Six mathematical routes through the 33 approved Rosenberg Cards. Shared Cards are reused across routes and count once in the Catalog. The source-only density extension is separate.
Representations without a unit equation, nonzero irreducibility and unitary equivalence set the language of the theorem. The singleton condition names one representative of the unique nonzero irreducible class.
A pure state detects a hypothetical nonzero kernel element and its GNS representation is irreducible. Unitary equivalence to the singleton representative then contradicts that kernel element, giving faithfulness and the related simplicity consequence.
Characters of a maximal abelian subalgebra of the unitization extend to pure states. A character distinct from the scalar character yields a joint unit eigenvector in the singleton model. Countability and an isolated non-scalar character then give a nonzero scalar-corner projection and its rank-one image.
One represented rank-one projection yields every rank-one operator using dense orbits. Closed-range and compact approximation arguments put all compact operators in the image of the original algebra.
The preimage ideal of compact operators and a separating character exclude a noncompact represented image. The contradiction uses the same projection while forcing the character to take both zero and one.
Faithfulness and both compact-image inclusions identify the nonunital compact-operator model in an unbundled sense. In the unital singleton case the identity becomes compact, yielding finite dimension and the exact full-operator image consequences.