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            "id": "the-different-and-the-discriminant",
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            "id": "the-dedekind-zeta-function-and-the-analytic-class-number-formula",
            "title": "The Dedekind zeta function and the analytic class number formula",
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            "id": "abelian-number-fields-and-dirichlet-l-functions-at-s-1",
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        "subject": "algebraic-geometry",
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            "id": "c-star-algebras-continuous-functional-calculus-automatic-continuity-positive-cones",
            "title": "C*-algebras: continuous functional calculus, automatic continuity, positive cones, approximate identities and quotients",
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            "id": "the-double-commutant-theorem",
            "title": "The double commutant theorem",
            "route": "https://kokunoyumeto.github.io/open-math-courses-public/courses/foundations-of-von-neumann-algebras/the-double-commutant-theorem.html",
            "editable_source": "https://kokunoyumeto.github.io/open-math-courses-public/courses/foundations-of-von-neumann-algebras/src/the-double-commutant-theorem.md"
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            "id": "kaplansky-s-density-theorem-and-its-consequences",
            "title": "Kaplansky's density theorem and its consequences",
            "route": "https://kokunoyumeto.github.io/open-math-courses-public/courses/foundations-of-von-neumann-algebras/kaplansky-s-density-theorem-and-its-consequences.html",
            "editable_source": "https://kokunoyumeto.github.io/open-math-courses-public/courses/foundations-of-von-neumann-algebras/src/kaplansky-s-density-theorem-and-its-consequences.md"
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          {
            "id": "abelian-operator-algebras",
            "title": "Abelian operator algebras",
            "route": "https://kokunoyumeto.github.io/open-math-courses-public/courses/foundations-of-von-neumann-algebras/abelian-operator-algebras.html",
            "editable_source": "https://kokunoyumeto.github.io/open-math-courses-public/courses/foundations-of-von-neumann-algebras/src/abelian-operator-algebras.md"
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            "id": "the-universal-enveloping-von-neumann-algebra-of-a-c-star-algebra-and-w-star-algebras",
            "title": "The universal enveloping von Neumann algebra of a C*-algebra, and W*-algebras",
            "route": "https://kokunoyumeto.github.io/open-math-courses-public/courses/foundations-of-von-neumann-algebras/the-universal-enveloping-von-neumann-algebra-of-a-c-star-algebra-and-w-star-algebras.html",
            "editable_source": "https://kokunoyumeto.github.io/open-math-courses-public/courses/foundations-of-von-neumann-algebras/src/the-universal-enveloping-von-neumann-algebra-of-a-c-star-algebra-and-w-star-algebras.md"
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            "id": "projections-and-types-of-von-neumann-algebras",
            "title": "Projections and types of von Neumann algebras",
            "route": "https://kokunoyumeto.github.io/open-math-courses-public/courses/foundations-of-von-neumann-algebras/projections-and-types-of-von-neumann-algebras.html",
            "editable_source": "https://kokunoyumeto.github.io/open-math-courses-public/courses/foundations-of-von-neumann-algebras/src/projections-and-types-of-von-neumann-algebras.md"
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            "id": "polar-decomposition-of-functionals-and-weak-compactness-in-preduals",
            "title": "Polar decomposition of functionals and weak compactness in preduals",
            "route": "https://kokunoyumeto.github.io/open-math-courses-public/courses/foundations-of-von-neumann-algebras/polar-decomposition-of-functionals-and-weak-compactness-in-preduals.html",
            "editable_source": "https://kokunoyumeto.github.io/open-math-courses-public/courses/foundations-of-von-neumann-algebras/src/polar-decomposition-of-functionals-and-weak-compactness-in-preduals.md"
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            "id": "completely-positive-maps",
            "title": "Completely positive maps",
            "route": "https://kokunoyumeto.github.io/open-math-courses-public/courses/foundations-of-von-neumann-algebras/completely-positive-maps.html",
            "editable_source": "https://kokunoyumeto.github.io/open-math-courses-public/courses/foundations-of-von-neumann-algebras/src/completely-positive-maps.md"
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            "id": "integral-representations-of-states",
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            "route": "https://kokunoyumeto.github.io/open-math-courses-public/courses/foundations-of-von-neumann-algebras/integral-representations-of-states.html",
            "editable_source": "https://kokunoyumeto.github.io/open-math-courses-public/courses/foundations-of-von-neumann-algebras/src/integral-representations-of-states.md"
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            "id": "polish-spaces-and-standard-borel-spaces",
            "title": "Polish spaces and standard Borel spaces",
            "route": "https://kokunoyumeto.github.io/open-math-courses-public/courses/foundations-of-von-neumann-algebras/polish-spaces-and-standard-borel-spaces.html",
            "editable_source": "https://kokunoyumeto.github.io/open-math-courses-public/courses/foundations-of-von-neumann-algebras/src/polish-spaces-and-standard-borel-spaces.md"
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            "id": "numerical-ranges-positive-matrices-and-co-souslin-sets",
            "title": "Numerical ranges, positive matrices and co-Souslin sets",
            "route": "https://kokunoyumeto.github.io/open-math-courses-public/courses/foundations-of-von-neumann-algebras/numerical-ranges-positive-matrices-and-co-souslin-sets.html",
            "editable_source": "https://kokunoyumeto.github.io/open-math-courses-public/courses/foundations-of-von-neumann-algebras/src/numerical-ranges-positive-matrices-and-co-souslin-sets.md"
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        "id": "measurable-fields-and-direct-integrals",
        "kind": "advanced",
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        "summary": "Fields of Hilbert spaces whose dimension may vary from point to point, their direct integrals, decomposable operators, vector-valued integration and preduals, the Borel structure on subspaces and von Neumann algebras, and direct integrals of von Neumann algebras with their commutants and centres.",
        "subject": "operator-algebras",
        "content_language": "en",
        "translations": [],
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            "id": "the-effros-borel-structure",
            "title": "The Effros Borel structure",
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            "editable_source": "https://kokunoyumeto.github.io/open-math-courses-public/courses/measurable-fields-and-direct-integrals/"
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            "id": "direct-integrals-of-von-neumann-algebras",
            "title": "Direct integrals of von Neumann algebras",
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        "id": "tensor-products-of-operator-algebras",
        "kind": "advanced",
        "title": {
          "en": "Tensor products of operator algebras"
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        "summary": "Spatial tensor products of von Neumann algebras on Hilbert spaces of any dimension: normal functionals, slice maps, the commutation theorem and coproducts. Tensor products of Banach and Hilbert spaces with the injective and projective norms, trace duality, and Jordan homomorphisms and isometries of C*-algebras.",
        "subject": "operator-algebras",
        "content_language": "en",
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            "title": "How the dual weight moves crossed-product generators",
            "route": "https://kokunoyumeto.github.io/open-math-courses-public/courses/OA-FLOW/",
            "editable_source": "https://kokunoyumeto.github.io/open-math-courses-public/courses/OA-FLOW/"
          },
          {
