{
  "schema": "oa-mod-selected-full-programme-proof-bodies/v1",
  "title": "Complete programme proofs included with the selected readings",
  "scope": "Exact selected proof inputs for bounded, spectral and concrete-predual foundations, preserving the full source/reader bodies. No full course or transitive theorem-closure claim. The same unchanged complete bodies also supply the exact selected Hilbert and scalar proof inputs for the closed-form lesson.",
  "provider_files": [
    {
      "path": "courses/foundations-of-von-neumann-algebras/src/hilbert-spaces-and-compact-operators.md",
      "sha256": "c37f79bc01c64ef1198571d05fb5aeb18a80c704afc642ab50f904fd88f62fd5",
      "bytes": 36935,
      "public_url": "https://kokunoyumeto.github.io/open-math-courses/courses/foundations-of-von-neumann-algebras/src/hilbert-spaces-and-compact-operators.md",
      "body_preservation": "Complete canonical public file copied without modification"
    },
    {
      "path": "courses/foundations-of-von-neumann-algebras/hilbert-spaces-and-compact-operators.html",
      "sha256": "96e743d357c26ed76a7cdb66d2b023a99e6005049dac4b735b1ce93000d5d13e",
      "bytes": 82325,
      "public_url": "https://kokunoyumeto.github.io/open-math-courses/courses/foundations-of-von-neumann-algebras/hilbert-spaces-and-compact-operators.html",
      "body_preservation": "Complete canonical public file copied without modification"
    },
    {
      "path": "courses/foundations-of-von-neumann-algebras/src/hahn-banach-baire-and-the-basic-theorems-on-banach-spaces.md",
      "sha256": "f09b763b00c87141faac24be5c8d4a4760c3439085288bcdb56263731281e850",
      "bytes": 42962,
      "public_url": "https://kokunoyumeto.github.io/open-math-courses/courses/foundations-of-von-neumann-algebras/src/hahn-banach-baire-and-the-basic-theorems-on-banach-spaces.md",
      "body_preservation": "Complete canonical public file copied without modification"
    },
    {
      "path": "courses/foundations-of-von-neumann-algebras/hahn-banach-baire-and-the-basic-theorems-on-banach-spaces.html",
      "sha256": "818dda13511ac9bb6e07f7f8d1c6e2bb9538b5c4003c8ee11c746d68af9611e9",
      "bytes": 100238,
      "public_url": "https://kokunoyumeto.github.io/open-math-courses/courses/foundations-of-von-neumann-algebras/hahn-banach-baire-and-the-basic-theorems-on-banach-spaces.html",
      "body_preservation": "Complete canonical public file copied without modification"
    },
    {
      "path": "courses/foundations-of-von-neumann-algebras/src/the-universal-enveloping-von-neumann-algebra-of-a-c-star-algebra-and-w-star-algebras.md",
      "sha256": "4de77e7e8e0e2ded632f7176ceb31036e9bdfd90a7e32156a59ff87afc0dccc3",
      "bytes": 165047,
      "public_url": "https://kokunoyumeto.github.io/open-math-courses/courses/foundations-of-von-neumann-algebras/src/the-universal-enveloping-von-neumann-algebra-of-a-c-star-algebra-and-w-star-algebras.md",
      "body_preservation": "Complete canonical public file copied without modification"
    },
    {
      "path": "courses/foundations-of-von-neumann-algebras/the-universal-enveloping-von-neumann-algebra-of-a-c-star-algebra-and-w-star-algebras.html",
      "sha256": "00a007f287d49f695fe799e65c40a7da651cb2aba9f9cbfa747097b7f957bdef",
      "bytes": 370516,
      "public_url": "https://kokunoyumeto.github.io/open-math-courses/courses/foundations-of-von-neumann-algebras/the-universal-enveloping-von-neumann-algebra-of-a-c-star-algebra-and-w-star-algebras.html",
      "body_preservation": "Complete canonical public file copied without modification"
    },
    {
      "path": "courses/foundations-of-von-neumann-algebras/src/the-double-commutant-theorem.md",
      "sha256": "826eb3ef54ab693b5e786295b279ae96694a50d551b641bfa87c9c8aa655605b",
      "bytes": 108587,
      "public_url": "https://kokunoyumeto.github.io/open-math-courses/courses/foundations-of-von-neumann-algebras/src/the-double-commutant-theorem.md",
      "body_preservation": "Complete canonical public file copied without modification"
    },
    {
      "path": "courses/foundations-of-von-neumann-algebras/the-double-commutant-theorem.html",
      "sha256": "ab55566322a7e50291443f2edf19c640e945eaeccc1164d3ae71739394d41875",
      "bytes": 251665,
      "public_url": "https://kokunoyumeto.github.io/open-math-courses/courses/foundations-of-von-neumann-algebras/the-double-commutant-theorem.html",
      "body_preservation": "Complete canonical public file copied without modification"
    },
    {
      "path": "courses/foundations-of-von-neumann-algebras/src/weak-topologies-tychonoff-banach-alaoglu-mazur-bipolars-krein-milman-and-eberlein-smulian.md",
      "sha256": "3b191ac5f149ab4b7705128185f944e4227b250290a5593a4932b68a0fb05df9",
      "bytes": 30325,
      "public_url": "https://kokunoyumeto.github.io/open-math-courses/courses/foundations-of-von-neumann-algebras/src/weak-topologies-tychonoff-banach-alaoglu-mazur-bipolars-krein-milman-and-eberlein-smulian.md",
      "body_preservation": "Complete canonical public file copied without modification"
    },
    {
      "path": "courses/foundations-of-von-neumann-algebras/weak-topologies-tychonoff-banach-alaoglu-mazur-bipolars-krein-milman-and-eberlein-smulian.html",
      "sha256": "4be2a48bff7b78aed3a9689b5bd87e32b64b85fba139b06847acbb4d27452131",
      "bytes": 71655,
      "public_url": "https://kokunoyumeto.github.io/open-math-courses/courses/foundations-of-von-neumann-algebras/weak-topologies-tychonoff-banach-alaoglu-mazur-bipolars-krein-milman-and-eberlein-smulian.html",
      "body_preservation": "Complete canonical public file copied without modification"
    },
    {
      "path": "courses/foundations-of-von-neumann-algebras/src/c-star-algebras-continuous-functional-calculus-automatic-continuity-positive-cones.md",
      "sha256": "217b49bb8e63bfbfe542eb957f3185b3f4af53f62676e1f3b8f7c071e996a8b9",
      "bytes": 154960,
      "public_url": "https://kokunoyumeto.github.io/open-math-courses/courses/foundations-of-von-neumann-algebras/src/c-star-algebras-continuous-functional-calculus-automatic-continuity-positive-cones.md",
      "body_preservation": "Complete canonical public file copied without modification"
    },
    {
      "path": "courses/foundations-of-von-neumann-algebras/c-star-algebras-continuous-functional-calculus-automatic-continuity-positive-cones.html",
