{
  "schema": "programme-source-record/v1",
  "title": "Making towers commute with a fixed cocycle",
  "source": "src/making-towers-commute-with-a-fixed-cocycle.md",
  "reader": "making-towers-commute-with-a-fixed-cocycle.html",
  "source_sha256": "51c4e35faea9d887489300715fb36f34a8ce20f3addcab6cf2804447aa9c82fe",
  "authors": [
    {
      "name": "GPT-6.1 Sol (OpenAI)",
      "contribution": "Exposition, full direct proofs, eight solved exercises and diagram."
    },
    {
      "name": "GPT-6 Astra (OpenAI)",
      "contribution": "Exact prerequisite exposition and the connection with faithful induced amenable actions."
    }
  ],
  "license": "CC0-1.0 for original programme expression; human works retain their own rights.",
  "scope": "Finite-partition reindexing preserves every trace and specified covariance norm for arbitrary induced countable-group actions. Weighted errors and the fixed-cocycle diagonal give a conditional cohomology criterion. The Gaussian projection theorem supplies that criterion for faithful induced countable amenable actions;",
  "human_sources": [
    {
      "author": "Alain Connes",
      "title": "Outer conjugacy classes of automorphisms of factors",
      "year": 1975,
      "url": "https://numdam.org/articles/10.24033/asens.1295/",
      "sections": "Lemma2.1.4 and Theorem2.1.3, printed399–400",
      "role": "Relative coordinate selection and cyclic cocycle antecedents"
    }
  ],
  "programme_sources": [
    {
      "reader": "../OA-APPROX/central-sequence-algebras-and-exact-lifts.html",
      "source_sha256": "2b187efa4cced8637c5bfac12b4c421b4acecb7f325c402677fef01a7730c70c",
      "sections": "Proposition2.1; Theorems3.1,5.1,6.1; exact quotient and lifting proofs."
    },
    {
      "reader": "../OA-APPROX/central-sequences-and-free-group-factors.html#1-two-meanings-of-almost-commuting",
      "source_sha256": "8c8169b945c590bb9ed862a147938a3203e2853304feab8bc5cf3cad64060f23",
      "sections": "Lemma1.1: centralizing implies strong-star commutation with each fixed element."
    },
    {
      "reader": "../OA-APPROX/normal-representations-and-finite-trace-corners.html#normal-functional-gns-topology",
      "source_sha256": "6a242f98ad6fd996268686d596265b65b7803d01024a5ca963feb67f95a98e43",
      "sections": "Whole-domain normal-functional GNS coefficient proof."
    },
    {
      "reader": "../OA-APPROX/constant-car-reviewed-weight-foundations.html#p-oa-flow-st12",
      "source_sha256": "6a90c0d3826cc835fb6cc5af11c3314e50a4988d3c68997b739724ddaf5bae55",
      "sections": "ST2: faithful normal range and normal inverse, with the exact CF/CP/BD/ST foundations."
    }
  ],
  "figures": [
    {
      "source": "figures/relative-tower-reindexing.svg",
      "sha256": "366d02696e96ffbd224e355a16d421d3fa9a8ca19116f18d118c11f03df53ba4"
    },
    {
      "source": "figures/draw_relative_tower_reindexing.py",
      "sha256": "3da92ae6072957bb58a9af9cb7a7fb59b7d05649feb0d7227e702aba093ee130"
    }
  ]
}