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A 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. ThenOnly 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.^[ref]

For an original example, take $u_R=20\mathrm{nm}$ and $s=40\mathrm{nm}$. Four comparisons give $u(L_X)=\sqrt{400+1600/4}\approx 28.3\mathrm{nm}$. 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, thenShared 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.^[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.

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 $X=f_X(R,\dots )$ and $Y=f_Y(R,\dots )$. 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,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.^[ref]

An original ratio-transfer model illustrates the difference: $X=q_{X}R+d_X$, $Y=q_{Y}R+d_Y$, 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 $\sqrt{20^2+40^2}\approx 44.72\mathrm{nm}$.

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, $c_{X}X+c_{Y}Y$ retains reference variance $(c_X{a}_X+c_Y{a}_Y{)}^2{u}^2(R)$.

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.^[ref] A reference-sensitivity cancellation check should accompany a claimed low-uncertainty contrast.