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Smooth anisotropic uniqueness in the Calderón problem from one boundary patch

OpenAI · September 24, 2026

Lean

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Abstract

We resolve the smooth anisotropic Calderón uniqueness problem with arbitrary same-patch measurements. A smooth Riemannian metric on a compact connected manifold of dimension at least three is determined, up to a diffeomorphism fixing the measured patch, by its zero-frequency Dirichlet-to-Neumann energy form with both input and observation on any nonempty open boundary patch.

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work in progress

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.

Comments and discussion

Formalization

  • GitHub user l-oksanen, openai-calderon (GitHub repository, 2026). An independent Lean comparator challenge for Theorem 1.1, specialized to n = 3, the unit ball and metrics Euclidean outside the ball; the main theorem in AnisotropicCalderon.lean is a statement with a sorry body, and the repo also has a converse test.

Possibly relevant literature (found by the agent, not checked)

News and blogs

  • Mikko Salo, Mathematics and AI (FAME Flagship, 2026-10-15). Identifies the release’s scalar and system versions of anisotropic Calderón recovery and its three-dimensional measurable-conductivity nonuniqueness claim as results of direct interest to inverse problems. [family-level]

Elsewhere

Independent sites; their content is not reviewed here.

Cite

@misc{OAI:Smooth-Anisotropic-Uniqueness-in-the-Calderon-Problem-from-One-Boundary-Patch-September-24-2026,
  author = {{OpenAI}},
  title = {{Smooth anisotropic uniqueness in the Calder\'on problem from one boundary patch}},
  howpublished = {OpenAI Math Release preprint
                  \href{https://github.com/openai/math/blob/main/preprints/Smooth-Anisotropic-Uniqueness-in-the-Calderon-Problem-from-One-Boundary-Patch-September-24-2026/paper.pdf}{OAI:Smooth-Anisotropic-Uniqueness-in-the-Calderon-Problem-from-One-Boundary-Patch-September-24-2026}},
  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.

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