= Gauge-block reference uncertainty floor
A \b[gauge-block reference uncertainty floor] is the component of a comparison's uncertainty that remains when repeated observations still rely on the same uncertain reference value. It is conditional on the stated reference and <measurement model>, rather than a universal lower bound on every method of measuring the block.
For a corrected comparison, write the unknown reference-temperature length as $L_X=L_R+\overline d$, where $L_R$ is the assigned reference length and $\overline d$ the mean of $n$ comparison differences. Suppose the reference has <standard uncertainty> $u_R$, the differences have independent repeatability scatter $s$, and these contributions are independent. Then
$$
u^2(L_X)=u_R^2+\frac{s^2}{n}.
$$
Only the repeated-comparison term decreases with $n$. The reference-value contribution is shared by all those observations; it is not newly sampled every time. NIST's comparison uncertainty combines information from the interferometric master history with comparison-process variability.<The Gauge Block Handbook by Ted Doiron and John S. Beers>[§4.4.3, Calculating the total uncertainty]{ref}
For an original example, take $u_R=20\,\mathrm{nm}$ and $s=40\,\mathrm{nm}$. Four comparisons give $u(L_X)=\sqrt{400+1600/4}\approx28.3\,\mathrm{nm}$. One hundred give about $20.4\,\mathrm{nm}$, and the limit in this model is 20 nm. Dividing the entire combined uncertainty by $\sqrt n$ would wrongly treat the reference assignment as independent repeatability noise.
The same shared reference can cancel from another measurand. If $L_X=L_R+d_X$ and $L_Y=L_R+d_Y$ use the same additive reference contribution, then
$$
L_X-L_Y=d_X-d_Y.
$$
<Shared gauge-block reference covariance> is what makes that cancellation consistent in <uncertainty propagation>. An absolute length and a length difference have different uncertainty budgets. Other shared terms, including temperature and comparator corrections, must be modeled on their own merits.<Guide to the expression of uncertainty in measurement by JCGM>[§5.2, correlated input quantities]{ref}
An independently improved reference assignment or additional independent calibration information can lower the floor; more repetitions with the unchanged reference cannot. State whether a certificate quotes standard or expanded uncertainty before combining values. Improved repeatability is useful, but does not establish improved absolute accuracy by itself.
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