-localization-triangle.md",
      "sha256": "d5e0f4079d0a4a0dedd36622b944e75d6072b3781976b6c8328271997ab1282a",
      "locus": "Section 5 and Exercises 4-5",
      "reader": "sections-with-support-and-the-localization-triangle.html#5-cup-products-and-support-pullback"
    },
    "licence": "CC0-1.0",
    "proof_status": "complete; writing-AI self-checked"
  },
  {
    "result_id": "SH-DC.perfect-criterion",
    "name": "Perfectness is equivalent to pseudo-coherence and locally finite Tor amplitude; amplitude[a,b] plus (a-1)-pseudo-coherence suffices for strict local models in[a,b]. Perfectness survives cones, tensor, retracts and pullback.",
    "statement": "Perfectness is equivalent to pseudo-coherence and locally finite Tor amplitude; amplitude[a,b] plus (a-1)-pseudo-coherence suffices for strict local models in[a,b]. Perfectness survives cones, tensor, retracts and pullback.",
    "conditions": "Arbitrary commutative unital ringed spaces; local intervals and covers may be nonuniform; finite projective sheaves are not asserted free unless stalk rings are local.",
    "writing_ai_selfcheck": true,
    "independent_review": false,
    "formalization": "not_formalized",
    "prerequisite_envelope_ids": [
      "SH-DC.modules-abelian",
      "SH-DC.filtered-colimits-sums",
      "SH-DC.pullback-adjunction",
      "SH-DC.extension-skyscraper",
      "SH-DC.homotopy-localization",
      "SH-DC.injective-embeddings",
      "SH-DC.flasque-acyclic",
      "SH-DC.K-injective-comparison",
      "SH-DC.bounded-derived",
      "SH-DC.Grothendieck-injective-embeddings",
      "SH-DC.unbounded-K-injective-existence",
      "SH-DC.unbounded-right-derived",
      "SH-DC.derived-products",
      "SH-DC.flat-module-sheaves",
      "SH-DC.K-flat-existence",
      "SH-DC.K-flat-calculus",
      "SH-DC.derived-tensor",
      "SH-DC.Tor-flatness",
      "SH-DC.derived-pull-push-adjunction",
      "SH-DC.local-cohomology-comparisons",
      "SH-DC.Leray-spectral-sequence",
      "SH-DC.internal-derived-Hom",
      "SH-DC.composition-dinaturality",
      "SH-DC.internal-Hom-examples",
      "SH-DC.localization-support",
      "SH-DC.Euclidean-point-support",
      "SH-DC.support-cup-pullback"
    ],
    "dependency_precision": "Conservative earlier-provider envelope; exact textual uses appear in programme_dependency_loci. Same-lesson preliminary lemmas are inside the full proof locus.",
    "programme_dependency_loci": [
      {
        "provider": "flat-modules-and-k-flat-resolutions",
        "locus": "Lemmas 1.2–1.3,2.2–2.4,4.1; Theorem 3.1",
        "correspondence": "exact_required_scope; stronger arbitrary-ring results specialize to commutative cases",
        "candidate_path": "course/src/flat-modules-and-k-flat-resolutions.md",
        "sha256": "7BCE2B46408B278A147517161F09AE29A7BC0273B4E271AA2F995966C1AA941D"
      },
      {
        "provider": "derived-tensor-products-and-tor-sheaves",
        "locus": "Theorems 2.2,3.3",
        "correspondence": "exact_required_scope; stronger arbitrary-ring results specialize to commutative cases",
        "candidate_path": "course/src/derived-tensor-products-and-tor-sheaves.md",
        "sha256": "B78C1AD9864901349E26C6EB3954B046E40B84E35E55EA6B3A71C553E185D292"
      },
      {
        "provider": "internal-derived-hom-and-ext-sheaves",
        "locus": "Lemma 1.1; Theorems 2.2,3.1; Section 4 canonical maps",
        "correspondence": "exact_required_scope; stronger arbitrary-ring results specialize to commutative cases",
        "candidate_path": "course/src/internal-derived-hom-and-ext-sheaves.md",
        "sha256": "B5B5E565B53E99059D5992F69DD576FA7D9FB771366C844AFEA3D09F1AA8C825"
      },
      {
        "provider": "derived-pullback-and-pushforward",
        "locus": "Theorem 1.1; Proposition 3.2",
        "correspondence": "exact_required_scope; stronger arbitrary-ring results specialize to commutative cases",
        "candidate_path": "course/src/derived-pullback-and-pushforward.md",
        "sha256": "FECE9386A64E3298CD133F9ED65D4D74AAC82B9A8B266FB5CCC6363C8C063AB2"
      },
      {
        "provider": "injective-modules-and-bounded-below-derived-functors",
        "locus": "Theorem 3.3",
        "correspondence": "exact_required_scope; stronger arbitrary-ring results specialize to commutative cases",
        "candidate_path": "course/src/injective-modules-and-bounded-below-derived-functors.md",
        "sha256": "B2DC989EFBC8D2D4004C8D24CBC3004A9E81DA485CD702BB7D293A7F8AF5EDBA"
      },
      {
        "provider": "complexes-cones-and-localization",
        "locus": "Cones, LES, localization roofs",
        "correspondence": "exact_required_scope; stronger arbitrary-ring results specialize to commutative cases",
        "candidate_path": "course/src/complexes-cones-and-localization.md",
        "sha256": "367394AE4A25DCF64DF87302304D8395AF84D92765E4E4E45A2333A0071C3862"
      },
      {
        "provider": "sections-with-support-and-the-localization-triangle",
        "locus": "Theorem 2.1",
        "correspondence": "exact_required_scope; stronger arbitrary-ring results specialize to commutative cases",
        "candidate_path": "course/src/sections-with-support-and-the-localization-triangle.md",
        "sha256": "058C339B17DE1604395E406A4905CE8CCB4A251AE6DB50EB4A71ACD714D67B38"
      }
    ],
    "proof": {
      "source": "src/perfect-complexes-and-duals.md",
      "sha256": "5d3a0bf017b53f29fb3668dabce086f51a909be2868ca3ea03928ff35ff94c97",
      "locus": "Propositions 2.1-2.2 and 3.1, Theorem 3.2, Corollary 3.3",
      "reader": "perfect-complexes-and-duals.html#3-tor-amplitude-and-the-perfectness-criterion"
    },
    "licence": "CC0-1.0",
    "proof_status": "complete; writing-AI self-checked"
  },
  {
    "result_id": "SH-DC.perfect-duality",
    "name": "Dualizable objects are exactly perfect objects; their dual is RHom(K,O), biduality and both coefficient tensor-Hom and evaluation-and-more maps are isomorphisms for perfect K.",
    "statement": "Dualizable objects are exactly perfect objects; their dual is RHom(K,O), biduality and both coefficient tensor-Hom and evaluation-and-more maps are isomorphisms for perfect K.",
    "conditions": "Arbitrary commutative unital ringed spaces, unbounded derived category; usual Koszul symmetry, derived tensors and Hom.",
    "writing_ai_selfcheck": true,
    "independent_review": false,
    "formalization": "not_formalized",
    "prerequisite_envelope_ids": [
      "SH-DC.modules-abelian",
      "SH-DC.filtered-colimits-sums",
      "SH-DC.pullback-adjunction",
      "SH-DC.extension-skyscraper",
      "SH-DC.homotopy-localization",
      "SH-DC.injective-embeddings",
      "SH-DC.flasque-acyclic",
      "SH-DC.K-injective-comparison",
      "SH-DC.bounded-derived",
      "SH-DC.Grothendieck-injective-embeddings",
      "SH-DC.unbounded-K-injective-existence",
      "SH-DC.unbounded-right-derived",
      "SH-DC.derived-products",
      "SH-DC.flat-module-sheaves",
      "SH-DC.K-flat-existence",
      "SH-DC.K-flat-calculus",
      "SH-DC.derived-tensor",
      "SH-DC.Tor-flatness",
      "SH-DC.derived-pull-push-adjunction",
      "SH-DC.local-cohomology-comparisons",
      "SH-DC.Leray-spectral-sequence",
      "SH-DC.internal-derived-Hom",
      "SH-DC.composition-dinaturality",
      "SH-DC.internal-Hom-examples",
      "SH-DC.localization-support",
      "SH-DC.Euclidean-point-support",
      "SH-DC.support-cup-pullback"
    ],
    "dependency_precision": "Conservative earlier-provider envelope; exact textual uses appear in programme_dependency_loci. Same-lesson preliminary lemmas are inside the full proof locus.",
    "programme_dependency_loci": [
      {
        "provider": "flat-modules-and-k-flat-resolutions",
        "locus": "Lemmas 1.2–1.3,2.2–2.4,4.1; Theorem 3.1",
        "correspondence": "exact_required_scope; stronger arbitrary-ring results specialize to commutative cases",
        "candidate_path": "course/src/flat-modules-and-k-flat-resolutions.md",
        "sha256": "7BCE2B46408B278A147517161F09AE29A7BC0273B4E271AA2F995966C1AA941D"
      },
      {
        "provider": "derived-tensor-products-and-tor-sheaves",
        "locus": "Theorems 2.2,3.3",
        "correspondence": "exact_required_scope; stronger arbitrary-ring results specialize to commutative cases",
        "candidate_path": "course/src/derived-tensor-products-and-tor-sheaves.md",
        "sha256": "B78C1AD9864901349E26C6EB3954B046E40B84E35E55EA6B3A71C553E185D292"
      },
      {
        "provider": "internal-derived-hom-and-ext-sheaves",
        "locus": "Lemma 1.1; Theorems 2.2,3.1; Section 4 canonical maps",
        "correspondence": "exact_required_scope; stronger arbitrary-ring results specialize to commutative cases",
        "candidate_path": "course/src/internal-derived-hom-and-ext-sheaves.md",
        "sha256": "B5B5E565B53E99059D5992F69DD576FA7D9FB771366C844AFEA3D09F1AA8C825"
      },
      {
        "provider": "derived-pullback-and-pushforward",
        "locus": "Theorem 1.1; Proposition 3.2",
        "correspondence": "exact_required_scope; stronger arbitrary-ring results specialize to commutative cases",
        "candidate_path": "course/src/derived-pullback-and-pushforward.md",
        "sha256": "FECE9386A64E3298CD133F9ED65D4D74AAC82B9A8B266FB5CCC6363C8C063AB2"
      },
      {
        "provider": "injective-modules-and-bounded-below-derived-functors",
        "locus": "Theorem 3.3",
        "correspondence": "exact_required_scope; stronger arbitrary-ring results specialize to commutative cases",
        "candidate_path": "course/src/injective-modules-and-bounded-below-derived-functors.md",
        "sha256": "B2DC989EFBC8D2D4004C8D24CBC3004A9E81DA485CD702BB7D293A7F8AF5EDBA"
      },
      {
        "provider": "complexes-cones-and-localization",
        "locus": "Cones, LES, localization roofs",
        "correspondence": "exact_required_scope; stronger arbitrary-ring results specialize to commutative cases",
        "candidate_path": "course/src/complexes-cones-and-localization.md",
        "sha256": "367394AE4A25DCF64DF87302304D8395AF84D92765E4E4E45A2333A0071C3862"
      },
      {
        "provider": "sections-with-support-and-the-localization-triangle",
        "locus": "Theorem 2.1",
        "correspondence": "exact_required_scope; stronger arbitrary-ring results specialize to commutative cases",
        "candidate_path": "course/src/sections-with-support-and-the-localization-triangle.md",
        "sha256": "058C339B17DE1604395E406A4905CE8CCB4A251AE6DB50EB4A71ACD714D67B38"
      }
    ],
    "proof": {
      "source": "src/perfect-complexes-and-duals.md",
      "sha256": "5d3a0bf017b53f29fb3668dabce086f51a909be2868ca3ea03928ff35ff94c97",
      "locus": "Theorems 4.1-4.2 and Exercises 1,3-5",
      "reader": "perfect-complexes-and-duals.html#4-duals-and-the-two-tensor-hom-maps"
    },
    "licence": "CC0-1.0",
    "proof_status": "complete; writing-AI self-checked"
  },
  {
    "result_id": "SH-DC.perfect-residue-examples",
