= Shared gauge-block reference covariance
\b[Shared gauge-block reference covariance] is the dependence between calibrated length estimates caused by using the same uncertain reference. It explains how a <gauge-block reference uncertainty floor> can remain in individual absolute lengths while canceling from their difference.
Let corrected estimates be $X=R+d_X$ and $Y=R+d_Y$. Suppose the shared reference error has variance $u_R^2$, the comparison errors have variances $u_X^2,u_Y^2$, and these three contributions are independent. The <covariance matrix> is
$$
\Sigma=u_R^2\begin{pmatrix}1&1\\1&1\end{pmatrix}
+\begin{pmatrix}u_X^2&0\\0&u_Y^2\end{pmatrix}.
$$
In particular, $\operatorname{cov}(X,Y)=u_R^2$. The covariance follows from the shared input, rather than from whether the two comparison readings happened at the same instant.
For the contrast $D=X-Y$,
$$
u^2(D)=u^2(X)+u^2(Y)-2\operatorname{cov}(X,Y)=u_X^2+u_Y^2.
$$
Both reference sensitivities are exactly one in this additive <measurement model>, so their contributions cancel. General <uncertainty propagation> includes precisely this covariance term.<Guide to the expression of uncertainty in measurement by JCGM>[§5.2, correlated input quantities]{ref}
An original example uses $u_R=20\,\mathrm{nm}$, $u_X=10\,\mathrm{nm}$ and $u_Y=15\,\mathrm{nm}$. The individual standard uncertainties are about 22.36 and 25 nm, while the difference has standard uncertainty $\sqrt{100+225}\approx18.03\,\mathrm{nm}$. Incorrectly treating the absolute length estimates as independent would give $\sqrt{500+625}\approx33.54\,\mathrm{nm}$ for the same difference.
For any <linear combination> $Z=c_XX+c_YY$, the reference contribution is $u_R^2(c_X+c_Y)^2$. It vanishes for a difference but remains $u_R^2$ in an equal-weight average, where $c_X=c_Y=1/2$. Repeatedly averaging values carrying the same reference error does not produce independent reference information.
<Unequal gauge-block reference sensitivities> change the cancellation. If the reference enters as $a_XR$ and $a_YR$, a difference retains reference variance $(a_X-a_Y)^2u_R^2$. Shared temperature, comparator or wringing contributions also need their own coefficients and dependence assumptions. NIST's uncertainty procedure separates the master history from comparison-process variability.<The Gauge Block Handbook by Ted Doiron and John S. Beers>[§4.4.3, Calculating the total uncertainty]{ref}
Retain the calibration lineage and input relationships with the estimates. A pair of certificate uncertainty numbers alone does not establish their covariance or justify setting it to zero.
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