            "id": "OA-FLOW-L15",
            "title": "Changing the Hilbert space of a regular crossed product",
            "route": "https://kokunoyumeto.github.io/open-math-courses-public/courses/OA-FLOW/",
            "editable_source": "https://kokunoyumeto.github.io/open-math-courses-public/courses/OA-FLOW/"
          },
          {
            "id": "OA-FLOW-L16",
            "title": "Recovering multiplicity from a Weyl system",
            "route": "https://kokunoyumeto.github.io/open-math-courses-public/courses/OA-FLOW/",
            "editable_source": "https://kokunoyumeto.github.io/open-math-courses-public/courses/OA-FLOW/"
          },
          {
            "id": "OA-FLOW-L17",
            "title": "Locating frequencies with Fourier filters",
            "route": "https://kokunoyumeto.github.io/open-math-courses-public/courses/OA-FLOW/",
            "editable_source": "https://kokunoyumeto.github.io/open-math-courses-public/courses/OA-FLOW/"
          },
          {
            "id": "OA-FLOW-L18",
            "title": "Detecting semifinite corners through central translation",
            "route": "https://kokunoyumeto.github.io/open-math-courses-public/courses/OA-FLOW/OA-FLOW-L18.html",
            "editable_source": "https://kokunoyumeto.github.io/open-math-courses-public/courses/OA-FLOW/"
          },
          {
            "id": "OA-FLOW-L19",
            "title": "Recovering a spectral coordinate from a scaled trace",
            "route": "https://kokunoyumeto.github.io/open-math-courses-public/courses/OA-FLOW/OA-FLOW-L19.html",
            "editable_source": "https://kokunoyumeto.github.io/open-math-courses-public/courses/OA-FLOW/"
          },
          {
            "id": "OA-FLOW-L20",
            "title": "Comparing dual weights through matrix corners",
            "route": "https://kokunoyumeto.github.io/open-math-courses-public/courses/OA-FLOW/",
            "editable_source": "https://kokunoyumeto.github.io/open-math-courses-public/courses/OA-FLOW/"
          },
          {
            "id": "OA-FLOW-L21",
            "title": "The coefficient algebra as the value space of a weight",
            "route": "https://kokunoyumeto.github.io/open-math-courses-public/courses/OA-FLOW/",
            "editable_source": "https://kokunoyumeto.github.io/open-math-courses-public/courses/OA-FLOW/"
          },
          {
            "id": "OA-FLOW-L22",
            "title": "Averaging a dual action and recovering its coefficients",
            "route": "https://kokunoyumeto.github.io/open-math-courses-public/courses/OA-FLOW/",
            "editable_source": "https://kokunoyumeto.github.io/open-math-courses-public/courses/OA-FLOW/"
          },
          {
            "id": "OA-FLOW-L23",
            "title": "Commuting with the original algebra inside its core",
            "route": "https://kokunoyumeto.github.io/open-math-courses-public/courses/OA-FLOW/",
            "editable_source": "https://kokunoyumeto.github.io/open-math-courses-public/courses/OA-FLOW/"
          },
          {
            "id": "OA-FLOW-L24",
            "title": "Recovering a group action from its integrated operators",
            "route": "https://kokunoyumeto.github.io/open-math-courses-public/courses/OA-FLOW/OA-FLOW-L24.html",
            "editable_source": "https://kokunoyumeto.github.io/open-math-courses-public/courses/OA-FLOW/src/24-group-representation-completions.md"
          },
          {
            "id": "OA-FLOW-L25",
            "title": "Recovering covariance with nonunital coefficients",
            "route": "https://kokunoyumeto.github.io/open-math-courses-public/courses/OA-FLOW/OA-FLOW-L25.html",
            "editable_source": "https://kokunoyumeto.github.io/open-math-courses-public/courses/OA-FLOW/src/25-cstar-covariant-representations.md"
          },
          {
            "id": "OA-FLOW-L26",
            "title": "Realizing central cocycles by spectral averaging",
            "route": "https://kokunoyumeto.github.io/open-math-courses-public/courses/OA-FLOW/",
            "editable_source": "https://kokunoyumeto.github.io/open-math-courses-public/courses/OA-FLOW/"
          },
          {
            "id": "OA-FLOW-L27",
            "title": "Recovering a weight from dual invariance",
            "route": "https://kokunoyumeto.github.io/open-math-courses-public/courses/OA-FLOW/",
            "editable_source": "https://kokunoyumeto.github.io/open-math-courses-public/courses/OA-FLOW/"
          },
          {
            "id": "OA-FLOW-L28",
            "title": "Removing noncentral cocycles from a trace-scaling action",
            "route": "https://kokunoyumeto.github.io/open-math-courses-public/courses/OA-FLOW/",
            "editable_source": "https://kokunoyumeto.github.io/open-math-courses-public/courses/OA-FLOW/"
          },
          {
            "id": "OA-FLOW-L29",
            "title": "Recovering a crossed product from its eigenunitaries",
            "route": "https://kokunoyumeto.github.io/open-math-courses-public/courses/OA-FLOW/OA-FLOW-L29.html",
            "editable_source": "https://kokunoyumeto.github.io/open-math-courses-public/courses/OA-FLOW/"
          },
          {
            "id": "OA-FLOW-L30",
            "title": "Recognizing the continuous core of a fixed algebra",
            "route": "https://kokunoyumeto.github.io/open-math-courses-public/courses/OA-FLOW/",
            "editable_source": "https://kokunoyumeto.github.io/open-math-courses-public/courses/OA-FLOW/"
          },
          {
            "id": "OA-FLOW-L31",
            "title": "Haar coordinates and integration after recognition",
            "route": "https://kokunoyumeto.github.io/open-math-courses-public/courses/OA-FLOW/OA-FLOW-L31.html",
            "editable_source": "https://kokunoyumeto.github.io/open-math-courses-public/courses/OA-FLOW/"
          },
          {
            "id": "OA-FLOW-L32",
            "title": "Dual weights and their supports",
            "route": "https://kokunoyumeto.github.io/open-math-courses-public/courses/OA-FLOW/OA-FLOW-L32.html",
            "editable_source": "https://kokunoyumeto.github.io/open-math-courses-public/courses/OA-FLOW/"
          },
          {
            "id": "OA-FLOW-L33",
            "title": "The weight after crossing twice",
            "route": "https://kokunoyumeto.github.io/open-math-courses-public/courses/OA-FLOW/OA-FLOW-L33.html",
            "editable_source": "https://kokunoyumeto.github.io/open-math-courses-public/courses/OA-FLOW/"
          },
          {
            "id": "OA-FLOW-L34",
            "title": "Derivatives of actions: domains, perturbations, and implementation",
            "route": "https://kokunoyumeto.github.io/open-math-courses-public/courses/OA-FLOW/OA-FLOW-L34.html",
            "editable_source": "https://kokunoyumeto.github.io/open-math-courses-public/courses/OA-FLOW/"
          },
          {
            "id": "OA-FLOW-L35",
            "title": "Innerness and spectral tails",
            "route": "https://kokunoyumeto.github.io/open-math-courses-public/courses/OA-FLOW/OA-FLOW-L35.html",
            "editable_source": "https://kokunoyumeto.github.io/open-math-courses-public/courses/OA-FLOW/"
          }
        ],
        "source_snapshot": "801f186833b9811fa826575ef0717e78a5047ab0",
        "portable_editions": []
      },
      {
        "id": "type-iii-factors",
        "kind": "advanced",
        "title": {
          "en": "Type III factors: central sequences, full factors and almost periodic states"
        },
        "summary": "This course studies factors of type III through bounded sequences that asymptotically commute with the whole algebra and its normal states. It shows that a factor with separable predual is full, meaning that its inner automorphisms form a closed subgroup, exactly when all such sequences are trivial; it builds the invariants Sd and τ from almost periodic weights and the modular group; and it constructs full factors of type III₁ that have no almost periodic weight and are not crossed products of a semifinite algebra by a discrete abelian group. Basic references are [Connes 1974], [Connes 1973], [Connes 1976] and [Takesaki II].",
        "subject": "operator-algebras",
        "content_language": "en",
        "translations": [],
        "route": "https://kokunoyumeto.github.io/open-math-courses-public/courses/type-iii-factors/",
        "status": "draft",
        "status_note": "Spot-checked by Claude Opus 5.5 in a separate session.",
