Reliable predictions from small cell populations: preparation dependence, measurement and certified count bounds

Prediction from small cell populations depends on how the population was prepared and how it is observed, not only on the dynamics of individual lineages. We study upper prediction regions $\{0,\dots,k\}$ for the cell count at a fixed horizon in two-type branching models with inherited states, and ask which observations and assumptions justify a given cutoff $k$. First, we construct two founder preparations, one drawing a growth class independently for each founder and one giving every founder in a well the same class, that have identical observation laws for every single-founder experiment yet different minimal $95\%$ two-founder cutoffs, six and seven. Ordered-endpoint error accounting yields finite-sample marginal coverage for a data-selected cutoff without independence between calibration and the future population; an exact binomial calculation attains coverage at least $245/256$ from $640$ independent calibration wells, and a continuous preparation family identifies $\rho=3/5$ as the dependence level at which the minimal cutoff changes. Second, independent cell detection can erase or reverse the signal of a low-count statistic at efficiency one half, while a bounded generating-function statistic separates the preparations for every positive efficiency; thinning-invariant moment identities identify an unknown constant efficiency together with the preparation, and a feasible-set rule with bounded statistics keeps the coverage guarantee when the efficiency is unresolved. Third, one-sided stochastic comparisons give uniform recorded-count bounds under demographic, founder and false-object uncertainty: recorded count at most four with probability at least $0.9503$ for all birth rates up to $0.105$ and all death rates at least $0.05$ per time unit at horizon seven, and at most five with probability at least $0.9567$ for arbitrary death and switching rates when births are at most $0.1$. Fourth, for a reference inherited-state rate class with per-type $\ell^1$ uncertainty, a finite successful-history certificate proves recorded count at most three with probability at least $0.955935$, with three the smallest endpoint valid over the class. We separate theorems checked in Lean 4 from conventional extensions and exact computations, and report a documented lineage-data analysis that does not establish biological coverage. The results give explicit conditions for trustworthy count predictions and for measurements that justify smaller prediction regions.