    "name": "Smooth analytic surface residue has strict Koszul model and Tor amplitude[-2,0]; dual-number residue is pseudo-coherent with infinite negative tensor cohomology and is not perfect or dualizable.",
    "statement": "Smooth analytic surface residue has strict Koszul model and Tor amplitude[-2,0]; dual-number residue is pseudo-coherent with infinite negative tensor cohomology and is not perfect or dualizable.",
    "conditions": "Holomorphic-function sheaf on C^2 at origin for smooth example; one-point dual-number ring for singular example.",
    "writing_ai_selfcheck": true,
    "independent_review": false,
    "formalization": "not_formalized",
    "prerequisite_envelope_ids": [
      "SH-DC.modules-abelian",
      "SH-DC.filtered-colimits-sums",
      "SH-DC.pullback-adjunction",
      "SH-DC.extension-skyscraper",
      "SH-DC.homotopy-localization",
      "SH-DC.injective-embeddings",
      "SH-DC.flasque-acyclic",
      "SH-DC.K-injective-comparison",
      "SH-DC.bounded-derived",
      "SH-DC.Grothendieck-injective-embeddings",
      "SH-DC.unbounded-K-injective-existence",
      "SH-DC.unbounded-right-derived",
      "SH-DC.derived-products",
      "SH-DC.flat-module-sheaves",
      "SH-DC.K-flat-existence",
      "SH-DC.K-flat-calculus",
      "SH-DC.derived-tensor",
      "SH-DC.Tor-flatness",
      "SH-DC.derived-pull-push-adjunction",
      "SH-DC.local-cohomology-comparisons",
      "SH-DC.Leray-spectral-sequence",
      "SH-DC.internal-derived-Hom",
      "SH-DC.composition-dinaturality",
      "SH-DC.internal-Hom-examples",
      "SH-DC.localization-support",
      "SH-DC.Euclidean-point-support",
      "SH-DC.support-cup-pullback",
      "SH-DC.holomorphic-coordinate-germs"
    ],
    "dependency_precision": "Conservative earlier-provider envelope; exact textual uses appear in programme_dependency_loci. Same-lesson preliminary lemmas are inside the full proof locus.",
    "programme_dependency_loci": [
      {
        "provider": "flat-modules-and-k-flat-resolutions",
        "locus": "Lemmas 1.2–1.3,2.2–2.4,4.1; Theorem 3.1",
        "correspondence": "exact_required_scope; stronger arbitrary-ring results specialize to commutative cases",
        "candidate_path": "course/src/flat-modules-and-k-flat-resolutions.md",
        "sha256": "7BCE2B46408B278A147517161F09AE29A7BC0273B4E271AA2F995966C1AA941D"
      },
      {
        "provider": "derived-tensor-products-and-tor-sheaves",
        "locus": "Theorems 2.2,3.3",
        "correspondence": "exact_required_scope; stronger arbitrary-ring results specialize to commutative cases",
        "candidate_path": "course/src/derived-tensor-products-and-tor-sheaves.md",
        "sha256": "B78C1AD9864901349E26C6EB3954B046E40B84E35E55EA6B3A71C553E185D292"
      },
      {
        "provider": "internal-derived-hom-and-ext-sheaves",
        "locus": "Lemma 1.1; Theorems 2.2,3.1; Section 4 canonical maps",
        "correspondence": "exact_required_scope; stronger arbitrary-ring results specialize to commutative cases",
        "candidate_path": "course/src/internal-derived-hom-and-ext-sheaves.md",
        "sha256": "B5B5E565B53E99059D5992F69DD576FA7D9FB771366C844AFEA3D09F1AA8C825"
      },
      {
        "provider": "derived-pullback-and-pushforward",
        "locus": "Theorem 1.1; Proposition 3.2",
        "correspondence": "exact_required_scope; stronger arbitrary-ring results specialize to commutative cases",
        "candidate_path": "course/src/derived-pullback-and-pushforward.md",
        "sha256": "FECE9386A64E3298CD133F9ED65D4D74AAC82B9A8B266FB5CCC6363C8C063AB2"
      },
      {
        "provider": "injective-modules-and-bounded-below-derived-functors",
        "locus": "Theorem 3.3",
        "correspondence": "exact_required_scope; stronger arbitrary-ring results specialize to commutative cases",
        "candidate_path": "course/src/injective-modules-and-bounded-below-derived-functors.md",
        "sha256": "B2DC989EFBC8D2D4004C8D24CBC3004A9E81DA485CD702BB7D293A7F8AF5EDBA"
      },
      {
        "provider": "complexes-cones-and-localization",
        "locus": "Cones, LES, localization roofs",
        "correspondence": "exact_required_scope; stronger arbitrary-ring results specialize to commutative cases",
        "candidate_path": "course/src/complexes-cones-and-localization.md",
        "sha256": "367394AE4A25DCF64DF87302304D8395AF84D92765E4E4E45A2333A0071C3862"
      },
      {
        "provider": "sections-with-support-and-the-localization-triangle",
        "locus": "Theorem 2.1",
        "correspondence": "exact_required_scope; stronger arbitrary-ring results specialize to commutative cases",
        "candidate_path": "course/src/sections-with-support-and-the-localization-triangle.md",
        "sha256": "058C339B17DE1604395E406A4905CE8CCB4A251AE6DB50EB4A71ACD714D67B38"
      },
      {
        "provider": "SH-DC.holomorphic-coordinate-germs",
        "candidate_path": "course/src/derived-tensor-products-and-tor-sheaves.md",
        "locus": "Section 4, Lemma (holomorphic coordinate germs), full proof",
        "anchor": "holomorphic-coordinate-germs",
        "sha256": "B78C1AD9864901349E26C6EB3954B046E40B84E35E55EA6B3A71C553E185D292",
        "correspondence": "Holomorphic coordinate germ calculation, exact curve/surface scope."
      }
    ],
    "proof": {
      "source": "src/perfect-complexes-and-duals.md",
      "sha256": "5d3a0bf017b53f29fb3668dabce086f51a909be2868ca3ea03928ff35ff94c97",
      "locus": "Section 5",
      "reader": "perfect-complexes-and-duals.html#5-residues-and-finiteness-examples"
    },
    "licence": "CC0-1.0",
    "proof_status": "complete; writing-AI self-checked"
  },
  {
    "result_id": "SH-DC.projection-formula",
    "name": "Canonical projection map is invertible for perfect coefficients and arbitrary unbounded E; for a homeomorphism onto a closed subset it is invertible for all coefficients. Finite locally projective coefficients give all higher direct-image identities.",
    "statement": "Canonical projection map is invertible for perfect coefficients and arbitrary unbounded E; for a homeomorphism onto a closed subset it is invertible for all coefficients. Finite locally projective coefficients give all higher direct-image identities.",
    "conditions": "Any commutative unital ringed-space morphism; perfectness local and nonuniform; closed-case coefficient homomorphism arbitrary and need not be flat.",
    "writing_ai_selfcheck": true,
    "independent_review": false,
    "formalization": "not_formalized",
    "prerequisite_envelope_ids": [
      "SH-DC.modules-abelian",
      "SH-DC.filtered-colimits-sums",
      "SH-DC.pullback-adjunction",
      "SH-DC.extension-skyscraper",
      "SH-DC.homotopy-localization",
      "SH-DC.injective-embeddings",
      "SH-DC.flasque-acyclic",
      "SH-DC.K-injective-comparison",
      "SH-DC.bounded-derived",
      "SH-DC.Grothendieck-injective-embeddings",
      "SH-DC.unbounded-K-injective-existence",
      "SH-DC.unbounded-right-derived",
      "SH-DC.derived-products",
      "SH-DC.flat-module-sheaves",
      "SH-DC.K-flat-existence",
      "SH-DC.K-flat-calculus",
      "SH-DC.derived-tensor",
      "SH-DC.Tor-flatness",
      "SH-DC.derived-pull-push-adjunction",
      "SH-DC.local-cohomology-comparisons",
      "SH-DC.Leray-spectral-sequence",
      "SH-DC.internal-derived-Hom",
      "SH-DC.composition-dinaturality",
      "SH-DC.internal-Hom-examples",
      "SH-DC.localization-support",
      "SH-DC.Euclidean-point-support",
      "SH-DC.support-cup-pullback",
      "SH-DC.perfect-criterion",
      "SH-DC.perfect-duality",
      "SH-DC.perfect-residue-examples"
    ],
    "dependency_precision": "Conservative earlier-provider envelope; exact textual uses appear in programme_dependency_loci. Same-lesson preliminary lemmas are inside the full proof locus.",
    "programme_dependency_loci": [
      {
        "provider": "derived-pullback-and-pushforward",
        "locus": "Theorem 1.1; Propositions 1.2,3.1–3.2,4.2; Theorem 2.2",
        "correspondence": "exact_required_scope; stronger arbitrary-ring results specialize to commutative cases",
        "candidate_path": "course/src/derived-pullback-and-pushforward.md",
        "sha256": "FECE9386A64E3298CD133F9ED65D4D74AAC82B9A8B266FB5CCC6363C8C063AB2"
      },
      {
        "provider": "perfect-complexes-and-duals",
        "locus": "Lemma 1.1; Proposition 2.2; Corollary 3.3; Theorems 4.1–4.2",
        "correspondence": "exact_required_scope; stronger arbitrary-ring results specialize to commutative cases",
        "use_scope": "Exact textual uses in the complete lesson; conservative prerequisite envelope remains distinct.",
        "candidate_path": "course/src/perfect-complexes-and-duals.md",
        "sha256": "5D3A0BF017B53F29FB3668DABCE086F51A909BE2868CA3EA03928FF35FF94C97"
      },
      {
        "provider": "derived-tensor-products-and-tor-sheaves",
        "locus": "Theorems 1.2,2.2",
        "correspondence": "exact_required_scope; stronger arbitrary-ring results specialize to commutative cases",
        "candidate_path": "course/src/derived-tensor-products-and-tor-sheaves.md",
        "sha256": "B78C1AD9864901349E26C6EB3954B046E40B84E35E55EA6B3A71C553E185D292"
      },
      {
        "provider": "sheaves-of-modules-on-a-ringed-space",
        "locus": "Theorem 4.4",
        "correspondence": "exact_required_scope; stronger arbitrary-ring results specialize to commutative cases",
        "candidate_path": "course/src/sheaves-of-modules-on-a-ringed-space.md",
        "sha256": "22FCF2E4E2E126FE922EBD68FD81F3DEDC581ED2B70CBB60A49250E21DA7BABE"
      },
      {
        "provider": "sections-with-support-and-the-localization-triangle",
        "locus": "Proposition 1.1",
        "correspondence": "exact_required_scope; stronger arbitrary-ring results specialize to commutative cases",
        "candidate_path": "course/src/sections-with-support-and-the-localization-triangle.md",
        "sha256": "058C339B17DE1604395E406A4905CE8CCB4A251AE6DB50EB4A71ACD714D67B38"
      },
      {
        "provider": "injective-modules-and-bounded-below-derived-functors",
        "locus": "Theorem 3.3",
        "correspondence": "exact_required_scope; stronger arbitrary-ring results specialize to commutative cases",
        "candidate_path": "course/src/injective-modules-and-bounded-below-derived-functors.md",
        "sha256": "B2DC989EFBC8D2D4004C8D24CBC3004A9E81DA485CD702BB7D293A7F8AF5EDBA"
      },
      {
        "provider": "k-injective-resolutions-in-grothendieck-categories",
        "locus": "Theorem 4.1",
        "correspondence": "exact_required_scope; stronger arbitrary-ring results specialize to commutative cases",
        "candidate_path": "course/src/k-injective-resolutions-in-grothendieck-categories.md",
        "sha256": "75118C6AB567222AEB5E42C60AA5DC206466FFF9EA8A9F193EA1B598DECD760B"
      },
      {
        "provider": "complexes-cones-and-localization",
        "correspondence": "exact_required_scope",
        "locus": "Lemma 4.1; Theorems 4.2, 5.1; Proposition 5.2",
        "use_scope": "Exact textual uses in the complete lesson; conservative prerequisite envelope remains distinct.",
