Identification Capacity and Rate-Query Tradeoffs in Classification Systems

We study zero-error class identification under constrained observations with three resources: tag rate L (bits per entity), identification cost W (attribute queries), and distortion D (misidentification probability). If the attribute-profile map π is not injective on classes, then attribute-only observation cannot identify class identity with zero error.

Let A_π:= maxᵤ|{c: π(c) = u}| be collision multiplicity. Any D = 0 scheme must satisfy L ≥ log₂A_π, and this bound is tight. In maximal-barrier domains (A_π = k), the nominal point (L, W, D) = (⌈log₂k⌉, O(1), 0) is the unique Pareto-optimal zero-error solution under the stated observation model: no other feasible combination achieves D = 0. Without tags (L = 0), zero-error identification requires W = Ω(d) queries, where d is the distinguishing dimension. Minimal sufficient query sets form the bases of a matroid, making d well-defined as an invariant of the observation structure rather than of a particular query strategy. We also prove exact finite-block confusability laws: at blocklength t, zero-error feasibility is equivalent to a tag alphabet threshold A_πᵗ, yielding exact linear log-bit scaling.

Conceptually, the framework extends classical rate-distortion accounting by treating query cost as an additional interactive resource (structurally analogous to perception-constraint extensions). We give instantiations across databases, biology, software runtimes, and model registries, and machine-check the main claims in Lean 4 (8092 lines, 352 theorem/lemma statements, 0 sorry).