Exposure and amplitude limits for eradicating populations with inherited cellular states

How much intervention is necessary to eradicate a finite population whose cells inherit a molecular state, when the intervention may be chosen using the entire population history? We prove two independent obstructions for finite-type branching populations under bounded predictable feedback. The first is an exposure floor: a baseline extinction supersolution together with a componentwise bound on the intervention generator produces a family of supersolutions indexed by the remaining budget, and the resulting population barrier retains founder-count dependence without assuming that families are independent under feedback; for $n$ identical founders it gives survival probability at least $1-(1-\eta e^{-cB})^n$. The second is an amplitude floor: a single supersolution valid across the whole admitted action range bounds survival away from zero for every policy, at every budget and every horizon. Necessary and sufficient exposures match at order $\log(n/\delta)$ above the amplitude threshold, and an outward-rounded Taylor calculation certifies a finite-time policy at exposure $11.6$ with risk below $0.01$ from one founder of the two-site molecular source. We then locate the amplitude threshold exactly in that source as a function of memory size: the critical intrinsic erasure is bracketed to six decimals for $2\le N\le 8$ sites and rises from $.0927$ to $.4506$, so at five to eight sites no policy of amplitude $.29$ eradicates, whatever its budget or duration; explicit rational supersolutions bound the surviving probability below by $.17$, $.46$, $.61$ and $.70$ at $N=5,\dots,8$. A second actuator repairs this: added death on protected states has an $N$-uniform threshold $b-d_{\mathrm{prot}}$, gives a dimension-free sufficient exposure $1.45\log(n/\delta)$, and attains the exposure floor up to a factor below $1.54$. We also record a preparation-dependent failure of monotonicity in erasure, and conditional concentration and administered-amount lower bounds. Probability proofs are conventional; source certificates and the finite-time enclosure are exact finite computations; selected algebra is checked in Lean.