{
  "schema": "programme-proof-companion/v1",
  "title": "Crossed-product recognition and injective cores",
  "source": "foundations/crossed-product-and-injective-core-foundations.md",
  "reader": "crossed-product-and-injective-core-foundations.html",
  "license": "CC0-1.0",
  "scope": "Crossed-product recognition, corners, trace normalization, factorial-centralizer spectra and the injective type III-lambda reduction. PC9 proves norm-one retractions and injectivity; PC10 proves invariant-state finite classification; PC11 supplies tracial quotients, normal constant embeddings, finite unitary lifting and bounded trace expectations without factoriality or separability assumptions.",
  "authors": [
    {
      "name": "Claude Opus 5.5 (Anthropic)",
      "contribution": "Earlier programme recognition, averaging, corner and classification exposition."
    },
    {
      "name": "GPT-6 Astra (OpenAI)",
      "contribution": "Companion exposition, exact prerequisite interfaces, trace normalization and PC7–PC8 factorial-centralizer spectral proof. Full unital C*-algebra norm-one retraction proof and the representation-independent injectivity criterion. Invariant-state finite classification and exact finite-model, balanced-coupling and dyadic uniqueness synthesis. Tracial specialization, whole-domain normal embedding and bounded expectations with finite central comparison."
    }
  ],
  "programme_sources": [
    {
      "title": "Full factors without almost periodic weights",
      "sections": "Lemma 8.1"
    },
    {
      "title": "Classification of injective factors",
      "sections": "Lemma 2.2(e,f), Propositions 3.1–3.2, Theorems 4.1/4.3, Corollary 4.2, Example 4.4, Lemmas 8.1–8.2 and Theorem 8.3"
    },
    {
      "title": "Trace existence and central averaging",
      "source": "trace-existence-foundations.md",
      "sections": "S1, FCT7, GS1, PC-4"
    },
    {
      "title": "Discrete decomposition",
      "source": "../../OA-FLOW/src/OA-FLOW-DD.md",
      "sections": "DD-0–DD-5"
    },
    {
      "title": "The full weight spectral identity and the modular intersection comparison",
      "source": "../../OA-FLOW/src/OA-FLOW-MG.md",
      "sections": "MG-0–MG-3",
      "source_sha256": "96d8c542900309077367f97df94d6242096f1692298c8a89315d78d6dd79c94d"
    },
    {
      "title": "The real Connes spectrum: orbit supports, corners and cocycles",
      "source": "../../OA-FLOW/src/OA-FLOW-CS.md",
      "sections": "CS-0 and CS-3",
      "source_sha256": "871cccf60847c9557677eb511a6563dcb6d4e845bb237613c649c1af0eb737d5"
    },
    {
      "title": "Completely positive finite models",
      "source": "../../OA-APPROX/src/completely-positive-finite-models.md",
      "sections": "Section 2, Lemmas 2.1–2.2 and Theorem 2.3: full operator-system extension and preservation of unit value.",
      "source_sha256": "0025cfd141c2bdb20f2d8017ae5efaa84dac3331a1b5244a61c019be754988a7"
    },
    {
      "title": "Finite models of a von Neumann algebra",
      "source": "../../OA-APPROX/src/semidiscrete-finite-models.md",
      "sections": "Section 4, Proposition 4.1, Q1–Q3 and Corollary 4.3: positivity, block preservation, projection approximation and complete positivity.",
      "source_sha256": "b12d55f83be8f8e11432979a53369c7bf50f22f60a1e54fbfab46f25b27cec5d"
    },
    {
      "title": "Averaging, crossed products, and injectivity",
      "source": "../../OA-APPROX/src/averaging-crossed-products-injectivity.md",
      "sections": "Normal representation comparison, full standard-pair construction, Proposition 1.1, Lemma 1.2 and Theorem 1.3. BG-1–BG-6 and the concrete-predual/faithful-image predecessors retain their scope.",
      "source_sha256": "15bcdfd52bcfc0c4176d96360de0ae6bcfbac017f091b51245f890e7eb95dac8"
    },
    {
      "title": "Finite tensor norms and universal biduals",
      "source": "../../OA-APPROX/components/abstract-tensor-and-universal-bidual-foundations.md",
      "sections": "Scalar positivity and universal-bidual sections: state coefficients, predual restriction, product, normal extension and central quotient.",
      "source_sha256": "869584d4b31a23a3c8e7b2d300bb0e17bf2f37e6b3ed119c42aba8da798843c0"
    },
    {
      "title": "Hypertraces and finite injective algebras",
      "source": "../../OA-APPROX/src/hypertraces-finite-injectivity.md",
      "sections": "Sections1–5: scalar separation, compactness, finite-target extension, finite-rank/spatial constructions, trace representation, DEN and the specified-trace SC and M1–M6 matrix construction. No general finite/semifinite or later averaging theorem is needed.",
      "source_sha256": "e46b2f5788a6b05e4ca126c3fbe64da7ffbaa7a53cca8aa5def6cc792515b07f"
    },
    {
      "title": "Tracial adjoints and rational matrix models",
      "source": "../../OA-APPROX/src/tracial-adjoints-and-rational-matrix-models.md",
      "sections": "P00–P06, Lemma1.1 and Theorems2.1–3.1: trace adjoint, weighted Hilbert estimate and full rational channels.",
      "source_sha256": "9d193155b8f74765e0dd7cedd22dcc00ae5c10214f0d65a71d4146c9a70b5e57"