        "candidate_path": "course/src/complexes-cones-and-localization.md",
        "sha256": "367394AE4A25DCF64DF87302304D8395AF84D92765E4E4E45A2333A0071C3862"
      }
    ],
    "proof": {
      "source": "src/projection-formula-and-base-change.md",
      "sha256": "afeb289acd142e469f5683c985f314014e92ed8ae8ed3e4c5d1ff3bf02582674",
      "locus": "Theorems 1.2,2.1,3.1 and Exercises 1-4",
      "reader": "projection-formula-and-base-change.html#2-the-general-map-and-perfect-coefficients"
    },
    "licence": "CC0-1.0",
    "proof_status": "complete; writing-AI self-checked"
  },
  {
    "result_id": "SH-DC.base-change-map",
    "name": "Unbounded canonical base-change map exists for every commutative square, agrees with flat chain-model construction, and respects identity, horizontal/vertical pasting and projection maps.",
    "statement": "Unbounded canonical base-change map exists for every commutative square, agrees with flat chain-model construction, and respects identity, horizontal/vertical pasting and projection maps.",
    "conditions": "Commutative unital ringed spaces; no Cartesian requirement for construction; Tag02N7 ordinary-pullback case requires BOTH horizontal maps flat and asserts a map, not a universal isomorphism.",
    "writing_ai_selfcheck": true,
    "independent_review": false,
    "formalization": "not_formalized",
    "prerequisite_envelope_ids": [
      "SH-DC.modules-abelian",
      "SH-DC.filtered-colimits-sums",
      "SH-DC.pullback-adjunction",
      "SH-DC.extension-skyscraper",
      "SH-DC.homotopy-localization",
      "SH-DC.injective-embeddings",
      "SH-DC.flasque-acyclic",
      "SH-DC.K-injective-comparison",
      "SH-DC.bounded-derived",
      "SH-DC.Grothendieck-injective-embeddings",
      "SH-DC.unbounded-K-injective-existence",
      "SH-DC.unbounded-right-derived",
      "SH-DC.derived-products",
      "SH-DC.flat-module-sheaves",
      "SH-DC.K-flat-existence",
      "SH-DC.K-flat-calculus",
      "SH-DC.derived-tensor",
      "SH-DC.Tor-flatness",
      "SH-DC.derived-pull-push-adjunction",
      "SH-DC.local-cohomology-comparisons",
      "SH-DC.Leray-spectral-sequence",
      "SH-DC.internal-derived-Hom",
      "SH-DC.composition-dinaturality",
      "SH-DC.internal-Hom-examples",
      "SH-DC.localization-support",
      "SH-DC.Euclidean-point-support",
      "SH-DC.support-cup-pullback",
      "SH-DC.perfect-criterion",
      "SH-DC.perfect-duality",
      "SH-DC.perfect-residue-examples"
    ],
    "dependency_precision": "Conservative earlier-provider envelope; exact textual uses appear in programme_dependency_loci. Same-lesson preliminary lemmas are inside the full proof locus.",
    "programme_dependency_loci": [
      {
        "provider": "derived-pullback-and-pushforward",
        "locus": "Theorem 1.1; Propositions 1.2,3.1–3.2,4.2; Theorem 2.2",
        "correspondence": "exact_required_scope; stronger arbitrary-ring results specialize to commutative cases",
        "candidate_path": "course/src/derived-pullback-and-pushforward.md",
        "sha256": "FECE9386A64E3298CD133F9ED65D4D74AAC82B9A8B266FB5CCC6363C8C063AB2"
      },
      {
        "provider": "perfect-complexes-and-duals",
        "locus": "Lemma 1.1; Proposition 2.2; Corollary 3.3; Theorems 4.1–4.2",
        "correspondence": "exact_required_scope; stronger arbitrary-ring results specialize to commutative cases",
        "use_scope": "Exact textual uses in the complete lesson; conservative prerequisite envelope remains distinct.",
        "candidate_path": "course/src/perfect-complexes-and-duals.md",
        "sha256": "5D3A0BF017B53F29FB3668DABCE086F51A909BE2868CA3EA03928FF35FF94C97"
      },
      {
        "provider": "derived-tensor-products-and-tor-sheaves",
        "locus": "Theorems 1.2,2.2",
        "correspondence": "exact_required_scope; stronger arbitrary-ring results specialize to commutative cases",
        "candidate_path": "course/src/derived-tensor-products-and-tor-sheaves.md",
        "sha256": "B78C1AD9864901349E26C6EB3954B046E40B84E35E55EA6B3A71C553E185D292"
      },
      {
        "provider": "sheaves-of-modules-on-a-ringed-space",
        "locus": "Theorem 4.4",
        "correspondence": "exact_required_scope; stronger arbitrary-ring results specialize to commutative cases",
        "candidate_path": "course/src/sheaves-of-modules-on-a-ringed-space.md",
        "sha256": "22FCF2E4E2E126FE922EBD68FD81F3DEDC581ED2B70CBB60A49250E21DA7BABE"
      },
      {
        "provider": "sections-with-support-and-the-localization-triangle",
        "locus": "Proposition 1.1",
        "correspondence": "exact_required_scope; stronger arbitrary-ring results specialize to commutative cases",
        "candidate_path": "course/src/sections-with-support-and-the-localization-triangle.md",
        "sha256": "058C339B17DE1604395E406A4905CE8CCB4A251AE6DB50EB4A71ACD714D67B38"
      },
      {
        "provider": "injective-modules-and-bounded-below-derived-functors",
        "locus": "Theorem 3.3",
        "correspondence": "exact_required_scope; stronger arbitrary-ring results specialize to commutative cases",
        "candidate_path": "course/src/injective-modules-and-bounded-below-derived-functors.md",
        "sha256": "B2DC989EFBC8D2D4004C8D24CBC3004A9E81DA485CD702BB7D293A7F8AF5EDBA"
      },
      {
        "provider": "k-injective-resolutions-in-grothendieck-categories",
        "locus": "Theorem 4.1",
        "correspondence": "exact_required_scope; stronger arbitrary-ring results specialize to commutative cases",
        "candidate_path": "course/src/k-injective-resolutions-in-grothendieck-categories.md",
        "sha256": "75118C6AB567222AEB5E42C60AA5DC206466FFF9EA8A9F193EA1B598DECD760B"
      },
      {
        "provider": "complexes-cones-and-localization",
        "correspondence": "exact_required_scope",
        "locus": "Lemma 4.1; Theorems 4.2, 5.1; Proposition 5.2",
        "use_scope": "Exact textual uses in the complete lesson; conservative prerequisite envelope remains distinct.",
        "candidate_path": "course/src/complexes-cones-and-localization.md",
        "sha256": "367394AE4A25DCF64DF87302304D8395AF84D92765E4E4E45A2333A0071C3862"
      }
    ],
    "proof": {
      "source": "src/projection-formula-and-base-change.md",
      "sha256": "afeb289acd142e469f5683c985f314014e92ed8ae8ed3e4c5d1ff3bf02582674",
      "locus": "Theorem 4.1 and Section 5, Exercises 5-6",
      "reader": "projection-formula-and-base-change.html#4-base-change-flat-models-and-compatibility"
    },
    "licence": "CC0-1.0",
    "proof_status": "complete; writing-AI self-checked"
  },
  {
    "result_id": "SH-DC.costalk-adjunction",
    "name": "Proposition 5.3: stalk–skyscraper–costalk adjunction; closed-point support identification",
    "statement": "Proposition 5.3: stalk–skyscraper–costalk adjunction; closed-point support identification",
    "conditions": "Any topological space, any point, associative unital coefficients; closedness required only for support identification.",
    "writing_ai_selfcheck": true,
    "independent_review": false,
    "formalization": "not_formalized",
    "prerequisite_envelope_ids": [
      "SH-DC.extension-skyscraper"
    ],
    "dependency_precision": "Exact earlier proof providers; conditional/external-example boundary stated explicitly.",
    "programme_dependency_loci": [
      {
        "provider": "SH-DC.extension-skyscraper",
        "locus": "Lemmas 5.1 and 5.2",
        "correspondence": "used with the stated conditions",
        "candidate_path": "course/src/sheaves-of-modules-on-a-ringed-space.md",
        "sha256": "22FCF2E4E2E126FE922EBD68FD81F3DEDC581ED2B70CBB60A49250E21DA7BABE",
        "result_id": "SH-DC.extension-skyscraper",
        "anchor": "5-extension-by-zero-and-skyscrapers"
      }
    ],
    "proof": {
      "source": "src/sheaves-of-modules-on-a-ringed-space.md",
      "sha256": "22fcf2e4e2e126fe922ebd68fd81f3dedc581ed2b70cbb60a49250e21da7babe",
      "locus": "Proposition 5.3: stalk–skyscraper–costalk adjunction; closed-point support identification",
      "reader": "sheaves-of-modules-on-a-ringed-space.html#the-right-adjoint-of-a-skyscraper"
    },
    "licence": "CC0-1.0",
    "proof_status": "complete; writing-AI self-checked"
  },
  {
    "result_id": "SH-DC.semifree-K-projective",
    "name": "Theorem 3.4: unbounded semifree, K-projective and K-flat module resolutions and Hom comparison",
    "statement": "Theorem 3.4: unbounded semifree, K-projective and K-flat module resolutions and Hom comparison",
    "conditions": "Any associative unital ring; acyclic right modules test K-flatness; not a general sheaf K-projectivity claim.",
    "writing_ai_selfcheck": true,
    "independent_review": false,
    "formalization": "not_formalized",
    "prerequisite_envelope_ids": [
      "SH-DC.K-flat-existence",
      "SH-DC.homotopy-localization"
    ],
    "dependency_precision": "Exact earlier proof providers; conditional/external-example boundary stated explicitly.",
    "programme_dependency_loci": [
      {
        "provider": "SH-DC.K-flat-existence",
        "locus": "Lemmas 2.1-2.4, Theorem 3.1 and Proposition 4.3",
        "correspondence": "used with the stated conditions",
        "candidate_path": "course/src/flat-modules-and-k-flat-resolutions.md",
        "sha256": "7BCE2B46408B278A147517161F09AE29A7BC0273B4E271AA2F995966C1AA941D",
        "result_id": "SH-DC.K-flat-existence",
        "anchor": "3-a-surjective-resolution-by-attaching-cycles"
      },
      {
        "provider": "SH-DC.homotopy-localization",
        "locus": "Theorems 3.1, 4.2, 5.1 and Proposition 5.2",
        "correspondence": "used with the stated conditions",
        "candidate_path": "course/src/complexes-cones-and-localization.md",
        "sha256": "367394AE4A25DCF64DF87302304D8395AF84D92765E4E4E45A2333A0071C3862",
        "result_id": "SH-DC.homotopy-localization",
        "anchor": "3-the-triangulated-structure-of-the-homotopy-category"
      }
    ],
    "proof": {
      "source": "src/flat-modules-and-k-flat-resolutions.md",
      "sha256": "7bce2b46408b278a147517161f09ae29a7bc0273b4e271aa2f995966c1aa941d",
      "locus": "Theorem 3.4: unbounded semifree, K-projective and K-flat module resolutions and Hom comparison",
      "reader": "flat-modules-and-k-flat-resolutions.html#semifree-resolutions-over-a-ring"
    },
    "licence": "CC0-1.0",
    "proof_status": "complete; writing-AI self-checked"
  },
  {
    "result_id": "SH-DC.finite-dimensional-acyclic-models",
    "name": "Proposition 5.5: unbounded complexes of acyclic objects compute RF",
    "statement": "Proposition 5.5: unbounded complexes of acyclic objects compute RF",
    "conditions": "Grothendieck source; additive left-exact F into abelian target; uniform finite R^qF bound on all objects.",
    "writing_ai_selfcheck": true,
    "independent_review": false,
    "formalization": "not_formalized",
    "prerequisite_envelope_ids": [
      "SH-DC.unbounded-K-injective-existence",
      "SH-DC.unbounded-right-derived",
