= Common-zero cancellation in a drag interaction contrast
\b[Common-zero cancellation in a drag interaction contrast] follows from the coefficients summing to zero. Order the readings as $\mathbf d=(D_{00},D_{10},D_{01},D_{11})^T$ and put $\mathbf a=(1,-1,-1,1)^T$. If
$$
\mathbf d=\boldsymbol\mu+Z\mathbf1+\boldsymbol\epsilon,
$$
then
$$
\mathbf a^T\mathbf d=\mathbf a^T\boldsymbol\mu+\mathbf a^T\boldsymbol\epsilon,
$$
because $\mathbf a^T\mathbf1=0$. The cancellation is pointwise and does not require $Z$ to be independent of the residual errors. It requires the same additive offset to enter all four readings on the same scale.
For finite <covariance>, the residual contribution to the contrast's <variance> is $\mathbf a^T\Sigma_\epsilon\mathbf a$. Four independent residual errors with <standard deviation> $0.5\,\mathrm N$ give \b[contrast standard deviation $1.0\,\mathrm N$], however large the exactly common zero error is. A shared gain error instead multiplies a nonzero true contrast and generally survives.
Run order can also reject some changing zeros. With equally spaced run times $\mathbf t=(0,1,2,3)^T$ in the displayed configuration order, $\mathbf a^T\mathbf t=0$. An additive linear drift $r\mathbf t$ therefore cancels. A quadratic drift $c\mathbf t^{\circ2}$ does not: $\mathbf a^T\mathbf t^{\circ2}=4$, leaving $4c$. This <drift rejection by contrast orthogonality> concerns the actual time assignments; a different order can change it.
Normalization can defeat the common-scale assumption. If every dimensional reading has offset $Z$ but coefficients divide the last reading by twice the <dynamic pressure> used for the others, the scaled offset no longer has the form $Z\mathbf1$. Keeping raw loads and reference readings allows the <factorial contrast for aerodynamic interference> and its <measurement uncertainty> to be reconstructed consistently.
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