    },
    {
      "title": "Trace-preserving finite models",
      "source": "../../OA-APPROX/src/trace-preserving-finite-models.md",
      "sections": "Complete direct finite-model chain: density perturbation, N1–N3 normal unital correction, Theorems2.1–3.1 and unitary replacement Lemma4.1/Corollary4.2.",
      "source_sha256": "ff97291160cc1140a9412625da731d3f33e98ee1c24fcda3f8edb0b3306375a7"
    },
    {
      "title": "Balancing Kraus families and unitary couplings",
      "source": "../../OA-APPROX/src/balancing-kraus-families-and-unitary-couplings.md",
      "sections": "All direct proofs: both norm-averaging arguments, both positive-defect proofs, balanced reconstruction and coupling energy. Arbitrary finite-algebra defect scope and factor-only reconstruction are distinguished.",
      "source_sha256": "08a2622611d647e16121717693c334185889d0d80e0fabdcc2445d34b7dc2c26"
    },
    {
      "title": "Unitary couplings and finite injective factors",
      "source": "../../OA-APPROX/src/unitary-couplings-and-finite-injective-factors.md",
      "sections": "All direct proofs: phase sampling, radial and matrix clipping, trace Lipschitz bound, finite polar lift, central ultrapower comparison, Theorem5.1 and Corollary5.2.",
      "source_sha256": "a222d82b4ce4d4fc21dde6cfa5d0ee37d9c29cad6f6b51b354c9039ae8ed2ef7"
    },
    {
      "title": "Local approximation and the hyperfinite finite factor",
      "source": "../../OA-APPROX/src/hyperfinite-finite-factors.md",
      "sections": "PB1–PB2; full Lemmas1.1,3.1,4.1 and Theorem5.1/Corollary5.2: dyadic cuts, exact containment, generating sequence, normal trace-GNS identification and finite corners. Later subgroup and automorphism modules are not used.",
      "source_sha256": "427d7f9fbcf09d6d3f9a3b3bd3e54ce04d36316b843b1cc5ac5d235d658df13f"
    },
    {
      "title": "Finite CP reconstruction inside the algebra",
      "source": "../../OA-APPROX/src/properly-infinite-injective-algebras-and-dyadic-approximation.md",
      "sections": "Lemma4.1, finite internal Kraus formula only; no properly infinite one-isometry or approximation theorem is used.",
      "source_sha256": "3022cb708f9dd6353d8df15491f4b954f0b804de9f1427c8928f6937489f698a"
    },
    {
      "title": "Trace Hilbert spaces and finite trace expectations",
      "source": "../../OA-APPROX/components/invariant-state-trace-hilbert-inputs.md",
      "sections": "Full TL1–TL6 directly read; TL3 supplies the normal trace-preserving expectation and nearest L2 projection, with the measurable-operator and onto L1-predual predecessor retained.",
      "source_sha256": "5286d059384ef245d57099085bc2dfb33edde078166567c7e23bf65aea22e044"
    },
    {
      "title": "Multiplier ultraproducts and normal embeddings",
      "source": "../../OA-APPROX/src/multiplier-ultraproducts-and-normal-embeddings.md",
      "sections": "Complete lesson, all proofs and14solutions read. PC11 uses Lemmas2.1,3.1 and Theorem3.2 at faithful-normal-state generality; tracial specialization and whole-domain normal embedding are explicit.",
      "source_sha256": "a5611492f7d6026b8fc6f647253d2b1d20085409ce45bef4e8c9368171ebb80b"
    },
    {
      "title": "Quotient norms, trace completions and exact coordinate relations",
      "source": "../../OA-APPROX/components/central-sequence-quotient-and-lifting-inputs.md",
      "sections": "CQ1–7 complete direct proofs read; PC11 uses CQ1–3: complete C*-quotient, faithful trace GNS and complete trace ball.",
      "source_sha256": "feaaa6af00567b05376625d4d12910b033d6f3c9f28f6102cc9cdba490c8ec25"
    },
    {
      "title": "Normal-functional and faithful-state foundations",
      "source": "../../OA-APPROX/components/central-sequence-foundations.md",
      "sections": "NT1–2: concrete-predual decomposition and module bounds, faithful-state bounded topology and Hilbert convex separation.",
      "source_sha256": "2e9196153b7aeab948551247c9244cdc7d7d7a166cd40b50a093713764a2e08e"
    },
    {
      "title": "Bounded calculus, preduals, density and faithful representations",
      "source": "../../OA-APPROX/components/constant-car-reviewed-weight-foundations.md",
      "sections": "Full retained CF1–10, CP1–6, BD1–5, ST1–2 and ST-C read. No whole-compilation or modular-weight proof clearance is asserted.",
      "source_sha256": "6a90c0d3826cc835fb6cc5af11c3314e50a4988d3c68997b739724ddaf5bae55"
    },
    {
      "title": "Bounded tracial commutant and conditional expectation",
      "source": "../../OA-APPROX/components/hyperfinite-finite-algebra-and-vacuum-inputs.md",
      "sections": "HF0 complete proof, including bounded-vector separation, arbitrary subalgebra expectation, whole-domain normality and intrinsic trace-predual completion.",
      "source_sha256": "a53a8e1f0432c730b2a4157baa65fe140b3beca9346cd68b82ef401f81cc74b1"
    }
  ],
  "human_sources": "Exact bibliographic citations and mathematical roles appear in Sources and authorship. Human-source expression retains its own rights.",
  "source_sha256": "6c7edcf0b8972682b29fadb4a1d69012d43cd17e1332879ccf0563e7c2eaa17c"
}