arXiv2026
We study the descriptive complexity of algebraic and analytic properties by presenting structures as quotients of fixed generators. For separable Banach-type structures, admissible kernels form Polish spaces and quotient norms are continuous in the Wijsman topology. For countable algebraic structures, congruence spaces are compact and atomic predicates are clopen. In the unital $C^*$-algebra coding, commutativity, MF-ness, stable finiteness and existence of a tracial state are $Π^0_1$-complete. AF-ness, nuclearity, simplicity, property~(SP), approximate divisibility, fixed real- and stable-rank bounds, and fixed nuclear-dimension and decomposition-rank bounds are $Π^0_2$-complete. Finiteness of each of these four ranks is $Σ^0_3$-complete. Finite dimensionality and existence of a non-zero finite-dimensional representation are $Σ^0_2$-complete. Residual finite dimensionality is $Π^0_3$-complete, already among AF algebras, while quasidiagonality is $Π^0_3$. For countable abelian groups, slenderness is $Π^0_3$-complete even within the torsion-free locus. We also obtain a $Σ^0_3$ upper bound for uniformly open multiplication in Banach algebras. An internal $K_0$ presentation has $F_σ$ membership coordinates and is $Σ^0_3$-measurable. We establish Borel tensor-ideal assignments, continuity for a fixed nuclear tensor factor, and a $G_δ$ bound for absorption of a strongly self-absorbing algebra. Finally, separability of the dual is $Π^1_1$-complete in the full unital $C^*$-quotient coding, already among commutative AF quotients; superatomicity of countable Boolean algebras is also $Π^1_1$-complete.