Corollary: A smeared edge failure and its order

There are a smooth scalar ${\varphi }_0$ and a supported observation test $f$ for which

No correction starting at ${\kappa }^2$, or at positive $\hslash$-order in a formal double expansion, cancels this order-$\kappa$, order-${\hslash }^0$ coefficient while the lower-order prescription is held fixed.

Proof. At a time strictly after $\epsilon$, (eq:edge) implies that $w^1[K\xi ]$ is nonzero somewhere. Choose ${\varphi }_0=x^1$ and a nonnegative bump $f$ where that component has a fixed nonzero sign. The chart reserve may be chosen in advance to contain the compact wave support and the resulting test's truncated past.

At the flat metric the first relational variation under the simultaneous first-order field pullback is precisely (eq:smearededge). Formal coefficients of different powers cannot cancel it. A universal extension to all such parameters is therefore excluded without changing the lower-order data.