        "authorship": "Claude Opus 5.5 (Anthropic)",
        "licence": {
          "spdx": "CC0-1.0",
          "scope": "The lesson text, written for this course.",
          "exceptions": []
        },
        "lessons": [
          {
            "id": "ultraproducts-and-the-asymptotic-centralizer",
            "title": "Ultraproducts and the asymptotic centralizer",
            "route": "https://kokunoyumeto.github.io/open-math-courses-public/courses/type-iii-factors/ultraproducts-and-the-asymptotic-centralizer.html",
            "editable_source": "https://kokunoyumeto.github.io/open-math-courses-public/courses/type-iii-factors/src/ultraproducts-and-the-asymptotic-centralizer.md"
          },
          {
            "id": "full-factors",
            "title": "Full factors",
            "route": "https://kokunoyumeto.github.io/open-math-courses-public/courses/type-iii-factors/full-factors.html",
            "editable_source": "https://kokunoyumeto.github.io/open-math-courses-public/courses/type-iii-factors/src/full-factors.md"
          },
          {
            "id": "almost-periodic-weights-and-the-invariant-sd",
            "title": "Almost periodic weights and the invariant Sd",
            "route": "https://kokunoyumeto.github.io/open-math-courses-public/courses/type-iii-factors/almost-periodic-weights-and-the-invariant-sd.html",
            "editable_source": "https://kokunoyumeto.github.io/open-math-courses-public/courses/type-iii-factors/src/almost-periodic-weights-and-the-invariant-sd.md"
          },
          {
            "id": "full-factors-without-almost-periodic-weights",
            "title": "Full factors without almost periodic weights",
            "route": "https://kokunoyumeto.github.io/open-math-courses-public/courses/type-iii-factors/full-factors-without-almost-periodic-weights.html",
            "editable_source": "https://kokunoyumeto.github.io/open-math-courses-public/courses/type-iii-factors/src/full-factors-without-almost-periodic-weights.md"
          }
        ],
        "source_snapshot": "801f186833b9811fa826575ef0717e78a5047ab0",
        "portable_editions": []
      },
      {
        "id": "injective-factors",
        "kind": "advanced",
        "title": {
          "en": "Injective factors"
        },
        "summary": "This course classifies the injective factors with separable predual that are not of type III₁: the hyperfinite factor is the only injective factor of type II₁, there is only one of type II∞ and one of each type IIIλ with 0 < λ < 1, and the injective factors of type III₀ are the Krieger factors, classified by their flow of weights. It also shows that injectivity, approximate finite dimensionality, semidiscreteness and property P coincide for factors with separable predual (for type III₁ with the help of a later uniqueness theorem, used without proof), building on trace inequalities, property Γ and approximately inner and centrally trivial automorphisms. Basic references are [Connes 1976], [Connes 1973], [Takesaki III], [Brown–Ozawa] and [Connes 1994].",
        "subject": "operator-algebras",
        "content_language": "en",
        "translations": [],
        "route": "https://kokunoyumeto.github.io/open-math-courses-public/courses/injective-factors/",
        "status": "draft",
        "status_note": "Spot-checked by Claude Opus 5.5 in a separate session.",
        "authorship": "Claude Opus 5.5 (Anthropic)",
        "licence": {
          "spdx": "CC0-1.0",
          "scope": "The lesson text, written for this course.",
          "exceptions": []
        },
        "lessons": [
          {
            "id": "trace-inequalities-for-finite-von-neumann-algebras",
            "title": "Trace inequalities for finite von Neumann algebras",
            "route": "https://kokunoyumeto.github.io/open-math-courses-public/courses/injective-factors/trace-inequalities-for-finite-von-neumann-algebras.html",
            "editable_source": "https://kokunoyumeto.github.io/open-math-courses-public/courses/injective-factors/src/trace-inequalities-for-finite-von-neumann-algebras.md"
          },
          {
            "id": "property-gamma-and-the-algebra-generated-by-a-factor-and-its-commutant",
            "title": "Property Γ and the algebra generated by a factor and its commutant",
            "route": "https://kokunoyumeto.github.io/open-math-courses-public/courses/injective-factors/property-gamma-and-the-algebra-generated-by-a-factor-and-its-commutant.html",
            "editable_source": "https://kokunoyumeto.github.io/open-math-courses-public/courses/injective-factors/src/property-gamma-and-the-algebra-generated-by-a-factor-and-its-commutant.md"
          },
          {
            "id": "approximately-inner-and-centrally-trivial-automorphisms",
            "title": "Approximately inner and centrally trivial automorphisms",
            "route": "https://kokunoyumeto.github.io/open-math-courses-public/courses/injective-factors/approximately-inner-and-centrally-trivial-automorphisms.html",
            "editable_source": "https://kokunoyumeto.github.io/open-math-courses-public/courses/injective-factors/src/approximately-inner-and-centrally-trivial-automorphisms.md"
          },
          {
            "id": "injective-von-neumann-algebras",
            "title": "Injective von Neumann algebras",
            "route": "https://kokunoyumeto.github.io/open-math-courses-public/courses/injective-factors/injective-von-neumann-algebras.html",
            "editable_source": "https://kokunoyumeto.github.io/open-math-courses-public/courses/injective-factors/src/injective-von-neumann-algebras.md"
          },
          {
            "id": "uniqueness-of-the-injective-ii1-factor",
            "title": "Uniqueness of the injective II₁ factor",
            "route": "https://kokunoyumeto.github.io/open-math-courses-public/courses/injective-factors/uniqueness-of-the-injective-ii1-factor.html",
            "editable_source": "https://kokunoyumeto.github.io/open-math-courses-public/courses/injective-factors/src/uniqueness-of-the-injective-ii1-factor.md"
          },
          {
            "id": "classification-of-injective-factors",
            "title": "Classification of injective factors",
            "route": "https://kokunoyumeto.github.io/open-math-courses-public/courses/injective-factors/classification-of-injective-factors.html",
            "editable_source": "https://kokunoyumeto.github.io/open-math-courses-public/courses/injective-factors/src/classification-of-injective-factors.md"
          }
        ],
        "source_snapshot": "801f186833b9811fa826575ef0717e78a5047ab0",
        "portable_editions": []
      },
      {
        "id": "entropy-of-automorphisms",
        "kind": "advanced",
        "title": {
          "en": "Entropy of automorphisms"
        },
        "summary": "These lessons define an entropy for automorphisms of operator algebras, the noncommutative counterpart of the Kolmogorov–Sinai entropy of ergodic theory. They start with automorphisms that preserve a trace, where the entropy tells apart the shifts of the hyperfinite II₁ factor built from matrices of different sizes, then treat automorphisms that preserve a state on a C*-algebra or a von Neumann algebra, and end with the entropy of lattice translations in quantum spin chains. Basic references are [Connes–Størmer 1975], [Connes–Narnhofer–Thirring 1987], [Neshveyev–Størmer 2006] and [Ohya–Petz 1993].",
        "subject": "operator-algebras",
        "content_language": "en",
        "translations": [],
        "route": "https://kokunoyumeto.github.io/open-math-courses-public/courses/entropy-of-automorphisms/",
        "status": "draft",
        "status_note": "Spot-checked by Claude Opus 5.5 in a separate session.",
        "authorship": "Claude Opus 5.5 (Anthropic)",
        "licence": {
          "spdx": "CC0-1.0",
          "scope": "The lesson text, written for this course.",
          "exceptions": []
        },
        "lessons": [
          {
            "id": "entropy-of-finite-dimensional-subalgebras",
            "title": "Entropy of finite-dimensional subalgebras",
            "route": "https://kokunoyumeto.github.io/open-math-courses-public/courses/entropy-of-automorphisms/entropy-of-finite-dimensional-subalgebras.html",
            "editable_source": "https://kokunoyumeto.github.io/open-math-courses-public/courses/entropy-of-automorphisms/src/entropy-of-finite-dimensional-subalgebras.md"
          },
          {
            "id": "the-entropy-of-a-trace-preserving-automorphism",
