= Logarithmic tail-duration change at the zero-extension boundary
{title2=$L=\tau\max(0,\log R)$}
\b[Logarithmic tail-duration change at the zero-extension boundary] describes a recording requirement whose input ratio crosses the point where no further acquisition is needed. For fixed decay time $\tau>0$, put $R=A\tau/\varepsilon$ and $x=\log R$. The physical extension is
$$
L(x)=\tau\max(0,x).
$$
Here the logarithm applies to the input ratio, not to a zero output duration. <Logarithmic condition numbers of a tail-acquisition requirement> applies only where $L>0$.
At $x=0$, the left derivative is zero and the right derivative is $\tau$. The <directional derivative of a clipped logarithmic acquisition time> along a signed input change $h$ is
$$
\lim_{s\downarrow0}\frac{L(sh)-L(0)}s=\tau\max(0,h).
$$
Thus no single ordinary derivative describes threshold-crossing perturbations. Nevertheless, the duration is stable in absolute units:
$$
|L(R_2)-L(R_1)|\leq\tau\left|\log(R_2/R_1)\right|.
$$
This follows because the positive-part map has <Lipschitz continuity> with constant one.
For an original interval example, take $\tau=25\,\mathrm{ns}$ and a justified input-ratio range $0.99\leq R\leq1.02$. The corresponding duration range is $0\leq L\leq25\log1.02\approx0.4951\,\mathrm{ns}$. Its lower part is clipped at zero. Linearizing only on the zero-duration side would incorrectly predict no possible extension even though the allowed set crosses the threshold. A relative error divided by a nominal $L=0$ is undefined, while this absolute bound remains meaningful.
If $\tau$ is uncertain too, optimize the full expression over the justified joint parameter set rather than treating this fixed-$\tau$ inequality as a complete uncertainty analysis. The tail model must bound optical power throughout the missed interval. NIST's calibration apparatus measured detector/cable/oscilloscope response separately, illustrating why a recorded electrical decay cannot automatically certify such an optical envelope.<Calibration Service for Low-level Pulsed-Laser Radiometers at 1.06 µm: Pulse Energy and Peak Power by Rodney W. Leonhardt>[Appendix C.1–C.2, printed pp. 36–37]{ref} The clipping analysis is an original property of the stated conditional acquisition model.
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