Calderon's uniqueness question for bounded measurable scalar conductivities in dimension three

Theorem 1.1: there are real $\gamma_0,\gamma_1\in L^\infty(B(0,3))$ with $c\le\gamma_j\le C$, both equal to 1 near the boundary, differing on a set of positive measure, whose full weak Dirichlet-to-Neumann operators $H^{1/2}\to H^{-1/2}$ coincide. A corollary places $2^m$ such conductivities in any bounded Lipschitz domain in $\mathbb R^3$. The construction nests scalar blocks with toroidal interfaces. This refutes the uniqueness assertion quantified over all $n\ge3$. Not shown: counterexamples in each dimension $n\ge4$, control of the contrast, or anything about conductivities with some derivative regularity (where uniqueness is known).

Theorem 1.1: there are real $\gamma_0,\gamma_1\in L^\infty(B(0,3))$ with $c\le\gamma_j\le C$, both equal to 1 near the boundary, differing on a set of positive measure, whose full weak Dirichlet-to-Neumann operators $H^{1/2}\to H^{-1/2}$ coincide. A corollary places $2^m$ such conductivities in any bounded Lipschitz domain in $\mathbb R^3$. The construction nests scalar blocks with toroidal interfaces. This refutes the uniqueness assertion quantified over all $n\ge3$. Not shown: counterexamples in each dimension $n\ge4$, control of the contrast, or anything about conductivities with some derivative regularity (where uniqueness is known).