Identification of Latent Group Effects under Conditional Calibration

We study identification of a structural group effect when the group indicator G∈0,1 is unobserved, but the analyst observes a calibrated probability score p satisfying E[Gmid p,X]=p. Under a constant-coefficient structural mean model, the latent-group coefficient τ is point-identified by a closed-form ratio of observable moments whose denominator is the residual score variance V*=E[(p-E[pmid X])2]. Identification fails exactly when the score is a deterministic function of X; we construct an explicit continuum of observationally equivalent models showing the failure is genuine. The marginal latent mean gap decomposes as τ plus a compositional term that is itself identified in closed form, and we characterise when the two coincide. The oracle estimator is √n-consistent and asymptotically normal with a closed-form sandwich variance. Under calibration error bounded by δ, the bias obeys a sharp bound proportional to δ/V*, and hard-threshold classification attenuates the estimated gap. Monte Carlo experiments confirm the theory, including the variance-weighted estimand under heterogeneous effects.