A recurring statistical structure appears, under different names, across reliability, single-cell and spatial genomics, differential privacy, programmatic weak supervision, and electronic-health-record phenotyping. In each, a latent quantity is observed only through a coarsening: a candidate set, a dropout, a noised release, a vote, a diagnostic code. Two results recur. The first is seam-free across all six domains: an augmented-candidate-set rank condition decides when the latent parameter is identifiable, and a singleton candidate set (a report that pins the latent quantity to a point) restores identifiability by restoring column rank. The singleton wears a different costume in each field (spike-ins, single-cell-resolution probes, gold labels, non-private releases, chart review) but is one mechanism, stated once. The second recurring result is a consistency identity at the maximum-likelihood fit: under the coarsening-at-random conditions C1, C2, C3, the face-value likelihood can be maximized without modeling the coarsening, and the fit reproduces the empirical mean of a coarsening-sufficient statistic at the optimum. This identity has been stated five separate times (cell-total consistency in single-cell data, its spot-level variant in spatial deconvolution, release consistency for differential privacy, agreement consistency for weak supervision, and code-frequency consistency for phenotyping); we state it once and recover each named version as a special case, and we locate its boundary exactly. In a regular exponential family the identity is an exact finite-sample equality; in a location family it is exact for a single report and a population first-moment identity otherwise, with the finite-sample sample-mean form holding exactly only for the Gaussian kernel (which itself sits in the exponential-family regime). The empirical weight is carried by the sibling papers, which are cited rather than re-derived.
Alexander Towell (Wed,) studied this question.