{
  "schema": "lesson-sources-and-authorship/v1",
  "title": "Classifying modular cyclic returns",
  "source": "src/classifying-modular-cyclic-returns.md",
  "source_sha256": "d890a787672682bd02cdd5766129ae723f21c667d96d1bf52e6bdb8d5ccbcc0f",
  "authors": [
    "GPT-6.1 Sol (OpenAI)",
    "GPT-6 Astra (OpenAI)"
  ],
  "human_sources": [
    {
      "authors": "Colin E. Sutherland and Masamichi Takesaki",
      "title": "Actions of Discrete Amenable Groups and Groupoids on von Neumann Algebras",
      "year": 1985,
      "journal": "Publications of the Research Institute for Mathematical Sciences",
      "volume": "21",
      "pages": "1087–1120",
      "url": "https://ems.press/journals/prims/articles/3261",
      "exact_locator": "Theorem 1.2, p.1092; Section 4, p.1107",
      "role": "Inner subgroup, full characteristic invariant and trace-modulus classification; one-point specialization in Proposition 1.1."
    },
    {
      "author": "Masamichi Takesaki",
      "title": "Theory of Operator Algebras III",
      "year": 2003,
      "url": "https://doi.org/10.1007/978-3-662-10453-8",
      "exact_locator": "Definition XVIII.1.12 and Theorem XVIII.1.13, pp.310–313",
      "role": "Modular-return classification and finite-factor/trace-flow models."
    },
    {
      "authors": "Yasuyuki Kawahigashi, Colin E. Sutherland and Masamichi Takesaki",
      "title": "The structure of the automorphism group of an injective factor and the cocycle conjugacy of discrete abelian group actions",
      "year": 1992,
      "url": "https://doi.org/10.1007/BF02392758",
      "exact_locator": "Lemma 18, p.123; the type III-lambda argument, pp.123–124",
      "role": "Eigenvector and discrete-core antecedents of the centrally trivial converse used from the preceding programme lesson."
    },
    {
      "author": "Alain Connes",
      "title": "Outer conjugacy classes of automorphisms of factors",
      "year": 1975,
      "url": "https://numdam.org/articles/10.24033/asens.1295/",
      "role": "Finite-factor constructions and obstruction theory; exact proof inputs are specified in the linked finite-factor lessons."
    }
  ],
  "classical_locators": [
    "Sutherland–Takesaki 1985, Theorem 1.2, p.1092 and Section 4, p.1107",
    "Takesaki III, XVIII.1.12–1.13, pp.310–313",
    "Kawahigashi–Sutherland–Takesaki 1992, Lemma 18, p.123"
  ],
  "scope": "Outer conjugacy for the hyperfinite type III-lambda factor, under the stated generalized-trace, centrally trivial, finite-model and semifinite-comparison foundations. The lesson computes the inner-return lattice and its characters, proves the normal cocycle lift and gives seven solved exercises.",
  "license": "CC0-1.0",
  "license_scope": "Original programme expression only; cited works and separately licensed components retain their terms."
}