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Gauge block wringing joins suitably finished measuring faces by bringing them into intimate contact with a controlled sliding or twisting motion. The resulting adhesion lets gauge blocks form a composite length standard or contact an auxiliary plane for interferometry. The contact layer is part of the length convention: a calibrated block length includes one wringing-layer contribution.^[ref]

Adhesion is not explained adequately by saying that air was removed and atmospheric pressure alone holds the blocks together. NIST's handbook describes multiple surface- and fluid-dependent contributions, including surface tension, and warns that film thickness varies with surface condition and time. It does not supply a universal predictive film model.^[ref]

An original bookkeeping model explains the length convention. Let $g_i$ be a block's geometric face separation and let $w_i$ be the effective layer contribution in its calibration, so its defined length is $l_i=g_i+w_i$. If a stack of $n$ blocks is wrung to an auxiliary plane, it has $n-1$ internal contacts and one base contact. Its top height can be writtenThus compatible contacts can make defined lengths additive without treating every interface as an additional correction beyond the certificates. Differences between calibration and use remain a wringing-film uncertainty.

For three stipulated blocks each carrying an effective $10\mathrm{nm}$ calibration contribution, three equivalent contacts contribute the same $30\mathrm{nm}$ already counted in their defined lengths. If one actual contact contributes $25\mathrm{nm}$ instead, the height in this simplified model differs by $15\mathrm{nm}$. These chosen values illustrate the convention; they are not a measured stack or a universal film specification.

Experimentally, NIST describes viewing a block wrung to a transparent optical flat: a uniform contact appearance supports a satisfactory wring, while coloured or bright regions indicate poor contact. Rewringing and checking the optical result tests an interface contribution that merely adding nominal lengths would miss.^[ref] Cleanliness, surface damage and thermal expansion during handling still affect the comparison. A visibly adhering stack is not, by itself, proof of the required dimensional accuracy.

A 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 $\mathbf{\delta }=({\delta }_1,\dots ,{\delta }_n{)}^T$ and its covariance matrix be $\Sigma$. For a correction $\Delta ={\mathbf{c}}^T\mathbf{\delta }$, propagation givesThe 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.^[ref]

A simple original model writes ${\delta }_j=b+{ϵ}_j$, where one common contribution $b$ has variance $u_b^2$ and independent contact contributions have variance $u_{ϵ}^2$. ThenFor three equal-sign contacts with $u_b=3\mathrm{nm}$ and $u_{ϵ}=2\mathrm{nm}$, the variance is $93\mathrm{n}{\mathrm{m}}^2$. For a difference of two contacts with coefficients $(1,-1)$, the common contribution cancels and the variance is only $8\mathrm{n}{\mathrm{m}}^2$. 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 isPerfectly 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.^[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.