A partial-data configuration outside the full-data theorem

The set $\mathrm{\Gamma }$ is nonempty because it contains the north pole and is open in the sphere because it is the inverse image of $(1\mathrm{/}2,\mathrm{\infty })$ under the continuous coordinate $x\mapsto x_3$. It is proper because the south pole is not in it. A nonzero smooth bump supported in a small lower cap defines boundary data outside $C_c^{\mathrm{\infty }}(\mathrm{\Gamma })$, so the allowed inputs form a proper subset of the full boundary-data space.

By [F1] and [F2], the full-data theorem requires the map on every $H^{1\mathrm{/}2}(\mathrm{\partial }\mathrm{\Omega })$ input and the corresponding full boundary output. The stated protocol restricts both input and output to $\mathrm{\Gamma }$, so it does not satisfy that hypothesis. This classifies the configuration as outside the theorem's scope and asserts no uniqueness or nonuniqueness conclusion.