Proposition: Full weighting changes the free sheet

Multiplication of the entire signed Einstein–Hilbert density by ${\sigma }_{\epsilon }$ does not retain the XI-A/XI-B lower-order model. Its metric Euler derivative at the flat background contains

This expression is nonzero for the declared ramp.

Proof. The compact-variation calculation proved in Lemma lem:weighted-curvature below gives the displayed expression at $q=\eta$. Its spatial components are $-2{\delta }^{ij}{\sigma }_{\epsilon }^{\prime \prime }/\kappa$. The ramp has derivative zero at both endpoints but changes from zero to one, so ${\sigma }_{\epsilon }^{\prime \prime }$ cannot vanish identically. Thus flat space is not stationary even with zero scalar input. In particular, the original zero-background free sheet is not preserved.