            "title": "The entropy of a trace-preserving automorphism",
            "route": "https://kokunoyumeto.github.io/open-math-courses-public/courses/entropy-of-automorphisms/the-entropy-of-a-trace-preserving-automorphism.html",
            "editable_source": "https://kokunoyumeto.github.io/open-math-courses-public/courses/entropy-of-automorphisms/src/the-entropy-of-a-trace-preserving-automorphism.md"
          },
          {
            "id": "entropy-defect-and-abelian-models",
            "title": "Entropy defect and abelian models",
            "route": "https://kokunoyumeto.github.io/open-math-courses-public/courses/entropy-of-automorphisms/entropy-defect-and-abelian-models.html",
            "editable_source": "https://kokunoyumeto.github.io/open-math-courses-public/courses/entropy-of-automorphisms/src/entropy-defect-and-abelian-models.md"
          },
          {
            "id": "dynamical-entropy-of-c-star-algebras-and-von-neumann-algebras",
            "title": "Dynamical entropy of C*-algebras and von Neumann algebras",
            "route": "https://kokunoyumeto.github.io/open-math-courses-public/courses/entropy-of-automorphisms/dynamical-entropy-of-c-star-algebras-and-von-neumann-algebras.html",
            "editable_source": "https://kokunoyumeto.github.io/open-math-courses-public/courses/entropy-of-automorphisms/src/dynamical-entropy-of-c-star-algebras-and-von-neumann-algebras.md"
          },
          {
            "id": "entropy-of-lattice-translations-in-quantum-spin-chains",
            "title": "Entropy of lattice translations in quantum spin chains",
            "route": "https://kokunoyumeto.github.io/open-math-courses-public/courses/entropy-of-automorphisms/entropy-of-lattice-translations-in-quantum-spin-chains.html",
            "editable_source": "https://kokunoyumeto.github.io/open-math-courses-public/courses/entropy-of-automorphisms/src/entropy-of-lattice-translations-in-quantum-spin-chains.md"
          }
        ],
        "source_snapshot": "801f186833b9811fa826575ef0717e78a5047ab0",
        "portable_editions": []
      },
      {
        "id": "noncommutative-integration",
        "kind": "advanced",
        "title": {
          "en": "Noncommutative integration and spatial theory"
        },
        "summary": "This course develops integration on spaces that ordinary measure theory cannot see, such as the space of orbits of an ergodic group action or the leaf space of a foliation. It starts with the spatial derivative, which compares a weight on a von Neumann algebra with a weight on its commutant, and then builds transverse measures on measured groupoids, the von Neumann algebra of random operators, and the weights, formal dimension and index that come with them. Basic references are [Connes 1979], [Connes 1980a], [Connes 1982] and [Connes 1994].",
        "subject": "operator-algebras",
        "content_language": "en",
        "translations": [],
        "route": "https://kokunoyumeto.github.io/open-math-courses-public/courses/noncommutative-integration/",
        "status": "draft",
        "status_note": "Spot-checked by Claude Opus 5.5 in a separate session.",
        "authorship": "Claude Opus 5.5 (Anthropic)",
        "licence": {
          "spdx": "CC0-1.0",
          "scope": "The lesson text, written for this course.",
          "exceptions": []
        },
        "lessons": [
          {
            "id": "the-spatial-derivative",
            "title": "The spatial derivative",
            "route": "https://kokunoyumeto.github.io/open-math-courses-public/courses/noncommutative-integration/the-spatial-derivative.html",
            "editable_source": "https://kokunoyumeto.github.io/open-math-courses-public/courses/noncommutative-integration/src/the-spatial-derivative.md"
          },
          {
            "id": "measured-groupoids-and-transverse-measures",
            "title": "Measured groupoids and transverse measures",
            "route": "https://kokunoyumeto.github.io/open-math-courses-public/courses/noncommutative-integration/measured-groupoids-and-transverse-measures.html",
            "editable_source": "https://kokunoyumeto.github.io/open-math-courses-public/courses/noncommutative-integration/src/measured-groupoids-and-transverse-measures.md"
          },
          {
            "id": "square-integrable-representations-and-random-operators",
            "title": "Square-integrable representations and random operators",
            "route": "https://kokunoyumeto.github.io/open-math-courses-public/courses/noncommutative-integration/square-integrable-representations-and-random-operators.html",
            "editable_source": "https://kokunoyumeto.github.io/open-math-courses-public/courses/noncommutative-integration/src/square-integrable-representations-and-random-operators.md"
          },
          {
            "id": "weights-on-random-operators-and-formal-dimension",
            "title": "Weights on random operators and formal dimension",
            "route": "https://kokunoyumeto.github.io/open-math-courses-public/courses/noncommutative-integration/weights-on-random-operators-and-formal-dimension.html",
            "editable_source": "https://kokunoyumeto.github.io/open-math-courses-public/courses/noncommutative-integration/src/weights-on-random-operators-and-formal-dimension.md"
          }
        ],
        "source_snapshot": "801f186833b9811fa826575ef0717e78a5047ab0",
        "portable_editions": []
      },
      {
        "id": "NCG-CYCLIC",
        "kind": "advanced",
        "title": {
          "en": "Cyclic cohomology, connections and transverse geometry"
        },
        "summary": "Traces and cyclic cohomology, connections and Chern forms on modules over an algebra with a group action, a pseudodifferential calculus for actions of Euclidean space, cyclic forms that extend to norm completions, and the fundamental class of an action on a manifold.",
        "subject": "noncommutative-geometry",
        "content_language": "en",
        "translations": [],
        "route": "https://kokunoyumeto.github.io/open-math-courses-public/courses/NCG-CYCLIC/",
        "status": "draft",
        "status_note": "Self-checked by its author, GPT-6.1 Sol; spot-checked by GPT-6 Astra.",
        "authorship": "GPT-6.1 Sol (OpenAI), in Codex, at the Ultra setting (maximum reasoning effort)",
        "licence": {
          "spdx": "CC0-1.0",
          "scope": "The lesson text, written for this course.",
          "exceptions": []
        },
        "lessons": [
          {
            "id": "NCGCY-L01",
            "title": "Cyclic cohomology: traces, differentials and symmetry",
            "route": "https://kokunoyumeto.github.io/open-math-courses-public/courses/NCG-CYCLIC/",
            "editable_source": "https://kokunoyumeto.github.io/open-math-courses-public/courses/NCG-CYCLIC/"
          },
          {
            "id": "NCGCY-L02",
            "title": "Connections and curvature from symmetries of an algebra",
            "route": "https://kokunoyumeto.github.io/open-math-courses-public/courses/NCG-CYCLIC/",
            "editable_source": "https://kokunoyumeto.github.io/open-math-courses-public/courses/NCG-CYCLIC/"
          },
          {
            "id": "NCGCY-L03",
            "title": "Frequency calculus for an action of Euclidean space",
            "route": "https://kokunoyumeto.github.io/open-math-courses-public/courses/NCG-CYCLIC/",
            "editable_source": "https://kokunoyumeto.github.io/open-math-courses-public/courses/NCG-CYCLIC/"
          },
          {
            "id": "NCGCY-L04",
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        "summary": "Hilbert C*-modules, adjointable and compact operators, finite projective modules, tensor products, stabilization, Fredholm index, regular operators and continuous fields; completely positive maps, imprimitivity bimodules, induced representations, stable isomorphism and Morita equivalence in K-theory and noncommutative tori.",
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          "spdx": "CC0-1.0",
          "scope": "Original lesson text written for this course.",
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            "The linked projection proof by Marina Prokhorova, From graph to Riesz continuity, retains CC BY 4.0; its text is not reproduced. The field-to-projection reduction is original CC0 writing.",
            "Other linked works and mathematical rendering assets retain their own licences."
