A quadratic coefficient is not separately the full gauge identity

Thus linear gauge transformations of the quadratic coefficient couple to the nonlinear transformation of the linear coefficient. Dropping the latter is not a valid quadratic gauge audit.

As a direct first-order check,

makes (eq:linear-explicit) zero. For a pure spatial gauge perturbation, $Q_{1,ij}={\partial }_i{\xi }_j+{\partial }_j{\xi }_i$, so

These checks fix the displacement signs.