      "SH-DC.bounded-derived"
    ],
    "dependency_precision": "Exact earlier proof providers; conditional/external-example boundary stated explicitly.",
    "programme_dependency_loci": [
      {
        "provider": "SH-DC.unbounded-K-injective-existence",
        "locus": "Proposition 1.1, Lemmas 3.1-3.2 and Theorem 4.1",
        "correspondence": "used with the stated conditions",
        "candidate_path": "course/src/k-injective-resolutions-in-grothendieck-categories.md",
        "sha256": "75118C6AB567222AEB5E42C60AA5DC206466FFF9EA8A9F193EA1B598DECD760B",
        "result_id": "SH-DC.unbounded-K-injective-existence",
        "anchor": "4-constructing-the-unbounded-resolution"
      },
      {
        "provider": "SH-DC.unbounded-right-derived",
        "locus": "Theorem 5.1 and Proposition 5.2",
        "correspondence": "used with the stated conditions",
        "candidate_path": "course/src/k-injective-resolutions-in-grothendieck-categories.md",
        "sha256": "75118C6AB567222AEB5E42C60AA5DC206466FFF9EA8A9F193EA1B598DECD760B",
        "result_id": "SH-DC.unbounded-right-derived",
        "anchor": "5-derived-functors-adjoints-and-products"
      },
      {
        "provider": "SH-DC.bounded-derived",
        "locus": "Theorem 4.1",
        "correspondence": "used with the stated conditions",
        "candidate_path": "course/src/injective-modules-and-bounded-below-derived-functors.md",
        "sha256": "B2DC989EFBC8D2D4004C8D24CBC3004A9E81DA485CD702BB7D293A7F8AF5EDBA",
        "result_id": "SH-DC.bounded-derived",
        "anchor": "4-right-derived-functors-on-bounded-below-complexes"
      }
    ],
    "proof": {
      "source": "src/k-injective-resolutions-in-grothendieck-categories.md",
      "sha256": "75118c6ab567222aeb5e42c60aa5dc206466fff9ea8a9f193ea1b598decd760b",
      "locus": "Proposition 5.5: unbounded complexes of acyclic objects compute RF",
      "reader": "k-injective-resolutions-in-grothendieck-categories.html#finite-cohomological-dimension-and-acyclic-models"
    },
    "licence": "CC0-1.0",
    "proof_status": "complete; writing-AI self-checked"
  },
  {
    "result_id": "SH-DC.Koszul-regular-sequence",
    "name": "Proposition 4.1: regular-sequence Koszul resolution and explicit Tor/Ext vector-space calculations",
    "statement": "Proposition 4.1: regular-sequence Koszul resolution and explicit Tor/Ext vector-space calculations",
    "conditions": "Commutative ring; regularity on specified M for vanishing, on R for quotient resolution; no Yoneda algebra claim.",
    "writing_ai_selfcheck": true,
    "independent_review": false,
    "formalization": "not_formalized",
    "prerequisite_envelope_ids": [
      "SH-DC.derived-tensor",
      "SH-DC.K-flat-calculus",
      "SH-DC.semifree-K-projective"
    ],
    "dependency_precision": "Exact earlier proof providers; conditional/external-example boundary stated explicitly.",
    "programme_dependency_loci": [
      {
        "provider": "SH-DC.derived-tensor",
        "locus": "Lemmas 1.1 and 2.1, Theorems 1.2 and 2.2",
        "correspondence": "used with the stated conditions",
        "candidate_path": "course/src/derived-tensor-products-and-tor-sheaves.md",
        "sha256": "B78C1AD9864901349E26C6EB3954B046E40B84E35E55EA6B3A71C553E185D292",
        "result_id": "SH-DC.derived-tensor",
        "anchor": "1-the-two-tensor-comparisons"
      },
      {
        "provider": "SH-DC.K-flat-calculus",
        "locus": "Lemmas 2.1-2.4, Corollary 4.4 and Lemma 4.5",
        "correspondence": "used with the stated conditions",
        "candidate_path": "course/src/flat-modules-and-k-flat-resolutions.md",
        "sha256": "7BCE2B46408B278A147517161F09AE29A7BC0273B4E271AA2F995966C1AA941D",
        "result_id": "SH-DC.K-flat-calculus",
        "anchor": "2-k-flat-complexes-and-their-closure-properties"
      },
      {
        "provider": "SH-DC.semifree-K-projective",
        "candidate_path": "course/src/flat-modules-and-k-flat-resolutions.md",
        "locus": "Theorem 3.4: unbounded semifree, K-projective and K-flat module resolutions and Hom comparison",
        "sha256": "7BCE2B46408B278A147517161F09AE29A7BC0273B4E271AA2F995966C1AA941D"
      }
    ],
    "proof": {
      "source": "src/derived-tensor-products-and-tor-sheaves.md",
      "sha256": "b78c1ad9864901349e26c6eb3954b046e40b84e35e55ea6b3a71c553e185d292",
      "locus": "Proposition 4.1: regular-sequence Koszul resolution and explicit Tor/Ext vector-space calculations",
      "reader": "derived-tensor-products-and-tor-sheaves.html#koszul-models-for-several-equations"
    },
    "licence": "CC0-1.0",
    "proof_status": "complete; writing-AI self-checked"
  },
  {
    "result_id": "SH-DC.bar-resolution",
    "name": "Proposition 4.2: bar free resolution and Tor/Ext computation",
    "statement": "Proposition 4.2: bar free resolution and Tor/Ext computation",
    "conditions": "Any associative unital algebra over a field; module version, k-linear contraction.",
    "writing_ai_selfcheck": true,
    "independent_review": false,
    "formalization": "not_formalized",
    "prerequisite_envelope_ids": [
      "SH-DC.semifree-K-projective",
      "SH-DC.derived-tensor"
    ],
    "dependency_precision": "Exact earlier proof providers; conditional/external-example boundary stated explicitly.",
    "programme_dependency_loci": [
      {
        "provider": "SH-DC.semifree-K-projective",
        "locus": "Theorem 3.4: unbounded semifree, K-projective and K-flat module resolutions and Hom comparison",
        "correspondence": "used with the stated conditions",
        "candidate_path": "course/src/flat-modules-and-k-flat-resolutions.md",
        "sha256": "7BCE2B46408B278A147517161F09AE29A7BC0273B4E271AA2F995966C1AA941D",
        "result_id": "SH-DC.semifree-K-projective",
        "anchor": "semifree-resolutions-over-a-ring"
      },
      {
        "provider": "SH-DC.derived-tensor",
        "locus": "Lemmas 1.1 and 2.1, Theorems 1.2 and 2.2",
        "correspondence": "used with the stated conditions",
        "candidate_path": "course/src/derived-tensor-products-and-tor-sheaves.md",
        "sha256": "B78C1AD9864901349E26C6EB3954B046E40B84E35E55EA6B3A71C553E185D292",
        "result_id": "SH-DC.derived-tensor",
        "anchor": "1-the-two-tensor-comparisons"
      }
    ],
    "proof": {
      "source": "src/derived-tensor-products-and-tor-sheaves.md",
      "sha256": "b78c1ad9864901349e26c6eb3954b046e40b84e35e55ea6b3a71c553e185d292",
      "locus": "Proposition 4.2: bar free resolution and Tor/Ext computation",
      "reader": "derived-tensor-products-and-tor-sheaves.html#a-bar-model-without-commutativity"
    },
    "licence": "CC0-1.0",
    "proof_status": "complete; writing-AI self-checked"
  },
  {
    "result_id": "SH-DC.skyscraper-derived-Hom",
    "name": "Proposition 3.2: internal derived Hom into skyscrapers",
    "statement": "Proposition 3.2: internal derived Hom into skyscrapers",
    "conditions": "Arbitrary point, commutative ringed space, both inputs unbounded; special target, no general stalk Hom isomorphism.",
    "writing_ai_selfcheck": true,
    "independent_review": false,
    "formalization": "not_formalized",
    "prerequisite_envelope_ids": [
      "SH-DC.costalk-adjunction",
      "SH-DC.internal-derived-Hom",
      "SH-DC.K-injective-comparison"
    ],
    "dependency_precision": "Exact earlier proof providers; conditional/external-example boundary stated explicitly.",
    "programme_dependency_loci": [
      {
        "provider": "SH-DC.costalk-adjunction",
        "locus": "Proposition 5.3: stalk–skyscraper–costalk adjunction; closed-point support identification",
        "correspondence": "used with the stated conditions",
        "candidate_path": "course/src/sheaves-of-modules-on-a-ringed-space.md",
        "sha256": "22FCF2E4E2E126FE922EBD68FD81F3DEDC581ED2B70CBB60A49250E21DA7BABE",
        "result_id": "SH-DC.costalk-adjunction",
        "anchor": "the-right-adjoint-of-a-skyscraper"
      },
      {
        "provider": "SH-DC.internal-derived-Hom",
        "locus": "Lemmas 1.1 and 2.1-2.3; Theorems 2.2 and 3.1; Section 3",
        "correspondence": "used with the stated conditions",
        "candidate_path": "course/src/internal-derived-hom-and-ext-sheaves.md",
        "sha256": "B5B5E565B53E99059D5992F69DD576FA7D9FB771366C844AFEA3D09F1AA8C825",
        "result_id": "SH-DC.internal-derived-Hom",
        "anchor": "2-constructing-internal-derived-hom"
      },
      {
        "provider": "SH-DC.K-injective-comparison",
        "locus": "Lemmas 3.1-3.2, Theorem 3.3 and Corollary 3.4",
        "correspondence": "used with the stated conditions",
        "candidate_path": "course/src/injective-modules-and-bounded-below-derived-functors.md",
        "sha256": "B2DC989EFBC8D2D4004C8D24CBC3004A9E81DA485CD702BB7D293A7F8AF5EDBA",
        "result_id": "SH-DC.K-injective-comparison",
        "anchor": "3-complexes-that-admit-comparison-maps"
      }
    ],
    "proof": {
      "source": "src/internal-derived-hom-and-ext-sheaves.md",
      "sha256": "b5b5e565b53e99059d5992f69dd576fa7d9fb771366c844afea3d09f1aa8c825",
      "locus": "Proposition 3.2: internal derived Hom into skyscrapers",
      "reader": "internal-derived-hom-and-ext-sheaves.html#internal-hom-into-a-skyscraper"
    },
    "licence": "CC0-1.0",
    "proof_status": "complete; writing-AI self-checked"
  },
  {
    "result_id": "SH-DC.derived-costalk-and-closed-coefficients",
    "name": "Derived costalk identification; Proposition 1.2 closed coefficient right adjoint",
    "statement": "Derived costalk identification; Proposition 1.2 closed coefficient right adjoint",
    "conditions": "Costalk for any point; support identification closed point; changed-coefficient formula for commutative ringed closed embedding; unbounded.",
    "writing_ai_selfcheck": true,
    "independent_review": false,
    "formalization": "not_formalized",
    "prerequisite_envelope_ids": [
      "SH-DC.costalk-adjunction",
      "SH-DC.localization-support",
      "SH-DC.internal-derived-Hom"
    ],
    "dependency_precision": "Exact earlier proof providers; conditional/external-example boundary stated explicitly.",
    "programme_dependency_loci": [
      {
        "provider": "SH-DC.costalk-adjunction",
        "locus": "Proposition 5.3: stalk–skyscraper–costalk adjunction; closed-point support identification",
        "correspondence": "used with the stated conditions",
        "candidate_path": "course/src/sheaves-of-modules-on-a-ringed-space.md",
        "sha256": "22FCF2E4E2E126FE922EBD68FD81F3DEDC581ED2B70CBB60A49250E21DA7BABE",
        "result_id": "SH-DC.costalk-adjunction",
        "anchor": "the-right-adjoint-of-a-skyscraper"
      },
      {
        "provider": "SH-DC.localization-support",
        "locus": "Proposition 1.1, Theorem 2.1, Proposition 2.2 and Proposition 3.1",
        "correspondence": "used with the stated conditions",
        "candidate_path": "course/src/sections-with-support-and-the-localization-triangle.md",
        "sha256": "058C339B17DE1604395E406A4905CE8CCB4A251AE6DB50EB4A71ACD714D67B38",
