Simpson Reversal as Exact Latent-Stratum Nonidentifiability and Recovery — Sharp Identification Frontiers in Binary Mixture Models

This paper develops a complete impossibility/recovery theory for Simpson reversal in binary mixture models. The setting is a binary treatment, binary outcome, and unobserved binary group indicator, where the observable data consists only of the marginal 2x2 table. The paper proves the nonidentifiability result in its strongest correct form: for every interior observable law with nonzero marginal association, the fiber of latent two-group decompositions contains both an all-positive and an all-negative subgroup-effect model. Consequently no function of the observable table can determine the sign of subgroup contrasts, even when asking only whether all subgroup effects are simultaneously positive or simultaneously negative. The paper then derives the exact algebraic decomposition of the marginal risk difference for an arbitrary finite latent stratum: the marginal contrast equals the treatment-weighted average of within-group contrasts plus a confounding-capacity term equal to the covariance between treatment propensity and baseline risk, divided by the marginal treatment variance. For two groups this yields an explicit confounding-capacity bound. Under a common subgroup effect assumption and primitive quantitative bounds on the spread of treatment propensities and baseline risks, the exact identified set for the common subgroup effect is proved to be a closed interval centered at the marginal risk difference, with radius equal to the product of the two bound parameters divided by four times the marginal treatment variance. The sign of the common subgroup effect is identified if and only if the marginal risk difference exceeds this confounding-capacity radius. The result is extended to heterogeneous effects via a bounded oscillation budget, yielding robust uniform sign certification. Finally, an exact proxy identifiability theorem is proved: if a binary proxy for the latent stratum is observed with a known channel matrix and is conditionally independent of treatment and outcome given the stratum, then the full latent parameter vector is exactly identified if and only if the proxy channel matrix is invertible. The paper thus moves from the classical impossibility observation to a mathematically precise impossibility/recovery theory with an explicit identification frontier.