= Channel switching under uncertainty in a combined receiver horizon
<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 $H=\max_jH_j$, 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 gives
$$
\Pr(H\le h)=\Pr(H_1\le h,H_2\le h)
=\Phi\!\left(\frac{h-\mu}{s}\right)^2,
$$
where $\Phi$ is the standard <normal distribution> <cumulative distribution function>. This square is already enough to show that $H$ is not normal: $\Pr(H\le\mu)=1/4$.
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, giving
$$
EH=\mu+\frac{s}{\sqrt\pi},\qquad
\operatorname{Var}(H)=s^2\left(1-\frac1\pi\right).
$$
For $\mu=100\,\mathrm{ns}$ 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.<SciPy 1.16.2 documentation>[scipy.stats.norm, loc, scale, cdf and ppf]{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.<Guidelines for Evaluating and Expressing the Uncertainty of NIST Measurement Results by Barry N. Taylor and Chris E. Kuyatt>[Appendix A, equation A-3; §§6.2–6.3]{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.
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