Conservative molecular inheritance and population risk: extinction and variance ordering, approximation limits, and regrowth

When a dividing cell distributes a conserved set of molecular marks between its daughters, the two sister states are dependent even though their later lineages evolve independently. We study a finite reader–writer chromatin motif with $N$ modifiable sites, explicit cell division and a state-dependent death hazard, and compare the exact complementary (conservative) allocation of old marks with the common simplification that draws the two daughters independently from the same marginal law. The two models have identical type-resolved mean dynamics. We prove that, for every $N$, every choice of nonnegative rate constants, constant division rate and any death hazard that is monotone in the natural order on mark states, complementary allocation gives the smaller extinction probability, the smaller population generating function at every time, and the smaller population variance; under irreducibility the extinction inequality is strict at every founder state whenever survival is possible. An exact identity writes the extinction difference as the conditional covariance of daughter continuation risks propagated by a nonnegative response operator, and positive-vector bounds make it quantitative. The raw pair laws are far apart in total variation, so any accuracy of the independence approximation is observable-specific. Exact rational certificates and a validated Taylor integration give, for a two-site source, a difference of more than $0.098$ in the probability of ever reaching $300$ cells and of more than $0.084$ in reaching $300$ cells by time $300$. For a $16$-site source with a sigmoidal hazard, a near-balanced founder has an extinction gap above $0.0128$ uniformly over the slope interval $[7,9]$, while the all-active founder has a gap below $3.6\times10^{-4}$. For a fixed smooth readout we prove that survival from balanced founders tends to zero as $N\to\infty$ at an explicit polynomial rate, so the certified finite-size contrast is a finite-size effect; a discontinuous readout behaves differently and is left open, as is state-dependent division, for which we exhibit an exact obstruction to the proof. We also derive what paired-clone and barcode observations can and cannot identify. Probability arguments are conventional; selected finite algebra is checked in Lean, and all numerical claims are exact rational comparisons. The examples are mechanistic and are not clinically calibrated.