{
  "schema": "lesson-sources-and-authorship/v1",
  "title": "Lattice derivatives and weighted orbit matching",
  "source": "src/lattice-derivatives-and-weighted-orbit-matching.md",
  "source_sha256": "52db33d5e736818b64bb45fc7aebd15485cfceaf45ab106551bfd28eff795d15",
  "authors": [
    "GPT-6.1 Sol (OpenAI)",
    "GPT-6 Astra (OpenAI)"
  ],
  "human_sources": [
    {
      "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",
      "exact_locator": "Theorem 1.3.2 and Remark 1.3.3, pp.152–153; Theorems 2.4.1, 3.2.1 and 3.4.1",
      "role": "Modular periods, bounded perturbations and factorial centralizers."
    },
    {
      "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",
      "exact_locator": "Lemma 3.1.4(2), p.20",
      "role": "Equal-mass full-pseudogroup matching."
    },
    {
      "author": "Robert T. Powers",
      "title": "Representations of uniformly hyperfinite algebras and their associated von Neumann rings",
      "year": 1967,
      "url": "https://www.mathnet.ru/eng/mat520",
      "edition": "Russian translation by A. G. Stepin, Matematika 13(4) (1969),94–124",
      "role": "Historical product-factor construction; the finite-state model is proved in the preceding programme 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": "Lemmas XVIII.1.5–1.6, pp.300–302; Definition XVIII.1.7 and Lemma XVIII.1.8, pp.302–304",
      "role": "Lattice probability, ergodic kernel, finite seed matching, weighted arrays and interval packing."
    }
  ],
  "scope": "Homogeneous orbit matching, weighted binary matrix arrays and quantitative partition approximation with compatible refinement. For type III_lambda relations the lattice probability and ergodic kernel follow under the stated modular and relation-algebra foundations. Six solved exercises distinguish set approximation from orbit exhaustion.",
  "license": "CC0-1.0",
  "license_scope": "Original programme expression only; cited works and separately licensed components retain their terms."
}