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Active-set clipping in cost-weighted baseline-tolerance allocation enforces the reference caps that an interior solution of cost-weighted allocation of baseline-response tolerances may violate. Keep the parent's stipulated logarithmic effort modelHere $B_j{\delta }_j$ is a physical bias contribution in the common units of $E$. If $E\ge {\sum }_j{B}_j{\delta }_j^{\mathrm{ref}}$, no tightening is needed. Otherwise the budget binds, and the Karush-Kuhn-Tucker conditions giveThe positive Lagrange multiplier $\lambda$ prices scarce bias budget. For a set $A$ of capped components and remaining free set $F$, stationarity givesprovided the denominator is positive and the resulting choices agree with the proposed capped/free status. The monotonicity of the multiplier equation for capped tolerances permits a scalar search; strict convexity of the objective gives a unique allocation.

For an original cap-activating version of the parent's apparatus example, let $E=10\mathrm{nV}$, $a_1=1$, $a_2=3$, $B_1=0.01\mathrm{V}$, and $B_2=100\mathrm{V}{\mathrm{s}}^{-1}$. Set reference bias contributions $B_1{\delta }_1^{\mathrm{ref}}=1\mathrm{nV}$ and $B_2{\delta }_2^{\mathrm{ref}}=20\mathrm{nV}$. The unconstrained shares $2.5$ and $7.5\mathrm{nV}$ violate the first cap. Capping it leaves $9\mathrm{nV}$ for the second component, soMerely clipping the first share and leaving the second at $7.5\mathrm{nV}$ wastes $1.5\mathrm{nV}$ of permitted bias and unnecessarily increases the chosen effort. Reallocation lowers that effort by $3\log (9/7.5)\approx 0.547$ units.

The physical motivation remains leakage from residual offset and drift in a pulsed-radiometer baseline, documented in NIST's calibration apparatus.^[ref] The logarithmic cost and the numerical caps here are illustrative assumptions, not reported apparatus constants.

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 tolerance-budget residual bound for multiplier bisection controls the physical bias left unused by a numerical allocation, rather than only the numerical error in its Lagrange multiplier. Use $Q(\lambda )={\sum }_j\min (c_j,a_j/\lambda )$ from monotonicity of the multiplier equation for capped tolerances. Suppose a positive bracket $[\ell ,u]$ satisfies $Q(\ell )\ge E\ge Q(u)$. Choosing the upper endpoint gives a feasible allocation, with residualChoosing the lower endpoint can violate the budget. The direction of monotonicity therefore matters even when both endpoints are numerically close.

For an a priori bound, put $A={\sum }_j{a}_j$. On the entire bracket, each summand obeys Lipschitz continuity with constant at most $a_j/{\ell }^2$, including across its breakpoint. HenceAfter $m$ bisections of an original bracket $[{\ell }_0,u_0]$ with ${\ell }_0>0$, the corresponding worst-case bound is $A(u_0-{\ell }_0)/(2^m{\ell }_0^2)$. This is conservative; evaluating the two endpoint budgets can give a much tighter residual certificate.

For an original stopping example, use $(a_j)=(1,2,1)$, $(c_j)=(1,1,4)$ and $E=3$ in common normalized bias units. The bracket $[0.99,1.03]$ straddles the root. Its feasible upper endpoint leaves $3-Q(1.03)\approx 0.05825$ unused. One bisection evaluates $Q(1.01)<3$ and replaces the upper endpoint, reducing that unused amount to about $0.01980$. A residual allowance of $0.03$ is now met, despite a breakpoint lying inside the bracket.

This gives a thresholded stopping of a capped-tolerance multiplier search in the same units as the permitted bias. Floating-point comparisons also need an allowance for rounding if they are to serve as hard certificates. The underlying bias coefficients and cost model remain assumptions of cost-weighted allocation of baseline-response tolerances; numerical precision cannot validate missing physical terms. NIST's pulse-radiometer account motivates the offset/leakage constraints rather than prescribing this numerical stopping rule.^[ref]