{
  "schema": "programme-source-record/v1",
  "title": "Realizing flows with skew odometers",
  "source": "src/realizing-flows-with-skew-odometers.md",
  "reader": "realizing-flows-with-skew-odometers.html",
  "source_sha256": "037f3c2f20d07da68a3d668446cd1b42658f5f108b4519ecd3bb8f931d897638",
  "authors": [
    {
      "name": "GPT-6.1 Sol (OpenAI)",
      "contribution": "Mathematical exposition, figure and seven solved exercises."
    },
    {
      "name": "GPT-6 Astra (OpenAI)",
      "contribution": "Source comparison and exposition of the coordinate-design and matrix-model arguments."
    }
  ],
  "license": "CC0-1.0 for original programme expression; cited human works retain their rights.",
  "scope": "Five-digit realization of a prescribed properly ergodic real flow, its invariant coordinate, compatible matrix approximation and abelian crossed-product model; seven solved exercises.",
  "human_sources": [
    {
      "author": "Masamichi Takesaki",
      "title": "Theory of Operator Algebras III",
      "year": 2003,
      "url": "https://doi.org/10.1007/978-3-662-10453-8",
      "scope": "XVIII.2, Lemmas2.3–2.4 and Theorem2.5, pp.317–321.",
      "role": "Classical five-digit, finite-permutation and two-dense-group realization construction."
    },
    {
      "authors": [
        "Edwin Hewitt",
        "Leonard J. Savage"
      ],
      "title": "Symmetric measures on Cartesian products",
      "year": 1955,
      "role": "Historical attribution for the independent-digit zero-one law; Lemma4.1 supplies the proof used here."
    },
    {
      "author": "Wolfgang Krieger",
      "title": "On ergodic flows and the isomorphism of factors",
      "year": 1976,
      "url": "https://doi.org/10.1007/BF01360278",
      "role": "Associated-flow realization and classification."
    },
    {
      "authors": [
        "Alain Connes",
        "Masamichi Takesaki"
      ],
      "title": "The flow of weights on factors of type III",
      "year": 1977,
      "url": "https://doi.org/10.2748/tmj/1178240493",
      "correction_url": "https://doi.org/10.2748/tmj/1178229923",
      "role": "Flow-of-weights context; the actual programme core proofs are separately identified."
    }
  ],
  "programme_sources": [
    {
      "source": "src/comparing-trace-scaling-flows-over-their-centres.md",
      "source_sha256": "ccde0b545b719064909d352ebbdc723e5767bcf77665f90d368fe9aee2603e5f"
    }
  ],
  "external_programme_sources": [
    {
      "source": "https://github.com/KokunoYumeto/open-math-courses/blob/eb9f99919b1d0508b690a71f6958f87884d3e9a4/docs/courses/OA-ERGODIC/src/measurable-actions-and-compact-models.md",
      "source_sha256": "fa839f846200dae73aea75ed3e9191c8aed860ce5e21c0bf33e65eb2a5c9d769",
      "scope": "Theorem0.1, Lemma1.1, Lemma2.1 and Theorem2.2: densities, joint versions and continuity."
    },
    {
      "source": "https://github.com/KokunoYumeto/open-math-courses/blob/eb9f99919b1d0508b690a71f6958f87884d3e9a4/docs/courses/OA-ERGODIC/src/towers-and-odometer-orbits.md",
      "source_sha256": "16855baf037d0e9a2585dc7a958426bb647dafb3ebe97bdedb0c05bb2abb27f6",
      "scope": "Sections1–3: nonsingular towers, cyclic boundary estimates and the tail-intersection finite exhaustion."
    },
    {
      "source": "https://github.com/KokunoYumeto/open-math-courses/blob/eb9f99919b1d0508b690a71f6958f87884d3e9a4/docs/courses/OA-ERGODIC/src/matching-sets-and-nonsingular-dyadic-arrays.md",
      "source_sha256": "8de836438694382c5f3c7044ff5c55ddef642aa39e6bf345fc67631def4cc8e7",
      "scope": "Sections1–5 through Corollary5.3: matching, general balancing, exact refinement and strongly dense matrix inclusions; earlier finite-measure balancing remains a transitive input."
    },
    {
      "source": "https://github.com/KokunoYumeto/open-math-courses/blob/eb9f99919b1d0508b690a71f6958f87884d3e9a4/docs/courses/OA-ERGODIC/src/finite-orbit-classes-and-matrix-blocks.md",
      "source_sha256": "a60918bc033ad5f7b7c32c4c237146d080b0e3e7fb41facd63729de777eb580a",
      "scope": "Sections1–5: finite selectors, measurable matrix fields and the distinction between stagewise and compatible approximation."
    },
    {
      "source": "https://github.com/KokunoYumeto/open-math-courses/blob/eb9f99919b1d0508b690a71f6958f87884d3e9a4/docs/courses/OA-ERGODIC/src/orbits-stabilizers-and-relation-algebras.md",
      "source_sha256": "7da7bc7a6ab9672e824032e160a25fc85fac89558c71d289d326bfd5c334b140",
      "scope": "Sections3–4: partial orbit operators, normal diagonal, maximal abelianness and the center."
    },
    {
      "source": "https://github.com/KokunoYumeto/open-math-courses/blob/eb9f99919b1d0508b690a71f6958f87884d3e9a4/docs/courses/OA-FLOW/src/OA-FLOW-LFT.md",
      "source_sha256": "4aaf34d4505c803c5e178c79b3e94ce59218fa66349d7d3a88fce5cad79836e2",
      "scope": "Setting and Sections1–5: full lift, trace, fixed algebra, onto Fourier core identification and normalization."
    },
    {
      "source": "https://github.com/KokunoYumeto/open-math-courses/blob/eb9f99919b1d0508b690a71f6958f87884d3e9a4/docs/courses/OA-FLOW/src/OA-FLOW-ZDC.md",
      "source_sha256": "7136d488d023a539642e99c2ec177f6d42a103f1affdd50212b5ffc2e0b1eff2",
      "scope": "Section2: properly ergodic center criterion; the cited modular invariant, duality and type foundations remain transitive inputs."
    }
  ],
  "figures": [
    {
      "path": "figures/skew-odometer-flow-coordinate.svg",
      "sha256": "ce86e6bda74e1231691793dce9f5bc91ad349b9dfaabbef0fac2f485c9963222"
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      "sha256": "c6ad6607548943d998199488e4902d45b481496f30120ee2610b5721b6941cb5"
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