A Sharp Regularity Threshold for Uniqueness in Riemannian Calder\'on-type Problems

We prove a sharp regularity threshold for uniqueness in two anisotropic Calder\'on-type inverse problems in dimension $n\ge 3$. The main setting is the Riemannian Schr\"odinger problem with fixed scalar potential: for a prescribed nonconstant analytic function $V$, we study whether the Dirichlet-to-Neumann map of $-\Delta_g+V$ on a domain $\Omega\subset\mathbb{R}^n$ determines the unknown metric $g$. The natural gauge is the group of boundary-fixing diffeomorphisms preserving $V$. We show that, while analytic metrics are uniquely determined modulo this gauge by a minor adaptation of the Lassas--Uhlmann reconstruction theorem, uniqueness fails densely in every non-analytic Gevrey class $G^\sigma$, $\sigma>1$. In fact, our counterexamples are not isometric in the sense that they are not connected by the pushforward of any diffeomorphism of $\overline\Omega$. We also prove the analogous sharp threshold for the anisotropic Calder\'on problem at fixed nonzero frequency, thereby upgrading the previously known finite-regularity counterexamples to Gevrey and $C^\infty$ regularity. The two constructions use different scalar mechanisms: for fixed potentials, the nonconstant potential itself provides a local coordinate, while at nonzero frequency one uses a compactly supported prescribed-Jacobian lemma in Gevrey spaces. Thus analyticity is the exact threshold for uniqueness in both problems.