= Response-normalized tolerance for baseline-feasibility residuals
{title2=$\eta=\max_j|r_j|/\delta_j$}
A \b[response-normalized tolerance for baseline-feasibility residuals] measures a pulse estimator's constraint errors against physically chosen response limits. For residual $\mathbf r=F^T\widehat{\mathbf a}-\mathbf b$, assign positive tolerances $\delta_j$ in the corresponding response units and calculate
$$
\eta=\max_j\frac{|r_j|}{\delta_j}.
$$
The criterion $\eta\leq1$ means every specified response limit is satisfied. <Numerical feasibility residual for baseline constraints> explains why a raw unscaled norm alone has no universal physical interpretation.
These tolerances can enforce a deterministic bias budget. If signal and background coefficients are bounded by $|\theta|\leq B_s$ and $|\beta_j|\leq B_j$, then
$$
|\operatorname{Bias}(\widehat\theta)|\leq B_s|r_s|+\sum_jB_j|r_j|
\leq B_s\delta_s+\sum_jB_j\delta_j.
$$
The second inequality follows when the normalized test passes. The <allocation of a deterministic bias budget among baseline constraints> therefore connects a numerical acceptance criterion to a stated physical error allowance.
For an original example, bound the signal amplitude by $1\,\mathrm{mV}$, constant background by $20\,\mathrm{mV}$ and drift rate by $100\,\mathrm{V/s}$. Choose signal-response tolerance $2\times10^{-6}$, constant-response tolerance $2\times10^{-7}$ and linear-time-response tolerance $4\times10^{-11}\,\mathrm s$. Their maximum bias contributions are respectively $2$, $4$ and $4\,\mathrm{nV}$, totaling $10\,\mathrm{nV}$. If all actual residuals are half these tolerances, $\eta=0.5$ and the corresponding bias bound is $5\,\mathrm{nV}$.
The normalized test can be invariant to constraint units. Multiplying one equation and its required response by a nonzero factor $q_j$ multiplies its residual by $q_j$; replacing its tolerance by $|q_j|\delta_j$ keeps the ratio unchanged. The associated coefficient bound changes inversely, so the physical product $B_j\delta_j$ is unchanged as well.
NIST's before/after pulse baselines supply a measurement setting where residual offset and drift responses matter.<Calibration Service for Low-level Pulsed-Laser Radiometers at 1.06 µm: Pulse Energy and Peak Power by Rodney W. Leonhardt>[§3.5, printed p. 8; §7.2, Leakage Effect]{ref} The example bounds and tolerances are stipulated, not specifications of that apparatus. This deterministic leakage test is separate from noise <variance>, which needs the actual measurement <covariance>.<Guidelines for Evaluating and Expressing the Uncertainty of NIST Measurement Results by Barry N. Taylor and Chris E. Kuyatt>[Appendix A, equation A-3]{ref} Backgrounds outside the assumed family also need separate treatment.
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