{
  "schema": "exact.hp.public-facing-card-candidate.v1",
  "stable_card_id": "a1397f366783063d30db5ab13e31d3e5339a0d31518fb16b965d5299478b8142",
  "content": {
    "assumptions": "The CAR algebra C, the distinguished root pure state, the selected atomic GNS representation Π on Hₐₜ, and the one homogeneity-based shell family have already been fixed by the construction. There is no additional assumption of simplicity, capture of representations, or the KOS theorem.",
    "conclusion": "The abbreviation denotes the concrete closed unital *-subalgebra A of B(Hₐₜ). Its carrier is used as a C*-algebra in later declarations. This declaration defines A; it does not by itself assert all the later counterexample properties.",
    "content_completeness": "CARD_CONTENT_COMPLETE",
    "definition_explanation": "The Hilbert space Hₐₜ is the Hilbert direct sum of the selected pure-state GNS spaces, including the designated root component. The operators Lᵢ are the full shell-link unitaries obtained from the shell data, not the individual partial-isometry elements of C. The algebra is generated inside an existing bounded-operator algebra; no universal completion is postulated.",
    "display_title": "The fixed CAR-based C*-algebra",
    "language_tag": "en",
    "lean_realization_notes": "Source kind: abbrev. Exact name: AtomicCounterexampleAlgebra. The elaborated value has type StarSubalgebra ℂ B(Hₐₜ); its use as a carrier employs the standard coercion. The chosen family is homogeneityShellFamily and the target is ShellFamilyTarget of that family. “Fixed” refers to one source-defined classical choice, not uniqueness of all possible presentations. The counterexample properties are established by the counterexample theorem for A; they are not inputs to this abbreviation.",
    "natural_language_statement": "Let C be the completed CAR algebra and Π its atomic representation on Hₐₜ. Use the source-defined family of homogeneity automorphisms and shell data to choose shell-link unitaries Lᵢ on Hₐₜ. Define A = C*(Π(C) ∪ {Lᵢ : i ∈ I}) ⊆ B(Hₐₜ), where C*(…) means the norm-closed unital *-algebra generated by the displayed operators.",
    "one_sentence_role": "Defines the one algebra used throughout the irreducible, separable, tracial, and density results.",
    "proof_steps": [],
    "proof_summary": null
  },
  "card_ref": {
    "record_type": "LFH_DECLARATION_CARD",
    "revision": 2,
    "sha256": "ba2e8276346975fcb757ba309080b548e3825fd7a9e5f505c0947c847694f385",
    "uid": "lfh:lfh-declaration-card:sha256:a1397f366783063d30db5ab13e31d3e5339a0d31518fb16b965d5299478b8142"
  },
  "exposition_ref": {
    "record_type": "LFH_CARD_EXPOSITION",
    "revision": 2,
    "sha256": "a5bfc1e3ffa72a83a9fc18fc5eff9287d254f1f6c457d91462ad12de6cb4c896",
    "uid": "lfh:lfh-card-exposition:sha256:41eec3a3c98a9cb06d27838ef416ef2f04066072d1d7231ea4b9a31567f0c948"
  },
  "source_binding": {
    "declaration_kind_from_source": "abbrev",
    "qualified_name": "MathlibAnnex.CStarAlgebra.CAR.AtomicCounterexampleAlgebra",
    "source_commit": "437e6e46228bbb8e91211ebded349d7a30020e73",
    "source_lines": [
      78,
      79
    ],
    "source_path": "MathlibAnnex/Analysis/CStarAlgebra/CAR/AtomicCounterexample.lean",
    "source_repository": "r-tanaka-math/mathlib-annex",
    "source_url": "https://github.com/r-tanaka-math/mathlib-annex/blob/437e6e46228bbb8e91211ebded349d7a30020e73/MathlibAnnex/Analysis/CStarAlgebra/CAR/AtomicCounterexample.lean#L78-L79"
  },
  "citations": [
    {
      "citation_kind": "SOURCE_ATTRIBUTION",
      "target": {
        "kind": "EXACT_PROVIDER_LOCATOR",
        "module_name": "MathlibAnnex.Analysis.CStarAlgebra.CAR.AtomicCounterexample",
        "provider": "PROJECT",
        "provider_source_sha256": "5622f3f3cf8c469547e3e2584e0134f5204f202fa7a520217a5abcdb9a2a49e1",
        "qualified_name": "MathlibAnnex.CStarAlgebra.CAR.homogeneityShellFamily"
      }
    },
    {
      "citation_kind": "SOURCE_ATTRIBUTION",
      "target": {
        "kind": "EXACT_PROVIDER_LOCATOR",
        "module_name": "MathlibAnnex.Analysis.CStarAlgebra.CAR.AtomicEndpoint",
        "provider": "PROJECT",
        "provider_source_sha256": "6d85e71586556b28ff438f1b4f9807e475cf6fbb14dcdf11ef40d32fad653ee8",
        "qualified_name": "MathlibAnnex.CStarAlgebra.CAR.ShellFamilyTarget"
      }
    }
  ],
  "presentation": {
