= Mismatch remaining after common RC gain cancellation
The <mismatch remaining after common RC gain cancellation> is the timing error caused by unequal gains in the amplitude and threshold paths after their common multiplicative part has canceled. It depends on their <ratio>, so improving a shared gain alone does not remove it.
For the zero-baseline <RC threshold-crossing time>, let $a=\theta/H$ with $0<a<1$. If the amplitude is multiplied by $G_H>0$ and the threshold by $G_\theta>0$, define the residual relative error by $G_\theta/G_H=1+\epsilon$. The exact result from <ratiometric cancellation of RC timing gain errors> is
$$
\Delta t=-\tau\log\left(1-\frac{a\epsilon}{1-a}\right),\qquad
0<a(1+\epsilon)<1.
$$
Here $\tau$ is the <time constant>. As $a(1+\epsilon)$ approaches one, the crossing is arbitrarily late; at or above one there is no finite rising crossing. An error budget must check this existence condition as well as a small timing residual.
For a new example, set $H=1\,\mathrm V$, $\theta=0.8\,\mathrm V$ and $\tau=1\,\mathrm{\mu s}$. A one-percent positive mismatch gives $\Delta t=-\log(0.96)\,\mathrm{\mu s}\approx40.822\,\mathrm{ns}$; a one-percent negative mismatch gives $-\log(1.04)\,\mathrm{\mu s}\approx-39.221\,\mathrm{ns}$. The first-order estimate is $\Delta t\simeq\tau a\epsilon/(1-a)$, or $\pm40\,\mathrm{ns}$ here. The unequal exact magnitudes are a consequence of the <logarithm>, not an asymmetric tolerance specification.
If $|\epsilon|\le e<(1-a)/a$, monotonicity gives the exact worst absolute error
$$
|\Delta t|\le-\tau\log\left(1-\frac{ae}{1-a}\right).
$$
Thus a $10\,\mathrm{ns}$ bound in this example requires $e\le(1-a)(1-e^{-0.01})/a\approx0.00248754$, about $0.249\%$. This is a bound on the gain <ratio>, not on either gain separately.
For small fractional path errors $\eta_H,\eta_\theta$, $\epsilon\simeq\eta_\theta-\eta_H$ and
$$
\operatorname{Var}(\epsilon)\simeq u_\theta^2+u_H^2-2\operatorname{Cov}(\eta_\theta,\eta_H).
$$
A perfectly shared component has zero contribution to that difference. This is the <law of propagation of uncertainty> applied to the two gains; it does not cancel a change in $\tau$, an additive threshold error or a nonzero unscaled baseline.<Guidelines for Evaluating and Expressing the Uncertainty of NIST Measurement Results by Barry N. Taylor and Chris E. Kuyatt>[Appendix A.3, covariance terms in equation A-3]{ref}
A proposed check varies the common gain while holding its <ratio> fixed, then varies only one path. Recording the two <electric potential differences> and the crossing on an <oscilloscope> distinguishes the exact common-gain invariance from the surviving mismatch.
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