        "result_id": "SH-DC.localization-support",
        "anchor": "1-closed-supports-and-their-adjunction"
      },
      {
        "provider": "SH-DC.internal-derived-Hom",
        "locus": "Lemmas 1.1 and 2.1-2.3; Theorems 2.2 and 3.1; Section 3",
        "correspondence": "used with the stated conditions",
        "candidate_path": "course/src/internal-derived-hom-and-ext-sheaves.md",
        "sha256": "B5B5E565B53E99059D5992F69DD576FA7D9FB771366C844AFEA3D09F1AA8C825",
        "result_id": "SH-DC.internal-derived-Hom",
        "anchor": "2-constructing-internal-derived-hom"
      }
    ],
    "proof": {
      "source": "src/sections-with-support-and-the-localization-triangle.md",
      "sha256": "d5e0f4079d0a4a0dedd36622b944e75d6072b3781976b6c8328271997ab1282a",
      "locus": "Derived costalk identification; Proposition 1.2 closed coefficient right adjoint",
      "reader": "sections-with-support-and-the-localization-triangle.html#derived-costalks-and-changes-of-closed-coefficients"
    },
    "licence": "CC0-1.0",
    "proof_status": "complete; writing-AI self-checked"
  },
  {
    "result_id": "SH-DC.split-resolution-towers",
    "name": "Lemmas 1.1,2.1; Propositions1.2,2.2: strict extension, split resolution towers and K-injective limit",
    "statement": "Lemmas 1.1,2.1; Propositions1.2,2.2: strict extension, split resolution towers and K-injective limit",
    "conditions": "Complete locally small abelian category with enough injectives; no AB4* needed for target K-injectivity.",
    "writing_ai_selfcheck": true,
    "independent_review": false,
    "formalization": "not_formalized",
    "prerequisite_envelope_ids": [
      "SH-DC.bounded-derived",
      "SH-DC.K-injective-comparison",
      "SH-DC.derived-products"
    ],
    "dependency_precision": "Exact earlier proof providers; conditional/external-example boundary stated explicitly.",
    "programme_dependency_loci": [
      {
        "provider": "SH-DC.bounded-derived",
        "locus": "Theorem 4.1",
        "correspondence": "used with the stated conditions",
        "candidate_path": "course/src/injective-modules-and-bounded-below-derived-functors.md",
        "sha256": "B2DC989EFBC8D2D4004C8D24CBC3004A9E81DA485CD702BB7D293A7F8AF5EDBA",
        "result_id": "SH-DC.bounded-derived",
        "anchor": "4-right-derived-functors-on-bounded-below-complexes"
      },
      {
        "provider": "SH-DC.K-injective-comparison",
        "locus": "Lemmas 3.1-3.2, Theorem 3.3 and Corollary 3.4",
        "correspondence": "used with the stated conditions",
        "candidate_path": "course/src/injective-modules-and-bounded-below-derived-functors.md",
        "sha256": "B2DC989EFBC8D2D4004C8D24CBC3004A9E81DA485CD702BB7D293A7F8AF5EDBA",
        "result_id": "SH-DC.K-injective-comparison",
        "anchor": "3-complexes-that-admit-comparison-maps"
      },
      {
        "provider": "SH-DC.derived-products",
        "locus": "Propositions 5.3-5.4",
        "correspondence": "used with the stated conditions",
        "candidate_path": "course/src/k-injective-resolutions-in-grothendieck-categories.md",
        "sha256": "75118C6AB567222AEB5E42C60AA5DC206466FFF9EA8A9F193EA1B598DECD760B",
        "result_id": "SH-DC.derived-products",
        "anchor": "5-derived-functors-adjoints-and-products"
      }
    ],
    "proof": {
      "source": "src/inverse-limits-and-unbounded-resolutions.md",
      "sha256": "91ae732a4b5c0cccc3c21fb1c9bff4fdec904bd7e5e0fb1f1bc17abab9948872",
      "locus": "Lemmas 1.1,2.1; Propositions1.2,2.2: strict extension, split resolution towers and K-injective limit",
      "reader": "inverse-limits-and-unbounded-resolutions.html#1-building-a-compatible-tower"
    },
    "licence": "CC0-1.0",
    "proof_status": "complete; writing-AI self-checked"
  },
  {
    "result_id": "SH-DC.AB4-star-finite-resolution",
    "name": "Lemma3.1,Corollary3.2,Theorem3.3: finite higher-product bound and unbounded DG-injective resolution",
    "statement": "Lemma3.1,Corollary3.2,Theorem3.3: finite higher-product bound and unbounded DG-injective resolution",
    "conditions": "Complete abelian category, enough injectives, AB4*-k for some finite k; no generator or AB5 assumption.",
    "writing_ai_selfcheck": true,
    "independent_review": false,
    "formalization": "not_formalized",
    "prerequisite_envelope_ids": [
      "SH-DC.split-resolution-towers"
    ],
    "dependency_precision": "Exact earlier proof providers; conditional/external-example boundary stated explicitly.",
    "programme_dependency_loci": [
      {
        "provider": "SH-DC.split-resolution-towers",
        "locus": "Lemmas 1.1,2.1; Propositions1.2,2.2: strict extension, split resolution towers and K-injective limit",
        "correspondence": "used with the stated conditions",
        "candidate_path": "course/src/inverse-limits-and-unbounded-resolutions.md",
        "sha256": "91AE732A4B5C0CCCC3C21FB1C9BFF4FDEC904BD7E5E0FB1F1BC17ABAB9948872",
        "result_id": "SH-DC.split-resolution-towers",
        "anchor": "1-building-a-compatible-tower"
      }
    ],
    "proof": {
      "source": "src/inverse-limits-and-unbounded-resolutions.md",
      "sha256": "91ae732a4b5c0cccc3c21fb1c9bff4fdec904bd7e5e0fb1f1bc17abab9948872",
      "locus": "Lemma3.1,Corollary3.2,Theorem3.3: finite higher-product bound and unbounded DG-injective resolution",
      "reader": "inverse-limits-and-unbounded-resolutions.html#3-finite-higher-product-dimension"
    },
    "licence": "CC0-1.0",
    "proof_status": "complete; writing-AI self-checked"
  },
  {
    "result_id": "SH-DC.CE-product-totalization",
    "name": "Theorem4.1: unbounded Cartan–Eilenberg product totalization",
    "statement": "Theorem4.1: unbounded Cartan–Eilenberg product totalization",
    "conditions": "Same finite AB4*-k hypothesis; CE injective cycle/boundary/cohomology rows; full horseshoe and finite elimination given.",
    "writing_ai_selfcheck": true,
    "independent_review": false,
    "formalization": "not_formalized",
    "prerequisite_envelope_ids": [
      "SH-DC.AB4-star-finite-resolution"
    ],
    "dependency_precision": "Exact earlier proof providers; conditional/external-example boundary stated explicitly.",
    "programme_dependency_loci": [
      {
        "provider": "SH-DC.AB4-star-finite-resolution",
        "locus": "Lemma3.1,Corollary3.2,Theorem3.3: finite higher-product bound and unbounded DG-injective resolution",
        "correspondence": "used with the stated conditions",
        "candidate_path": "course/src/inverse-limits-and-unbounded-resolutions.md",
        "sha256": "91AE732A4B5C0CCCC3C21FB1C9BFF4FDEC904BD7E5E0FB1F1BC17ABAB9948872",
        "result_id": "SH-DC.AB4-star-finite-resolution",
        "anchor": "3-finite-higher-product-dimension"
      }
    ],
    "proof": {
      "source": "src/inverse-limits-and-unbounded-resolutions.md",
      "sha256": "91ae732a4b5c0cccc3c21fb1c9bff4fdec904bd7e5e0fb1f1bc17abab9948872",
      "locus": "Theorem4.1: unbounded Cartan–Eilenberg product totalization",
      "reader": "inverse-limits-and-unbounded-resolutions.html#4-products-and-cartan-eilenberg-totalization"
    },
    "licence": "CC0-1.0",
    "proof_status": "complete; writing-AI self-checked"
  },
  {
    "result_id": "SH-DC.diagonal-obstruction",
    "name": "Proposition5.1: precise torsion-product obstruction to diagonal resolution",
    "statement": "Proposition5.1: precise torsion-product obstruction to diagonal resolution",
    "conditions": "Exact quotient and product formula plus augmentation H_R^0 isomorphism and torsion higher H_R^j; non-torsion infinite product. Conditional theorem fully proved; concrete Nagata category is separately cited further reading.",
    "writing_ai_selfcheck": true,
    "independent_review": false,
    "formalization": "not_formalized",
    "prerequisite_envelope_ids": [
      "SH-DC.split-resolution-towers"
    ],
    "dependency_precision": "Exact earlier proof providers; conditional/external-example boundary stated explicitly.",
    "programme_dependency_loci": [
      {
        "provider": "SH-DC.split-resolution-towers",
        "locus": "Lemmas 1.1,2.1; Propositions1.2,2.2: strict extension, split resolution towers and K-injective limit",
        "correspondence": "used with the stated conditions",
        "candidate_path": "course/src/inverse-limits-and-unbounded-resolutions.md",
        "sha256": "91AE732A4B5C0CCCC3C21FB1C9BFF4FDEC904BD7E5E0FB1F1BC17ABAB9948872",
        "result_id": "SH-DC.split-resolution-towers",
        "anchor": "1-building-a-compatible-tower"
      }
    ],
    "proof": {
      "source": "src/inverse-limits-and-unbounded-resolutions.md",
      "sha256": "91ae732a4b5c0cccc3c21fb1c9bff4fdec904bd7e5e0fb1f1bc17abab9948872",
      "locus": "Proposition5.1: precise torsion-product obstruction to diagonal resolution",
      "reader": "inverse-limits-and-unbounded-resolutions.html#5-how-an-obstruction-survives"
    },
    "licence": "CC0-1.0",
    "proof_status": "complete; writing-AI self-checked"
  },
  {
    "result_id": "SH-DC.finite-coresolution-product-bound",
    "name": "Proposition6.1: length-n product-acyclic coresolution implies AB4*-n",
    "statement": "Proposition6.1: length-n product-acyclic coresolution implies AB4*-n",
    "conditions": "Objectwise exact identity coresolution, all families of F_i values product-acyclic.",
    "writing_ai_selfcheck": true,
    "independent_review": false,
    "formalization": "not_formalized",
    "prerequisite_envelope_ids": [
      "SH-DC.bounded-derived"
    ],
    "dependency_precision": "Exact earlier proof providers; conditional/external-example boundary stated explicitly.",
    "programme_dependency_loci": [
      {
        "provider": "SH-DC.bounded-derived",
        "locus": "Theorem 4.1",
        "correspondence": "used with the stated conditions",
        "candidate_path": "course/src/injective-modules-and-bounded-below-derived-functors.md",
        "sha256": "B2DC989EFBC8D2D4004C8D24CBC3004A9E81DA485CD702BB7D293A7F8AF5EDBA",
        "result_id": "SH-DC.bounded-derived",
        "anchor": "4-right-derived-functors-on-bounded-below-complexes"
      }
    ],
    "proof": {
      "source": "src/inverse-limits-and-unbounded-resolutions.md",
      "sha256": "91ae732a4b5c0cccc3c21fb1c9bff4fdec904bd7e5e0fb1f1bc17abab9948872",
      "locus": "Proposition6.1: length-n product-acyclic coresolution implies AB4*-n",
      "reader": "inverse-limits-and-unbounded-resolutions.html#6-finite-coresolutions-as-a-source-of-the-hypothesis"
    },
    "licence": "CC0-1.0",
    "proof_status": "complete; writing-AI self-checked"
  },
  {
    "result_id": "SH-DC.affine-QCoh",
    "name": "Exact affine QCoh/module equivalence, universal Hom adjunction, exact localization, finite principal Čech complex, tensor/pullback/pushforward and flat base change.",
    "statement": "Exact affine QCoh/module equivalence, universal Hom adjunction, exact localization, finite principal Čech complex, tensor/pullback/pushforward and flat base change.",
