Prediction coverage in inherited-state branching models: an exact mean-closure counterexample, a repaired count threshold, and the effect of founder number

Low-cell-count proliferation assays are often summarized by a scalar birth–death model whose per-cell rates are read off the expected phenotypic composition of the population. We show that such a mean-composition closure can reproduce the exact expected cell count, and even the exact expected birth and death fluxes, while assigning misleading probabilities to future counts. The witness is an explicit two-type branching process with positive switching in both directions, one founder of random type, and an observation horizon of one resistant doubling time. The scalar closure assigns probability at least $332/343>0.9679$ to the counts $\{0,1,2\}$, whereas the inherited-state process assigns them probability at most $1-L<0.95$ with $L=0.05044\ldots$ an explicit rational number. Both inequalities are proved analytically: the scalar tail is bounded through an explicit box for the mean and the accumulated-birth parameter, certified by one positive polynomial identity, and the actual tail is bounded from below by seven no-switch histories. Widening the region to $\{0,\dots,3\}$ restores coverage at least $H=0.9713\ldots$, and three is the smallest upper endpoint that reaches $95\%$ for this preparation. A variance comparison ($\operatorname{Var}Z_T\le 11/16<4/5<\operatorname{Var}N_T$) shows that the discrepancy does not disappear with many independent founders: the limiting actual coverage of the scalar central $95\%$ interval is below $0.9308$, and numerically about $0.849$. We give a general identity that expresses the variance defect of any mean-composition closure as an integrated covariance between population size and composition-weighted net growth, and closed low-count backward equations that compute corrected thresholds without solving the joint population law. Numerical checks reproduce the witness, scan the switching rates, exhibit the same mechanism with positive birth, death and day-scale switching in both states, and retain a source-linked preparation in which the closure performs well. The tail obstruction, the analytic box, the explicit scalar law and the repaired threshold are verified in Lean 4; the two process-identification bridges are proved conventionally and are identified as such. The witness is a synthetic persistent-state regime, not a fitted experiment.