{
  "schema": "programme-source-record/v1",
  "title": "Reconstructing continuous comparisons from return arrows",
  "source": "src/reconstructing-continuous-comparisons-from-return-arrows.md",
  "reader": "reconstructing-continuous-comparisons-from-return-arrows.html",
  "source_sha256": "67dca53aae2bb03a809339b68315be2224aa3a5b6f9faf575aec9fa15647c53e",
  "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": "Exact reconstruction from labelled return arrows for varying separable factor fields, with semifinite characteristic comparison and trace normalization under the stated 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 return-arrow reduction and the vertical comparison formula are the classical construction of Sutherland–Takesaki [ST89], Section 1.3 and the proof of Theorem 4.2 (pp.430–431). The return invariant and its semifinite classification use [ST], Theorem 1.2 and Section 4. This lesson retains every isotropy label, writes the composition and range-unitary identities explicitly, and integrates both the standard-form map and its inverse. The scalar and positive gauges use the preceding programme comparison theorem."
    },
    {
      "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/continuous-return-reconstruction.svg",
      "sha256": "ef2a15dba073592c994eb2e8bc6a197a125e52dd4216140459e52044a66e210b"
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    {
      "path": "figures/continuous-return-reconstruction.png",
      "sha256": "a7fff8ea96cddd385d4b62b5c8709849115828c5a0c0409a7cd74d9ae64a40ad"
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      "path": "figures/draw_continuous_return_reconstruction.py",
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    {
      "source": "src/from-global-characteristic-invariants-to-return-data.md",
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      "source": "src/gluing-comparisons-of-isotropy-actions.md",
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