 Unobserved delayed-pulse counterexample to a fitted tail bound

An unobserved delayed-pulse counterexample to a fitted tail bound constructs two admissible optical inputs giving the same finite measurement record but different missed energies. The future-domain validity of a fitted power-law optical envelope article gives a nonnegative, continuously differentiable delayed pulse with adjustable energy. The counterexample persists through a calibrated causal detector, not just an ideal instantaneous power readout.

Let $P_1=P_0+q$, where the additional pulse has support $[S,S+w]$ after the record ends at $T<S$. A linear receiver with impulse response $h$ and responsivity $g$ measures $y_i=g(P_i\ast h)$. Causality means $h(t)=0$ for $t<0$. Therefore, for every recorded time $t\le T$,since every argument $t-u$ is negative. This causal indistinguishability of delayed optical inputs on a finite record holds even when the impulse response is known exactly and the recorded data are noiseless.

For an original example, record through $T=80\mathrm{ns}$ and place the smooth sine-squared pulse of the parent article at $S=150\mathrm{ns}$, with width $w=40\mathrm{ns}$ and peak $B=0.020\mathrm{W}$. Its extra optical energy is $Bw/2=0.40\mathrm{nJ}$. A chosen causal exponential receiver with decay time $5\mathrm{ns}$ produces no difference in the initial record, although it will respond when the later pulse arrives. If its impulse response has unit area, the complete later output-area difference is $g\times 0.40\mathrm{nJ}$ in the receiver's output-area units.

Any rule using only the common initial record must return the same proposed bound for both inputs. Without a justified restriction on later illumination, the delayed pulse energy can be chosen larger than any proposed finite guarantee. More accurate calibration of that same initial record cannot resolve this information gap.

NIST characterized its detector, cable and oscilloscope together with a short known optical input before correcting pulse measurements.^[ref] Such response calibration addresses receiver distortion. Source shutoff, exclusion of later pulses or a future-valid rate envelope supplies the additional physical information needed for a missed-energy guarantee. The delayed pulse here is a mathematical construction, not a reported defect of that apparatus.