= Wringing-film covariance model
A \b[wringing-film covariance model] specifies how uncertain effective contact-layer contributions vary together in a <gauge block wringing> measurement. Individual contact uncertainties do not determine the uncertainty of a stack unless their dependence is also specified.
Let the vector of contact differences be $\boldsymbol\delta=(\delta_1,\ldots,\delta_n)^T$ and its <covariance matrix> be $\Sigma$. For a correction $\Delta=\mathbf c^T\boldsymbol\delta$, propagation gives
$$
u^2(\Delta)=\mathbf c^T\Sigma\mathbf c.
$$
The coefficients encode the actual geometry and calibration subtraction. They need not all be positive. This is the general correlated-input law applied to <wringing-film uncertainty>.<Guide to the expression of uncertainty in measurement by JCGM>[§5.2, propagation with correlated inputs]{ref}
A simple original model writes $\delta_j=b+\epsilon_j$, where one common contribution $b$ has variance $u_b^2$ and independent contact contributions have variance $u_\epsilon^2$. Then
$$
\Sigma=u_b^2\mathbf1\mathbf1^T+u_\epsilon^2I,
\qquad
u^2(\Delta)=u_b^2\left(\sum_jc_j\right)^2+u_\epsilon^2\sum_jc_j^2.
$$
For three equal-sign contacts with $u_b=3\,\mathrm{nm}$ and $u_\epsilon=2\,\mathrm{nm}$, the variance is $93\,\mathrm{nm^2}$. For a difference of two contacts with coefficients $(1,-1)$, the common contribution cancels and the variance is only $8\,\mathrm{nm^2}$. These are chosen covariance structures, not universal properties of wrung surfaces.
Calibration and use can share only part of their contact history. If $n$ calibration contacts share effect $b_c$ and $n$ use contacts share effect $b_u$, their common-effect correction is $n(b_u-b_c)$. Its variance is
$$
n^2\left[u^2(b_u)+u^2(b_c)-2\operatorname{cov}(b_u,b_c)\right].
$$
Perfectly shared equal contributions cancel; independent preparation effects do not. Assuming either relation requires evidence about preparation, reference surfaces and measurement history.
Repeated readouts of one unchanged wring mostly probe readout scatter. Rewringing samples a different level of variability; changes of surface preparation or elapsed time may add another. The NIST handbook documents that contact layers depend on conditions and can evolve after thermal equilibration.<The Gauge Block Handbook by Ted Doiron and John S. Beers>[Appendix B, Physics of Wringing Films]{ref} A <nested wringing-repeatability study> can help separate those levels, but repeatability data alone do not identify every shared calibration contribution.
Retain the covariance assumptions with the result and test whether the proposed matrix is a valid <positive semidefinite matrix>. A small combined uncertainty obtained by assigning cancelling correlations is meaningful only when those correlations describe the physical comparison.
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