Reserve capacity and reliable eradication in inherited-state populations

We study finite-time eradication of a branching population with inherited cellular states under a shared intervention that also damages a finite regenerative reserve. The mission requires separate bounds on target survival and on reserve loss at every time during treatment, not merely at the deadline. Our main probabilistic tool is an excursion bound anchored at a freely chosen return level below capacity. It yields an explicit finite-product reserve certificate, it accommodates incomplete initial filling, and we show it is asymptotically exact at high renewal. Composing it with source-specific weighted drift inequalities for a six-state inheritance model and a bounded administration course with uncertain clearance, we prove that a single course, uniform over independent one-percent relative errors in the target rates, gives target-survival probability below $0.00856$ and reserve-loss probability below $7\times10^{-6}$ from a $400$-unit reserve that need only be filled to $278$. The same course still meets both one-percent mission tolerances from any initial filling of $209$ units, nine above the failure threshold, and a $232$-unit reserve suffices at the same proportions. A capacity law with an explicit integer-rounding correction quantifies how additional reserve replaces fast renewal, while a complementary fluid-limit statement shows that capacity cannot compensate for an inadequate equilibrium reserve. We also prove the exact high-renewal limit $r_{\mathrm{eff}}^{\,d}\,\mathbb P_K(\tau_H\le T)\to(Km)^{d+1}T/d!$ for buffer $d$, and we state the limitations imposed by baseline loss and delayed replenishment. The probability arguments are conventional proofs; the decisive finite source and numerical inequalities are checked in Lean. The results are sufficient guarantees for an explicit synthetic source, not an optimal control frontier or a clinically calibrated regimen.