      "sha256": "a6a3cf847a88331dae92e314c86692cf0a885bfbb329add6715f40ad7d7d47ed",
      "bytes": 358543,
      "public_url": "https://kokunoyumeto.github.io/open-math-courses/courses/foundations-of-von-neumann-algebras/c-star-algebras-continuous-functional-calculus-automatic-continuity-positive-cones.html",
      "body_preservation": "Complete canonical public file copied without modification"
    },
    {
      "path": "courses/harmonic-analysis-on-locally-compact-groups/reader/dependencies/foundations-of-von-neumann-algebras/hilbert-spaces-and-compact-operators.html",
      "sha256": "a0fbe32c9fd02a5e39936d838ca815d9ce0e1f4ad495c3083a67b6fb3c9c7377",
      "bytes": 79850,
      "public_url": "https://kokunoyumeto.github.io/open-math-courses/courses/harmonic-analysis-on-locally-compact-groups/reader/dependencies/foundations-of-von-neumann-algebras/hilbert-spaces-and-compact-operators.html",
      "body_preservation": "Exact saved anonymous response; complete reader/source unchanged"
    },
    {
      "path": "courses/harmonic-analysis-on-locally-compact-groups/reader/dependencies/function-algebras-and-approximation/the-stone-weierstrass-theorem-for-functions-vanishing-at-infinity.html",
      "sha256": "e34d6db731ca9e1d4a2a0e12c3a58a2be28739734886de36be61c60b330b0197",
      "bytes": 283361,
      "public_url": "https://kokunoyumeto.github.io/open-math-courses/courses/harmonic-analysis-on-locally-compact-groups/reader/dependencies/function-algebras-and-approximation/the-stone-weierstrass-theorem-for-functions-vanishing-at-infinity.html",
      "body_preservation": "Exact saved anonymous response; complete reader/source unchanged"
    },
    {
      "path": "courses/harmonic-analysis-on-locally-compact-groups/reader/haar-measure-on-locally-compact-groups.html",
      "sha256": "0a2b8bf367407652ae7214c207933b3c04d9e92e1ddea9a7437ab610a622bc79",
      "bytes": 320318,
      "public_url": "https://kokunoyumeto.github.io/open-math-courses/courses/harmonic-analysis-on-locally-compact-groups/reader/haar-measure-on-locally-compact-groups.html",
      "body_preservation": "Exact saved anonymous response; complete reader/source unchanged"
    },
    {
      "path": "courses/harmonic-analysis-on-locally-compact-groups/reader/measure-and-hilbert-space-tools.html",
      "sha256": "289ab310ddb0239ea5df935bc2b8e110f619e025f4e933e940c52dff786e2929",
      "bytes": 52033,
      "public_url": "https://kokunoyumeto.github.io/open-math-courses/courses/harmonic-analysis-on-locally-compact-groups/reader/measure-and-hilbert-space-tools.html",
      "body_preservation": "Exact saved anonymous response; complete reader/source unchanged"
    },
    {
      "path": "courses/harmonic-analysis-on-locally-compact-groups/src/dependencies/foundations-of-von-neumann-algebras/src/hilbert-spaces-and-compact-operators.md",
      "sha256": "ee431aa610e598b2ac40bcc48e4d00eb5d666842e5d9e1937be2c3173f6fdbe3",
      "bytes": 36138,
      "public_url": "https://kokunoyumeto.github.io/open-math-courses/courses/harmonic-analysis-on-locally-compact-groups/src/dependencies/foundations-of-von-neumann-algebras/src/hilbert-spaces-and-compact-operators.md",
      "body_preservation": "Exact saved anonymous response; complete reader/source unchanged"
    },
    {
      "path": "courses/harmonic-analysis-on-locally-compact-groups/src/dependencies/function-algebras-and-approximation/src/the-stone-weierstrass-theorem-for-functions-vanishing-at-infinity.md",
      "sha256": "e2d19f41c0f9aab7a562ebdd668587360405d3475fa3d8834f9e9d165168f4d2",
      "bytes": 125564,
      "public_url": "https://kokunoyumeto.github.io/open-math-courses/courses/harmonic-analysis-on-locally-compact-groups/src/dependencies/function-algebras-and-approximation/src/the-stone-weierstrass-theorem-for-functions-vanishing-at-infinity.md",
      "body_preservation": "Exact saved anonymous response; complete reader/source unchanged"
    },
    {
      "path": "courses/harmonic-analysis-on-locally-compact-groups/src/haar-measure-on-locally-compact-groups.md",
      "sha256": "c3d938e2f10056cc392605c1b811fbc9399eca53104287036a43605c0bd17ad0",
      "bytes": 143387,
      "public_url": "https://kokunoyumeto.github.io/open-math-courses/courses/harmonic-analysis-on-locally-compact-groups/src/haar-measure-on-locally-compact-groups.md",
      "body_preservation": "Exact saved anonymous response; complete reader/source unchanged"
    },
    {
      "path": "courses/harmonic-analysis-on-locally-compact-groups/src/measure-and-hilbert-space-tools.md",
      "sha256": "ee22f2f44dd2c089b2bfb10d25fd5b36ebd9d7c245ea3206a1abb71e393e2e12",
      "bytes": 29235,
      "public_url": "https://kokunoyumeto.github.io/open-math-courses/courses/harmonic-analysis-on-locally-compact-groups/src/measure-and-hilbert-space-tools.md",
      "body_preservation": "Exact saved anonymous response; complete reader/source unchanged"
    }
  ],
  "primary_scalar_proof_ranges": [
    {
      "item": "Theorem 2.2, complete six-step RMK proof",
      "source": "src/haar-measure-on-locally-compact-groups.md",
      "lines": [
        55,
        78
      ],
      "range_sha256_LF_normalized": "145e240babbb41861eec7c779d7ef3d6932075e21e19d9428efa9748fee891c7",
      "reader_url": "https://kokunoyumeto.github.io/open-math-courses/courses/harmonic-analysis-on-locally-compact-groups/reader/haar-measure-on-locally-compact-groups.html#OA-FND-HM-01"
    },
    {
      "item": "Proposition 2.3, including finite-Borel compact inner approximation",
      "source": "src/haar-measure-on-locally-compact-groups.md",
      "lines": [
        80,
        88
      ],
      "range_sha256_LF_normalized": "fa76d74a3cbaa05a861f0822d45a9583967d26e6fb0fe2659dbf02677cf3d48c",
      "reader_url": "https://kokunoyumeto.github.io/open-math-courses/courses/harmonic-analysis-on-locally-compact-groups/reader/haar-measure-on-locally-compact-groups.html#OA-FND-HM-01"
    },
    {
      "item": "Theorem 1.1, Caratheodory",
      "source": "src/measure-and-hilbert-space-tools.md",
      "lines": [
        16,
        30
      ],
      "range_sha256_LF_normalized": "66dddeec238cfb9574adb998ad910aa7a4178f2d740e6da442b150720e11f066",
