{
  "schema": "programme-source-record/v1",
  "title": "From global characteristic invariants to return data",
  "source": "src/from-global-characteristic-invariants-to-return-data.md",
  "reader": "from-global-characteristic-invariants-to-return-data.html",
  "source_sha256": "708754115e5968e3755a54daa73154b4816415499c6f7b795a4f8ab1af35d708",
  "authors": [
    {
      "name": "GPT-6.1 Sol (OpenAI)",
      "contribution": "Mathematical exposition, figure and eight solved exercises."
    },
    {
      "name": "GPT-6 Astra (OpenAI)",
      "contribution": "Source comparison, classical construction attribution and exact prerequisite exposition."
    }
  ],
  "license": "CC0-1.0 for original programme expression; human works retain their rights.",
  "scope": "The global characteristic criterion for integer–real actions, including properly ergodic, free transitive and periodic transitive centres, under the stated factor and measured-relation foundations.",
  "human_sources": [
    {
      "authors": "Colin E. Sutherland and Masamichi Takesaki",
      "title": "Actions of discrete amenable groups on injective factors of type III-lambda, lambda not one",
      "year": 1989,
      "url": "https://msp.org/pjm/1989/137-2/pjm-v137-n2-p08-s.pdf",
      "scope": "Section1.3 and Assumption4.1/Theorem4.2,printed411–415and427–431.",
      "role": "The global criterion and the reduction through a countable section are due to Sutherland–Takesaki [ST], Assumption 4.1 and Theorem 4.2 (pp.427–431). Their proof identifies the stabilizers and inner kernel, compares the return characteristic and trace classes, and reconstructs the global comparison. The present cyclic specialization proves the three centre-flow alternatives, repairs the trace and the implementing pair on invariant conull domains, and supplies a transverse invariant mean. The common scalar quotient in Section 4 retains the relative phase of the two implementers. These details explain the classical classification mechanism."
    },
    {
      "authors": "Colin E. Sutherland and Masamichi Takesaki",
      "title": "Actions of Discrete Amenable Groups and Groupoids on von Neumann Algebras",
      "year": 1985,
      "url": "https://ems.press/journals/prims/articles/3261",
      "scope": "Theorem1.2 and Section4; exact preceding programme comparison retained."
    }
  ],
  "external_programme_sources": [
    {
      "source": "https://github.com/KokunoYumeto/open-math-courses/blob/0ce79518696ecb9eb6694fd92be9338da034e464/docs/courses/OA-ERGODIC/src/localizing-factor-actions-and-uniform-cocycles.md",
      "source_sha256": "a2d5b1f419b9cef10b3180ccaa9ed13deaaf794c576e5775a01249f4ee903fec",
      "scope": "Theorem1.2 (Haar essential-value repair), Theorem3.1 (strict versions), Lemma4.2 (automatic continuity), Lemma5.1 and Theorem5.2 (constant-field localization); path-space foundations retained."
    },
    {
      "source": "https://github.com/KokunoYumeto/open-math-courses/blob/0ce79518696ecb9eb6694fd92be9338da034e464/docs/courses/OA-ERGODIC/src/strict-variable-fields-and-ancillary-conjugacy.md",
      "source_sha256": "38cc254cc5156190b106daee52818701372e60a12425748b83b83ab3bd6f07e8",
      "scope": "Section2 and Theorem3.1: strict standard-form localization of varying separable factors; central decomposition and canonical standard form retained."
    },
    {
      "source": "https://github.com/KokunoYumeto/open-math-courses/blob/0ce79518696ecb9eb6694fd92be9338da034e464/docs/courses/OA-ERGODIC/src/variable-factor-fields-and-measurable-conjugacy.md",
      "source_sha256": "f0e55022243d4049a161e5531d19fbd18619fdf051a40764d9a69e8ff3128982",
      "scope": "Theorem1.1, Theorem3.1 and Lemma3.2: two-sided spatial integration and Borel/continuous cocycle reconstruction."
    },
    {
      "source": "https://github.com/KokunoYumeto/open-math-courses/blob/0ce79518696ecb9eb6694fd92be9338da034e464/docs/courses/OA-ERGODIC/src/measurable-actions-and-compact-models.md",
      "source_sha256": "fa839f846200dae73aea75ed3e9191c8aed860ce5e21c0bf33e65eb2a5c9d769",
      "scope": "Lemma3.1 and Theorem4.1: countable smoothing, compact metrizable point model, normal identification and nonsingular action."
    },
    {
      "source": "https://github.com/KokunoYumeto/open-math-courses/blob/0ce79518696ecb9eb6694fd92be9338da034e464/docs/courses/OA-ERGODIC/src/invariant-means-on-measured-relations.md",
      "source_sha256": "18ecee4c642980674c895cdd35093e3c995c533038da614d07f11889c62a5a7c",
      "scope": "Theorem6.1 and Corollary6.2: measured invariant mean to finite exhaustion."
    },
    {
      "source": "https://github.com/KokunoYumeto/open-math-courses/blob/0ce79518696ecb9eb6694fd92be9338da034e464/docs/courses/injective-factors/src/classification-of-injective-factors.md",
      "source_sha256": "fc38d5a7fee2e52cdaadfe2068a8f140be0dd38ea474a29251337ef8c350d76f",
      "scope": "Lemma6.1 and Theorem6.2: measurable finite-factor isomorphisms and a full finite corner; the stated decomposition, selection and projection foundations remain."
    },
    {
      "source": "https://github.com/KokunoYumeto/open-math-courses/blob/0ce79518696ecb9eb6694fd92be9338da034e464/docs/courses/OA-FLOW/src/OA-FLOW-L38.md",
      "source_sha256": "350b25105a0094f16c253ae39fcf5934ec29d2ad951303e1d5525d035ffd8099",
      "scope": "The complete suspension theorem, through (S32): separated crossings, two-sided returns, partial-translation product class, nonsingular and aperiodic return map."
    }
  ],
  "figures": [
    {
      "path": "figures/global-return-gauge.svg",
      "sha256": "1404b2313d83a3c24b696f266b2393a52b9ad9e3ae36a1038bcc8f847e30e95b"
    },
    {
      "path": "figures/global-return-gauge.png",
      "sha256": "2386388f5a8470a08d0fb8740162f1303dedb763579b7f630c42a3b55ea961d7"
    },
    {
      "path": "figures/draw_global_return_gauge.py",
      "sha256": "cff50a2e3c4d3d35c6a103f36062b68058684afdc01d7c99b1a6805c5d663ab9"
    }
  ],
  "programme_sources": [
    {
      "source": "src/characteristic-classification-of-semifinite-actions.md",
      "source_sha256": "7a2af11e0a9c8b1997d63bcc8c5254a3eb10a6c3807c737c94427163b6d53dfa"
    },
    {
      "source": "src/reconstructing-continuous-comparisons-from-return-arrows.md",
      "source_sha256": "67dca53aae2bb03a809339b68315be2224aa3a5b6f9faf575aec9fa15647c53e"
    },
    {
      "source": "src/relative-character-eigenunitaries.md",
      "source_sha256": "6ee2affcf01d49ac479aa58d9d0ee07924f4102bb0b9fc0bb966ea2157f3b939"
    }
  ]
}