A missing-angle covector has no data-side canonical pair

The defining function $F(\theta ,s,x)=s-x\cdot \theta$ is real analytic and has ${\mathrm{\partial }}_{s}F=1$. Thus $dF$ is surjective at every point of $Z_{\mathrm{\Theta }}$, and [F6] makes $Z_{\mathrm{\Theta }}$ an analytic hypersurface of dimension $3$.

Use a local angular coordinate $\alpha$ on $S^1$, so ${\mathrm{\partial }}_{\alpha }\theta ={\theta }^{\perp }$. The defining function $F(\theta ,s,x)=s-x\cdot \theta$ has differential $dF=ds-(x\cdot {\theta }^{\perp })d\alpha -\theta \cdot dx\mathrm{.}$ By [F4], a conormal covector is $\tau dF$ for a nonzero scalar $\tau$. Matching its data components and object component with the sign convention [F3] gives ${\eta }_s=\tau ,{\eta }_{\alpha }=-\tau x\cdot {\theta }^{\perp },\xi =\tau \theta \mathrm{.}$ Thus any object covector in the relation is parallel to the measured normal $\theta$.

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 $\dot{x}=(\dot{s}-x\cdot \dot{\theta })\theta$; then ${\pi }_G$ has a tangent lift and is a submersion. Given any object tangent vector $\dot{x}$, choose $\dot{\theta }=0$, $\dot{s}=\dot{x}\cdot \theta$; hence ${\pi }_X$ is a submersion. Both maps are analytic restrictions of product projections and $4>3>2\ge 2$. The displayed analytic total density is nowhere zero in these incidence coordinates. For compact $K\subset {\mathbb{R}}^2$ and compact $L\subset \mathrm{\Theta }\times \mathbb{R}$, the preimage of $L$ under ${\pi }_G{\mathrm{\mid }}_{{\pi }_X^{-1}(K)}$ is closed in ${\mathrm{pr}}_1(L)\times K$, which is compact because ${\mathrm{pr}}_1(L)⋐\mathrm{\Theta }$. Thus the restricted projection is proper and [F5] applies.

If $(\theta ,s,\eta ;x_0,{\xi }_0)\in C_{Z_{\mathrm{\Theta }}}$, step 1.2 implies ${\xi }_0\mathrm{/}\mathrm{\mid }{\xi }_0\mathrm{\mid }=\theta$ or $-\theta$. Since $\mathrm{\Theta }=-\mathrm{\Theta }$, this would place ${\xi }_0\mathrm{/}\mathrm{\mid }{\xi }_0\mathrm{\mid }$ in $\mathrm{\Theta }$, contradicting the hypothesis. No such data covector exists.