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    "assumptions": "Let C be the completed CAR algebra, φ₀ its distinguished pure state, φᵢ one selected pure state in each GNS-equivalence class, and eₙ the fixed root shell projections. Fix a representative shell family F: for each i, an automorphism αᵢ and elements wᵢ,ₙ of C satisfying φᵢ ∘ αᵢ = φ₀, wᵢ,ₙ* wᵢ,ₙ = αᵢ(eₙ), and wᵢ,ₙ wᵢ,ₙ* = eₙ. At the root class, α₀ is the identity and w₀,ₙ = eₙ. Let A_F ⊆ B(Hₐₜ) be the norm-closed unital *-algebra generated by the atomic image of C and the shell-link unitaries constructed from F; write j_F: C → A_F for the source map. States are continuous positive complex-linear functionals normalized by φ(1) = 1.",
    "conclusion": "There exists exactly one state φ: A_F → ℂ satisfying φ(ab) = φ(ba) for all a, b ∈ A_F. It is the constructed extension of the normalized CAR trace.",
    "content_completeness": "CARD_CONTENT_COMPLETE",
    "definition_explanation": null,
    "display_title": "Uniqueness of the tracial state",
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    "lean_realization_notes": "The theorem states unique existence of a state satisfying the trace identity, not uniqueness of all states. Its displayed conclusion does not itself assert faithfulness. The latter is proved separately and included for the fixed A in the counterexample theorem with its faithful separable tracial model.",
    "natural_language_statement": "For every representative shell family F, the associated C*-algebra A_F has exactly one tracial state.",
    "one_sentence_role": "Proves uniqueness among normalized positive traces on each shell-family algebra.",
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        "step_id": "1",
        "text": "Use the separate theorem proving traciality of the constructed extension."
      },
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        "step_id": "2",
        "text": "Restrict an arbitrary tracial state along j_F and apply uniqueness of the CAR trace."
      },
      {
        "step_id": "3",
        "text": "Use the uniqueness theorem for extensions of the CAR trace to identify that state on all of A_F."
      }
    ],
    "proof_summary": "The constructed extension is tracial. Any tracial state on A_F restricts to the unique normalized CAR trace, so uniqueness of the state extension identifies it with the constructed extension."
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    "assumptions": "Let C be the completed CAR algebra, φ₀ its distinguished pure state, φᵢ one selected pure state in each GNS-equivalence class, and eₙ the fixed root shell projections. Fix a representative shell family F: for each i, an automorphism αᵢ and elements wᵢ,ₙ of C satisfying φᵢ ∘ αᵢ = φ₀, wᵢ,ₙ* wᵢ,ₙ = αᵢ(eₙ), and wᵢ,ₙ wᵢ,ₙ* = eₙ. At the root class, α₀ is the identity and w₀,ₙ = eₙ. Let A_F ⊆ B(Hₐₜ) be the norm-closed unital *-algebra generated by the atomic image of C and the shell-link unitaries constructed from F; write j_F: C → A_F for the source map. States are continuous positive complex-linear functionals normalized by φ(1) = 1.",
    "conclusion": "There exists exactly one state φ: A_F → ℂ satisfying φ(ab) = φ(ba) for all a, b ∈ A_F. It is the constructed extension of the normalized CAR trace.",
    "content_completeness": "CARD_CONTENT_COMPLETE",
    "definition_explanation": null,
    "display_title": "Uniqueness of the tracial state",
    "language_tag": "en",
    "lean_realization_notes": "The theorem states unique existence of a state satisfying the trace identity, not uniqueness of all states. Its displayed conclusion does not itself assert faithfulness. The latter is proved separately and included for the fixed A in the counterexample theorem with its faithful separable tracial model.",
    "natural_language_statement": "For every representative shell family F, the associated C*-algebra A_F has exactly one tracial state.",
    "one_sentence_role": "Proves uniqueness among normalized positive traces on each shell-family algebra.",
    "proof_steps": [
      {
        "step_id": "1",
        "text": "Use the separate theorem proving traciality of the constructed extension."
      },
      {
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        "text": "Restrict an arbitrary tracial state along j_F and apply uniqueness of the CAR trace."
      },
      {
        "step_id": "3",
        "text": "Use the uniqueness theorem for extensions of the CAR trace to identify that state on all of A_F."
      }
    ],
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    "source_signature": {
      "text": "/-- The existing target has exactly one tracial state. Faithfulness is proved\nseparately above; uniqueness ranges over all states, not just selected extensions. -/\ntheorem existsUnique_tracial_state (family : RepresentativeShellFamily) :\n    ∃! φ : ShellFamilyTarget family →L[ℂ] ℂ,\n      φ ∈ MathlibAnnex.Analysis.CStarAlgebra.stateSpace (ShellFamilyTarget family) ∧\n        ∀ a b, φ (a * b) = φ (b * a) := by\n  refine ⟨traceExtension family,\n    ⟨traceExtension_mem_stateSpace family, traceExtension_mul_comm family⟩, ?_⟩\n  intro φ hφ\n  exact eq_traceExtension_of_mem_stateSpace_of_mul_comm family φ hφ.1 hφ.2"
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