Identifiability under imperfect coarsening: sensitivity bounds and the singleton sample complexity that restores them

Coarsening at random (the conditions C1, C2, C3) makes a latent parameter identifiable and the face-value maximum-likelihood fit consistent, but the symmetry condition C2 is the exception rather than the rule: real coarsening is informative. This paper develops the theory for imperfect coarsening. We parametrize the violation of C2 by a tilt of magnitude delta and prove a sensitivity bound: the asymptotic bias of the face-value MLE is, at leading order, linear in delta with an explicit constant set by the covariance of the face-value score with the tilt, so the latent parameter is partially identified over a delta-ball that contracts to a point as C2 is approached. We then prove a restoration theorem: a singleton report (one that pins the latent value, hence classical internal validation) restores point identification, and the number of singletons needed to recover the r confounded directions is of order r over gamma-squared, where gamma is the domain identification margin. The two results unify the reliability sensitivity bands, the single-cell spike-in bias, the weak-supervision gold-set sample complexity, and the differential-privacy mechanism partition as one structured-coarsening sensitivity-and-restoration theory, and place the coarsening program in the lineage of missing-not-at-random sensitivity analysis, measurement-error double sampling, and verification-bias correction. It is the C2-violation companion to the coarsening-at-random synthesis.