= Time-dependent decay-rate envelope
{title2=$P(t)\leq A e^{-\int_b^t k(s)\,ds}$}
A \b[time-dependent decay-rate envelope] bounds a nonnegative signal using a lower bound on its instantaneous decay rate. If $P$ is <absolutely continuous> on finite intervals, $P(b)\leq A$ and $P'\leq-k(t)P$ almost everywhere for locally integrable $k(t)\geq0$, then
$$
P(t)\leq A\exp\!\left[-\int_b^t k(s)\,ds\right].
$$
The <differential inequality for an exponential tail envelope> article proves this comparison using an <integrating factor> and gives a piecewise-constant example. A strictly positive constant lower bound on $k$ gives exponential decay, but a positive rate at every instant need not supply such a lower bound.
An original example takes $k(t)=\alpha/(t-b+t_0)$ with $t_0>0$ and $\alpha>0$. Its integrated rate is logarithmic, so
$$
P(t)\leq A\left(1+\frac{t-b}{t_0}\right)^{-\alpha}.
$$
This <power-law tail from a decaying rate envelope> tends to zero for every positive $\alpha$. Its envelope energy is nevertheless finite only when $\alpha>1$, in which case
$$
U_{\rm tail}\leq\frac{At_0}{\alpha-1}.
$$
For original chosen $A=0.20\,\mathrm W$, $t_0=5\,\mathrm{ns}$ and $\alpha=2$, the allowance is $1.0\,\mathrm{nJ}$. After a further $45\,\mathrm{ns}$ it is at most $0.10\,\mathrm{nJ}$. The remaining energy decreases algebraically rather than exponentially.
At $\alpha=1$ the integrated envelope diverges logarithmically despite its power tending to zero. This does not prove that every signal below it has infinite energy; it means these assumptions alone supply no finite all-time energy bound. Taking the envelope itself gives a mathematical counterexample to the claim that pointwise decay implies integrable energy.
An experimental rate law therefore needs both temporal validity and an integrability check before it bounds missed optical energy. A measured detector tail is not automatically the true power profile; NIST's primary calibration procedure measures the receiver response separately.<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 power-law example is an independent comparison model, not a reported tail law of that apparatus.
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