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    "conclusion": "𝔠 = ℵ₁ ⇔ there exists a Naimark counterexample of exact norm density ℵ₁ in the stated sense.",
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    "natural_language_statement": "The continuum hypothesis holds if and only if there exists a nontrivial, possibly nonunital complex C*-algebra of norm density ℵ₁ with one nonzero irreducible representation class and a displayed irreducible model that is not the full compact-operator algebra.",
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    "assumptions": "This is a biconditional with no extra hypothesis. CH means 𝔠 = ℵ₁, where 𝔠 = 2^ℵ₀. The existence predicate is the one described in the counterexample existence theorem, with density ℵ₁, and with carriers and comparison spaces in Type. No faithful separable representation is assumed for an arbitrary witness.",
    "conclusion": "𝔠 = ℵ₁ ⇔ there exists a Naimark counterexample of exact norm density ℵ₁ in the stated sense.",
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    "lean_realization_notes": "CH is a proposition in the biconditional, not an added axiom of the development. This is an object-level equivalence, not a forcing construction or a metatheoretic independence certificate. The supporting obstruction has broader universe parameters; the final biconditional here is its closed universe-zero form.",
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    "one_sentence_role": "Relates CH to the general density-existence predicate, not only to separably represented examples.",
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      "text": "/-- CH is equivalent to the existence of an ordinary possibly nonunital\nNaimark counterexample of exact norm density aleph one. Carriers in this closed\nexistence statement lie in Type; the reverse obstruction is universe-polymorphic. -/\ntheorem continuum_eq_aleph_one_iff_existsNaimarkCounterexampleOfDensity :\n    (Cardinal.continuum : Cardinal.{0}) = Cardinal.aleph 1 ↔\n      ExistsNaimarkCounterexampleOfDensity (Cardinal.aleph 1 : Cardinal.{0}) := by\n  constructor\n  · intro hCH\n    rw [← hCH]\n    exact existsNaimarkCounterexampleOfDensity_continuum\n  · exact continuum_eq_aleph_one_of_existsNaimarkCounterexampleOfDensity"
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