= Solution
For each <polynomial> $g$, take the <real analytic> adjoint solution on the common cap from the preceding part. On the flat boundary face the full <boundary jet> of $u$ vanishes; on the curved face that of $v$ vanishes. The <complete boundary-jet condition for formal adjoints>, proved under its corrected hypothesis, therefore applies. The two corner points have zero boundary measure, and the cap is a bounded piecewise smooth domain. It follows that
$$
\int_{\omega_\varepsilon}ug=\int_{\omega_\varepsilon}uP^*v=\int_{\omega_\varepsilon}(Pu)v=0.
$$
By the <Weierstrass approximation theorem>, <polynomials> uniformly approximate $\bar u$ on the compact cap closure. Taking the limit gives $\int_{\omega_\varepsilon}|u|^2=0$. Continuity then gives $u=0$ pointwise in the cap. For real $u$, simply approximate $u$ itself.
A positive cap alone is not a neighborhood of the origin. Apply the same argument after reflecting $x\mapsto-x$; the reflected operator remains <real analytic> and non-characteristic. Shrinking the two parameters to their minimum proves vanishing on
$$
\boxed{\{(x,y):|x|+y^2<\varepsilon_*,\ \varepsilon_*>0\}.}
$$
This set is open and contains the origin, including the flat face where the value data vanish. The argument is the <parabolic-cap proof of Holmgren uniqueness>, and it establishes <Holmgren uniqueness theorem> for nonanalytic $C^m$ solutions here.
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