= Tolerance-budget residual bound for multiplier bisection
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 residual
$$
0\le E-Q(u)\le Q(\ell)-Q(u).
$$
Choosing 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. Hence
$$
Q(\ell)-Q(u)\le\frac{A}{\ell^2}(u-\ell).
$$
After $m$ 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 $3-Q(1.03)\approx0.05825$ 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.<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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