How many cells resolve a perturbation direction? A closed-form, control-aware cell quota for single-cell perturbation screens

Single-cell perturbation screens increasingly infer mechanism from the direction of a treatment's displacement in a shared transcriptomic embedding, yet no sample-size rule states how many cells that direction needs. We derive a closed-form, anisotropic cell quota for the treated cells required to hold the angular error of an estimated perturbation direction at a chosen tolerance. Because noise aligned with the effect changes only its length, the quota depends on the noise perpendicular to the effect, on the effect magnitude and on the control-pool size, and it diverges below a magnitude floor set by the control alone. The deterministic core is machine-checked in Lean 4. Across four public atlases (six screens) reprocessed through one embedding, the per-cell noise variance is near 1.0 on chemical and genome-wide genetic platforms. On Tahoe-100M, referenced to its plate-shared DMSO vehicle of about 3,100 cells, 55% of 56,827 conditions are control-pool-limited and only 11% are over-sampled at a 5.7-degree tolerance. Genome-wide CRISPRi knockdowns are instead detection- and depth-limited. A downsample-and-measure test on every atlas recovers the parameter-free angular-error slope. The calculator turns a pilot covariance, an effect magnitude and a control design into a per-condition cell budget and flags conditions that no treated depth can resolve.