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Cost-weighted allocation of baseline-response tolerances chooses how tightly to enforce each baseline constraint when different tolerances require different effort. The allocation of a deterministic bias budget among baseline constraints gives the physical restriction ${\sum }_j{B}_j{\delta }_j\le E$. An optimization also needs an explicit cost model; the bias bound alone does not choose a unique allocation.

For one illustrative model, let $0<{\delta }_j\le {\delta }_j^{\mathrm{ref}}$ and minimizeTighter tolerances cost more, and the logarithm compares quantities in matching units. This is a stipulated convex effort model, not an empirical law for every numerical solver or measurement procedure.

When the budget binds and no reference upper bound is active, a constraint multiplier gives $-a_j/{\delta }_j+\lambda B_j=0$. ThereforeThe positive curvature $a_j/{\delta }_j^2$ establishes the unique interior minimum. A more expensive-to-tighten condition receives more of the allowed physical bias, while its coefficient bound converts that share to response units.

For an original two-constraint example, take $E=10\mathrm{nV}$, offset bound $B_1=10\mathrm{mV}$, drift-rate bound $B_2=100V/s$ and costs $a_1=1$, $a_2=3$. The optimum allocates $2.5$ and $7.5\mathrm{nV}$, giving ${\delta }_1=2.5\times 10^{-7}$ and ${\delta }_2=7.5\times 10^{-11}\mathrm{s}$. If their reference limits are $10^{-5}$ and $10^{-8}\mathrm{s}$, both are interior. Compared with equal physical bias shares, this optimum reduces the chosen cost by $\log (1/2)+3\log (3/2)\approx 0.523$ cost units.

Active-set clipping in cost-weighted baseline-tolerance allocation is needed if an interior solution exceeds a reference limit. If all reference limits already satisfy the bias budget, no tightening is needed in this cost model. Otherwise cap active components and redistribute the remaining budget among the others.

NIST's pulse-energy baselines motivate controlling residual offset and drift responses.^[ref] The selected tolerances control deterministic leakage within the stated coefficient bounds. Noise variance and unmodelled background shapes remain separate parts of the measurement analysis.

A response-normalized tolerance for baseline-feasibility residuals measures a pulse estimator's constraint errors against physically chosen response limits. For residual $\mathbf{r}=F^T\hat{\mathbf{a}}-\mathbf{b}$, assign positive tolerances ${\delta }_j$ in the corresponding response units and calculateThe criterion $\eta \le 1$ 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 $|\theta |\le B_s$ and $|{\beta }_j|\le B_j$, thenThe 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 $100V/s$. Choose signal-response tolerance $2\times 10^{-6}$, constant-response tolerance $2\times 10^{-7}$ and linear-time-response tolerance $4\times 10^{-11}\mathrm{s}$. 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.^[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.^[ref] Backgrounds outside the assumed family also need separate treatment.