This paper studies hidden-subgroup causal inference under binary treatment and binary outcome when the stratifying variable is latent. Under consistency and latent ignorability, the paper establishes a sharp impossibility theorem: every interior observable 2x2 law of (X,Y) admits one latent causal decomposition with all subgroup average treatment effects positive and another with all subgroup effects negative. Hence the noiseless aggregate table does not identify even the uniform sign of hidden-subgroup causal effects, and every n-sample test based only on the aggregate has minimax error at least 1/2 on the nonidentified interior class. For arbitrary latent group structure, the paper derives an exact causal decomposition of the aggregate contrast into a treatment-weighted causal term and a confounding covariance term. Under oscillation budgets on treatment propensity and baseline risk, the confounding term is bounded by a sharp radius attained by a binary two-point extremizer on the central strip. For every finite number of latent groups sharing a common subgroup effect, the exact identified interval is derived in closed form, with explicit boundary-truncation formulas on the budget-saturating branch where positivity and outcome constraints bind. The multi-valued treatment case is handled by exact arm-versus-control binary reduction; the bounded-outcome case extends the population-level theory verbatim. A proximal extension with one observed proxy, bounded selection distortion, and controlled proxy misspecification establishes a finite-support extremal representation theorem: the sharp identified set for any finite family of effect functionals is attained by finite-support latent laws. In the known-channel setting, an invertible proxy channel identifies the latent cell vectors exactly. Under epsilon-level misspecification, recovery error scales at rate kappa(M) times epsilon. The paper closes by formalizing the information hierarchy via Blackwell comparison: the experiment based on (X,Y,Z) strictly Blackwell-dominates the aggregate (X,Y) experiment for the latent-sign decision problem, with an explicit exponential finite-sample separation rate governed by the Chernoff information between the proxy-resolved joint laws. A worked empirical illustration calibrates the identification frontier to the Dobbie-Goldin-Yang pretrial release study.
Kevin Fathi (Tue,) studied this question.