      "reader_url": "https://kokunoyumeto.github.io/open-math-courses/courses/harmonic-analysis-on-locally-compact-groups/reader/measure-and-hilbert-space-tools.html#1-from-an-outer-measure-to-a-measure"
    },
    {
      "item": "Nonnegative integration, measurable sequential limits, Theorem 2.1 MCT and countable integral sums",
      "source": "src/measure-and-hilbert-space-tools.md",
      "lines": [
        36,
        57
      ],
      "range_sha256_LF_normalized": "393e5e9c36173e1f7722445861deb8802816ab900f41c705b3f68d951f24380a",
      "reader_url": "https://kokunoyumeto.github.io/open-math-courses/courses/harmonic-analysis-on-locally-compact-groups/reader/measure-and-hilbert-space-tools.html#2-integration-and-convergence-without-countability-assumptions"
    },
    {
      "item": "Theorem 2.2, Fatou and real/complex sequential DCT",
      "source": "src/measure-and-hilbert-space-tools.md",
      "lines": [
        59,
        69
      ],
      "range_sha256_LF_normalized": "266821b21803eef0c6bfc40a84b6817f83dd7a71ba64307f8f94262506eb3dc5",
      "reader_url": "https://kokunoyumeto.github.io/open-math-courses/courses/harmonic-analysis-on-locally-compact-groups/reader/measure-and-hilbert-space-tools.html#2-integration-and-convergence-without-countability-assumptions"
    },
    {
      "item": "L^p convention and Theorem 3.1 Holder/Minkowski",
      "source": "src/measure-and-hilbert-space-tools.md",
      "lines": [
        75,
        86
      ],
      "range_sha256_LF_normalized": "a6836154a0f24e576daaae78836ecf076b6cd57e104d8b235b78b1232c58712c",
      "reader_url": "https://kokunoyumeto.github.io/open-math-courses/courses/harmonic-analysis-on-locally-compact-groups/reader/measure-and-hilbert-space-tools.html#3-the-complete-spaces-of-integrable-functions"
    },
    {
      "item": "Theorem 3.2 completeness/simple density and linear-first L2",
      "source": "src/measure-and-hilbert-space-tools.md",
      "lines": [
        88,
        105
      ],
      "range_sha256_LF_normalized": "685d147df65ca8cfbc27473809fbac874ede1e2ef3d0bddd1298b71054e43bfb",
      "reader_url": "https://kokunoyumeto.github.io/open-math-courses/courses/harmonic-analysis-on-locally-compact-groups/reader/measure-and-hilbert-space-tools.html#3-the-complete-spaces-of-integrable-functions"
    }
  ],
  "topology_support": "Stone–Weierstrass dependency source lines 130–171: Proposition 4.2(2), Theorem 5.1 and Corollary 5.2; source/reader included unchanged",
  "credits": "Original and revision programme AI credits and human research references remain in the full files. This collection claims no human authorship or review. Pinned mathlib human contributors and Apache-2.0 formal-source terms remain in the scalar companion and proof records.",
  "component_terms": [
    {
      "component": "Independently written programme provider prose",
      "license": "CC0-1.0",
      "notices": [
        "courses/foundations-of-von-neumann-algebras/source-notices.html",
        "licensing.html"
      ]
    },
    {
      "component": "Spectral lesson and five original bridges",
      "license": "CC0-1.0"
    },
    {
      "component": "Haar historical/complementary source notice",
      "notice": "courses/harmonic-analysis-on-locally-compact-groups/src/licenses/design-science-license.txt",
      "scope": "Retained owner notice; does not override CC0 original provider prose"
    },
    {
      "component": "MathJax",
      "license": "Apache-2.0",
      "notice": "assets/mathjax/LICENSE"
    },
    {
      "component": "MathJax fonts",
      "notice": "assets/mathjax/FONT-LICENSES.txt"
    }
  ],
  "offline_layout": "Original course-relative source/reader/assets paths retained. An archive proof menu identifies the included complete bodies; other full-reader navigation can remain outside this selected offline subset.",
  "paid_only_construction_reliance": false,
  "human_PDFs_redistributed": false,
  "closed_form_proof_ranges": [
    {
      "id": "OA-FND-HS-01",
      "source": "courses/foundations-of-von-neumann-algebras/src/hilbert-spaces-and-compact-operators.md",
      "lines": [
        22,
        83
      ],
      "contract": "Positivity/polarization and Hilbert completion; bounded isometry extension on arbitrary Hilbert spaces.",
      "reader": "https://kokunoyumeto.github.io/open-math-courses/courses/foundations-of-von-neumann-algebras/hilbert-spaces-and-compact-operators.html#oa-fnd-hs-01",
      "license": "CC0-1.0",
      "raw_source_sha256": "c37f79bc01c64ef1198571d05fb5aeb18a80c704afc642ab50f904fd88f62fd5",
      "range_sha256_LF_with_final_newline": "5a58f40a883b44b19a32dce42348ae24781f339aa58794de40ac8fe2c57d7863",
      "verified_at": "2026-10-04T03:27:24.447458+00:00"
    },
    {
      "id": "OA-FND-HS-02",
      "source": "courses/foundations-of-von-neumann-algebras/src/hilbert-spaces-and-compact-operators.md",
      "lines": [
        84,
        115
      ],
      "exact_theorems": "Theorems 2.1, 2.2, 2.3",
      "contract": "Projection, orthogonal double complement and Riesz representation, linear-first, arbitrary H.",
      "reader": "https://kokunoyumeto.github.io/open-math-courses/courses/foundations-of-von-neumann-algebras/hilbert-spaces-and-compact-operators.html#oa-fnd-hs-02",
      "license": "CC0-1.0",
      "raw_source_sha256": "c37f79bc01c64ef1198571d05fb5aeb18a80c704afc642ab50f904fd88f62fd5",
      "range_sha256_LF_with_final_newline": "4da0acf479b6c4c00703ebcff6c184888f9f95c986f66b511007a7f1ec72bc69",
      "verified_at": "2026-10-04T03:27:24.447458+00:00"
    },
    {
      "id": "OA-FND-HS-03",
      "source": "courses/foundations-of-von-neumann-algebras/src/hilbert-spaces-and-compact-operators.md",
      "lines": [
        116,
        142
      ],
      "exact_theorems": "Theorem 3.1 and Corollary 3.2",
      "contract": "Bounded sesquilinear-form representation and adjoints. Rectangular adjoint is the same Riesz proof with target vector fixed; explicit bridge supplied in proposal.",
      "reader": "https://kokunoyumeto.github.io/open-math-courses/courses/foundations-of-von-neumann-algebras/hilbert-spaces-and-compact-operators.html#oa-fnd-hs-03",
      "license": "CC0-1.0",
      "raw_source_sha256": "c37f79bc01c64ef1198571d05fb5aeb18a80c704afc642ab50f904fd88f62fd5",
