= Mixed fourth moment of thermal inputs
The <mixed fourth moment of thermal inputs> measures how squared temperature and expansion-coefficient deviations occur together. For centered inputs $A=a-\mathbb E a$ and $T=\theta-\mathbb E\theta$, define
$$
M_{22}=\mathbb E[A^2T^2].
$$
It has total input degree four. It is not the fourth moment of the product $AT$, which would involve $\mathbb E[A^4T^4]$. For a centered thermal product, the exact <variance> is
$$
\operatorname{Var}(AT)=M_{22}-[\mathbb E(AT)]^2.
$$
Consequently individual input variances and their <covariance> do not in general determine the product variance.
For jointly normal centered inputs, write $A=s_aU$ and $T=s_T(\rho U+\sqrt{1-\rho^2}V)$ using independent standard <normal distributions>. Expanding and using $\mathbb E U^4=3$, $\mathbb E U^2V^2=1$ and vanishing odd terms gives
$$
M_{22}=s_a^2s_T^2(1+2\rho^2),\qquad
\operatorname{Var}(AT)=s_a^2s_T^2(1+\rho^2).
$$
This is a specific joint-Gaussian calculation, consistent with the <Isserlis theorem>; normal-looking marginals alone do not justify it.
An original bounded counterexample illustrates the extra information. Let $X$ equal $-\sqrt2,0,+\sqrt2$ with probabilities $1/4,1/2,1/4$, and let an independent sign $B$ take $\pm1$ equally. Put $A=s_aX$ and $T=s_TBX$. Both standardized marginals have mean zero and variance one, and $\mathbb E(AT)=0$, but $M_{22}=2s_a^2s_T^2$. Replacing the second standardized input by an independent copy of $X$ preserves both marginals and zero covariance while changing $M_{22}$ to $s_a^2s_T^2$.
For an illustrative length correction $d=LAT$, take $L=100\,\mathrm{mm}$, $s_a=10^{-6}\,\mathrm{K^{-1}}$ and $s_T=0.1\,\mathrm K$. The two bounded models give standard deviations approximately14.14nm and10nm. Their factor-of-two variance difference comes entirely from joint occurrence, rather than changed marginal uncertainties.
NIST's thermal correction combines block length, expansion coefficient and temperature difference, providing the physical setting for this joint-input issue.<The Gauge Block Handbook by Ted Doiron and John S. Beers>[§5.4.3.1, equation5.8]{ref} Joint observations and the way coefficient or thermometer corrections are assigned must support the dependence model. Increasing the number of simulated draws cannot recover an unmeasured mixed moment. The GUM explicitly calls for higher-order treatment when a nonlinear response makes first-order propagation inadequate.<Guide to the expression of uncertainty in measurement by JCGM>[§5.1.2, nonlinear contributions]{ref}
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