 Common-zero cancellation in a drag interaction contrast

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
then
because ${\mathbf{a}}^{T}1=0$. 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 ${\mathbf{a}}^T{\Sigma }_{ϵ}\mathbf{a}$. Four independent residual errors with standard deviation $0.5\mathrm{N}$ give 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 $c{\mathbf{t}}^{\circ 2}$ does not: ${\mathbf{a}}^T{\mathbf{t}}^{\circ 2}=4$, 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 $Z1$. Keeping raw loads and reference readings allows the factorial contrast for aerodynamic interference and its measurement uncertainty to be reconstructed consistently.