      "range_sha256_LF_with_final_newline": "737c3933e865d00463cf81242f65ba119fc1419e6ebc4e51f41d6bb043e792a7",
      "verified_at": "2026-10-04T03:27:24.447458+00:00"
    },
    {
      "id": "OA-MOD-BK-01",
      "source": "courses/OA-MOD/src/bounded-operator-kernel.md",
      "lines": [
        9,
        34
      ],
      "contract": "Concrete positivity, kernel square-root identity and closure(range T)=(ker T*)-perp.",
      "reader": "https://kokunoyumeto.github.io/open-math-courses/courses/OA-MOD/bounded-operator-kernel.html#OA-MOD-BK-01",
      "raw_source_sha256": "04e0495ab7ec51fdc6b6f07a5dc835dfa623680103fc9f5bfc0cf30c9204fd3b",
      "range_sha256_LF_with_final_newline": "9ffb23f424a25f12ab166dcce1dcbf3ec98aaf938fc2e90c9ffe32a789346312",
      "verified_at": "2026-10-04T03:27:24.447458+00:00"
    },
    {
      "id": "OA-MOD-BK-07",
      "source": "courses/OA-MOD/src/bounded-operator-kernel.md",
      "lines": [
        322,
        396
      ],
      "specific_lines": [
        372,
        395
      ],
      "contract": "Rectangular bounded polar decomposition C=U|C|=|C*|U; U unitary when C is injective with dense range. Construction uses bounded square roots and extension of an isometry, with no closed-form/unbounded-polar theorem.",
      "reader": "https://kokunoyumeto.github.io/open-math-courses/courses/OA-MOD/bounded-operator-kernel.html#OA-MOD-BK-07",
      "raw_source_sha256": "04e0495ab7ec51fdc6b6f07a5dc835dfa623680103fc9f5bfc0cf30c9204fd3b",
      "range_sha256_LF_with_final_newline": "a9f26b3e6ea01c9ae06068b0de1a9127a72b096ea1614600fdb5f87c30be36f4",
      "verified_at": "2026-10-04T03:27:24.447458+00:00"
    },
    {
      "id": "OA-MOD-SK-04",
      "source": "courses/OA-MOD/src/spectral-calculus-kernel.md",
      "lines": [
        70,
        110
      ],
      "contract": "Bounded Borel calculus, finite vector measures, spectral projections and uniqueness; arbitrary cyclic-sum index.",
      "reader": "https://kokunoyumeto.github.io/open-math-courses/courses/OA-MOD/spectral-calculus-kernel.html#OA-MOD-SK-04",
      "raw_source_sha256": "7a6a51a63cc5a4493d6ff428f2ca9731a57d26219c03207b522d328a69b092e0",
      "range_sha256_LF_with_final_newline": "c59feeb2a0e3aed5c609f7f0224de9f43c1c14a1dc59148a77ccc74ed2f344d2",
      "verified_at": "2026-10-04T03:27:24.447458+00:00"
    },
    {
      "id": "OA-MOD-SK-05",
      "source": "courses/OA-MOD/src/spectral-calculus-kernel.md",
      "lines": [
        112,
        156
      ],
      "contract": "Closedness, dense squared-integral domains, adjoints and self-adjointness for real functions; actual product domains; graph cutoffs.",
      "reader": "https://kokunoyumeto.github.io/open-math-courses/courses/OA-MOD/spectral-calculus-kernel.html#OA-MOD-SK-05",
      "raw_source_sha256": "7a6a51a63cc5a4493d6ff428f2ca9731a57d26219c03207b522d328a69b092e0",
      "range_sha256_LF_with_final_newline": "7ba262113155431a5aebad74ef86149096e58903c09e1d6ab60fb7bd1606b8b9",
      "verified_at": "2026-10-04T03:27:24.447458+00:00"
    },
    {
      "id": "OA-MOD-SK-06",
      "source": "courses/OA-MOD/src/spectral-calculus-kernel.md",
      "lines": [
        158,
        207
      ],
      "contract": "Existence and uniqueness of arbitrary self-adjoint spectral resolution via Cayley transform; positive support.",
      "reader": "https://kokunoyumeto.github.io/open-math-courses/courses/OA-MOD/spectral-calculus-kernel.html#OA-MOD-SK-06",
      "raw_source_sha256": "7a6a51a63cc5a4493d6ff428f2ca9731a57d26219c03207b522d328a69b092e0",
      "range_sha256_LF_with_final_newline": "f6e3282e4d4d32a36cdc9ae898f9a99a120597052731c619d48ea1f8c37b54e9",
      "verified_at": "2026-10-04T03:27:24.447458+00:00"
    },
    {
      "id": "OA-MOD-SK-07",
      "source": "courses/OA-MOD/src/spectral-calculus-kernel.md",
      "lines": [
        209,
        261
      ],
      "contract": "Changes of variable; unique positive square root; exact form pairing; inverse domains equal actual ranges; logarithms and powers.",
      "reader": "https://kokunoyumeto.github.io/open-math-courses/courses/OA-MOD/spectral-calculus-kernel.html#OA-MOD-SK-07",
      "raw_source_sha256": "7a6a51a63cc5a4493d6ff428f2ca9731a57d26219c03207b522d328a69b092e0",
      "range_sha256_LF_with_final_newline": "aebaf6ea4ab8152fed6e14139c54c69c847d8ec98861a26dda7112adbbc35a7e",
      "verified_at": "2026-10-04T03:27:24.447458+00:00"
    },
    {
      "id": "OA-MOD-SK-08",
      "source": "courses/OA-MOD/src/spectral-calculus-kernel.md",
      "lines": [
        263,
        288
      ],
      "contract": "Unitary/antiunitary transport with domains, reduction and bounded Borel membership/affiliation.",
      "reader": "https://kokunoyumeto.github.io/open-math-courses/courses/OA-MOD/spectral-calculus-kernel.html#OA-MOD-SK-08",
      "raw_source_sha256": "7a6a51a63cc5a4493d6ff428f2ca9731a57d26219c03207b522d328a69b092e0",
      "range_sha256_LF_with_final_newline": "1c0f3c225b2013caad735c74bd0c9c3b4faf18e02d034c2d92b7bdca3731584e",
      "verified_at": "2026-10-04T03:27:24.447458+00:00"
    },
    {
      "id": "OA-MOD-SK-09",
      "source": "courses/OA-MOD/src/spectral-calculus-kernel.md",
      "lines": [
        290,
        357
      ],
      "specific_lines": [
        290,
        301
      ],
      "contract": "Scalar spectral DCT/cutoffs and strong continuous functional-calculus convergence on bounded self-adjoint nets. Line 301 supplies the Bernstein proof of compact-interval polynomial approximation.",
      "reader": "https://kokunoyumeto.github.io/open-math-courses/courses/OA-MOD/spectral-calculus-kernel.html#OA-MOD-SK-09",
      "raw_source_sha256": "7a6a51a63cc5a4493d6ff428f2ca9731a57d26219c03207b522d328a69b092e0",
      "range_sha256_LF_with_final_newline": "0f00291b170a3f8de1d403e9168aae5144286eee6b32e6c35dfb42b89b42ee7f",
      "verified_at": "2026-10-04T03:27:24.447458+00:00"
    },
    {
      "id": "MEASURE-TOOLS-MCT-DCT",
      "source": "courses/harmonic-analysis-on-locally-compact-groups/src/measure-and-hilbert-space-tools.md",
      "lines": [
        36,
        69
      ],
      "exact_theorems": "Theorem 2.1 (48–57), Theorem 2.2 (59–69)",
      "contract": "Sequential monotone and real/complex dominated convergence for arbitrary measures; use on finite scalar spectral measures. No general net DCT.",
