The correct switching support

Without an observation plateau the shorter statement is false. For example, with $m=0$, $\varphi =t$, and an above-ramp observation outside $\mathrm{supp}g$, a compact ghost supported there gives the nonzero local dressing variation, even though it misses both the ramp and $\mathrm{supp}\mathrm{d}g$.

In a spatially constant massless diagnostic, $c=2\vartheta t{\partial }_t$ gives

Thus the scalar-response coefficient, after the ramp, is

A sufficiently large spatial cutoff realizes this formula in an observation dependence cylinder. It is a diagnostic of the different frame, not (eq:completeaffine).

Finally,

This is XI-B's Cauchy-uniqueness lemma XIB. The plateau identity is nontrivial for off-shell ghost test parameters; its restriction to compact plateau free ghosts has only the zero parameter. Unrestricted free ghosts, including the affine witness, are not thereby made closed.