Task-Completed Identifiability: Recovering Task-Relevant Factors for Selective Use

Representations can preserve all information needed for prediction while still mixing the latent factors that a task should use separately. A particularly important case arises when environmental variation alone cannot distinguish task-relevant factors, even though task responses provide complementary constraints on the same latent structure. We propose Task-Completed Identifiability (TCI), which formalizes when these two information sources jointly remove such relevant-factor ambiguities within a shared generative model. We derive the Task-Relevant Rank Condition (TRC), a projected rank condition on the first- and second-order derivatives of environmental and task-response contrasts. Under the stated assumptions, TRC identifies task-relevant factors and the common task–factor correspondence up to permutation and factor-wise invertible transformations, without requiring individual identification of factors irrelevant to all tasks. We further construct settings in which either information source alone admits genuine mixing among relevant factors, whereas their combination resolves it. To estimate the shared model, we introduce TC-iVAE, which couples a response-free inference branch with task-conditioned variational branches through shared latent coordinates. A composite-ELBO decomposition connects exact observable-law fit and posterior approximation to the population identification result. Experiments show that task responses can resolve relevant-factor mixtures left unresolved by environmental variation, while preserving predictive information. The resulting representations support more selective factor-wise editing, with substantially reduced changes in non-target factors. Together, these results distinguish preserving task information from identifying the factors needed to use that information selectively.