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The input-time horizon of a delayed optical receiver record is the latest physical input time that could influence the recorded output, given the receiver's causal onset delay. If $h(t)=0$ for $t<d$ and the last output time is $T$, convolution depends only on input timesThus $T-d$ is a necessary timing limit on information in that record. Recoverability within the limit additionally depends on bandwidth, noise, sampling and the response's invertibility. Causal indistinguishability of delayed optical inputs on a finite record shows why later input changes cannot be inferred from the existing output samples.

For several channels with last recorded times $T_j$ and known onset delays $d_j$, the latest possible input horizon isAny input perturbation starting strictly after $H$ is invisible in every stored channel. A channel's latest output timestamp alone is insufficient to compare its input horizon with another channel's.

For an original example, channel 1 records through $120\mathrm{ns}$ with delay $15\mathrm{ns}$, while channel 2 records through $100\mathrm{ns}$ with delay $2\mathrm{ns}$. Their input limits are $105$ and $98\mathrm{ns}$; the combined limit is $105\mathrm{ns}$. Knowing both response functions exactly still supplies no observation of an input pulse beginning later than that limit.

A receiver-latency uncertainty in the inferred input-time horizon also changes the timing claim. If channel 1's delay is known only to lie between $14$ and $16\mathrm{ns}$, its possible horizon ranges from $104$ to $106\mathrm{ns}$. An input beginning at $104.5\mathrm{ns}$ could affect the record for one allowed delay but arrive too late for another. Using the conservative endpoint $104\mathrm{ns}$ avoids claiming timing access that depends on an unresolved latency. It still does not guarantee successful reconstruction of every earlier input.

NIST characterized detector, cable and oscilloscope as a combined response chain using a short known optical pulse.^[ref] Such a calibration must relate output timing to the physical input time origin. The delays above are illustrative, not quoted apparatus values. De-embedding known latency aligns the record's time reference; it cannot extend observations into an unrecorded future interval.

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.