Analytic wavefront recovery under Bolker

Choose a smooth compact object cutoff $\rho$ equal to one near $x_0$ and supported in the small object neighborhood supplied by [F4]. Write $f=\rho f+(1-\rho )f$. The second summand is compactly supported away from $x_0$. By [F2] its support has no relation point over $(z_0,{\zeta }_0)$, because that full fiber consists of the supplied point over $x_0$. Hence [F3] gives $(z_0,{\zeta }_0)\mathrm{\notin }{\mathrm{WF}}_A(R((1-\rho )f))$. This includes all incidence points outside the local coordinate patch; it is not inferred merely from local injectivity.

By hypothesis the Gaussian data transform of $Rf$ is exponentially small on some data parameter neighborhood. Subtract the exponential estimate from step 1.1, using the same data chart and a common smaller cutoff neighborhood by [F5]. Linearity gives the same estimate for $R(\rho f)$ at the supplied data covector. The differential clause of [F2] supplies the local Bolker hypothesis in [F4]. Since $\rho f$ is supported in its chosen small object neighborhood, [F4] yields $(x_0,{\xi }_0)\mathrm{\notin }{\mathrm{WF}}_A(\rho f)$.

The localized distributions $\rho f$ and $f$ agree on a neighborhood of $x_0$. Their Gaussian transforms with any common cutoff supported there agree exactly; all other cutoff choices differ only by the exponential Gaussian tails in [F5]. Thus step 2.1 gives $(x_0,{\xi }_0)\mathrm{\notin }{\mathrm{WF}}_A(f)$, as asserted. The two suppliers account separately for global fiber uniqueness and local differential injectivity, and all errors are controlled in the finite number of derivatives needed by the compact distribution. No stability or choice assumption has been added.