    "conditions": "Commutative rings; arbitrary modules; affine equivalence is abelian, not a global unbounded equivalence.",
    "writing_ai_selfcheck": true,
    "independent_review": false,
    "formalization": "not_formalized",
    "prerequisite_envelope_ids": [
      "SH-DC.sheafification-stalks",
      "SH-DC.modules-abelian",
      "SH-DC.pullback-adjunction"
    ],
    "dependency_precision": "Exact earlier provider results; named loci and current hashes resolved below.",
    "programme_dependency_loci": [
      {
        "provider": "SH-DC.sheafification-stalks",
        "locus": "Lemma 1.3, full proof",
        "correspondence": "stated hypotheses",
        "candidate_path": "course/src/sheaves-of-modules-on-a-ringed-space.md",
        "sha256": "22FCF2E4E2E126FE922EBD68FD81F3DEDC581ED2B70CBB60A49250E21DA7BABE",
        "result_id": "SH-DC.sheafification-stalks",
        "anchor": "sheafification-and-the-stalk-test"
      },
      {
        "provider": "SH-DC.modules-abelian",
        "locus": "Theorem 2.1",
        "correspondence": "stated hypotheses",
        "candidate_path": "course/src/sheaves-of-modules-on-a-ringed-space.md",
        "sha256": "22FCF2E4E2E126FE922EBD68FD81F3DEDC581ED2B70CBB60A49250E21DA7BABE",
        "result_id": "SH-DC.modules-abelian",
        "anchor": "2-the-abelian-category-of-module-sheaves"
      },
      {
        "provider": "SH-DC.pullback-adjunction",
        "locus": "Theorems 4.1 and 4.4; Lemma 4.3",
        "correspondence": "stated hypotheses",
        "candidate_path": "course/src/sheaves-of-modules-on-a-ringed-space.md",
        "sha256": "22FCF2E4E2E126FE922EBD68FD81F3DEDC581ED2B70CBB60A49250E21DA7BABE",
        "result_id": "SH-DC.pullback-adjunction",
        "anchor": "module-pullback-and-extension-of-scalars"
      }
    ],
    "proof": {
      "source": "src/quasi-coherent-sheaves-and-concentrated-maps.md",
      "sha256": "1f1d5d47f4405d0514d9c2649d592e83af28e6568ef23d7b910f5f96989463cb",
      "locus": "Lemma 1.1; Theorem 1.2; Proposition 1.3",
      "reader": "quasi-coherent-sheaves-and-concentrated-maps.html#1-affine-sheaves-and-localization"
    },
    "licence": "CC0-1.0",
    "proof_status": "complete; writing-AI self-checked"
  },
  {
    "result_id": "SH-DC.scheme-geometry",
    "name": "Affine fibre products, separation and cover stability",
    "statement": "Affine fibre products have tensor-product rings for arbitrary test schemes; affine closed subschemes are affine, closed immersions are stable under base change, affine opens intersect affinely on a separated scheme; quasi-compact and quasi-separated morphisms are stable under arbitrary base change, with finite affine covers and their intersections preserved.",
    "conditions": "Arbitrary schemes and commutative unital rings; no noetherian, flatness or separatedness hypothesis except where expressly stated.",
    "writing_ai_selfcheck": true,
    "independent_review": false,
    "formalization": "not_formalized",
    "prerequisite_envelope_ids": [
      "SH-DC.affine-QCoh"
    ],
    "programme_dependency_loci": [
      {
        "provider": "SH-DC.affine-QCoh",
        "candidate_path": "course/src/quasi-coherent-sheaves-and-concentrated-maps.md",
        "locus": "Lemma 1.1; Theorem 1.2; Proposition 1.3",
        "sha256": "1F1D5D47F4405D0514D9C2649D592E83AF28E6568EF23D7B910F5F96989463CB"
      }
    ],
    "proof": {
      "source": "src/quasi-coherent-sheaves-and-concentrated-maps.md",
      "sha256": "1f1d5d47f4405d0514d9c2649d592e83af28e6568ef23d7b910f5f96989463cb",
      "locus": "Lemma 1.4, full proof",
      "reader": "quasi-coherent-sheaves-and-concentrated-maps.html#1-affine-sheaves-and-localization"
    },
    "licence": "CC0-1.0",
    "proof_status": "complete; writing-AI self-checked"
  },
  {
    "result_id": "SH-DC.Gabber-QCoh",
    "name": "QCoh of every scheme is Grothendieck; explicit small submodules, generator, enough injectives, quasi-coherent right adjoint and all limits.",
    "statement": "QCoh of every scheme is Grothendieck; explicit small submodules, generator, enough injectives, quasi-coherent right adjoint and all limits.",
    "conditions": "Any scheme; limits are in QCoh, not identified with all-module-sheaf products.",
    "writing_ai_selfcheck": true,
    "independent_review": false,
    "formalization": "not_formalized",
    "prerequisite_envelope_ids": [
      "SH-DC.affine-QCoh",
      "SH-DC.filtered-colimits-sums",
      "SH-DC.Grothendieck-injective-embeddings"
    ],
    "dependency_precision": "Exact earlier provider results; named loci and current hashes resolved below.",
    "programme_dependency_loci": [
      {
        "provider": "SH-DC.affine-QCoh",
        "locus": "Lemma 1.1; Theorem 1.2; Proposition 1.3",
        "correspondence": "stated hypotheses",
        "candidate_path": "course/src/quasi-coherent-sheaves-and-concentrated-maps.md",
        "sha256": "1F1D5D47F4405D0514D9C2649D592E83AF28E6568EF23D7B910F5F96989463CB",
        "result_id": "SH-DC.affine-QCoh",
        "anchor": "1-affine-sheaves-and-localization"
      },
      {
        "provider": "SH-DC.filtered-colimits-sums",
        "locus": "Lemma 1.1 and Theorem 3.1",
        "correspondence": "stated hypotheses",
        "candidate_path": "course/src/sheaves-of-modules-on-a-ringed-space.md",
        "sha256": "22FCF2E4E2E126FE922EBD68FD81F3DEDC581ED2B70CBB60A49250E21DA7BABE",
        "result_id": "SH-DC.filtered-colimits-sums",
        "anchor": "3-limits-colimits-stalks-and-sections-of-sums"
      },
      {
        "provider": "SH-DC.Grothendieck-injective-embeddings",
        "locus": "Lemmas 2.1-2.3 and Theorem 2.4",
        "correspondence": "stated hypotheses",
        "candidate_path": "course/src/k-injective-resolutions-in-grothendieck-categories.md",
        "sha256": "75118C6AB567222AEB5E42C60AA5DC206466FFF9EA8A9F193EA1B598DECD760B",
        "result_id": "SH-DC.Grothendieck-injective-embeddings",
        "anchor": "2-sizes-and-functorial-injective-embeddings"
      }
    ],
    "proof": {
      "source": "src/quasi-coherent-sheaves-and-concentrated-maps.md",
      "sha256": "1f1d5d47f4405d0514d9c2649d592e83af28e6568ef23d7b910f5f96989463cb",
      "locus": "Theorem 2.1; Lemma 2.2",
      "reader": "quasi-coherent-sheaves-and-concentrated-maps.html#2-small-submodules-and-enough-injectives"
    },
    "licence": "CC0-1.0",
    "proof_status": "complete; writing-AI self-checked"
  },
  {
    "result_id": "SH-DC.QCoh-affine-cover-product-bound",
    "name": "Theorem6.2: n+1 affine cover of qc semi-separated scheme gives QCoh AB4*-n",
    "statement": "Theorem6.2: n+1 affine cover of qc semi-separated scheme gives QCoh AB4*-n",
    "conditions": "QCoh category; finite affine cover and affine intersections; ordinary all-module-sheaf products/direct images excluded.",
    "writing_ai_selfcheck": true,
    "independent_review": false,
    "formalization": "not_formalized",
    "prerequisite_envelope_ids": [
      "SH-DC.finite-coresolution-product-bound",
      "SH-DC.affine-QCoh",
      "SH-DC.Gabber-QCoh"
    ],
    "dependency_precision": "Exact earlier proof providers; conditional/external-example boundary stated explicitly.",
    "programme_dependency_loci": [
      {
        "provider": "SH-DC.finite-coresolution-product-bound",
        "locus": "Proposition6.1: length-n product-acyclic coresolution implies AB4*-n",
        "correspondence": "full internal provider",
        "candidate_path": "course/src/inverse-limits-and-unbounded-resolutions.md",
        "sha256": "91AE732A4B5C0CCCC3C21FB1C9BFF4FDEC904BD7E5E0FB1F1BC17ABAB9948872",
        "result_id": "SH-DC.finite-coresolution-product-bound",
        "anchor": "6-finite-coresolutions-as-a-source-of-the-hypothesis"
      },
      {
        "provider": "SH-DC.affine-QCoh",
        "locus": "Lemma 1.1; Theorem 1.2; Proposition 1.3",
        "correspondence": "full internal provider",
        "candidate_path": "course/src/quasi-coherent-sheaves-and-concentrated-maps.md",
        "sha256": "1F1D5D47F4405D0514D9C2649D592E83AF28E6568EF23D7B910F5F96989463CB",
        "result_id": "SH-DC.affine-QCoh",
        "anchor": "1-affine-sheaves-and-localization"
      },
      {
        "provider": "SH-DC.Gabber-QCoh",
        "locus": "Theorem 2.1; Lemma 2.2",
        "correspondence": "full internal provider",
        "candidate_path": "course/src/quasi-coherent-sheaves-and-concentrated-maps.md",
        "sha256": "1F1D5D47F4405D0514D9C2649D592E83AF28E6568EF23D7B910F5F96989463CB",
        "result_id": "SH-DC.Gabber-QCoh",
        "anchor": "2-small-submodules-and-enough-injectives"
      }
    ],
    "proof": {
      "source": "src/inverse-limits-and-unbounded-resolutions.md",
      "sha256": "91ae732a4b5c0cccc3c21fb1c9bff4fdec904bd7e5e0fb1f1bc17abab9948872",
      "locus": "Theorem6.2: n+1 affine cover of qc semi-separated scheme gives QCoh AB4*-n",
      "reader": "inverse-limits-and-unbounded-resolutions.html#6-finite-coresolutions-as-a-source-of-the-hypothesis"
    },
    "licence": "CC0-1.0",
    "proof_status": "complete; writing-AI self-checked"
  },
  {
    "result_id": "SH-DC.affine-Cech-cohomology",
    "name": "Basis Čech criterion, affine QCoh acyclicity, finite affine-intersection Čech comparison and m-affine vanishing in degrees >=m.",
    "statement": "Basis Čech criterion, affine QCoh acyclicity, finite affine-intersection Čech comparison and m-affine vanishing in degrees >=m.",
    "conditions": "Principal basis on affines; finite affine cover with affine multiple intersections for the global comparison.",
    "writing_ai_selfcheck": true,
    "independent_review": false,
    "formalization": "not_formalized",
    "prerequisite_envelope_ids": [
      "SH-DC.affine-QCoh",
      "SH-DC.extension-skyscraper",
      "SH-DC.injective-embeddings",
      "SH-DC.bounded-derived",
      "SH-DC.homotopy-localization"
    ],
    "dependency_precision": "Exact earlier provider results; named loci and current hashes resolved below.",
    "programme_dependency_loci": [
      {
        "provider": "SH-DC.affine-QCoh",
        "locus": "Lemma 1.1; Theorem 1.2; Proposition 1.3",
        "correspondence": "stated hypotheses",
        "candidate_path": "course/src/quasi-coherent-sheaves-and-concentrated-maps.md",
        "sha256": "1F1D5D47F4405D0514D9C2649D592E83AF28E6568EF23D7B910F5F96989463CB",
        "result_id": "SH-DC.affine-QCoh",
        "anchor": "1-affine-sheaves-and-localization"
      },
      {
        "provider": "SH-DC.extension-skyscraper",
        "locus": "Lemmas 5.1 and 5.2",
        "correspondence": "stated hypotheses",
        "candidate_path": "course/src/sheaves-of-modules-on-a-ringed-space.md",
        "sha256": "22FCF2E4E2E126FE922EBD68FD81F3DEDC581ED2B70CBB60A49250E21DA7BABE",
        "result_id": "SH-DC.extension-skyscraper",
        "anchor": "5-extension-by-zero-and-skyscrapers"
      },
      {
        "provider": "SH-DC.injective-embeddings",
        "locus": "Lemmas 2.1-2.2 and Theorem 2.3",
        "correspondence": "stated hypotheses",
        "candidate_path": "course/src/injective-modules-and-bounded-below-derived-functors.md",
