= Thresholded stopping of a capped-tolerance multiplier search
<Thresholded stopping of a capped-tolerance multiplier search> ends a numerical allocation when a feasible physical budget is sufficiently well used. With the decreasing budget function $Q(\lambda)$ in <tolerance-budget residual bound for multiplier bisection>, an exact upper bracket endpoint $u$ satisfies $Q(u)\le E$. A residual requirement is $E-Q(u)\le\zeta$, where $\zeta>0$ has the same physical units as the permitted deterministic bias $E$. A small change in $u$ alone does not establish that condition.
Numerical evaluation also needs a bound. Suppose the reported value $\overline Q$ has validated absolute error $|Q-\overline Q|\le\epsilon_Q$. Both feasibility and the residual requirement are certified if
$$
\overline Q+\epsilon_Q\le E,\qquad
E-(\overline Q-\epsilon_Q)\le\zeta.
$$
Equivalently, the acceptable reported value lies in
$$
E-\zeta+\epsilon_Q\le\overline Q\le E-\epsilon_Q.
$$
This window is empty if $\zeta<2\epsilon_Q$. Iterating the multiplier more accurately cannot by itself overcome an evaluation-error bound larger than the desired physical certificate.
For an original numerical certificate, let $E=10\,\mathrm{nV}$, $\zeta=0.10\,\mathrm{nV}$ and $\epsilon_Q=0.02\,\mathrm{nV}$. Values $9.92\le\overline Q\le9.98\,\mathrm{nV}$ are certifiable. A reported $9.95\,\mathrm{nV}$ places the actual budget in $[9.93,9.97]\,\mathrm{nV}$, so it is feasible and leaves at most $0.07\,\mathrm{nV}$ unused. A reported $9.99\,\mathrm{nV}$ is closer to ten but could represent $10.01\,\mathrm{nV}$ and therefore does not certify feasibility.
The evaluation allowance must itself be justified, for example by validated arithmetic or an explicit forward-error analysis. Uncertain physical coefficients also need their stipulated worst-case bounds before the numerical sum can certify the physical error. NIST's pulse-radiometer discussion motivates the offset and leakage terms; it does not provide the hypothetical numerical tolerances above.<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}
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