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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 $y_j=P\ast h_j$ only for $t\le T$. Every perturbation $q$ supported strictly after $T$ satisfiesfor 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.^[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.

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 $0\le u\le T$. Choose a later time $S>T$ and define an additional pulseBoth $P_0$ and $P_1=P_0+q$ are nonnegative and continuously differentiable, and they agree exactly on the recorded interval. Yet the unobserved extra energy isThus 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 $A{t}_0/(1+T/t_0)=1/11\mathrm{nJ}\approx 0.0909\mathrm{nJ}$, 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.^[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.