        "sha256": "B2DC989EFBC8D2D4004C8D24CBC3004A9E81DA485CD702BB7D293A7F8AF5EDBA",
        "result_id": "SH-DC.injective-embeddings",
        "anchor": "2-functorial-injective-embeddings"
      },
      {
        "provider": "SH-DC.bounded-derived",
        "locus": "Theorem 4.1",
        "correspondence": "stated hypotheses",
        "candidate_path": "course/src/injective-modules-and-bounded-below-derived-functors.md",
        "sha256": "B2DC989EFBC8D2D4004C8D24CBC3004A9E81DA485CD702BB7D293A7F8AF5EDBA",
        "result_id": "SH-DC.bounded-derived",
        "anchor": "4-right-derived-functors-on-bounded-below-complexes"
      },
      {
        "provider": "SH-DC.homotopy-localization",
        "locus": "Theorems 3.1, 4.2, 5.1 and Proposition 5.2",
        "correspondence": "stated hypotheses",
        "candidate_path": "course/src/complexes-cones-and-localization.md",
        "sha256": "367394AE4A25DCF64DF87302304D8395AF84D92765E4E4E45A2333A0071C3862",
        "result_id": "SH-DC.homotopy-localization",
        "anchor": "3-the-triangulated-structure-of-the-homotopy-category"
      }
    ],
    "proof": {
      "source": "src/quasi-coherent-sheaves-and-concentrated-maps.md",
      "sha256": "1f1d5d47f4405d0514d9c2649d592e83af28e6568ef23d7b910f5f96989463cb",
      "locus": "Lemma 3.1; Theorem 3.2",
      "reader": "quasi-coherent-sheaves-and-concentrated-maps.html#3-affine-cohomology-and-finite-bounds"
    },
    "licence": "CC0-1.0",
    "proof_status": "complete; writing-AI self-checked"
  },
  {
    "result_id": "SH-DC.concentrated-cohomology-bound",
    "name": "Finite QCoh cohomological bound for qcqs schemes, localization, quasi-coherence of higher images of a concentrated map, and a bound preserved by affine base change.",
    "statement": "Finite QCoh cohomological bound for qcqs schemes, localization, quasi-coherence of higher images of a concentrated map, and a bound preserved by affine base change.",
    "conditions": "QCoh inputs; locally uniform on affine target opens. No bound on all module sheaves inferred.",
    "writing_ai_selfcheck": true,
    "independent_review": false,
    "formalization": "not_formalized",
    "prerequisite_envelope_ids": [
      "SH-DC.affine-Cech-cohomology",
      "SH-DC.flasque-acyclic",
      "SH-DC.Leray-spectral-sequence",
      "SH-DC.scheme-geometry"
    ],
    "dependency_precision": "Exact earlier provider results; named loci and current hashes resolved below.",
    "programme_dependency_loci": [
      {
        "provider": "SH-DC.affine-Cech-cohomology",
        "locus": "Lemma 3.1; Theorem 3.2",
        "correspondence": "stated hypotheses",
        "candidate_path": "course/src/quasi-coherent-sheaves-and-concentrated-maps.md",
        "sha256": "1F1D5D47F4405D0514D9C2649D592E83AF28E6568EF23D7B910F5F96989463CB",
        "result_id": "SH-DC.affine-Cech-cohomology",
        "anchor": "3-affine-cohomology-and-finite-bounds"
      },
      {
        "provider": "SH-DC.flasque-acyclic",
        "locus": "Lemmas 1.1-1.2 and Proposition 1.3",
        "correspondence": "stated hypotheses",
        "candidate_path": "course/src/injective-modules-and-bounded-below-derived-functors.md",
        "sha256": "B2DC989EFBC8D2D4004C8D24CBC3004A9E81DA485CD702BB7D293A7F8AF5EDBA",
        "result_id": "SH-DC.flasque-acyclic",
        "anchor": "1-extending-maps-and-extending-sections"
      },
      {
        "provider": "SH-DC.Leray-spectral-sequence",
        "locus": "Proposition 3.1 and Exercise 3 solution",
        "correspondence": "stated hypotheses",
        "candidate_path": "course/src/derived-pullback-and-pushforward.md",
        "sha256": "FECE9386A64E3298CD133F9ED65D4D74AAC82B9A8B266FB5CCC6363C8C063AB2",
        "result_id": "SH-DC.Leray-spectral-sequence",
        "anchor": "6-exercises-with-checked-solutions"
      },
      {
        "provider": "SH-DC.scheme-geometry",
        "candidate_path": "course/src/quasi-coherent-sheaves-and-concentrated-maps.md",
        "locus": "Lemma 1.4, full proof",
        "sha256": "1F1D5D47F4405D0514D9C2649D592E83AF28E6568EF23D7B910F5F96989463CB"
      }
    ],
    "proof": {
      "source": "src/quasi-coherent-sheaves-and-concentrated-maps.md",
      "sha256": "1f1d5d47f4405d0514d9c2649d592e83af28e6568ef23d7b910f5f96989463cb",
      "locus": "Proposition 3.3",
      "reader": "quasi-coherent-sheaves-and-concentrated-maps.html#3-affine-cohomology-and-finite-bounds"
    },
    "licence": "CC0-1.0",
    "proof_status": "complete; writing-AI self-checked"
  },
  {
    "result_id": "SH-DC.cohomology-continuity",
    "name": "Cohomology on qcqs schemes and higher images of concentrated maps commute with filtered colimits and sums of arbitrary module sheaves.",
    "statement": "Cohomology on qcqs schemes and higher images of concentrated maps commute with filtered colimits and sums of arbitrary module sheaves.",
    "conditions": "Small filtered diagrams; sections comparison uses quasi-compact intersections; no finite cohomological dimension asserted.",
    "writing_ai_selfcheck": true,
    "independent_review": false,
    "formalization": "not_formalized",
    "prerequisite_envelope_ids": [
      "SH-DC.affine-Cech-cohomology",
      "SH-DC.filtered-colimits-sums",
      "SH-DC.Grothendieck-injective-embeddings",
      "SH-DC.scheme-geometry"
    ],
    "dependency_precision": "Exact earlier provider results; named loci and current hashes resolved below.",
    "programme_dependency_loci": [
      {
        "provider": "SH-DC.affine-Cech-cohomology",
        "locus": "Lemma 3.1; Theorem 3.2",
        "correspondence": "stated hypotheses",
        "candidate_path": "course/src/quasi-coherent-sheaves-and-concentrated-maps.md",
        "sha256": "1F1D5D47F4405D0514D9C2649D592E83AF28E6568EF23D7B910F5F96989463CB",
        "result_id": "SH-DC.affine-Cech-cohomology",
        "anchor": "3-affine-cohomology-and-finite-bounds"
      },
      {
        "provider": "SH-DC.filtered-colimits-sums",
        "locus": "Lemma 1.1 and Theorem 3.1",
        "correspondence": "stated hypotheses",
        "candidate_path": "course/src/sheaves-of-modules-on-a-ringed-space.md",
        "sha256": "22FCF2E4E2E126FE922EBD68FD81F3DEDC581ED2B70CBB60A49250E21DA7BABE",
        "result_id": "SH-DC.filtered-colimits-sums",
        "anchor": "3-limits-colimits-stalks-and-sections-of-sums"
      },
      {
        "provider": "SH-DC.Grothendieck-injective-embeddings",
        "locus": "Lemmas 2.1-2.3 and Theorem 2.4",
        "correspondence": "stated hypotheses",
        "candidate_path": "course/src/k-injective-resolutions-in-grothendieck-categories.md",
        "sha256": "75118C6AB567222AEB5E42C60AA5DC206466FFF9EA8A9F193EA1B598DECD760B",
        "result_id": "SH-DC.Grothendieck-injective-embeddings",
        "anchor": "2-sizes-and-functorial-injective-embeddings"
      },
      {
        "provider": "SH-DC.scheme-geometry",
        "candidate_path": "course/src/quasi-coherent-sheaves-and-concentrated-maps.md",
        "locus": "Lemma 1.4, full proof",
        "sha256": "1F1D5D47F4405D0514D9C2649D592E83AF28E6568EF23D7B910F5F96989463CB"
      }
    ],
    "proof": {
      "source": "src/quasi-coherent-sheaves-and-concentrated-maps.md",
      "sha256": "1f1d5d47f4405d0514d9c2649d592e83af28e6568ef23d7b910f5f96989463cb",
      "locus": "Proposition 3.4",
      "reader": "quasi-coherent-sheaves-and-concentrated-maps.html#3-affine-cohomology-and-finite-bounds"
    },
    "licence": "CC0-1.0",
    "proof_status": "complete; writing-AI self-checked"
  },
  {
    "result_id": "SH-DC.unbounded-affine-concentrated",
    "name": "D(A) equals D_qc(Spec A); concentrated Rf_* preserves D_qc, has a local finite QCoh bound and commutes with sums; arbitrary Lf* preserves D_qc.",
    "statement": "D(A) equals D_qc(Spec A); concentrated Rf_* preserves D_qc, has a local finite QCoh bound and commutes with sums; arbitrary Lf* preserves D_qc.",
    "conditions": "Unbounded inputs; split towers checked on affines, no exact all-sheaf product assumption; no global D(QCoh)=D_qc assertion.",
    "writing_ai_selfcheck": true,
    "independent_review": false,
    "formalization": "not_formalized",
    "prerequisite_envelope_ids": [
      "SH-DC.affine-QCoh",
      "SH-DC.concentrated-cohomology-bound",
      "SH-DC.cohomology-continuity",
      "SH-DC.split-resolution-towers",
      "SH-DC.derived-pull-push-adjunction",
      "SH-DC.semifree-K-projective"
    ],
    "dependency_precision": "Exact earlier provider results; named loci and current hashes resolved below.",
    "programme_dependency_loci": [
      {
        "provider": "SH-DC.affine-QCoh",
        "locus": "Lemma 1.1; Theorem 1.2; Proposition 1.3",
        "correspondence": "stated hypotheses",
        "candidate_path": "course/src/quasi-coherent-sheaves-and-concentrated-maps.md",
        "sha256": "1F1D5D47F4405D0514D9C2649D592E83AF28E6568EF23D7B910F5F96989463CB",
        "result_id": "SH-DC.affine-QCoh",
        "anchor": "1-affine-sheaves-and-localization"
      },
      {
        "provider": "SH-DC.concentrated-cohomology-bound",
        "locus": "Proposition 3.3",
        "correspondence": "stated hypotheses",
        "candidate_path": "course/src/quasi-coherent-sheaves-and-concentrated-maps.md",
        "sha256": "1F1D5D47F4405D0514D9C2649D592E83AF28E6568EF23D7B910F5F96989463CB",
        "result_id": "SH-DC.concentrated-cohomology-bound",
        "anchor": "3-affine-cohomology-and-finite-bounds"
      },
      {
        "provider": "SH-DC.cohomology-continuity",
        "locus": "Proposition 3.4",
        "correspondence": "stated hypotheses",
        "candidate_path": "course/src/quasi-coherent-sheaves-and-concentrated-maps.md",
        "sha256": "1F1D5D47F4405D0514D9C2649D592E83AF28E6568EF23D7B910F5F96989463CB",
        "result_id": "SH-DC.cohomology-continuity",
        "anchor": "3-affine-cohomology-and-finite-bounds"
      },
      {
        "provider": "SH-DC.split-resolution-towers",
        "locus": "Lemmas 1.1,2.1; Propositions1.2,2.2: strict extension, split resolution towers and K-injective limit",
        "correspondence": "stated hypotheses",
        "candidate_path": "course/src/inverse-limits-and-unbounded-resolutions.md",
        "sha256": "91AE732A4B5C0CCCC3C21FB1C9BFF4FDEC904BD7E5E0FB1F1BC17ABAB9948872",
        "result_id": "SH-DC.split-resolution-towers",
        "anchor": "1-building-a-compatible-tower"
      },
      {
        "provider": "SH-DC.derived-pull-push-adjunction",
        "locus": "Theorem 1.1, Proposition 1.2, Lemma 2.1, Theorem 2.2 and Proposition 3.1",
        "correspondence": "stated hypotheses",
        "candidate_path": "course/src/derived-pullback-and-pushforward.md",
        "sha256": "FECE9386A64E3298CD133F9ED65D4D74AAC82B9A8B266FB5CCC6363C8C063AB2",
        "result_id": "SH-DC.derived-pull-push-adjunction",
        "anchor": "1-derived-pullback-on-k-flat-models"
      },
      {
        "provider": "SH-DC.semifree-K-projective",
        "locus": "Theorem 3.4: unbounded semifree, K-projective and K-flat module resolutions and Hom comparison",
