Residual checks for any future replacement

A future rod replacement $Y_1\mapsto Y_1+Z_1$ on the same clock slices changes the quadratic lapse, with other data fixed, by

A claim of identity must either prove this expression vanishes on the named domain or supply the compensating changes in the other coefficients. Equality of selected source contractions is not that proof.

Equation (eq:rod-residual) is the present residual-check template. The prepared source $S_{\lambda }$ and the Cauchy residual of XIX-B are documentary warnings, not cosmological rod inputs.