= Causal indistinguishability of delayed optical inputs on a finite record
{title2=$(q*h_j)(t)=0\quad(t\leq T<\inf\operatorname{supp}q)$}
\b[Causal indistinguishability of delayed optical inputs on a finite record] means that an observation ending at a fixed time cannot discriminate inputs whose differences have not yet reached its receivers. The <unobserved delayed-pulse counterexample to a fitted tail bound> proves this for one calibrated <convolution> channel. The same information limit holds for any finite set of causal linear channels.
Let channel $j$ have response $h_j$ with $h_j(t)=0$ for $t<0$, and record $y_j=P*h_j$ only for $t\leq T$. Every perturbation $q$ supported strictly after $T$ satisfies
$$
(q*h_j)(t)=0\qquad(t\leq T)
$$
for every channel. Thus the finite-record observation operator has a whole family of invisible input directions. This is compatible with nonnegative optical power: choose any nonnegative delayed pulse and compare $P$ with $P+q$. No cancellation between unphysical negative powers is required.
Known propagation or detector latency can hide part of the input even before the record ends. If $h_j(t)=0$ for $t<d_j$, an input starting at $S$ produces no response before $S+d_j$. The <input-time horizon of a delayed optical receiver record> therefore depends on the channel's time reference, not just the last stored sample time.
For an original example, suppose two ideal delay channels measure $y_1(t)=P(t-2\,\mathrm{ns})$ and $y_2(t)=P(t-7\,\mathrm{ns})$, both recorded through $T=100\,\mathrm{ns}$. A pulse starting at input time $99\,\mathrm{ns}$ first appears at output times $101$ and $106\,\mathrm{ns}$. Both stored records are unchanged by that pulse, even though it began before the nominal record endpoint. Additional causal channels with smaller delays can extend the input horizon; none can observe an input that begins after their recording ends.
Offline processing of identical stored records still gives identical results, even if the processing filter itself uses samples on both sides of each stored time. To distinguish the inputs, obtain additional observations or supply independently justified restrictions on later illumination. Exact knowledge of the receiver response alone supplies neither.
NIST's pulse calibration characterized the detector, cable and oscilloscope as a combined response chain using 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 measurement procedure]{ref} Such characterization is important for aligning the physical input and recorded output times. The two pure delays above are chosen models, not measured delays of that apparatus. A future-valid optical envelope or documented source cutoff remains additional information needed for an all-future missed-energy bound.
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