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Nonuniqueness for bounded measurable scalar conductivities in three dimensions
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Abstract
We construct two distinct uniformly positive bounded measurable scalar conductivities on a ball in ℝ3 with the same full Dirichlet-to-Neumann operator. Both conductivities equal one near the boundary. This gives nonuniqueness in the scalar Calderón problem at bounded measurable regularity.
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Notes
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Collected automatically by AI tools (Claude Opus 5.5 and GPT-6 Astra, 2026-10-21) and not checked by a person. May be wrong or incomplete. Report an error.
Formalization
- OpenAI math release, Lean scope notes for family 365 (lean/docs/365.md, Oct 2026). The scope document covers only this paper's Theorem 1.1 on the ball B(0,3) ⊂ ℝ³, describing formalized scalar nonuniqueness with uniform positivity, agreement with one near the boundary, distinctness on a set of positive volume, identical full weak Dirichlet-to-Neumann operators, and unique weak solvability for every boundary trace.
- OpenAI math release, Conductivity.lean (comparator challenge statement) (lean/ComparatorChallenges, Oct 2026). The Lean statement of the nonuniqueness theorem used as the comparator target; its body here is
sorryby design. - OpenAI math release, OAI/Analysis/Conductivity/Main.lean (Lean solution module, Oct 2026). This module contains the proof of
main_nonuniquenessand has nosorryin the file itself; it imports roughly 15 further modules.
Possibly relevant literature (found by the agent, not checked)
- Santacesaria, Note on Calderón's inverse problem for measurable conductivities (Inverse Problems & Imaging 13 (2019) 149–157). It proposes a Clifford-algebra approach as a possible first step toward $L^\infty$ uniqueness in dimension ≥ 3, the question that Theorem 1.1 answers negatively.
- Greenleaf, Lassas, Uhlmann, On nonuniqueness for Calderón's inverse problem (arXiv:math/0302258, 2003; Math. Res. Letters). It constructs anisotropic conductivities, singular near an interior surface, with the same DN map as a homogeneous one, an earlier nonuniqueness mechanism distinct from the paper's bounded, uniformly positive scalar setting.
- Boaz Haberman and Daniel Tataru, Uniqueness in Calderon's problem with Lipschitz conductivities (arXiv:1108.6068; Duke Mathematical Journal 162, 2013). Its uniqueness results for arbitrary continuously differentiable conductivities and Lipschitz conductivities sufficiently close to the identity provide a regularity comparison for this paper’s claimed nonuniqueness with only bounded measurability.
Elsewhere
- VibeMathed: Problem record
- Emergent Mind: family 365
Independent sites; their content is not reviewed here.
Cite
@misc{OAI:Nonuniqueness-for-Bounded-Measurable-Scalar-Conductivities-in-Three-Dimensions-September-23-2026,
author = {{OpenAI}},
title = {{Nonuniqueness for Bounded Measurable Scalar Conductivities in Three Dimensions}},
howpublished = {OpenAI Math Release preprint
\href{https://github.com/openai/math/blob/main/preprints/Nonuniqueness-for-Bounded-Measurable-Scalar-Conductivities-in-Three-Dimensions-September-23-2026/paper.pdf}{OAI:Nonuniqueness-for-Bounded-Measurable-Scalar-Conductivities-in-Three-Dimensions-September-23-2026}},
year = {2026}
}Lean formalization
OpenAI provides Lean files for this family: lean/docs/365.md. The Lean statements have not been checked by people against the paper's claims.
Family 365: Joint metric and connection recovery from one boundary patch
Zero-frequency measurements on any nonempty open boundary patch determine a smooth metric and smooth unitary connection on a trivial Hermitian rank-two bundle over a compact connected manifold of dimension at least three, up to diffeomorphism and gauge fixed on that patch. Inputs and observations use the same patch. In contrast, distinct uniformly positive bounded measurable scalar conductivities on a three-dimensional ball can have identical full-boundary data.
Other papers in this family:
- Determination of a metric and a unitary connection from one boundary patch · October 5, 2026
- Smooth anisotropic uniqueness in the Calderón problem from one boundary patch · September 24, 2026