= Validated lower bound on a power-law tail exponent
{title2=$-P'/P\geq\alpha_{\min}/(u+t_0)$}
A \b[validated lower bound on a power-law tail exponent] is an exponent constraint that is justified over the future interval where it is used to bound unrecorded energy. A fitted slope over a finite observation window is not by itself such a guarantee. <Acquisition duration from a power-law tail-energy tolerance> needs a global optical-power envelope, including a lower exponent limit above one for its finite-energy guarantee.
One sufficient condition, for positive absolutely continuous power, is
$$
-\frac{P'(b+u)}{P(b+u)}\geq\frac{\alpha_{\min}}{u+t_0}\quad\text{for almost every }u\geq0,
\qquad P(b)\leq A.
$$
Integrating the logarithmic derivative gives $P(b+u)\leq A(1+u/t_0)^{-\alpha_{\min}}$. The <time-dependent decay-rate envelope> provides the equivalent integrating-factor argument when zeros of the power need handling. This is a sufficient condition, not a necessary characterization of every bounded tail.
For an original conditional estimation example, assume an exact power-law family with known $t_0=10\,\mathrm{ns}$ and initial power $A=0.10\,\mathrm W$. At $u=90\,\mathrm{ns}$, a measured ratio range $0.008\leq P(b+u)/P(b)\leq0.012$ yields
$$
\frac{-\log0.012}{\log10}\leq\alpha\leq\frac{-\log0.008}{\log10},
$$
approximately $1.9208\leq\alpha\leq2.0969$. Within that assumed family, the lower endpoint gives a conservative extension about $123.32\,\mathrm{ns}$ for a $0.10\,\mathrm{nJ}$ remaining-energy tolerance. A nominal exponent two would instead require $90\,\mathrm{ns}$.
The ratio interval constrains the exponent only conditional on the exact family, known time scale and suitable measurement uncertainty. It cannot exclude a later secondary pulse or a transition to a slower law after the observed window. <Future-domain validity of a fitted power-law optical envelope> therefore requires independent physical justification or explicitly limited claims. If the justified exponent set reaches one, this envelope alone supplies no finite all-time energy guarantee.
Finally, the measured trace must represent optical power rather than an uncorrected receiver tail. NIST measured the detector, cable and oscilloscope together using a short characterized optical input before correcting pulse measurements.<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, Figure C1 and response-measurement procedure]{ref} The exponent example is an original model calculation, not an observed law attributed to that apparatus.
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