= Common offset in RC timing
{title2=$t(H+B,\theta+B,V_b+B)=t(H,\theta,V_b)$}
A <common offset in RC timing> cancels only when every level used in the crossing model shares the same additive change. For high level $H$, threshold $\theta$, and initial baseline $V_b$, assume $V_b<\theta<H$ and fixed <time constant> $\tau$. The crossing time is
$$
t(H,\theta,V_b)=\tau\log\frac{H-V_b}{H-\theta}.
$$
Adding $B$ to all three levels leaves both differences unchanged, so the invariance in the header is exact. It describes a shared readout zero, or an equivalently translated step with a translated initial condition, rather than every possible physical offset experiment.
The three sensitivities make the cancellation visible locally:
$$
c_H=\tau\left(\frac1{H-V_b}-\frac1{H-\theta}\right),\quad
c_\theta=\frac{\tau}{H-\theta},\quad
c_b=-\frac{\tau}{H-V_b},\qquad c_H+c_\theta+c_b=0.
$$
A common random offset has <covariance> matrix $u_B^2\mathbf1\mathbf1^{\mathsf T}$ for these three inputs and contributes zero to the propagated timing <variance>. NIST's primary law retains exactly such <covariance> terms; setting them to zero would represent another measurement model.<Guidelines for Evaluating and Expressing the Uncertainty of NIST Measurement Results by Barry N. Taylor and Chris E. Kuyatt>[Appendix A.3, equation A-3]{ref}
For an original nonzero-baseline example, take $H=2\,\mathrm V$, $\theta=0.8\,\mathrm V$, $V_b=0.2\,\mathrm V$ and $\tau=0.6\,\mathrm{\mu s}$. The crossing is about $0.243279\,\mathrm{\mu s}$. Adding $20\,\mathrm{mV}$ to all three levels leaves it exactly unchanged. Adding the same offset only to $H$ and $\theta$ while keeping the baseline fixed instead increases it by about $6.630\,\mathrm{ns}$. The two-level sensitivity is $c_H+c_\theta=\tau/(H-V_b)$, predicting $6.667\,\mathrm{ns}$ to first order.
A calibration test must state whether the initial <capacitor> output, <electronic comparator> reference, input level and readout share the offset. Independent threshold errors, uncertainty of the <time constant>, and baseline memory remain even when this one common component cancels. The full distinction is developed in <correlated threshold and amplitude errors in RC timing>; a common gain is a separate invariance treated in <ratiometric cancellation of RC timing gain errors>.
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