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    "assumptions": "Let A ⊆ B(Hₐₜ) be the fixed unital C*-algebra obtained by adjoining the chosen shell-link unitaries to the atomic representation of the CAR algebra C, and then taking the norm-closed unital *-algebra they generate. Let πₐₜ be its distinguished atomic inclusion on Hₐₜ. This theorem concerns Hₐₜ, not the separable trace-GNS space Hτ.",
    "conclusion": "There is a norm-dense subset D ⊆ Hₐₜ of cardinality 𝔠, and every norm-dense E ⊆ Hₐₜ satisfies 𝔠 ≤ |E|. Thus dens(Hₐₜ) = 𝔠.",
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    "assumptions": "Let A ⊆ B(Hₐₜ) be the fixed unital C*-algebra obtained by adjoining the chosen shell-link unitaries to the atomic representation of the CAR algebra C, and then taking the norm-closed unital *-algebra they generate. Let πₐₜ be its distinguished atomic inclusion on Hₐₜ. This theorem concerns Hₐₜ, not the separable trace-GNS space Hτ.",
    "conclusion": "There is a norm-dense subset D ⊆ Hₐₜ of cardinality 𝔠, and every norm-dense E ⊆ Hₐₜ satisfies 𝔠 ≤ |E|. Thus dens(Hₐₜ) = 𝔠.",
    "content_completeness": "CARD_CONTENT_COMPLETE",
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    "display_title": "The atomic Hilbert space has norm density continuum",
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      "text": "theorem hasDensityCharacter_atomicCounterexampleHilbert :\n    HasDensityCharacter\n      (MathlibAnnex.CStarAlgebra.PureState.SelectedAtomicHilbert completedRootPureState)\n      Cardinal.continuum := by\n  let H := MathlibAnnex.CStarAlgebra.PureState.SelectedAtomicHilbert completedRootPureState\n  let pi : Representation AtomicCounterexampleAlgebra H :=\n    shellFamilyInclusion homogeneityShellFamily\n  have hirr : Representation.IsIrreducible pi :=\n    isIrreducible_shellFamilyInclusion homogeneityShellFamily\n  letI : Nontrivial H := Representation.nontrivial_of_isNonzero pi hirr.1\n  obtain ⟨x, hx⟩ : ∃ x : H, x ≠ 0 := exists_ne 0\n  let s : Set H := Set.range (fun a : AtomicCounterexampleAlgebra => pi a x)\n  have hs : Dense s := Representation.denseRange_orbitMap_of_isIrreducible pi hirr hx\n  have hupper' : Cardinal.lift.{0} (#s) ≤\n      Cardinal.lift.{0} (#AtomicCounterexampleAlgebra) := Cardinal.mk_range_le_lift\n  have hupper : #s ≤ Cardinal.continuum := by\n    simpa [cardinalMk_atomicCounterexampleAlgebra] using hupper'\n  refine ⟨⟨s, hs, le_antisymm hupper\n    (continuum_le_cardinalMk_dense_atomicCounterexampleHilbert s hs)⟩,\n    continuum_le_cardinalMk_dense_atomicCounterexampleHilbert⟩"
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