Trustworthy decisions in stochastic amplification assays: timing limits, measurement design, and effective capacity

Changing a readout time, adding an identity-sensitive measurement, increasing the reagent reserve, and improving the inference procedure solve different problems, and a laboratory that confuses them can spend its effort on a step that cannot succeed. We separate these problems for a finite pure-birth amplification source with Poisson loading, a latching detection threshold and an effective resource capacity. For the source at threshold five, every measurable randomized classifier of the complete crossing time that keeps blank error at most $1\%$ misses more than $5.55\%$ of loaded reactions; this all-rule obstruction, its concrete source identities and a source-preserving hit-and-identity repair with blank error below $0.03\%$ and miss below $4.22443\%$ are verified in Lean 4. We then prove, with ordinary mathematics, that for any fixed sequential source with shared kinetics a deadline is optimal among all crossing-time classifiers; this converts every all-deadline exclusion into an all-classifier exclusion and sharpens the certified bound to $6.9\%$. Using the holding-time representation of the crossing time we show that the obstruction is not a small-count artefact: for every integer threshold $h\ge10^{6}$ with the resource exhausted at detection, every crossing-time classifier at $1\%$ blank error misses more than $5.63\%$, an explicit identity measurement misses less than $4.13\%$, and doubling the capacity restores a usable deadline with errors below $0.9373\%$ and $4.8732\%$. The proof replaces earlier Chebyshev allowances by gamma Chernoff and weighted Bernstein bounds and reduces the sufficient threshold from $10^{8}$ to $10^{6}$. For the mechanism question we exhibit two capacities with exactly the same blank endpoint law, prove that calibrated intermediate observations recover capacity through clock-free ordered block ratios with strictly monotone, uniquely invertible ordering probabilities, extend the block theorem to non-exponential factorized holding times, and show that thresholds chosen as fractions of each reaction’s own plateau lose the leading capacity information. A signal-to-state interval construction and a censoring-safe confidence procedure that retains every incomplete observation complete an executable analysis. All performance statements are conditional mathematical results; no physical channel, calibration or biochemical resource is asserted to have been measured.