          ]
        },
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            "id": "hilbert-c-star-modules",
            "title": "Hilbert C*-modules",
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            "id": "imprimitivity-bimodules-and-morita-equivalence",
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            "title": "Morita invariance of K-theory and maps induced by correspondences",
            "route": "https://kokunoyumeto.github.io/open-math-courses-public/courses/hilbert-c-star-modules-and-morita-equivalence/morita-invariance-of-k-theory-and-correspondence-maps.html",
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        "id": "AG-LTF",
        "kind": "advanced",
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        "summary": "Adic coefficients and Frobenius traces, developed through Lefschetz formulas, L-functions, vanishing cycles and the Riemann hypothesis over finite fields.",
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        "content_language": "en",
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          "spdx": "CC0-1.0",
          "scope": "The original course text, proofs, examples and exercises. Linked references retain their own rights.",
          "exceptions": [
            "Linked prerequisite and source works retain their own rights."
          ]
        },
        "lessons": [
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            "id": "l-adic-sheaves-and-their-cohomology",
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            "id": "the-pro-etale-site-and-l-adic-complexes",
            "title": "The pro-étale site and ℓ-adic complexes",
            "route": "https://kokunoyumeto.github.io/open-math-courses-public/courses/AG-LTF/the-pro-etale-site-and-l-adic-complexes.html",
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            "id": "frobenius-morphisms-and-their-action-on-cohomology",
            "title": "Frobenius morphisms and their action on cohomology",
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            "editable_source": "https://kokunoyumeto.github.io/open-math-courses-public/courses/AG-LTF/src/frobenius-morphisms-and-their-action-on-cohomology.md"
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            "id": "traces-of-perfect-complexes-and-lefschetz-numbers",
            "title": "Traces of perfect complexes and Lefschetz numbers",
            "route": "https://kokunoyumeto.github.io/open-math-courses-public/courses/AG-LTF/traces-of-perfect-complexes-and-lefschetz-numbers.html",
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            "id": "cycle-classes-and-the-lefschetz-fixed-point-formula-for-curves",
            "title": "Cycle classes and the Lefschetz fixed-point formula for curves",
            "route": "https://kokunoyumeto.github.io/open-math-courses-public/courses/AG-LTF/cycle-classes-and-the-lefschetz-fixed-point-formula-for-curves.html",
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            "id": "the-trace-formula-for-curves",
            "title": "The trace formula for curves",
            "route": "https://kokunoyumeto.github.io/open-math-courses-public/courses/AG-LTF/the-trace-formula-for-curves.html",
            "editable_source": "https://kokunoyumeto.github.io/open-math-courses-public/courses/AG-LTF/src/the-trace-formula-for-curves.md"
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            "route": "https://kokunoyumeto.github.io/open-math-courses-public/courses/AG-LTF/the-trace-formula-in-all-dimensions.html",
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            "route": "https://kokunoyumeto.github.io/open-math-courses-public/courses/AG-LTF/l-functions-rationality-and-the-functional-equation.html",
            "editable_source": "https://kokunoyumeto.github.io/open-math-courses-public/courses/AG-LTF/src/l-functions-rationality-and-the-functional-equation.md"
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            "id": "the-fundamental-estimate",
            "title": "The fundamental estimate",
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            "title": "The Riemann hypothesis over finite fields",
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      {
        "id": "GL-DMOD",
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        "subject": "sheaves-geometry",
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        "status_note": "Self-checked by the writing AI, GPT-6.1 Sol (OpenAI). Independent AI review pending.",
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        "summary": "A graduate course on the analytic theory of the Riemann zeta function: Dirichlet series, prime-counting estimates, the Gamma function and the functional equation, entire-function products, growth and zeros, the prime number theorem and explicit formulas, mean values, critical-line zeros and short intervals.",
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            "editable_source": "https://kokunoyumeto.github.io/open-math-courses-public/courses/AG-MO/src/rational-maps-and-birational-morphisms.md"
          },
          {
            "id": "AG-MO-15",
            "title": "Effective Cartier divisors and invertible sheaves",
            "route": "https://kokunoyumeto.github.io/open-math-courses-public/courses/AG-MO/",
            "editable_source": "https://kokunoyumeto.github.io/open-math-courses-public/courses/AG-MO/src/effective-cartier-divisors-and-invertible-sheaves.md"
          },
          {
            "id": "AG-MO-16",
            "title": "Blowing up",
            "route": "https://kokunoyumeto.github.io/open-math-courses-public/courses/AG-MO/",
            "editable_source": "https://kokunoyumeto.github.io/open-math-courses-public/courses/AG-MO/src/blowing-up.md"
          },
          {
            "id": "AG-MO-17",
            "title": "Weil divisors and the class group",
            "route": "https://kokunoyumeto.github.io/open-math-courses-public/courses/AG-MO/",
            "editable_source": "https://kokunoyumeto.github.io/open-math-courses-public/courses/AG-MO/src/weil-divisors-and-the-class-group.md"
          }
        ],
        "source_snapshot": "801f186833b9811fa826575ef0717e78a5047ab0",
        "portable_editions": []
      },
      {
        "id": "AG-SS",
        "kind": "advanced",
        "title": {
          "en": "Sheaves and schemes"
        },
        "summary": "The language of schemes: sheaves on topological spaces, ringed spaces and their modules, affine schemes and the equivalence between modules and quasi-coherent sheaves, schemes glued from affine pieces, the functor of points and representability, fibre products, local properties, quasi-coherent sheaves and closed subschemes, and Proj with projective space and projective bundles. The course cites the site's lesson on sheaves of modules for the homological facts it proves and supplies the sheaf-theoretic background that lesson assumes. Lessons prove what they teach; the Stacks project (read in its Clanker Stacks edition) is cited by tag.",
        "subject": "algebraic-geometry",
        "content_language": "en",
        "translations": [],
        "route": "https://kokunoyumeto.github.io/open-math-courses-public/courses/AG-SS/",
        "status": "draft",
        "status_note": "Self-checked by the writing AI, GPT-6.1 Sol (OpenAI), in Codex at Ultra.",
        "authorship": "GPT-6.1 Sol (OpenAI), in Codex, at the Ultra setting (maximum reasoning effort)",
        "licence": {
          "spdx": "CC0-1.0",
          "scope": "The independently authored course text, original figures and figure-generation sources.",
          "exceptions": [
            "Referenced works and font notices retain their own licences. No source-book text or figures are reproduced."
