Redundant Regulatory Programs Can Check Cell Groups They Locate Poorly

Curated regulatory programs, such as pathway gene sets and regulons, are built into single-cell models, yet what they add beyond an expression encoder is unclear. We find that, used as linear features next to an encoder, programs estimate the encoder-defined state poorly but can test whether a group of cells fits the relations among programs seen in known cells. A geometric theorem explains why: the set of consistent states is wide along directions that program directions combine into only at high cost, whereas the threshold for detecting a shift of one program grows only with the cost of combining the others into it, which redundancy keeps small. Our conformally calibrated, state-invariant statistic, HellyCell, flags a group when no state fits all programs and returns a witness of at most $d+1$ programs that no state satisfies jointly ($d$: state dimension); if data are exchangeable and few programs shift, a flag whose witness misses every shifted program has probability at most the level $\alpha$. Confidence sets from programs alone are 3.7 to 8.1 times wider than the encoder's with least squares, and 19.6 to 24.7 times as the intersections of intervals that the check uses. Estimated on pilot cells, the part of a mean shift that no change of state explains predicts which of 154 real perturbations are detected in held-out groups of ten cells (rank correlation 0.89; 0.91 in the largest screen, against 0.81 for the raw shift): interferon responses are detected, most single-gene CRISPR perturbations are not. Baarda's $w$-test has more power on sparse or strong shifts, so HellyCell trades power for witnesses; invariance to the state fails under new sequencing protocols.