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Allocation of a deterministic bias budget among baseline constraints assigns how much residual signal-response and background-leakage error each constraint may contribute. Let the allowed total bias be $E>0$, and let coefficient bounds $B_j>0$ multiply response errors $r_j$. Choose nonnegative fractions $f_j$ with ${\sum }_j{f}_j\le 1$ and setIf every computed response obeys $|r_j|\le {\delta }_j$, the triangle inequality gives $|\mathrm{Bias}|\le {\sum }_j{B}_j{\delta }_j\le E$. The response-normalized tolerance for baseline-feasibility residuals checks those component limits. Positive tolerances require positive allocated fractions; a zero allocation demands exact cancellation in the stated model.

For an original example, take $E=12\mathrm{nV}$ and bounds of $2\mathrm{mV}$ for signal amplitude, $10\mathrm{mV}$ for constant background and $100V/s$ for linear drift. Allocate fractions $1/6$, $1/3$ and $1/2$. The respective bias shares are $2$, $4$ and $6\mathrm{nV}$, giving tolerancesEqual bias shares would instead allow $4\mathrm{nV}$ each, relaxing the signal-response tolerance to $2\times 10^{-6}$ while tightening the drift tolerance to $4\times 10^{-11}\mathrm{s}$. Both allocations guarantee the same total bound when their assumptions hold; their numerical and experimental demands differ.

An allocation should reflect attainable constraint accuracy and the cost of tightening each response. Cost-weighted allocation of baseline-response tolerances can formalize that tradeoff. Equal numerical tolerances across differently dimensioned conditions do not mean equal physical bias contributions. If a bound $B_j$ is uncertain, use a justified upper bound or include its uncertainty explicitly before calling the total a guarantee.

NIST's pulse-energy apparatus used before/after measurements for baseline and drift correction.^[ref] The bounds above are illustrative design choices. This budget controls deterministic leakage within the assumed background family. It is separate from measurement-noise variance, whose covariance contributions must still be propagated.^[ref]

A numerical feasibility residual for baseline constraints measures how closely computed pulse-estimator weights satisfy their requested responses. For $F^T\mathbf{a}=\mathbf{b}$, evaluateThis is not the same diagnostic as the projected unattainable response in null-space feasibility test for baseline-estimator constraints. That test asks whether a solution exists at the chosen matrix rank; the residual checks the weights actually returned.

Small numerical components need physical scaling. If the measurement model is $\mathbf{x}=\theta \mathbf{s}+{\sum }_j{\beta }_j{\mathbf{h}}_j+\mathbf{ϵ}$, an estimator intended to preserve the signal and reject the backgrounds instead has biasThus an unweighted Euclidean residual norm is not a universal error bound. Coefficient magnitudes and the units of each constraint matter.

For an original example, suppose the constant-background residual is $r_0=10^{-6}$ and the linear-time residual is $r_1=10^{-12}\mathrm{s}$. A background offset $1\mathrm{mV}$ contributes $1\mathrm{nV}$ through the first residual. A drift rate $1mV/\mu \mathrm{s}=10^{3}V/s$ also contributes $1\mathrm{nV}$ through the second. The much smaller raw numerical value is not a much smaller physical leakage. Expressing time in microseconds changes $r_1$ to $10^{-6}\mu \mathrm{s}$ without changing that leakage.

A response-normalized tolerance for baseline-feasibility residuals can specify allowed response errors ${\delta }_j>0$ in matching units and check $\eta ={\max }_j|r_j|/{\delta }_j\le 1$. When coefficient bounds $|\theta |\le B_s$, $|{\beta }_j|\le B_j$ are justified, the corresponding bias bound is $B_s|r_s|+{\sum }_j{B}_j|r_j|$. A noise-minimizing estimator still needs a separate variance calculation from the actual covariance.^[ref]

NIST's before/after pulse baselines provide the physical setting for rejecting background and drift.^[ref] Report the chosen constraint scaling and tolerated leakage. A small solver residual alone establishes neither low variance nor negligible bias for every possible background.