Task-Aware Identifiability: Observation Quotients, Statistical Geometry, and Representation Accessibility

Abstract

Intelligent systems often assume observations contain the information required for a task. This fails struc- turally when a declared mechanism assigns identical ob- servation laws to latent states the task must distin- guish. We develop a unified diagnostic framework con- necting observation-induced equivalence, task identifia- bility, local statistical geometry, representation accessi- bility, and achievable risk. Its organizing condition is a standard factorization property: the observation parti- tion refines the task partition, and conditionally iid rep- etitions of that fixed channel cannot recover an exactly collapsed distinction. On regular quotient strata, Fisher geometry and nuisance-efficient Fisher information char- acterize first-order local resolvability. For representa- tions, we distinguish law-valued encoder experiments from deterministic quotient embeddings and exact task sufficiency from scale-controlled decoder accessibility. We organize task risk into structural, finite-observation, representation-channel, and learning gaps. Controlled synthetic studies validate these layers; a physics-based synthetic near-field model yields sharply different range and angle information, while NYUv2 shows persistent, decoder-modulated task preferences among frozen en- coders under two pointwise probes. TAI thereby local- izes whether task-relevant information is absent at ob- servation, weakly resolved, lost or poorly accessible in representation, or left to the learner.