      "reader": "https://kokunoyumeto.github.io/open-math-courses/courses/harmonic-analysis-on-locally-compact-groups/reader/measure-and-hilbert-space-tools.html#2-integration-and-convergence-without-countability-assumptions",
      "license": "CC0-1.0",
      "raw_source_sha256": "ee22f2f44dd2c089b2bfb10d25fd5b36ebd9d7c245ea3206a1abb71e393e2e12",
      "range_sha256_LF_with_final_newline": "c7f587e9e93f437180c96b6cf12e9f0cd87cbe998db615ed029d251fd7a5343c",
      "verified_at": "2026-10-04T03:27:24.447458+00:00"
    },
    {
      "id": "OA-MOD-BK-02",
      "source": "courses/OA-MOD/src/bounded-operator-kernel.md",
      "lines": [
        35,
        117
      ],
      "contract": "Bicommutant theorem; required only for QF-04 optional algebra application.",
      "reader": "https://kokunoyumeto.github.io/open-math-courses/courses/OA-MOD/bounded-operator-kernel.html#OA-MOD-BK-02",
      "raw_source_sha256": "04e0495ab7ec51fdc6b6f07a5dc835dfa623680103fc9f5bfc0cf30c9204fd3b",
      "range_sha256_LF_with_final_newline": "241587d8fd8f6eaf7aa2acd3f4abe2e8edc33d881b4e28ac4c49bbda2af34a73",
      "verified_at": "2026-10-04T03:27:24.447458+00:00"
    },
    {
      "id": "OA-MOD-BK-08",
      "source": "courses/OA-MOD/src/bounded-operator-kernel.md",
      "lines": [
        397,
        410
      ],
      "contract": "Four-unitary linear span and unitary commutation test; only optional algebra application.",
      "reader": "https://kokunoyumeto.github.io/open-math-courses/courses/OA-MOD/bounded-operator-kernel.html#OA-MOD-BK-08",
      "raw_source_sha256": "04e0495ab7ec51fdc6b6f07a5dc835dfa623680103fc9f5bfc0cf30c9204fd3b",
      "range_sha256_LF_with_final_newline": "066ed5856199bcdc9bc094cb9c2a0dd8f6a2c91be4142775ee358498be69e4d0",
      "verified_at": "2026-10-04T03:27:24.447458+00:00"
    },
    {
      "id": "OA-FND-HB-01",
      "source": "courses/foundations-of-von-neumann-algebras/src/hahn-banach-baire-and-the-basic-theorems-on-banach-spaces.md",
      "lines": [
        30,
        46
      ],
      "exact_theorems": "Theorem 1.1",
      "contract": "Zorn maximal principle from choice, transitive input to SK-03. QF does not use a new Hahn–Banach assumption.",
      "reader": "https://kokunoyumeto.github.io/open-math-courses/courses/foundations-of-von-neumann-algebras/hahn-banach-baire-and-the-basic-theorems-on-banach-spaces.html#oa-fnd-hb-01",
      "license": "CC0-1.0",
      "raw_source_sha256": "f09b763b00c87141faac24be5c8d4a4760c3439085288bcdb56263731281e850",
      "range_sha256_LF_with_final_newline": "2e5ef455fd837f8dcc2a87e1868ecd8a21c081927f9bae52a7ad313a424e1d70",
      "verified_at": "2026-10-04T03:27:24.447458+00:00"
    }
  ],
  "tomita_proof_ranges": [
    {
      "id": "OA-FND-HS-01",
      "source": "courses/foundations-of-von-neumann-algebras/src/hilbert-spaces-and-compact-operators.md",
      "public_source_url": "https://kokunoyumeto.github.io/open-math-courses/courses/foundations-of-von-neumann-algebras/src/hilbert-spaces-and-compact-operators.md",
      "source_sha256": "c37f79bc01c64ef1198571d05fb5aeb18a80c704afc642ab50f904fd88f62fd5",
      "source_bytes": 36935,
      "lines_1_based": [
        22,
        83
      ],
      "range_sha256_LF_with_final_newline": "5a58f40a883b44b19a32dce42348ae24781f339aa58794de40ac8fe2c57d7863",
      "reader_url": "https://kokunoyumeto.github.io/open-math-courses/courses/foundations-of-von-neumann-algebras/hilbert-spaces-and-compact-operators.html#oa-fnd-hs-01",
      "full_reader_sha256": "96e743d357c26ed76a7cdb66d2b023a99e6005049dac4b735b1ce93000d5d13e",
      "full_reader_bytes": 82325,
      "contract": "Linear-first positivity/polarization/Cauchy–Schwarz, Hilbert completion and extension of bounded maps",
      "consumers": [
        "TC03 conjugate Hilbert construction",
        "TC05 graph norm",
        "TC08 isometry extension"
      ],
      "role": "direct core input",
      "source_snapshot_commit": "80c6b35cd62dcf39e803dade1f8ae4c11af12983",
      "license": "CC0-1.0"
    },
    {
      "id": "OA-FND-HS-02",
      "source": "courses/foundations-of-von-neumann-algebras/src/hilbert-spaces-and-compact-operators.md",
      "public_source_url": "https://kokunoyumeto.github.io/open-math-courses/courses/foundations-of-von-neumann-algebras/src/hilbert-spaces-and-compact-operators.md",
      "source_sha256": "c37f79bc01c64ef1198571d05fb5aeb18a80c704afc642ab50f904fd88f62fd5",
      "source_bytes": 36935,
      "lines_1_based": [
        84,
        115
      ],
      "range_sha256_LF_with_final_newline": "4da0acf479b6c4c00703ebcff6c184888f9f95c986f66b511007a7f1ec72bc69",
      "reader_url": "https://kokunoyumeto.github.io/open-math-courses/courses/foundations-of-von-neumann-algebras/hilbert-spaces-and-compact-operators.html#oa-fnd-hs-02",
      "full_reader_sha256": "96e743d357c26ed76a7cdb66d2b023a99e6005049dac4b735b1ce93000d5d13e",
      "full_reader_bytes": 82325,
      "contract": "Projection/double orthogonal complement and linear-first Riesz Theorems2.1–2.3, including zero functional/zero Hilbert",
      "consumers": [
        "TC02 orbit projection",
        "TC03 adjoint scalar test and graph double complement"
      ],
      "role": "direct core input",
      "source_snapshot_commit": "80c6b35cd62dcf39e803dade1f8ae4c11af12983",
      "license": "CC0-1.0"
    },
    {
      "id": "OA-FND-HS-03",
      "source": "courses/foundations-of-von-neumann-algebras/src/hilbert-spaces-and-compact-operators.md",
      "public_source_url": "https://kokunoyumeto.github.io/open-math-courses/courses/foundations-of-von-neumann-algebras/src/hilbert-spaces-and-compact-operators.md",
      "source_sha256": "c37f79bc01c64ef1198571d05fb5aeb18a80c704afc642ab50f904fd88f62fd5",
      "source_bytes": 36935,
      "lines_1_based": [
        116,
        142
      ],
      "range_sha256_LF_with_final_newline": "737c3933e865d00463cf81242f65ba119fc1419e6ebc4e51f41d6bb043e792a7",
      "reader_url": "https://kokunoyumeto.github.io/open-math-courses/courses/foundations-of-von-neumann-algebras/hilbert-spaces-and-compact-operators.html#oa-fnd-hs-03",
      "full_reader_sha256": "96e743d357c26ed76a7cdb66d2b023a99e6005049dac4b735b1ce93000d5d13e",
      "full_reader_bytes": 82325,
