= Unequal gauge-block reference sensitivities
\b[Unequal gauge-block reference sensitivities] occur when two length estimates respond differently to the same uncertain reference value. A reference contribution cancels from their difference only when its coefficients agree. Sharing a calibration source is necessary information, but does not by itself guarantee cancellation.
Let estimates have <measurement models> $X=f_X(R,\ldots)$ and $Y=f_Y(R,\ldots)$. At the stated input values define <sensitivity coefficients> $a_X=\partial f_X/\partial R$ and $a_Y=\partial f_Y/\partial R$. The reference contribution to $D=X-Y$ is, to first order,
$$
u_R^2(D)=(a_X-a_Y)^2u^2(R).
$$
This follows either by differentiating the contrast directly or by retaining <shared gauge-block reference covariance> in the two-output propagation. The general law includes both sensitivities and covariance.<Guide to the expression of uncertainty in measurement by JCGM>[§§5.1.3 and 5.2]{ref}
An original ratio-transfer model illustrates the difference: $X=q_XR+d_X$, $Y=q_YR+d_Y$, with fixed known $q_X,q_Y$. If $R=100\,\mathrm{mm}$, $u(R)=20\,\mathrm{nm}$, $q_X=1$ and $q_Y=2$, their length difference retains a 20 nm reference contribution. The reference error is shared, but appears twice as strongly in $Y$. Treating the reference contributions as independent would instead give $\sqrt{20^2+40^2}\approx44.72\,\mathrm{nm}$.
The contrast $2X-Y$ cancels this model's reference term because $2q_X-q_Y=0$. It is a different <measurand> from $X-Y$; cancellation is not permission to replace the quantity the experiment intended to determine. More generally, $c_XX+c_YY$ retains reference variance $(c_Xa_X+c_Ya_Y)^2u^2(R)$.
Nearly matched sensitivities can provide strong suppression. With $a_X=1$, $a_Y=0.98$ and the same 20 nm reference uncertainty, the residual reference contribution is 0.4 nm. That does not imply the total uncertainty is 0.4 nm: comparison scatter, coefficient uncertainty, temperature and wringing contributions remain.
If the transfer ratios are themselves uncertain, their derivatives introduce terms proportional to the reference length, along with any shared <covariance>. Determine them from the actual calibration model rather than assigning ideal coefficients solely to obtain cancellation. NIST separates reference-master history from comparator-process variability in its uncertainty calculation.<The Gauge Block Handbook by Ted Doiron and John S. Beers>[§4.4.3, Calculating the total uncertainty]{ref} A <reference-sensitivity cancellation check> should accompany a claimed low-uncertainty contrast.
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