    "assumptions": "The CAR algebra C, the distinguished root pure state, the selected atomic GNS representation Π on Hₐₜ, and the one homogeneity-based shell family have already been fixed by the construction. There is no additional assumption of simplicity, capture of representations, or the KOS theorem.",
    "conclusion": "The abbreviation denotes the concrete closed unital *-subalgebra A of B(Hₐₜ). Its carrier is used as a C*-algebra in later declarations. This declaration defines A; it does not by itself assert all the later counterexample properties.",
    "content_completeness": "CARD_CONTENT_COMPLETE",
    "definition_explanation": "The Hilbert space Hₐₜ is the Hilbert direct sum of the selected pure-state GNS spaces, including the designated root component. The operators Lᵢ are the full shell-link unitaries obtained from the shell data, not the individual partial-isometry elements of C. The algebra is generated inside an existing bounded-operator algebra; no universal completion is postulated.",
    "display_title": "The fixed CAR-based C*-algebra",
    "language_tag": "en",
    "lean_realization_notes": "Source kind: abbrev. Exact name: AtomicCounterexampleAlgebra. The elaborated value has type StarSubalgebra ℂ B(Hₐₜ); its use as a carrier employs the standard coercion. The chosen family is homogeneityShellFamily and the target is ShellFamilyTarget of that family. “Fixed” refers to one source-defined classical choice, not uniqueness of all possible presentations. The counterexample properties are established by the counterexample theorem for A; they are not inputs to this abbreviation.",
    "natural_language_statement": "Let C be the completed CAR algebra and Π its atomic representation on Hₐₜ. Use the source-defined family of homogeneity automorphisms and shell data to choose shell-link unitaries Lᵢ on Hₐₜ. Define A = C*(Π(C) ∪ {Lᵢ : i ∈ I}) ⊆ B(Hₐₜ), where C*(…) means the norm-closed unital *-algebra generated by the displayed operators.",
    "one_sentence_role": "Defines the one algebra used throughout the irreducible, separable, tracial, and density results.",
    "proof_steps": [],
    "proof_summary": null,
    "stable_card_id": "a1397f366783063d30db5ab13e31d3e5339a0d31518fb16b965d5299478b8142",
    "name": "MathlibAnnex.CStarAlgebra.CAR.AtomicCounterexampleAlgebra",
    "declaration_kind": "abbrev",
    "source_path": "MathlibAnnex/Analysis/CStarAlgebra/CAR/AtomicCounterexample.lean",
    "source_locator": {
      "start_line": 78,
      "end_line": 79
    },
    "source_signature": {
      "text": "/-- The single concrete C-star algebra used by every field of `AtomicCounterexampleEndpoint`. -/\nabbrev AtomicCounterexampleAlgebra := ShellFamilyTarget homogeneityShellFamily"
    },
    "card_ref": {
      "record_type": "LFH_DECLARATION_CARD",
      "revision": 2,
      "sha256": "ba2e8276346975fcb757ba309080b548e3825fd7a9e5f505c0947c847694f385",
      "uid": "lfh:lfh-declaration-card:sha256:a1397f366783063d30db5ab13e31d3e5339a0d31518fb16b965d5299478b8142"
    },
    "exposition_ref": {
      "record_type": "LFH_CARD_EXPOSITION",
      "revision": 2,
      "sha256": "a5bfc1e3ffa72a83a9fc18fc5eff9287d254f1f6c457d91462ad12de6cb4c896",
      "uid": "lfh:lfh-card-exposition:sha256:41eec3a3c98a9cb06d27838ef416ef2f04066072d1d7231ea4b9a31567f0c948"
    },
    "citations": [
      {
        "label": "MathlibAnnex.CStarAlgebra.CAR.homogeneityShellFamily",
        "text": "Exact source attribution.",
        "website_href": "../../sources/naimark20/8a66eab9b4f5dc708624def8/index.html#L72"
      },
      {
        "label": "MathlibAnnex.CStarAlgebra.CAR.ShellFamilyTarget",
        "text": "Exact source attribution.",
        "website_href": "../../sources/naimark20/ce288b9c4ba802b96de96500/index.html#L62"
      }
    ]
  },
  "website_metadata": {
    "publication_state": "PUBLIC_CURRENT",
    "publication_authorized": true,
    "first_publication_date": "2026-09-24",
    "this_revision_publication_date": "2026-09-24",
    "source_release": "v0.4.0",
    "source_exposition_correspondence": "APPROVED",
    "canonical_url": "https://exactmathematics.org/mathlibannex/cards/a1397f366783063d30db5ab13e31d3e5339a0d31518fb16b965d5299478b8142/",
    "catalog_current": true,
    "approval_scope": "APPROVED_SOURCE_EXPOSITION_CORRESPONDENCE",
    "publication_batch": "EM_NAIMARK_ARXIV_CARD20_PUBLICATION_R1"
  }
}