Theorem: Empty fixed-input jet for the new action

For every mass $m\ge 0$ and every switch of the declared plateau class,

The obstruction already occurs in the order-one compatibility condition, independently of the allowed leading metric data.

Proof. For $\nu =0$, equation (eq:late-compatibility) becomes $\frac{1}{2}{\partial }_{t}g=0$ above the ramp. This contradicts Lemma lem:late-switch.

More explicitly, choose $f\ge 0$ supported in $O$ with $\int f{\mathrm{d}}^{4}x=1$. Then

because the ramp defect is zero on $O$. No leading vacuum metric configuration can alter this pairing. Higher-order coefficients cannot cancel a nonzero lower-order compatibility coefficient.