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        "text": "Convert the proved singleton property to the general nonunital representation predicate."
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    "assumptions": "No CH assumption is made. The existence predicate permits nonunital C*-algebras and nonunital complex *-representations. In this closed statement, the algebra carrier, displayed Hilbert space, and comparison Hilbert spaces lie in Type (universe zero).",
    "conclusion": "The density-existence predicate holds at 𝔠. A witness is the fixed unital CAR-based algebra A with its atomic inclusion; its density is exactly 𝔠.",
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      "text": "/-- The fixed construction supplies an ordinary counterexample of exact\nnorm density continuum. This existence assertion has no CH premise. -/\ntheorem existsNaimarkCounterexampleOfDensity_continuum :\n    ExistsNaimarkCounterexampleOfDensity (Cardinal.continuum : Cardinal.{0}) := by\n  refine ⟨AtomicCounterexampleAlgebra, inferInstance, inferInstance, inferInstance,\n    inferInstance,\n    MathlibAnnex.CStarAlgebra.PureState.SelectedAtomicHilbert completedRootPureState,\n    inferInstance, inferInstance, inferInstance,\n    (shellFamilyInclusion homogeneityShellFamily).toNonUnitalStarAlgHom, ?_, ?_,\n    hasDensityCharacter_atomicCounterexampleAlgebra⟩\n  · exact\n      Representation.IsSingletonIrreducibleModelAmongNonUnital.isSingletonIrreducibleModel_toNonUnitalStarAlgHom\n        (isUniqueIrreducibleModelAmongNonUnital_shellFamilyInclusion.{0}\n          homogeneityShellFamily).2\n  · exact not_isCompactOperatorModel_shellFamilyTarget homogeneityShellFamily _"
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