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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. Let τC be the normalized CAR trace. The family parameter F is retained; it is not replaced by an arbitrary C*-algebra.",
    "conclusion": "There exists a unique continuous complex-linear functional φ: A_F → ℂ which is positive, satisfies φ(1) = 1, and obeys φ(j_F(c)) = τC(c) for every c ∈ C.",
    "content_completeness": "CARD_CONTENT_COMPLETE",
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    "lean_realization_notes": "The source theorem is family-parametric. Uniqueness is among all state extensions of this particular trace, not merely among tracial extensions, and not among all states on A_F. Traciality and faithfulness are separate conclusions elsewhere. Taking the fixed homogeneity family recovers the algebra A.",
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        "step_id": "3",
        "text": "Evaluate the equal vector functionals on each a ∈ A_F."
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    ],
    "proof_summary": "The constructed trace extension gives existence. Compare the cyclic GNS representation of any other state extension with the constructed tracial representation by a pointed unitary; equality of their vector states gives uniqueness."
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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. Let τC be the normalized CAR trace. The family parameter F is retained; it is not replaced by an arbitrary C*-algebra.",
    "conclusion": "There exists a unique continuous complex-linear functional φ: A_F → ℂ which is positive, satisfies φ(1) = 1, and obeys φ(j_F(c)) = τC(c) for every c ∈ C.",
    "content_completeness": "CARD_CONTENT_COMPLETE",
    "definition_explanation": null,
    "display_title": "Unique extension of the CAR trace among all states",
    "language_tag": "en",
    "lean_realization_notes": "The source theorem is family-parametric. Uniqueness is among all state extensions of this particular trace, not merely among tracial extensions, and not among all states on A_F. Traciality and faithfulness are separate conclusions elsewhere. Taking the fixed homogeneity family recovers the algebra A.",
    "natural_language_statement": "For every representative shell family F, the normalized trace τC on the CAR algebra C extends to exactly one state of the associated C*-algebra A_F.",
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    "proof_steps": [
      {
        "step_id": "1",
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      },
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        "text": "Apply the pointed-unitary comparison theorem to an arbitrary state extension and its cyclic GNS vector."
      },
      {
        "step_id": "3",
        "text": "Evaluate the equal vector functionals on each a ∈ A_F."
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      "text": "/-- Existence and uniqueness are on the actual fixed target. -/\ntheorem existsUnique_state_extension_trace (family : RepresentativeShellFamily) :\n    ∃! φ : ShellFamilyTarget family →L[ℂ] ℂ,\n      φ ∈ MathlibAnnex.Analysis.CStarAlgebra.stateSpace (ShellFamilyTarget family) ∧\n        ∀ b, φ (shellFamilySourceHom family b) = trace b := by\n  refine ⟨traceExtension family,\n    ⟨traceExtension_mem_stateSpace family, traceExtension_shellFamilySourceHom family⟩, ?_⟩\n  intro φ hφ\n  exact eq_traceExtension_of_mem_stateSpace_of_apply_shellFamilySourceHom_eq_trace family φ hφ.1 hφ.2"
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