A visible Euclidean Radon conormal satisfies Bolker

The defining function $F(\theta ,s,x)=s-x\cdot \theta$ is analytic and has ${\mathrm{\partial }}_{s}F=1$. Thus [F6] makes $Z=F^{-1}(0)$ an analytic hypersurface of dimension $3$.

In a local angle coordinate $\alpha$ with ${\mathrm{\partial }}_{\alpha }\theta ={\theta }^{\perp }$, the defining function has differential $dF=ds-(x\cdot {\theta }^{\perp })d\alpha -\theta \cdot dx\mathrm{.}$ By [F2], the canonical relation consists of $\xi =\tau \theta ,{\eta }_s=\tau ,{\eta }_{\theta }=-\tau x\cdot {\theta }^{\perp },\tau \mathrm{\ne }0.$ These signs follow by matching $(\eta ,-\xi )=\tau dF$.

A tangent vector satisfies $\dot{s}-\dot{x}\cdot \theta -x\cdot \dot{\theta }=0$. Given any data tangent pair $(\dot{\theta },\dot{s})$, choose $\dot{x}=(\dot{s}-x\cdot \dot{\theta })\theta$; this lifts it to $TZ$, so ${\pi }_G$ is a submersion. Given any object tangent vector $\dot{x}$, take $\dot{\theta }=0$ and $\dot{s}=\dot{x}\cdot \theta$, so ${\pi }_X$ is a submersion. Since ${\pi }_X^{-1}(K)$ is parameterized by the compact set $S^1\times K$ for every compact $K\subset {\mathbb{R}}^2$, the restriction to $G$ is proper. The specified analytic total density is nowhere zero. The dimensions satisfy $4>3>2\ge 2$, so [F5] applies.

For fixed $(\theta ,s,{\eta }_s,{\eta }_{\theta })$ in the image, $\tau ={\eta }_s\mathrm{\ne }0$ and $x=s\theta -\frac{{\eta }_{\theta }}{\tau }{\theta }^{\perp },\xi =\tau \theta \mathrm{.}$ Thus the left fiber is a singleton. To check the differential, let a tangent variation be in the kernel of the left projection. Then $\dot{\theta }=0$, $\dot{s}=0$, $\dot{\tau }={\dot{\eta }}_s=0$, and $0=\dot{s}=\dot{x}\cdot \theta ,0={\dot{\eta }}_{\theta }=-\tau \dot{x}\cdot {\theta }^{\perp }\mathrm{.}$ Since $\tau \mathrm{\ne }0$ and $(\theta ,{\theta }^{\perp })$ is a basis, $\dot{x}=0$ and $\dot{\xi }=0$. Hence the left projection is an injective immersion, so [F3] gives global Bolker.

For the specified object covector, $\theta =\xi \mathrm{/}\mathrm{\mid }\xi \mathrm{\mid }$ and $\tau =\mathrm{\mid }\xi \mathrm{\mid }$ satisfy $\xi =\tau \theta$. With $s$ and $\eta$ chosen in the statement, step 1.2 supplies the remaining relation equations. Therefore the data-side canonical pair exists for every nonzero $\xi$.

The contrapositive of [F4] shows that $(x,\xi )\in {\mathrm{WF}}_A(f)$ forces $(\theta ,s;{\eta }_{\theta },{\eta }_s)\in {\mathrm{WF}}_A(Rf)$. Its hypotheses hold by the analytic nonvanishing density, properness and full Bolker calculations in steps 1.1–2.2.