Repeated selection of inherited chemical states: finite-population guarantees and a logarithmic horizon law

Serial transfer is the standard way to demonstrate selection in a population of chemical compartments, but each transfer discards most of the population, and a state that has become rare can be lost outright to sampling rather than to chemistry. We prove finite-population guarantees for an arbitrary prescribed number of such transfers in a fully specified four-species compartment model. The protocol grows a population on a shared precursor, samples exactly $M$ endpoint compartments uniformly and intact, recovers those same compartments, and refills in proportion to their actual retained sizes; a finite marked probability law retains every failure branch and every returned division phase, and nothing is conditioned on success or resampled. For every horizon $K$ and every confidence below one there are finite parameters under which both inherited states remain present at every census while the measured high-to-low compartment-count log odds exceed $Kg-\log2$, with $g=\frac35\log4-\frac{19}{500}-\log\frac{51}{49}=0.75377\ldots$. The mission transfer error is governed by a geometric series whose base is the per-cycle loss of minority share, and the attainable horizon for this event has order $\log M$. We then close part of the gap between the sufficient and necessary constants. A machine-checked exponential-generator inequality is lifted to a batch statement showing that both ancestral size totals grow by a factor at least $3/2$, which improves the geometric base from $204/49=4.1633$ to $136/49=2.7755$; a multiplicative concentration bound for weighted uniform sampling without replacement replaces a Chebyshev tail. Together these move the sufficient leading coefficient from $1/\log(204/49)=0.7011$ to $1/\log(136/49)=0.9796$ against a necessary $1/g=1.3267$, and reduce the population sufficient for a ten-cycle mission from $M=10^{13}$ to $M=4\times10^{9}$ at higher confidence. Machine-checked statements, conventional proofs and numerical diagnostics are separated throughout, and the optimal exponential rate remains open.