      "contract": "Bounded adjoints and scalar pairings; TC03 proves the unbounded conjugate-linear adjoint facts itself",
      "consumers": [
        "TC02 bounded adjoint invariance",
        "TC04 commutant pairing"
      ],
      "role": "direct core input",
      "source_snapshot_commit": "80c6b35cd62dcf39e803dade1f8ae4c11af12983",
      "license": "CC0-1.0"
    },
    {
      "id": "OA-FND-HS-04-SELECTED",
      "source": "courses/foundations-of-von-neumann-algebras/src/hilbert-spaces-and-compact-operators.md",
      "public_source_url": "https://kokunoyumeto.github.io/open-math-courses/courses/foundations-of-von-neumann-algebras/src/hilbert-spaces-and-compact-operators.md",
      "source_sha256": "c37f79bc01c64ef1198571d05fb5aeb18a80c704afc642ab50f904fd88f62fd5",
      "source_bytes": 36935,
      "lines_1_based": [
        143,
        170
      ],
      "range_sha256_LF_with_final_newline": "58610386d1fd0c406b40a99abed515e66d074cec06f949ed602871afbb6a8e30",
      "reader_url": "https://kokunoyumeto.github.io/open-math-courses/courses/foundations-of-von-neumann-algebras/hilbert-spaces-and-compact-operators.html#oa-fnd-hs-04",
      "full_reader_sha256": "96e743d357c26ed76a7cdb66d2b023a99e6005049dac4b735b1ce93000d5d13e",
      "full_reader_bytes": 82325,
      "contract": "Bessel, Parseval and finite-support expansion for arbitrary orthonormal index sets",
      "consumers": [
        "TC12 problem2 arbitrary-index finite-support model",
        "TC11 chosen countable matrix model"
      ],
      "role": "example-specific selected proof",
      "source_snapshot_commit": "80c6b35cd62dcf39e803dade1f8ae4c11af12983",
      "license": "CC0-1.0"
    },
    {
      "id": "OA-FND-HS-08",
      "source": "courses/foundations-of-von-neumann-algebras/src/hilbert-spaces-and-compact-operators.md",
      "public_source_url": "https://kokunoyumeto.github.io/open-math-courses/courses/foundations-of-von-neumann-algebras/src/hilbert-spaces-and-compact-operators.md",
      "source_sha256": "c37f79bc01c64ef1198571d05fb5aeb18a80c704afc642ab50f904fd88f62fd5",
      "source_bytes": 36935,
      "lines_1_based": [
        292,
        331
      ],
      "range_sha256_LF_with_final_newline": "88d08af1fbb52ce363d8d3de6fc9dd84710038dd751c40ecfd634760a8d716e3",
      "reader_url": "https://kokunoyumeto.github.io/open-math-courses/courses/foundations-of-von-neumann-algebras/hilbert-spaces-and-compact-operators.html#oa-fnd-hs-08",
      "full_reader_sha256": "96e743d357c26ed76a7cdb66d2b023a99e6005049dac4b735b1ce93000d5d13e",
      "full_reader_bytes": 82325,
      "contract": "Complete ell2(S) coordinate Hilbert spaces and tensor products; square-summable matrices give the Hilbert–Schmidt model",
      "consumers": [
        "TC11 infinite matrix realization",
        "TC12 problem2"
      ],
      "role": "example-specific complete input",
      "source_snapshot_commit": "80c6b35cd62dcf39e803dade1f8ae4c11af12983",
      "license": "CC0-1.0"
    },
    {
      "id": "OA-MOD-BK-02",
      "source": "courses/OA-MOD/src/bounded-operator-kernel.md",
      "public_source_url": "https://kokunoyumeto.github.io/open-math-courses/courses/OA-MOD/src/bounded-operator-kernel.md",
      "source_sha256": "04e0495ab7ec51fdc6b6f07a5dc835dfa623680103fc9f5bfc0cf30c9204fd3b",
      "source_bytes": 27670,
      "lines_1_based": [
        35,
        117
      ],
      "range_sha256_LF_with_final_newline": "241587d8fd8f6eaf7aa2acd3f4abe2e8edc33d881b4e28ac4c49bbda2af34a73",
      "reader_url": "https://kokunoyumeto.github.io/open-math-courses/courses/OA-MOD/bounded-operator-kernel.html#OA-MOD-BK-02",
      "full_reader_sha256": "64405e28af5f51be23ad3785e19b5454156904827d270295c4b813e24370211e",
      "full_reader_bytes": 54150,
      "contract": "Bicommutant theorem by finite-vector approximation on arbitrary Hilbert spaces, including H=0",
      "consumers": [
        "TC02 second cyclic/separating equivalence",
        "TC12 problem5 VN criterion"
      ],
      "role": "direct core input",
      "source_snapshot_commit": "80c6b35cd62dcf39e803dade1f8ae4c11af12983",
      "license": "CC0-1.0",
      "native_provider": {
        "path": "public/src/bounded-operator-kernel.md",
        "source_sha256": "69f02c394c533404ae4d337847212d5d3d3e7494e2914be149dafecef98b8052",
        "source_bytes": 26407,
        "id": "OA-MOD-BK-02",
        "byte_start": 3507,
        "byte_end_exclusive": 7747,
        "component_sha256": "01dcc8d858b495a37a63426a2e823cdad28a0277fdb5be7fd112ffbe42f4bac8",
        "component_bytes": 4240,
        "start_line_1_based": 37,
        "end_line_1_based": 120,
        "LF_range_sha256": "f73f6fbcfa8ee17c8c296f20fb645c5995c212a8b4875dea5ec039f990c75a2a"
      },
      "native_public_complete_proof_body_literal": false,
      "native_public_BK02_proof_equal_after_CRLF_and_manual_heading_anchor_projection": true
    },
    {
      "id": "OA-MOD-QF-03",
      "source": "courses/OA-MOD/src/closed-positive-forms.md",
      "public_source_url": "https://kokunoyumeto.github.io/open-math-courses/courses/OA-MOD/src/closed-positive-forms.md",
      "source_sha256": "6a1387411dd13152645cc16d8ed3d6f73988af10b2f9b366b1c14217911202d2",
      "source_bytes": 36768,
      "lines_1_based": [
        142,
        231
      ],
      "range_sha256_LF_with_final_newline": "d9d7c0ec77bb3139ad5ccdb81d18542776545efffd8691d4b00555aea12a726f",
      "reader_url": "https://kokunoyumeto.github.io/open-math-courses/courses/OA-MOD/closed-positive-forms.html#OA-MOD-QF-03",
      "full_reader_sha256": "0d0a7383df5bce3f475312afc03d29b45f881f8ccca4c3a3690e10f018c2aa60",
      "full_reader_bytes": 69978,
      "contract": "Closed nonnegative form representation, intrinsic operator-domain criterion and uniqueness, exact square-root domain and converse",
      "consumers": [
        "TC07 Delta=S*S exact composition domain",
        "TC08 polar uniqueness"
      ],
      "role": "direct core input",
      "source_snapshot_commit": "80c6b35cd62dcf39e803dade1f8ae4c11af12983",
      "license": "CC0-1.0",
      "native_provider": {
        "path": "public/src/closed-positive-forms.md",
        "source_sha256": "eb8483eb3d978280f52d937398d226c8a45e56ab21dd12216b26351e6e7d0589",
        "source_bytes": 37757,
        "id": "OA-MOD-QF-03",
        "byte_start": 11114,
