{
  "schema": "lesson-sources-and-authorship/v1",
  "title": "Weighted finite models and biased tail classification",
  "source": "src/weighted-finite-models-and-biased-tail-classification.md",
  "source_sha256": "0c704ba939abb50d3d2a414ddfec67c4e1f0d431d28921ef8f641a9cd372bb5e",
  "authors": [
    "GPT-6.1 Sol (OpenAI)",
    "GPT-6 Astra (OpenAI)"
  ],
  "human_sources": [
    {
      "author": "Sorin Popa",
      "title": "Lectures on ergodic theory of group actions (a von Neumann algebra approach)",
      "url": "https://www.math.ucla.edu/~popa/Books/OElectures.pdf",
      "role": "Measured-relation background; the Borel inverse in this lesson uses the stated standard-Borel theorem."
    },
    {
      "author": "Alain Connes",
      "title": "Une classification des facteurs de type III",
      "year": 1973,
      "url": "https://www.numdam.org/article/ASENS_1973_4_6_2_133_0.pdf",
      "role": "Modular invariants underlying the lattice normalization in the preceding lesson."
    },
    {
      "author": "Masamichi Takesaki",
      "title": "Theory of Operator Algebras III",
      "year": 2003,
      "url": "https://doi.org/10.1007/978-3-662-10453-8",
      "exact_locator": "Lemma XVIII.1.8(ii), pp.302–306; Lemma XVIII.1.9, p.306; Theorem XVIII.1.4(ii), p.300, proof p.307",
      "role": "Classical finite orbit approximation, base-cell refinement and biased-tail classification."
    }
  ],
  "scope": "Binomial level capacities retain complete finite orbit profiles. Exact weighted refinement controls both domain and range error, and a Borel address map identifies hyperfinite type III_lambda relations with the biased binary tail relation under the stated modular normalization and standard-Borel foundations. Six solved exercises accompany the construction.",
  "license": "CC0-1.0",
  "license_scope": "Original programme expression only; cited works and separately licensed components retain their terms."
}