          ]
        },
        "lessons": [
          {
            "id": "sheaves-on-topological-spaces",
            "title": "Sheaves on topological spaces",
            "route": "https://kokunoyumeto.github.io/open-math-courses-public/courses/AG-SS/sheaves-on-topological-spaces.html",
            "editable_source": "https://kokunoyumeto.github.io/open-math-courses-public/courses/AG-SS/src/sheaves-on-topological-spaces.md"
          },
          {
            "id": "ringed-spaces-and-sheaves-of-modules",
            "title": "Ringed spaces and sheaves of modules",
            "route": "https://kokunoyumeto.github.io/open-math-courses-public/courses/AG-SS/ringed-spaces-and-sheaves-of-modules.html",
            "editable_source": "https://kokunoyumeto.github.io/open-math-courses-public/courses/AG-SS/src/ringed-spaces-and-sheaves-of-modules.md"
          },
          {
            "id": "quasi-coherent-coherent-and-locally-free-modules",
            "title": "Quasi-coherent, coherent and locally free modules",
            "route": "https://kokunoyumeto.github.io/open-math-courses-public/courses/AG-SS/quasi-coherent-coherent-and-locally-free-modules.html",
            "editable_source": "https://kokunoyumeto.github.io/open-math-courses-public/courses/AG-SS/src/quasi-coherent-coherent-and-locally-free-modules.md"
          },
          {
            "id": "affine-schemes",
            "title": "Affine schemes",
            "route": "https://kokunoyumeto.github.io/open-math-courses-public/courses/AG-SS/affine-schemes.html",
            "editable_source": "https://kokunoyumeto.github.io/open-math-courses-public/courses/AG-SS/src/affine-schemes.md"
          },
          {
            "id": "schemes-gluing-and-immersions",
            "title": "Schemes, gluing and immersions",
            "route": "https://kokunoyumeto.github.io/open-math-courses-public/courses/AG-SS/schemes-gluing-and-immersions.html",
            "editable_source": "https://kokunoyumeto.github.io/open-math-courses-public/courses/AG-SS/src/schemes-gluing-and-immersions.md"
          },
          {
            "id": "the-functor-of-points-and-representability",
            "title": "The functor of points and representability",
            "route": "https://kokunoyumeto.github.io/open-math-courses-public/courses/AG-SS/the-functor-of-points-and-representability.html",
            "editable_source": "https://kokunoyumeto.github.io/open-math-courses-public/courses/AG-SS/src/the-functor-of-points-and-representability.md"
          },
          {
            "id": "fibre-products-and-base-change",
            "title": "Fibre products and base change",
            "route": "https://kokunoyumeto.github.io/open-math-courses-public/courses/AG-SS/fibre-products-and-base-change.html",
            "editable_source": "https://kokunoyumeto.github.io/open-math-courses-public/courses/AG-SS/src/fibre-products-and-base-change.md"
          },
          {
            "id": "properties-of-schemes",
            "title": "Properties of schemes",
            "route": "https://kokunoyumeto.github.io/open-math-courses-public/courses/AG-SS/properties-of-schemes.html",
            "editable_source": "https://kokunoyumeto.github.io/open-math-courses-public/courses/AG-SS/src/properties-of-schemes.md"
          },
          {
            "id": "quasi-coherent-sheaves-on-schemes",
            "title": "Quasi-coherent sheaves on schemes",
            "route": "https://kokunoyumeto.github.io/open-math-courses-public/courses/AG-SS/quasi-coherent-sheaves-on-schemes.html",
            "editable_source": "https://kokunoyumeto.github.io/open-math-courses-public/courses/AG-SS/src/quasi-coherent-sheaves-on-schemes.md"
          },
          {
            "id": "closed-subschemes-and-scheme-theoretic-images",
            "title": "Closed subschemes and scheme-theoretic images",
            "route": "https://kokunoyumeto.github.io/open-math-courses-public/courses/AG-SS/closed-subschemes-and-scheme-theoretic-images.html",
            "editable_source": "https://kokunoyumeto.github.io/open-math-courses-public/courses/AG-SS/src/closed-subschemes-and-scheme-theoretic-images.md"
          },
          {
            "id": "proj-of-a-graded-ring",
            "title": "Proj of a graded ring",
            "route": "https://kokunoyumeto.github.io/open-math-courses-public/courses/AG-SS/proj-of-a-graded-ring.html",
            "editable_source": "https://kokunoyumeto.github.io/open-math-courses-public/courses/AG-SS/src/proj-of-a-graded-ring.md"
          },
          {
            "id": "projective-space-relative-proj-and-maps-to-projective-space",
            "title": "Projective space, relative Proj and maps to projective space",
            "route": "https://kokunoyumeto.github.io/open-math-courses-public/courses/AG-SS/projective-space-relative-proj-and-maps-to-projective-space.html",
            "editable_source": "https://kokunoyumeto.github.io/open-math-courses-public/courses/AG-SS/src/projective-space-relative-proj-and-maps-to-projective-space.md"
          }
        ],
        "source_snapshot": "801f186833b9811fa826575ef0717e78a5047ab0",
        "portable_editions": []
      },
      {
        "id": "AG-QC",
        "kind": "advanced",
        "title": {
          "en": "Cohomology of quasi-coherent sheaves"
        },
        "summary": "Sheaf and Čech cohomology, affine and projective schemes, proper direct images, base change, formal geometry, Serre duality and GAGA.",
        "subject": "algebraic-geometry",
        "content_language": "en",
        "translations": [],
        "route": "https://kokunoyumeto.github.io/open-math-courses-public/courses/AG-QC/",
        "status": "draft",
        "status_note": "Self-checked by the writing AI, GPT-6.1 Sol (OpenAI), in Codex, at Ultra effort. No separate AI review is claimed.",
        "authorship": "GPT-6.1 Sol (OpenAI), in Codex, at the Ultra setting (maximum reasoning effort)",
        "licence": {
          "spdx": "CC0-1.0",
          "scope": "Independently written lesson prose, proofs, examples, exercises and original drawings. Cited works keep their own rights.",
          "exceptions": [
            "The preparation-and-division component in Complex analytic spaces and analytification, Section 2, adapts Jean-Pierre Demailly’s 21 June 2012 book, crediting C. L. Siegel, under the author’s explicit OpenContent permission to distribute and modify it on the web while retaining authorship. That component is outside the CC0 dedication; exact attribution and changes accompany the proof. Other cited books and MIT supplementary notes retain their own rights."
          ]
        },
        "lessons": [
          {
            "id": "cohomology-of-sheaves-on-ringed-spaces",
            "title": "Cohomology of sheaves on ringed spaces",
            "route": "https://kokunoyumeto.github.io/open-math-courses-public/courses/AG-QC/cohomology-of-sheaves-on-ringed-spaces.html",
            "editable_source": "https://kokunoyumeto.github.io/open-math-courses-public/courses/AG-QC/src/cohomology-of-sheaves-on-ringed-spaces.md"
          },
          {
            "id": "cech-cohomology",
            "title": "Čech cohomology",
            "route": "https://kokunoyumeto.github.io/open-math-courses-public/courses/AG-QC/cech-cohomology.html",
            "editable_source": "https://kokunoyumeto.github.io/open-math-courses-public/courses/AG-QC/src/cech-cohomology.md"
          },
          {
            "id": "affine-cohomology-and-serres-criterion",
            "title": "Cohomology of affine schemes and Serre's criterion",
            "route": "https://kokunoyumeto.github.io/open-math-courses-public/courses/AG-QC/affine-cohomology-and-serres-criterion.html",
            "editable_source": "https://kokunoyumeto.github.io/open-math-courses-public/courses/AG-QC/src/affine-cohomology-and-serres-criterion.md"
          },