        "byte_end_exclusive": 15125,
        "component_sha256": "031adc49831538583b3f36e9b55defba73f821f1587db344c063952cd1fb0fcc",
        "component_bytes": 4011,
        "start_line_1_based": 142,
        "end_line_1_based": 231,
        "LF_range_sha256": "031adc49831538583b3f36e9b55defba73f821f1587db344c063952cd1fb0fcc"
      },
      "native_public_complete_proof_body_literal": false,
      "native_public_QF03_proof_body_equal_except_relative_BK07_link": true,
      "QF_existing_current_binding": [
        {
          "path": "audit/closed-form-foundation-inputs-159.json",
          "sha256": "3991af2b81b4c5dcd440343f54c8cfcde649561178b876847a78e2a377b468f6",
          "bytes": 70508
        },
        {
          "path": "audit/closed-form-foundation-review-159.json",
          "sha256": "28fdd2ceef9a379005d5054c98c2268ebb86026748326a50b7dc8c1cc1e4bdd8",
          "bytes": 1052
        },
        {
          "path": "audit/closed-form-current-binding-159.json",
          "sha256": "1247a79a7a6c8185d1a438e7662ae453d21390082b97696f03503640b294fe98",
          "bytes": 4513
        }
      ]
    },
    {
      "id": "OA-MOD-SK-04",
      "source": "courses/OA-MOD/src/spectral-calculus-kernel.md",
      "public_source_url": "https://kokunoyumeto.github.io/open-math-courses/courses/OA-MOD/src/spectral-calculus-kernel.md",
      "source_sha256": "7a6a51a63cc5a4493d6ff428f2ca9731a57d26219c03207b522d328a69b092e0",
      "source_bytes": 38720,
      "lines_1_based": [
        70,
        110
      ],
      "range_sha256_LF_with_final_newline": "c59feeb2a0e3aed5c609f7f0224de9f43c1c14a1dc59148a77ccc74ed2f344d2",
      "reader_url": "https://kokunoyumeto.github.io/open-math-courses/courses/OA-MOD/spectral-calculus-kernel.html#OA-MOD-SK-04",
      "full_reader_sha256": "92cec1106cb92cf28f00fe21f675bf8b33c5c74401c9923f3f269a7ae1fba93f",
      "full_reader_bytes": 70711,
      "contract": "Projection-valued bounded Borel calculus, finite scalar measures and uniqueness",
      "consumers": [
        "TC07–10"
      ],
      "role": "direct core input",
      "source_snapshot_commit": "80c6b35cd62dcf39e803dade1f8ae4c11af12983",
      "license": "CC0-1.0",
      "native_provider": {
        "path": "public/src/spectral-calculus-kernel.md",
        "source_sha256": "b0e3221bb390aa0b11c6a7d7f5aa6a49e816cde8bac8dfcc47b5e757cc428d98",
        "source_bytes": 37209,
        "id": "OA-MOD-SK-04",
        "byte_start": 7047,
        "byte_end_exclusive": 11090,
        "component_sha256": "9492604adfa77d1d7e90f4cb2efc5407f1aee43fd36b0de87427a438a9826c66",
        "component_bytes": 4043,
        "start_line_1_based": 68,
        "end_line_1_based": 109,
        "LF_range_sha256": "9492604adfa77d1d7e90f4cb2efc5407f1aee43fd36b0de87427a438a9826c66"
      },
      "native_public_complete_proof_body_literal": true
    },
    {
      "id": "OA-MOD-SK-05",
      "source": "courses/OA-MOD/src/spectral-calculus-kernel.md",
      "public_source_url": "https://kokunoyumeto.github.io/open-math-courses/courses/OA-MOD/src/spectral-calculus-kernel.md",
      "source_sha256": "7a6a51a63cc5a4493d6ff428f2ca9731a57d26219c03207b522d328a69b092e0",
      "source_bytes": 38720,
      "lines_1_based": [
        112,
        156
      ],
      "range_sha256_LF_with_final_newline": "7ba262113155431a5aebad74ef86149096e58903c09e1d6ab60fb7bd1606b8b9",
      "reader_url": "https://kokunoyumeto.github.io/open-math-courses/courses/OA-MOD/spectral-calculus-kernel.html#OA-MOD-SK-05",
      "full_reader_sha256": "92cec1106cb92cf28f00fe21f675bf8b33c5c74401c9923f3f269a7ae1fba93f",
      "full_reader_bytes": 70711,
      "contract": "Unbounded squared-integral domains, adjoints, kernels, actual product domains and graph cutoffs",
      "consumers": [
        "TC07–10"
      ],
      "role": "direct core input",
      "source_snapshot_commit": "80c6b35cd62dcf39e803dade1f8ae4c11af12983",
      "license": "CC0-1.0",
      "native_provider": {
        "path": "public/src/spectral-calculus-kernel.md",
        "source_sha256": "b0e3221bb390aa0b11c6a7d7f5aa6a49e816cde8bac8dfcc47b5e757cc428d98",
        "source_bytes": 37209,
        "id": "OA-MOD-SK-05",
        "byte_start": 11090,
        "byte_end_exclusive": 14458,
        "component_sha256": "a76eecd27753d2e550ad86e33be3d6e4449b4005367d5fd8030ed9fabeb0e1d7",
        "component_bytes": 3368,
        "start_line_1_based": 110,
        "end_line_1_based": 155,
        "LF_range_sha256": "a76eecd27753d2e550ad86e33be3d6e4449b4005367d5fd8030ed9fabeb0e1d7"
      },
      "native_public_complete_proof_body_literal": true
    },
    {
      "id": "OA-MOD-SK-06",
      "source": "courses/OA-MOD/src/spectral-calculus-kernel.md",
      "public_source_url": "https://kokunoyumeto.github.io/open-math-courses/courses/OA-MOD/src/spectral-calculus-kernel.md",
      "source_sha256": "7a6a51a63cc5a4493d6ff428f2ca9731a57d26219c03207b522d328a69b092e0",
      "source_bytes": 38720,
      "lines_1_based": [
        158,
        207
      ],
      "range_sha256_LF_with_final_newline": "f6e3282e4d4d32a36cdc9ae898f9a99a120597052731c619d48ea1f8c37b54e9",
      "reader_url": "https://kokunoyumeto.github.io/open-math-courses/courses/OA-MOD/spectral-calculus-kernel.html#OA-MOD-SK-06",
      "full_reader_sha256": "92cec1106cb92cf28f00fe21f675bf8b33c5c74401c9923f3f269a7ae1fba93f",
      "full_reader_bytes": 70711,
      "contract": "Every self-adjoint operator has a unique spectral resolution, including zero Hilbert and positivity support",
      "consumers": [
        "TC07–10"
      ],
      "role": "direct core input",
      "source_snapshot_commit": "80c6b35cd62dcf39e803dade1f8ae4c11af12983",
      "license": "CC0-1.0",
      "native_provider": {
        "path": "public/src/spectral-calculus-kernel.md",
        "source_sha256": "b0e3221bb390aa0b11c6a7d7f5aa6a49e816cde8bac8dfcc47b5e757cc428d98",
        "source_bytes": 37209,
        "id": "OA-MOD-SK-06",
        "byte_start": 14458,
        "byte_end_exclusive": 19208,
        "component_sha256": "4794d105c2574652d9562143b4b7ff5ae9e1e3b369730b5f0afcb0de193bc2c3",
        "component_bytes": 4750,
        "start_line_1_based": 156,
        "end_line_1_based": 206,
        "LF_range_sha256": "4794d105c2574652d9562143b4b7ff5ae9e1e3b369730b5f0afcb0de193bc2c3"
