= Future-domain validity of a fitted power-law optical envelope
{title2=$P(u)\leq A(1+u/t_0)^{-\alpha}\quad(u\geq0)$}
\b[Future-domain validity of a fitted power-law optical envelope] means that the bound applies throughout the unobserved future interval where its integral is used, rather than merely matching a recorded waveform. The <validated lower bound on a power-law tail exponent> supplies a sufficient differential condition for such a bound. Agreement on a finite observation window alone does not establish it.
A concrete counterexample uses a fitted profile $P_0(u)=A(1+u/t_0)^{-2}$ observed only for $0\leq u\leq T$. Choose a later time $S>T$ and define an additional pulse
$$
q(u)=\begin{cases}
B\sin^2\bigl(\pi(u-S)/w\bigr),&S\leq u\leq S+w,\\
0,&\text{otherwise},
\end{cases}
\qquad B,w>0.
$$
Both $P_0$ and $P_1=P_0+q$ are nonnegative and continuously differentiable, and they agree exactly on the recorded interval. Yet the unobserved extra energy is
$$
\int_S^{S+w}q(u)\,du=\frac{Bw}{2}.
$$
Thus nonnegativity, this degree of smoothness and even a perfect finite-window fit cannot exclude an arbitrarily large later contribution.
For an original example, take $A=0.10\,\mathrm W$, $t_0=10\,\mathrm{ns}$ and $T=100\,\mathrm{ns}$. The fitted profile predicts remaining energy $At_0/(1+T/t_0)=1/11\,\mathrm{nJ}\approx0.0909\,\mathrm{nJ}$, below a $0.10\,\mathrm{nJ}$ tolerance. Put the later pulse at $S=300\,\mathrm{ns}$ with width $w=200\,\mathrm{ns}$ and peak $B=0.010\,\mathrm W$. It adds $1.0\,\mathrm{nJ}$, although all recorded data are unchanged. This <unobserved delayed-pulse counterexample to a fitted tail bound> violates the claimed missed-energy tolerance without altering the fit quality.
To use the envelope as a guarantee, state the physical assumptions that exclude such later input, justify the rate bound over the required future interval, or limit the claim to the interval actually validated. Recording a longer finite window improves evidence but does not alone supply an all-future theorem under only the assumptions used in this counterexample.
The waveform must also represent optical power rather than detector response. NIST's primary pulse calibration separately characterized the detector/cable/oscilloscope chain with a short optical input.<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} Correcting the receiver is necessary for interpreting the trace, but it does not rule out the additional source pulse constructed here. The example is a mathematical comparison, not a reported failure of that apparatus.
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