          {
            "id": "cohomology-of-projective-space",
            "title": "Cohomology of projective space",
            "route": "https://kokunoyumeto.github.io/open-math-courses-public/courses/AG-QC/cohomology-of-projective-space.html",
            "editable_source": "https://kokunoyumeto.github.io/open-math-courses-public/courses/AG-QC/src/cohomology-of-projective-space.md"
          },
          {
            "id": "serres-theorems-on-projective-schemes",
            "title": "Coherent sheaves on projective schemes: Serre's theorems",
            "route": "https://kokunoyumeto.github.io/open-math-courses-public/courses/AG-QC/serres-theorems-on-projective-schemes.html",
            "editable_source": "https://kokunoyumeto.github.io/open-math-courses-public/courses/AG-QC/src/serres-theorems-on-projective-schemes.md"
          },
          {
            "id": "proper-morphisms-and-coherent-direct-images",
            "title": "Coherence of higher direct images under proper morphisms",
            "route": "https://kokunoyumeto.github.io/open-math-courses-public/courses/AG-QC/proper-morphisms-and-coherent-direct-images.html",
            "editable_source": "https://kokunoyumeto.github.io/open-math-courses-public/courses/AG-QC/src/proper-morphisms-and-coherent-direct-images.md"
          },
          {
            "id": "euler-characteristics-and-hilbert-polynomials",
            "title": "Euler characteristics and Hilbert polynomials",
            "route": "https://kokunoyumeto.github.io/open-math-courses-public/courses/AG-QC/euler-characteristics-and-hilbert-polynomials.html",
            "editable_source": "https://kokunoyumeto.github.io/open-math-courses-public/courses/AG-QC/src/euler-characteristics-and-hilbert-polynomials.md"
          },
          {
            "id": "base-change-and-the-grothendieck-complex",
            "title": "Base change and the Grothendieck complex",
            "route": "https://kokunoyumeto.github.io/open-math-courses-public/courses/AG-QC/base-change-and-the-grothendieck-complex.html",
            "editable_source": "https://kokunoyumeto.github.io/open-math-courses-public/courses/AG-QC/src/base-change-and-the-grothendieck-complex.md"
          },
          {
            "id": "semicontinuity-and-grauerts-theorem",
            "title": "Semicontinuity and Grauert's theorem",
            "route": "https://kokunoyumeto.github.io/open-math-courses-public/courses/AG-QC/semicontinuity-and-grauerts-theorem.html",
            "editable_source": "https://kokunoyumeto.github.io/open-math-courses-public/courses/AG-QC/src/semicontinuity-and-grauerts-theorem.md"
          },
          {
            "id": "the-theorem-on-formal-functions",
            "title": "The theorem on formal functions",
            "route": "https://kokunoyumeto.github.io/open-math-courses-public/courses/AG-QC/the-theorem-on-formal-functions.html",
            "editable_source": "https://kokunoyumeto.github.io/open-math-courses-public/courses/AG-QC/src/the-theorem-on-formal-functions.md"
          },
          {
            "id": "zariski-connectedness-and-stein-factorization",
            "title": "Zariski's connectedness theorem and Stein factorization",
            "route": "https://kokunoyumeto.github.io/open-math-courses-public/courses/AG-QC/zariski-connectedness-and-stein-factorization.html",
            "editable_source": "https://kokunoyumeto.github.io/open-math-courses-public/courses/AG-QC/src/zariski-connectedness-and-stein-factorization.md"
          },
          {
            "id": "grothendiecks-existence-theorem",
            "title": "Grothendieck's existence theorem",
            "route": "https://kokunoyumeto.github.io/open-math-courses-public/courses/AG-QC/grothendiecks-existence-theorem.html",
            "editable_source": "https://kokunoyumeto.github.io/open-math-courses-public/courses/AG-QC/src/grothendiecks-existence-theorem.md"
          },
          {
            "id": "algebraization-of-formal-schemes",
            "title": "Algebraization of formal schemes",
            "route": "https://kokunoyumeto.github.io/open-math-courses-public/courses/AG-QC/algebraization-of-formal-schemes.html",
            "editable_source": "https://kokunoyumeto.github.io/open-math-courses-public/courses/AG-QC/src/algebraization-of-formal-schemes.md"
          },
          {
            "id": "ext-sheaves-and-serre-duality-on-projective-space",
            "title": "Ext sheaves and Serre duality on projective space",
            "route": "https://kokunoyumeto.github.io/open-math-courses-public/courses/AG-QC/ext-sheaves-and-serre-duality-on-projective-space.html",
            "editable_source": "https://kokunoyumeto.github.io/open-math-courses-public/courses/AG-QC/src/ext-sheaves-and-serre-duality-on-projective-space.md"
          },
          {
            "id": "dualizing-sheaves-and-serre-duality-for-projective-schemes",
            "title": "Dualizing sheaves and Serre duality for projective schemes",
            "route": "https://kokunoyumeto.github.io/open-math-courses-public/courses/AG-QC/dualizing-sheaves-and-serre-duality-for-projective-schemes.html",
            "editable_source": "https://kokunoyumeto.github.io/open-math-courses-public/courses/AG-QC/src/dualizing-sheaves-and-serre-duality-for-projective-schemes.md"
          },
          {
            "id": "the-right-adjoint-of-derived-pushforward",
            "title": "The right adjoint of derived pushforward",
            "route": "https://kokunoyumeto.github.io/open-math-courses-public/courses/AG-QC/the-right-adjoint-of-derived-pushforward.html",
            "editable_source": "https://kokunoyumeto.github.io/open-math-courses-public/courses/AG-QC/src/the-right-adjoint-of-derived-pushforward.md"
          },
          {
            "id": "complex-analytic-spaces-and-analytification",
            "title": "Complex analytic spaces and analytification",
            "route": "https://kokunoyumeto.github.io/open-math-courses-public/courses/AG-QC/complex-analytic-spaces-and-analytification.html",
            "editable_source": "https://kokunoyumeto.github.io/open-math-courses-public/courses/AG-QC/src/complex-analytic-spaces-and-analytification.md"
          },
          {
            "id": "serres-comparison-theorems-and-chows-theorem",
            "title": "Serre's comparison theorems and Chow's theorem",
            "route": "https://kokunoyumeto.github.io/open-math-courses-public/courses/AG-QC/serres-comparison-theorems-and-chows-theorem.html",
            "editable_source": "https://kokunoyumeto.github.io/open-math-courses-public/courses/AG-QC/src/serres-comparison-theorems-and-chows-theorem.md"
          }
        ],
        "source_snapshot": "801f186833b9811fa826575ef0717e78a5047ab0",
        "portable_editions": []
      },
      {
        "id": "NOE-INV",
        "kind": "advanced",
        "title": {
          "en": "Invariants and finiteness"
        },
        "summary": "Classical covariants and symbolic notation; Hilbert finite generation; finite groups, the Noether degree bound and Molien series; polarization and vector invariants; finiteness over every characteristic and Noetherian bases.",
        "subject": "algebra",
        "content_language": "en",
        "translations": [],
        "route": "https://kokunoyumeto.github.io/open-math-courses-public/courses/NOE-INV/",
        "status": "draft",
        "status_note": "Self-checked by the writing AI, GPT-6.1 Sol (OpenAI), in Codex at Ultra.",
        "authorship": "GPT-6.1 Sol (OpenAI), in Codex, at the Ultra setting (maximum reasoning effort)",
        "licence": {
          "spdx": "CC0-1.0",
          "scope": "Independently authored course text, proofs, exercises and original diagrams.",
          "exceptions": [
            "Linked historical works and references retain their own rights; MathJax and font assets retain their licences."
