Finite-resource amplification can eliminate diagnostic readout windows: certified small-threshold instances and large-threshold regimes

An amplification reaction may become sensitive enough only after its background-positive probability has already exceeded the permitted limit. We study this timing conflict for an explicit finite-resource birth–immigration source: $z$ active units grow at rate $(b+gz)(1-z/R)$ until an aggregate detection threshold $h$ is reached, a blank starts empty, and a loaded reaction starts with a Poisson number of units. A deadline is acceptable when the blank-positive probability is at most $1\%$ and the missed-call probability at most $5\%$. At threshold five, Poisson mean four, $b=0.01$ and $g=1$ per minute, capacities five, six and seven admit no acceptable deadline at all, capacity eight admits one but none on a $0.1$-minute observation grid, and capacity nine is the least grid-feasible capacity. Capacity ten is feasible at every deadline in $[3.15,3.30]$ minutes, and remains feasible for every fixed rate vector within $2\%$ of nominal, every loading mean at least $3.99$ and every actual deadline in $[3.19,3.21]$ minutes. These statements, including the source law, the monotonicity arguments and the rational exponential enclosures, are verified in Lean 4. A uniform rescaling of all rates cannot create a window; a state-dependent resource change can. We then analyse arbitrarily large thresholds through the independent holding times of the count chain and the beta–gamma algebra. If the capacity exceeds the threshold by a margin that grows without bound, depletion becomes a common additive delay and the undepleted design rules apply after re-centering. If the margin is a fixed number $m$ of units, the last births near exhaustion add an independent $\mathop{\mathrm{Gamma}}(m+1)$ delay in logarithmic time, and the limiting error tradeoff depends only on $b/g$, the loading mean and $m$: with these parameters, at most two units of headroom leave no window, three or more do. For every threshold $h\ge10^{8}$ we prove, with explicit constants and without any $h$-state computation, that capacity $h$ has no acceptable deadline while capacity $2h$ has one at an explicit time; the certified instances at threshold five are therefore not an artefact of small counts or of reactions that are positive at time zero. Numerical Laplace inversion at $h=10^{8}$ confirms the bounds. The large-threshold results are conventional proofs and are identified as such. The model is an effective one; no clinical calibration or specific molecular capacity intervention is established.