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Second-order thermal mismatch uncertainty retains product terms that a linear uncertainty propagation can miss when calculating differential thermal-expansion uncertainty. For known nominal length $L$, expansion-coefficient difference $a$ and common temperature offset $\theta$, the correction is $d=La\theta$. Both first derivatives vanish at $a=\theta =0$, but the product of uncertain inputs need not vanish.

An original exact calculation assumes independent random variables with means $a_0,{\theta }_0$ and variances ${\sigma }_a^2,{\sigma }_{\theta }^2$. Independence givesTo derive the variance, use $\mathbb{E}[a^2{\theta }^2]=\mathbb{E}[a^2]\mathbb{E}[{\theta }^2]$ and subtract $a_0^2{\theta }_0^2$. No normal-distribution assumption is required for this independent-input result. The last term is absent from first-order propagation.

For stipulated $L=100\mathrm{mm}$, $a_0={\theta }_0=0$, ${\sigma }_a=10^{-6}{\mathrm{K}}^{-1}$ and ${\sigma }_{\theta }=0.1\mathrm{K}$, the exact standard deviation is 10 nm. A zero central correction therefore does not establish zero uncertainty. The GUM specifically warns that significant nonlinearity requires higher-order terms.^[ref]

Correlated inputs introduce a further distinction between the central product and the output mean. If $(a,\theta )$ has a multivariate normal distribution with covariance $c$, expansion about its means and the Gaussian fourth-moment identity giveThe covariance term in the mean must not be silently discarded. With the same zero means and standard deviations as above, but correlation coefficient $1/2$, the mean correction is 5 nm and its standard deviation about 11.18 nm. These are probability-model calculations, not measured block specifications.

The formulas assume known $L$ and a justified joint input model. A thermal-product input-distribution check must address shared temperature calibration and coefficient estimation before assigning a correlation. NIST's physical comparison procedure treats thermal corrections through measured temperatures and material expansion properties.^[ref] Keeping the nonlinear contribution is useful only when the inputs describe that actual measurement.

Thermal-product dependence beyond covariance is the influence of higher joint moments on uncertainty in a nonlinear thermal expansion correction. Two input models can have identical means, variances and covariance yet predict different uncertainty in a product. A thermal-product input-distribution check must therefore examine how the temperature offset and expansion-coefficient error occur together.

Let $a={\mu }_a+A$ and $\theta ={\mu }_{\theta }+T$, where the centered variables have expected values zero, variances $s_a^2,s_{\theta }^2$ and covariance $c=\mathbb{E}[AT]$. Direct expansion givesThe third joint moments and the mixed fourth moment of thermal inputs are extra information. For centered inputs, only the last two terms remain. A first-order uncertainty propagation rule at zero input means would miss the product entirely; the GUM explicitly requires higher-order treatment when nonlinearity matters.^[ref]

An original counterexample keeps even the marginal shapes fixed. Let $Z$ have a standard normal distribution and let $S$ independently take values $+1$ and $-1$ with equal probabilities. Define $A=s_{a}Z$ and $T=s_{\theta }SZ$. Both have centered normal distributions, the quoted variances and zero covariance. They are not independent: their absolute standardized values are identical.

Their product is $s_a{s}_{\theta }S{Z}^2$. Its expected value is zero, while its variance is $3s_a^2{s}_{\theta }^2$, because $\mathbb{E}[Z^4]=3$. Independent centered normal inputs with the same individual variances instead give product variance $s_a^2{s}_{\theta }^2$. The joint model, rather than a marginal histogram or correlation coefficient alone, determines the difference.

For an illustrative length correction $d=La\theta$ with zero means, $L=100\mathrm{mm}$, $s_a=10^{-6}{\mathrm{K}}^{-1}$ and $s_{\theta }=0.1\mathrm{K}$, these models give standard deviations $\sqrt{3}\times 10\approx 17.32$ nm and 10 nm. This constructed dependence is a mathematical test case, not a proposed universal thermometer model.

In a physical gauge block study, track paired temperature and coefficient information, shared thermometer corrections and any procedure that estimates one input using the other. NIST's thermal correction explicitly combines block length, expansion coefficient and temperature difference.^[ref] Preserve that dependence when generating joint draws. JCGM 101 distinguishes model formulation from propagation and gives a multivariate Gaussian input model as a specific choice, not a consequence of normal marginals alone.^[ref] More simulation trials reduce numerical error conditional on the chosen model; they cannot identify unmeasured joint moments.