        "correspondence": "stated hypotheses",
        "candidate_path": "course/src/flat-modules-and-k-flat-resolutions.md",
        "sha256": "7BCE2B46408B278A147517161F09AE29A7BC0273B4E271AA2F995966C1AA941D",
        "result_id": "SH-DC.semifree-K-projective",
        "anchor": "semifree-resolutions-over-a-ring"
      }
    ],
    "proof": {
      "source": "src/quasi-coherent-sheaves-and-concentrated-maps.md",
      "sha256": "1f1d5d47f4405d0514d9c2649d592e83af28e6568ef23d7b910f5f96989463cb",
      "locus": "Lemma 4.1; Theorem 4.2",
      "reader": "quasi-coherent-sheaves-and-concentrated-maps.html#4-controlling-unbounded-complexes"
    },
    "licence": "CC0-1.0",
    "proof_status": "complete; writing-AI self-checked"
  },
  {
    "result_id": "SH-DC.all-module-finite-dimensional",
    "name": "Concentrated Rf_* of uniform finite dimension on all module sheaves has unbounded amplitude [0,d] and preserves sums, by a finite Godement coresolution and acyclic models.",
    "statement": "Concentrated Rf_* of uniform finite dimension on all module sheaves has unbounded amplitude [0,d] and preserves sums, by a finite Godement coresolution and acyclic models.",
    "conditions": "The uniform all-module bound is an additional hypothesis, not a consequence of being concentrated.",
    "writing_ai_selfcheck": true,
    "independent_review": false,
    "formalization": "not_formalized",
    "prerequisite_envelope_ids": [
      "SH-DC.finite-dimensional-acyclic-models",
      "SH-DC.extension-skyscraper",
      "SH-DC.flasque-acyclic",
      "SH-DC.cohomology-continuity"
    ],
    "dependency_precision": "Exact earlier provider results; named loci and current hashes resolved below.",
    "programme_dependency_loci": [
      {
        "provider": "SH-DC.finite-dimensional-acyclic-models",
        "locus": "Proposition 5.5: unbounded complexes of acyclic objects compute RF",
        "correspondence": "stated hypotheses",
        "candidate_path": "course/src/k-injective-resolutions-in-grothendieck-categories.md",
        "sha256": "75118C6AB567222AEB5E42C60AA5DC206466FFF9EA8A9F193EA1B598DECD760B",
        "result_id": "SH-DC.finite-dimensional-acyclic-models",
        "anchor": "finite-cohomological-dimension-and-acyclic-models"
      },
      {
        "provider": "SH-DC.extension-skyscraper",
        "locus": "Lemmas 5.1 and 5.2",
        "correspondence": "stated hypotheses",
        "candidate_path": "course/src/sheaves-of-modules-on-a-ringed-space.md",
        "sha256": "22FCF2E4E2E126FE922EBD68FD81F3DEDC581ED2B70CBB60A49250E21DA7BABE",
        "result_id": "SH-DC.extension-skyscraper",
        "anchor": "5-extension-by-zero-and-skyscrapers"
      },
      {
        "provider": "SH-DC.flasque-acyclic",
        "locus": "Lemmas 1.1-1.2 and Proposition 1.3",
        "correspondence": "stated hypotheses",
        "candidate_path": "course/src/injective-modules-and-bounded-below-derived-functors.md",
        "sha256": "B2DC989EFBC8D2D4004C8D24CBC3004A9E81DA485CD702BB7D293A7F8AF5EDBA",
        "result_id": "SH-DC.flasque-acyclic",
        "anchor": "1-extending-maps-and-extending-sections"
      },
      {
        "provider": "SH-DC.cohomology-continuity",
        "locus": "Proposition 3.4",
        "correspondence": "stated hypotheses",
        "candidate_path": "course/src/quasi-coherent-sheaves-and-concentrated-maps.md",
        "sha256": "1F1D5D47F4405D0514D9C2649D592E83AF28E6568EF23D7B910F5F96989463CB",
        "result_id": "SH-DC.cohomology-continuity",
        "anchor": "3-affine-cohomology-and-finite-bounds"
      }
    ],
    "proof": {
      "source": "src/quasi-coherent-sheaves-and-concentrated-maps.md",
      "sha256": "1f1d5d47f4405d0514d9c2649d592e83af28e6568ef23d7b910f5f96989463cb",
      "locus": "Proposition 4.3",
      "reader": "quasi-coherent-sheaves-and-concentrated-maps.html#4-controlling-unbounded-complexes"
    },
    "licence": "CC0-1.0",
    "proof_status": "complete; writing-AI self-checked"
  },
  {
    "result_id": "SH-DC.concentrated-projection",
    "name": "Canonical unbounded concentrated projection formula for E and K quasi-coherent-cohomology; alternatively arbitrary E if f has uniform finite all-module cohomological dimension.",
    "statement": "Canonical unbounded concentrated projection formula for E and K quasi-coherent-cohomology; alternatively arbitrary E if f has uniform finite all-module cohomological dimension.",
    "conditions": "K in D_qc with no perfection or boundedness restriction; f qcqs; specified alternative condition on E.",
    "writing_ai_selfcheck": true,
    "independent_review": false,
    "formalization": "not_formalized",
    "prerequisite_envelope_ids": [
      "SH-DC.projection-formula",
      "SH-DC.unbounded-affine-concentrated",
      "SH-DC.all-module-finite-dimensional",
      "SH-DC.semifree-K-projective",
      "SH-DC.derived-tensor"
    ],
    "dependency_precision": "Exact earlier provider results; named loci and current hashes resolved below.",
    "programme_dependency_loci": [
      {
        "provider": "SH-DC.projection-formula",
        "locus": "Theorems 1.2,2.1,3.1 and Exercises 1-4",
        "correspondence": "stated hypotheses",
        "candidate_path": "course/src/projection-formula-and-base-change.md",
        "sha256": "AFEB289ACD142E469F5683C985F314014E92ED8AE8ED3E4C5D1FF3BF02582674",
        "result_id": "SH-DC.projection-formula",
        "anchor": "2-the-general-map-and-perfect-coefficients"
      },
      {
        "provider": "SH-DC.unbounded-affine-concentrated",
        "locus": "Lemma 4.1; Theorem 4.2",
        "correspondence": "stated hypotheses",
        "candidate_path": "course/src/quasi-coherent-sheaves-and-concentrated-maps.md",
        "sha256": "1F1D5D47F4405D0514D9C2649D592E83AF28E6568EF23D7B910F5F96989463CB",
        "result_id": "SH-DC.unbounded-affine-concentrated",
        "anchor": "4-controlling-unbounded-complexes"
      },
      {
        "provider": "SH-DC.all-module-finite-dimensional",
        "locus": "Proposition 4.3",
        "correspondence": "stated hypotheses",
        "candidate_path": "course/src/quasi-coherent-sheaves-and-concentrated-maps.md",
        "sha256": "1F1D5D47F4405D0514D9C2649D592E83AF28E6568EF23D7B910F5F96989463CB",
        "result_id": "SH-DC.all-module-finite-dimensional",
        "anchor": "4-controlling-unbounded-complexes"
      },
      {
        "provider": "SH-DC.semifree-K-projective",
        "locus": "Theorem 3.4: unbounded semifree, K-projective and K-flat module resolutions and Hom comparison",
        "correspondence": "stated hypotheses",
        "candidate_path": "course/src/flat-modules-and-k-flat-resolutions.md",
        "sha256": "7BCE2B46408B278A147517161F09AE29A7BC0273B4E271AA2F995966C1AA941D",
        "result_id": "SH-DC.semifree-K-projective",
        "anchor": "semifree-resolutions-over-a-ring"
      },
      {
        "provider": "SH-DC.derived-tensor",
        "locus": "Lemmas 1.1 and 2.1, Theorems 1.2 and 2.2",
        "correspondence": "stated hypotheses",
        "candidate_path": "course/src/derived-tensor-products-and-tor-sheaves.md",
        "sha256": "B78C1AD9864901349E26C6EB3954B046E40B84E35E55EA6B3A71C553E185D292",
        "result_id": "SH-DC.derived-tensor",
        "anchor": "1-the-two-tensor-comparisons"
      }
    ],
    "proof": {
      "source": "src/quasi-coherent-sheaves-and-concentrated-maps.md",
      "sha256": "1f1d5d47f4405d0514d9c2649d592e83af28e6568ef23d7b910f5f96989463cb",
      "locus": "Theorem 5.1",
      "reader": "quasi-coherent-sheaves-and-concentrated-maps.html#5-the-unbounded-projection-formula"
    },
    "licence": "CC0-1.0",
    "proof_status": "complete; writing-AI self-checked"
  },
  {
    "result_id": "SH-DC.concentrated-flat-base-change",
    "name": "Ordinary QCoh flat base change and canonical unbounded flat base-change isomorphism for concentrated maps.",
    "statement": "Ordinary QCoh flat base change and canonical unbounded flat base-change isomorphism for concentrated maps.",
    "conditions": "Cartesian schemes, f qcqs, bottom map flat, E in D_qc; canonical mate explicitly identified.",
    "writing_ai_selfcheck": true,
    "independent_review": false,
    "formalization": "not_formalized",
    "prerequisite_envelope_ids": [
      "SH-DC.base-change-map",
      "SH-DC.unbounded-affine-concentrated",
      "SH-DC.concentrated-cohomology-bound",
      "SH-DC.affine-Cech-cohomology",
      "SH-DC.scheme-geometry"
    ],
    "dependency_precision": "Exact earlier provider results; named loci and current hashes resolved below.",
    "programme_dependency_loci": [
      {
        "provider": "SH-DC.base-change-map",
        "locus": "Theorem 4.1 and Section 5, Exercises 5-6",
        "correspondence": "stated hypotheses",
        "candidate_path": "course/src/projection-formula-and-base-change.md",
        "sha256": "AFEB289ACD142E469F5683C985F314014E92ED8AE8ED3E4C5D1FF3BF02582674",
        "result_id": "SH-DC.base-change-map",
        "anchor": "4-base-change-flat-models-and-compatibility"
      },
      {
        "provider": "SH-DC.unbounded-affine-concentrated",
        "locus": "Lemma 4.1; Theorem 4.2",
        "correspondence": "stated hypotheses",
        "candidate_path": "course/src/quasi-coherent-sheaves-and-concentrated-maps.md",
        "sha256": "1F1D5D47F4405D0514D9C2649D592E83AF28E6568EF23D7B910F5F96989463CB",
        "result_id": "SH-DC.unbounded-affine-concentrated",
        "anchor": "4-controlling-unbounded-complexes"
      },
      {
        "provider": "SH-DC.concentrated-cohomology-bound",
        "locus": "Proposition 3.3",
        "correspondence": "stated hypotheses",
        "candidate_path": "course/src/quasi-coherent-sheaves-and-concentrated-maps.md",
        "sha256": "1F1D5D47F4405D0514D9C2649D592E83AF28E6568EF23D7B910F5F96989463CB",
        "result_id": "SH-DC.concentrated-cohomology-bound",
        "anchor": "3-affine-cohomology-and-finite-bounds"
      },
      {
        "provider": "SH-DC.affine-Cech-cohomology",
        "locus": "Lemma 3.1; Theorem 3.2",
        "correspondence": "stated hypotheses",
        "candidate_path": "course/src/quasi-coherent-sheaves-and-concentrated-maps.md",
        "sha256": "1F1D5D47F4405D0514D9C2649D592E83AF28E6568EF23D7B910F5F96989463CB",
        "result_id": "SH-DC.affine-Cech-cohomology",
        "anchor": "3-affine-cohomology-and-finite-bounds"
      },
      {
        "provider": "SH-DC.scheme-geometry",
        "candidate_path": "course/src/quasi-coherent-sheaves-and-concentrated-maps.md",
        "locus": "Lemma 1.4, full proof",
        "sha256": "1F1D5D47F4405D0514D9C2649D592E83AF28E6568EF23D7B910F5F96989463CB"
      }
    ],
    "proof": {
      "source": "src/quasi-coherent-sheaves-and-concentrated-maps.md",
      "sha256": "1f1d5d47f4405d0514d9c2649d592e83af28e6568ef23d7b910f5f96989463cb",
      "locus": "Lemma 6.1; Theorem 6.2",
      "reader": "quasi-coherent-sheaves-and-concentrated-maps.html#6-flat-base-change"
    },
    "licence": "CC0-1.0",
    "proof_status": "complete; writing-AI self-checked"
  }
]