          ]
        },
        "lessons": [
          {
            "id": "NOE-INV-01",
            "title": "Covariants of binary and ternary forms",
            "route": "https://kokunoyumeto.github.io/open-math-courses-public/courses/NOE-INV/NOE-INV-01.html",
            "editable_source": "https://kokunoyumeto.github.io/open-math-courses-public/courses/NOE-INV/"
          },
          {
            "id": "NOE-INV-02",
            "title": "Forms in several variables and Hilbert finiteness",
            "route": "https://kokunoyumeto.github.io/open-math-courses-public/courses/NOE-INV/NOE-INV-02.html",
            "editable_source": "https://kokunoyumeto.github.io/open-math-courses-public/courses/NOE-INV/"
          },
          {
            "id": "NOE-INV-03",
            "title": "Invariants of finite groups and Noether's degree bound",
            "route": "https://kokunoyumeto.github.io/open-math-courses-public/courses/NOE-INV/NOE-INV-03.html",
            "editable_source": "https://kokunoyumeto.github.io/open-math-courses-public/courses/NOE-INV/"
          },
          {
            "id": "NOE-INV-04",
            "title": "Polarization and invariants of many vectors",
            "route": "https://kokunoyumeto.github.io/open-math-courses-public/courses/NOE-INV/NOE-INV-04.html",
            "editable_source": "https://kokunoyumeto.github.io/open-math-courses-public/courses/NOE-INV/"
          },
          {
            "id": "NOE-INV-05",
            "title": "Finiteness for finite groups in every characteristic",
            "route": "https://kokunoyumeto.github.io/open-math-courses-public/courses/NOE-INV/NOE-INV-05.html",
            "editable_source": "https://kokunoyumeto.github.io/open-math-courses-public/courses/NOE-INV/"
          }
        ],
        "source_snapshot": "801f186833b9811fa826575ef0717e78a5047ab0",
        "portable_editions": []
      },
      {
        "id": "NT-DIRL",
        "kind": "advanced",
        "title": {
          "en": "Dirichlet L-functions and primes in progressions"
        },
        "summary": "Dirichlet characters and L-functions, from Dirichlet's theorem to average distribution and the least prime in a progression. The course develops conductors, Gauss sums, character sums, functional equations, special values, zero-free regions, exceptional zeros, explicit formulas, the prime number theorem for progressions, Siegel and Siegel–Walfisz estimates, the large sieve, Vaughan's identity, the Bombieri–Vinogradov and Barban–Davenport–Halberstam theorems, and Linnik's theorem. The opening lessons prove the character constructions, quadratic reciprocity and elementary analytic prerequisites used in their arguments.",
        "subject": "number-theory",
        "content_language": "en",
        "translations": [],
        "route": "https://kokunoyumeto.github.io/open-math-courses-public/courses/NT-DIRL/",
        "status": "draft",
        "status_note": "Self-checked by the writing AI, GPT-6.1 Sol (OpenAI), in Codex at Ultra. No separate AI review is recorded.",
        "authorship": "GPT-6.1 Sol (OpenAI), in Codex, at the Ultra setting (maximum reasoning effort)",
        "licence": {
          "spdx": "CC0-1.0",
          "scope": "The independently authored mathematical text and original figures.",
          "exceptions": [
            "Bundled renderer dependencies and rendered font glyphs retain their own licences."
          ]
        },
        "lessons": [
          {
            "id": "dirichlet-characters",
            "title": "Dirichlet characters",
            "route": "https://kokunoyumeto.github.io/open-math-courses-public/courses/NT-DIRL/",
            "editable_source": "https://kokunoyumeto.github.io/open-math-courses-public/courses/NT-DIRL/"
          },
          {
            "id": "dirichlet-s-theorem-on-primes-in-arithmetic-progressions",
            "title": "Dirichlet's theorem on primes in arithmetic progressions",
            "route": "https://kokunoyumeto.github.io/open-math-courses-public/courses/NT-DIRL/",
            "editable_source": "https://kokunoyumeto.github.io/open-math-courses-public/courses/NT-DIRL/"
          },
          {
            "id": "gauss-sums",
            "title": "Gauss sums",
            "route": "https://kokunoyumeto.github.io/open-math-courses-public/courses/NT-DIRL/",
            "editable_source": "https://kokunoyumeto.github.io/open-math-courses-public/courses/NT-DIRL/"
          },
          {
            "id": "the-functional-equation-of-dirichlet-l-functions",
            "title": "The functional equation of Dirichlet L-functions",
            "route": "https://kokunoyumeto.github.io/open-math-courses-public/courses/NT-DIRL/",
            "editable_source": "https://kokunoyumeto.github.io/open-math-courses-public/courses/NT-DIRL/"
          },
          {
            "id": "values-of-dirichlet-l-functions-at-s-1",
            "title": "Values of Dirichlet L-functions at s = 1",
            "route": "https://kokunoyumeto.github.io/open-math-courses-public/courses/NT-DIRL/",
            "editable_source": "https://kokunoyumeto.github.io/open-math-courses-public/courses/NT-DIRL/"
          },
          {
            "id": "zero-free-regions-and-the-exceptional-zero",
            "title": "Zero-free regions and the exceptional zero",
            "route": "https://kokunoyumeto.github.io/open-math-courses-public/courses/NT-DIRL/",
            "editable_source": "https://kokunoyumeto.github.io/open-math-courses-public/courses/NT-DIRL/"
          },
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            "Intrinsic Christoffel symbols and the theorema egregium, in Submanifolds and hypersurfaces, The two explicitly delimited statement-and-proof blockquotes in Section 1: adapted from Matthias Weber, Differential Geometry of Curves and Surfaces, 7 January 2026, Theorems 3.5.3 and 3.5.6, printed pages294–295 and297–298 (PDF295–296 and298–299); retains CC-BY-NC-SA-4.0 (https://creativecommons.org/licenses/by-nc-sa/4.0/). Source and changes are credited at the component.",
            "Sodré’s uniform spectral lower bound, in Jacobi fields conjugate points and the morse index theorem, The complete attributed Theorem2.2 statement and proof blockquote in Section8: adapted from Eduardo V. Sodré, Revisiting Arnold’s topological proof of the Morse index theorem, arXiv:2307.00655v1,2July2023, Theorem2.2,p.5; retains CC-BY-4.0 (https://creativecommons.org/licenses/by/4.0/). Source and changes are credited at the component.",
            "Kamtue’s Bonnet–Myers diameter proof, in Comparison theorems cut loci and curvature and topology, The complete attributed Theorem 4.1 statement and proof blockquote in Section 4: adapted from Supanat Kamtue, Discrete curvatures motivated from Riemannian geometry and optimal transport: Bonnet-Myers-type diameter bounds and rigidity, Doctoral thesis, Durham University, 2021, Chapter 4, Theorem 4.1, printed pp. 29–31 (PDF 42–44); retains CC-BY-3.0 (https://creativecommons.org/licenses/by/3.0/). Source and changes are credited at the component."
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        "conditions": [
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        "conditions": [
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    "producer": {
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        "scope_assessment": "The finite basis-extension argument and its selected algebraic prerequisites use field operations and inversion of nonzero coefficients, not order or positivity. The Fields topic explicitly explains the applicable field generalization. This does not justify generalizing metric/Hermitian results by replacing symbols.",
        "integrity_finding": "In the second half of lm:AddVecLiSetIsLiIffVecNotInSpan, zero equals sum(c_i*s_i) plus c_(n+1)*v. Solving for v needs coefficients -c_i/c_(n+1), but the preserved source prints positive c_i/c_(n+1). Membership in the span remains the intended and valid conclusion after the sign is accounted for.",
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        "finding_id": "Jalur perluasan basis telah ditemukan secara tepat, tetapi belum diterima sebagai prasyarat dengan pembuktian lengkap. Argumen aljabarnya berlaku untuk medan sebarang; tanda minus yang hilang dalam lema yang dipakai harus dicatat, dan pembuktian tepatnya memerlukan jangkar pembaca publik."
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        "locus": "Proposition 2.1 and the orthogonal-complement argument immediately after it; Section 7 states the assumed fact",
        "required_statement": "Every finite-dimensional positive definite complex Hermitian space has an orthonormal basis and V=U direct-sum U-perp, with the inner product linear in its first variable.",
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        "finding_en": "Do not route the complex Hermitian result to a real Euclidean projection merely because both are called orthogonal projection. Locate and read the complex proof and its prerequisites before admitting this dependency.",
        "finding_id": "Jangan memetakan hasil Hermitian kompleks ke proyeksi Euklides riil hanya karena keduanya disebut proyeksi ortogonal. Temukan dan baca pembuktian kompleks beserta prasyaratnya sebelum menerima dependensi ini."
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    "not_claimed": [
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      "Every prerequisite of the representation-theory lesson is closed",
      "The phone's local provider records are already public or independently certified",
      "An English interface constitutes an Indonesian translation of the advanced lessons"
    ],
    "next_actions": [
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      "Bind the basis-extension proof to the actual native public reader anchor before admission",
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