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The directional derivative of a clipped logarithmic acquisition time describes the first-order recording extension when several envelope parameters change together at the acquisition threshold. For positive $A,\tau ,\epsilon$, writeThe logarithmic tail-duration change at the zero-extension boundary article treats a fixed decay time and logarithmic input coordinate. Here consider a boundary point $A\tau =\epsilon$ and an additive parameter direction $\mathbf{h}=(a,t,e)$.

The one-sided directional derivative uses positive step sizes:The logarithmic chain rule gives the expression inside the maximum. Differentiating the prefactor contributes zero at the boundary because the clipped logarithm there is zero.

For an original example, let $A=0.10\mathrm{W}$, $\tau =10\mathrm{ns}$ and $\epsilon =1.0\mathrm{nJ}$. Increase amplitude in direction $a=0.002\mathrm{W}$ while decreasing decay time in direction $t=-0.10\mathrm{ns}$, with $e=0$. The net fractional direction is $0.02-0.01=0.01$, so $D^{+}L=0.10\mathrm{ns}$ per unit step. Reversing the direction gives derivative zero, not $-0.10\mathrm{ns}$.

The derivative is positively homogeneous in the direction but is not linear, so no single gradient represents all threshold-crossing perturbations. Above threshold the usual smooth derivative applies; strictly below threshold all sufficiently small changes leave the extension zero. Gaussian input uncertainty at a clipped acquisition-time threshold can consequently produce a mass of outcomes exactly at zero rather than a symmetric distribution around a nominal duration. The directional calculation alone does not assign a confidence level.

This is a conditional optical-envelope calculation. Separating optical power from the receiver's electrical response remains necessary before its parameters describe missed light. NIST independently characterized the detector/cable/oscilloscope response in its pulse calibration procedure.^[ref] A derivative can describe parameter sensitivity accurately while the envelope itself still needs experimental justification.

Logarithmic condition numbers of a tail-acquisition requirement quantify the fractional response of a positive recording extension to fractional parameter changes. For $L=\tau \ell$ with $\ell =\log (A\tau /\epsilon )>0$, define the signed sensitivity ${\kappa }_x=\partial \log L/\partial \log x$. ThenTheir absolute values are componentwise relative condition numbers. Acquisition-time sensitivity to tail-model parameters gives the corresponding dimensional derivatives and uncertainty propagation.

These numbers depend on which coordinates are held fixed. If $M=A\tau$ denotes the initial tail-energy allowance, then $L=\tau \log (M/\epsilon )$. In coordinates $(M,\tau ,\epsilon )$, the decay-time sensitivity at fixed $M$ is exactly one, rather than $1+1/\ell$. This is the same physical formula: holding $M$ fixed while changing $\tau$ also changes $A$. Transforming the parameter covariance consistently leaves the propagated output uncertainty unchanged.^[ref]

An original boundary example takes $A\tau /\epsilon =1.01$ and $\tau =20\mathrm{ns}$. The required extension is only $L\approx 0.1990\mathrm{ns}$, but ${\kappa }_A\approx 100.50$. Increasing $A$ by $1\%$ multiplies the ratio by $1.01$, giving $1.0201=(1.01{)}^2$ and doubling the required extension exactly. Its relative change is large because the original extension was very small.

This does not mean the absolute requirement is singular there. In the active branch $\partial L/\partial \log (A\tau /\epsilon )=\tau$ remains finite. At and below the threshold, the physical extension is zero, so $\log L$ is undefined. A logarithmic tail-duration change at the zero-extension boundary must instead use the piecewise requirement or absolute changes. Large relative conditioning alone is not evidence of a large absolute recording-time risk.

Finally, conditioning describes the stated envelope model, not its experimental validity. The optical power tail must be distinguished from detector response, which NIST measured independently in its pulse calibration procedure.^[ref] A well-conditioned calculation based on an invalid optical envelope still gives an unjustified acquisition guarantee.