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Channel switching under uncertainty in a combined receiver horizon occurs when uncertainty changes which receiver has the latest accessible input time. For channel $j$, let $H_j=T_j-d_j$ be its last output time minus its effective input-to-output offset. The combined horizon is $H={\max }_j{H}_j$, as in receiver-latency uncertainty in the inferred input-time horizon. Away from ties a small perturbation may preserve the maximizing channel; near a tie it need not. Propagating only that channel's variance omits the switching operation.

An original two-channel model makes the effect exact. Suppose $H_1$ and $H_2$ are independent normal random variables with common mean $\mu$ and standard deviation $s>0$. Independence giveswhere $\Phi$ is the standard normal distribution cumulative distribution function. This square is already enough to show that $H$ is not normal: $\mathrm{Pr}(H\le \mu )=1/4$.

To calculate the resulting bias relative to the common nominal horizon, write $H=(H_1+H_2+|H_1-H_2|)/2$. The difference has normal distribution with variance $2s^2$, hence $E|H_1-H_2|=2s/\sqrt{\pi }$. The sum and difference are independent in this model, givingFor $\mu =100\mathrm{ns}$ and $s=1\mathrm{ns}$, the combined mean is about $100.564\mathrm{ns}$ and its standard deviation is about $0.826\mathrm{ns}$. The formulas concern the uncertainty distribution of the horizon, not a promise of extra observable future illumination. The normal cdf/ppf routines provide direct evaluation of the probabilities.^[ref]

Shared clock or calibration errors invalidate the independence product. The distribution of the maximum of correlated receiver horizons must retain the joint timing model; NIST's general uncertainty guidance retains input covariance and requires distributional justification for coverage.^[ref] For hard bounds instead, $H_1\in [99,101]\mathrm{ns}$ and $H_2\in [100,102]\mathrm{ns}$ imply $H\in [100,102]\mathrm{ns}$ without selecting a fixed winning channel. Neither calculation establishes invertibility of the physical receiver.

A common-clock contribution to a maximum receiver horizon is a residual timing error that survives in every channel's inferred input-time horizon. Write $H_j=h_j+C+{\epsilon }_j$, with common error $C$ independent of the remaining channel errors. Then the identityshows that selecting the latest channel does not average away the common term. With finite variances,Here $C$ means a component remaining after each channel's subtraction $T_j-d_j$. A clock contribution that already cancels inside that subtraction must not be added again; see receiver-latency uncertainty in the inferred input-time horizon.

For two equal nominal horizons $h_j=\mu$, take independent channel errors ${\epsilon }_j\sim N(0,s^2)$ and $EC=0$. The independent maximum moments derived in channel switching under uncertainty in a combined receiver horizon giveIf the common error is also normal, each channel has variance ${\sigma }_C^2+s^2$ and their correlation is ${\sigma }_C^2/({\sigma }_C^2+s^2)$, recovering the equal-channel model in distribution of the maximum of correlated receiver horizons.

For an original example, choose ${\sigma }_C=0.8\mathrm{ns}$ and $s=0.3\mathrm{ns}$. The combined mean is about $0.1693\mathrm{ns}$ above $\mu$, and its standard deviation is about $0.8375\mathrm{ns}$. Yet the difference between channel horizons has standard deviation only $\sqrt{2}s\approx 0.4243\mathrm{ns}$, because $C$ cancels there. Good agreement between channels therefore need not establish a precise absolute horizon.

A timing experiment needs an independently justified reference for the common component, in addition to channel-to-channel alignment observations. Shared-input covariance is explicitly retained in NIST's uncertainty law.^[ref] Repeated reads sharing the same clock realization cannot estimate away its absolute displacement simply by choosing the largest result.

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.