      },
      "native_public_complete_proof_body_literal": true
    },
    {
      "id": "OA-MOD-SK-07",
      "source": "courses/OA-MOD/src/spectral-calculus-kernel.md",
      "public_source_url": "https://kokunoyumeto.github.io/open-math-courses/courses/OA-MOD/src/spectral-calculus-kernel.md",
      "source_sha256": "7a6a51a63cc5a4493d6ff428f2ca9731a57d26219c03207b522d328a69b092e0",
      "source_bytes": 38720,
      "lines_1_based": [
        209,
        261
      ],
      "range_sha256_LF_with_final_newline": "aebaf6ea4ab8152fed6e14139c54c69c847d8ec98861a26dda7112adbbc35a7e",
      "reader_url": "https://kokunoyumeto.github.io/open-math-courses/courses/OA-MOD/spectral-calculus-kernel.html#OA-MOD-SK-07",
      "full_reader_sha256": "92cec1106cb92cf28f00fe21f675bf8b33c5c74401c9923f3f269a7ae1fba93f",
      "full_reader_bytes": 70711,
      "contract": "Positive square roots and uniqueness; inverse on actual range; powers, form pairing and bounded imaginary-power group",
      "consumers": [
        "TC07–10"
      ],
      "role": "direct core input",
      "source_snapshot_commit": "80c6b35cd62dcf39e803dade1f8ae4c11af12983",
      "license": "CC0-1.0",
      "native_provider": {
        "path": "public/src/spectral-calculus-kernel.md",
        "source_sha256": "b0e3221bb390aa0b11c6a7d7f5aa6a49e816cde8bac8dfcc47b5e757cc428d98",
        "source_bytes": 37209,
        "id": "OA-MOD-SK-07",
        "byte_start": 19208,
        "byte_end_exclusive": 22599,
        "component_sha256": "b1da647aa9e91839ea3b2859a85b231d45a159f4fdd51e146ddd4708f7d97a5b",
        "component_bytes": 3391,
        "start_line_1_based": 207,
        "end_line_1_based": 260,
        "LF_range_sha256": "b1da647aa9e91839ea3b2859a85b231d45a159f4fdd51e146ddd4708f7d97a5b"
      },
      "native_public_complete_proof_body_literal": true
    },
    {
      "id": "OA-MOD-SK-08",
      "source": "courses/OA-MOD/src/spectral-calculus-kernel.md",
      "public_source_url": "https://kokunoyumeto.github.io/open-math-courses/courses/OA-MOD/src/spectral-calculus-kernel.md",
      "source_sha256": "7a6a51a63cc5a4493d6ff428f2ca9731a57d26219c03207b522d328a69b092e0",
      "source_bytes": 38720,
      "lines_1_based": [
        263,
        288
      ],
      "range_sha256_LF_with_final_newline": "1c0f3c225b2013caad735c74bd0c9c3b4faf18e02d034c2d92b7bdca3731584e",
      "reader_url": "https://kokunoyumeto.github.io/open-math-courses/courses/OA-MOD/spectral-calculus-kernel.html#OA-MOD-SK-08",
      "full_reader_sha256": "92cec1106cb92cf28f00fe21f675bf8b33c5c74401c9923f3f269a7ae1fba93f",
      "full_reader_bytes": 70711,
      "contract": "Unitary/antiunitary spectral transport, conjugated scalar coefficients and transported domains",
      "consumers": [
        "TC07–10"
      ],
      "role": "direct core input",
      "source_snapshot_commit": "80c6b35cd62dcf39e803dade1f8ae4c11af12983",
      "license": "CC0-1.0",
      "native_provider": {
        "path": "public/src/spectral-calculus-kernel.md",
        "source_sha256": "b0e3221bb390aa0b11c6a7d7f5aa6a49e816cde8bac8dfcc47b5e757cc428d98",
        "source_bytes": 37209,
        "id": "OA-MOD-SK-08",
        "byte_start": 22599,
        "byte_end_exclusive": 25841,
        "component_sha256": "1b79bbcf387d631ab1f828d8f406fa92b00c50cc1609aa77162a2c6523576ea5",
        "component_bytes": 3242,
        "start_line_1_based": 261,
        "end_line_1_based": 287,
        "LF_range_sha256": "1b79bbcf387d631ab1f828d8f406fa92b00c50cc1609aa77162a2c6523576ea5"
      },
      "native_public_complete_proof_body_literal": true
    },
    {
      "id": "MEASURE-SEQUENTIAL-MCT-DCT",
      "source": "courses/harmonic-analysis-on-locally-compact-groups/src/measure-and-hilbert-space-tools.md",
      "public_source_url": "https://kokunoyumeto.github.io/open-math-courses/courses/harmonic-analysis-on-locally-compact-groups/src/measure-and-hilbert-space-tools.md",
      "source_sha256": "ee22f2f44dd2c089b2bfb10d25fd5b36ebd9d7c245ea3206a1abb71e393e2e12",
      "source_bytes": 29235,
      "lines_1_based": [
        36,
        69
      ],
      "range_sha256_LF_with_final_newline": "c7f587e9e93f437180c96b6cf12e9f0cd87cbe998db615ed029d251fd7a5343c",
      "reader_url": "https://kokunoyumeto.github.io/open-math-courses/courses/harmonic-analysis-on-locally-compact-groups/reader/measure-and-hilbert-space-tools.html#2-integration-and-convergence-without-countability-assumptions",
      "full_reader_sha256": "289ab310ddb0239ea5df935bc2b8e110f619e025f4e933e940c52dff786e2929",
      "full_reader_bytes": 52033,
      "contract": "Scalar sequential MCT/DCT without ambient sigma-finiteness; imaginary-power continuity uses finite vector measure and bound4",
      "consumers": [
        "TC10 strong continuity",
        "TC12 problem5 truncation"
      ],
      "role": "direct core input",
      "source_snapshot_commit": "80c6b35cd62dcf39e803dade1f8ae4c11af12983",
      "license": "CC0-1.0"
    },
    {
      "id": "MEASURE-L2-COMPLETENESS",
      "source": "courses/harmonic-analysis-on-locally-compact-groups/src/measure-and-hilbert-space-tools.md",
      "public_source_url": "https://kokunoyumeto.github.io/open-math-courses/courses/harmonic-analysis-on-locally-compact-groups/src/measure-and-hilbert-space-tools.md",
      "source_sha256": "ee22f2f44dd2c089b2bfb10d25fd5b36ebd9d7c245ea3206a1abb71e393e2e12",
      "source_bytes": 29235,
      "lines_1_based": [
        75,
        105
      ],
      "range_sha256_LF_with_final_newline": "c4d767e54610ccea1f9410469a3827889f9944ba321151d4c54631db5eef34c6",
      "reader_url": "https://kokunoyumeto.github.io/open-math-courses/courses/harmonic-analysis-on-locally-compact-groups/reader/measure-and-hilbert-space-tools.html#3-the-complete-spaces-of-integrable-functions",
      "full_reader_sha256": "289ab310ddb0239ea5df935bc2b8e110f619e025f4e933e940c52dff786e2929",
      "full_reader_bytes": 52033,
      "contract": "Hölder/Minkowski, Lp completeness/simple density and linear-first L2 pairing on arbitrary measure spaces",
      "consumers": [
        "TC12 probability-space solution"
      ],
      "role": "example-specific complete input",
      "source_snapshot_commit": "80c6b35cd62dcf39e803dade1f8ae4c11af12983",
      "license": "CC0-1.0"
    }
  ]
}
