= Parabolic-cap proof of Holmgren uniqueness
Near a non-characteristic plane $x=0$, use the small cap $\omega_\varepsilon=\{x>0,\ x+y^2<\varepsilon\}$. Its curved face is non-characteristic by continuity of the <principal symbol>. A <polynomial> coordinate change flattens that face. Uniform <real analytic> coefficient bounds and the <uniform Cauchy radius for polynomial forcing> solve $P^*v=g$ throughout a common cap, with zero <boundary jet> on the curved face. The <complete boundary-jet condition for formal adjoints> then gives $\int_{\omega_\varepsilon}ug=0$ for every <polynomial> $g$. Uniform approximation of $\bar u$ gives $\int|u|^2=0$. Repeating on the other side of the plane proves two-sided local uniqueness.
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