Finite-batch selection between inherited chemical states under a shared limiting resource

A chemical difference between compartments can change how fast they grow without changing how many compartments of each kind there are: in an asynchronously dividing population, accumulated size and offspring number are out of phase. We prove a quantitative finite-batch enrichment theorem for an explicit four-species count reaction network in which every compartment grows by drawing on one shared finite precursor pool while its resident feeds and removal channels remain maintained. Starting from any $h$ high-state and $\ell$ low-state newborns in explicit, nonempty, chemically certified regions around two stationary compositions, the high-state cell-count odds at the nutrient hitting time exceed $e^{s}h/\ell$ by the deadline $T$, except on an event of probability at most $\mathcal E_{\mathrm{chem}}(N,M,T)+\mathcal E_{\mathrm{sel}}(N,s)+\mathcal E_{\mathrm{time}}(N,\gamma,T)$, with all three terms explicit. At $s=0.1$ and $T=8/\gamma$ the error is at most $M(32+4/(21\gamma))e^{-cN}+e^{-19N/500000}+e^{-N/2500}$ with $c=1/(1.024\times10^{21})$, and a sharper retained bound is $O_{M,\gamma}(Ne^{-3cN})$. No fitness parameter is imposed: the growth advantage is the certified difference in resident concentration of the growth species, and chemical fidelity of every descendant is proved rather than assumed. The proof combines a size-position exponential that controls chemical recovery across divisions without any recovery clock, ancestral size totals that survive complementary division unchanged, and a single finite stopped law under which all failure branches are bounded without independence assumptions. For two founders the success event forces a high-state frequency of at least $4/7$. The finite stopped-law theorem and its quantitative consequences are verified in Lean 4; the identification with an ongoing nonexplosive count process up to the stopping event is a conventional localization argument. The copy numbers for which the bound is informative are astronomically large, and the result is a theorem about a specified effective model, not an experimental calibration.