Result 365, Partial differential equations

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.

Lean formalization Proof

The bigger picture

Why it matters

The manuscripts claim that static measurements on just one boundary patch can identify both interior geometry and a rule for transporting two-component complex fields. They also report that uniqueness fails for certain rough conductivities.

What changes?

For a compact connected smooth manifold with smooth boundary in dimension at least three, the joint claim assumes a smooth metric, defining lengths and angles, and a smooth unitary connection, defining norm-preserving transport of two-component complex fields on a trivial Hermitian rank-two bundle. Zero-frequency inputs and observations use the same arbitrary nonempty open boundary patch. Recovery is up to a diffeomorphism and unitary gauge, meaning coordinate and field-frame changes that are the identity on that patch.

What does that help mathematicians do?

The claimed uniqueness rules out additional smooth ambiguities beyond those coordinate and frame changes, even with access confined to one patch. The contrasting manuscript reports two distinct uniformly positive, bounded measurable scalar conductivities on a three-dimensional ball with identical full-boundary data; both equal one near the boundary. This counterexample would prevent extending conductivity uniqueness indiscriminately to rough coefficients, even by measuring the entire boundary.

Are there practical applications?

The immediate value is foundational for inverse boundary problems: understanding which interior structures boundary responses can identify. The reported results address identifiability, not a reconstruction procedure, stability under measurement error, or performance with finite data. They therefore clarify theoretical limits for recovering conductivity from boundary measurements without establishing a practical imaging method.

This section was generated by GPT-6 Astra Medium. This explanation is based on the result summary and manuscript abstracts below. This context is separate from OpenAI's source text.

3 manuscripts

Determination of a metric and a unitary connection from one boundary patch

October 5, 2026 50 pages

We prove that zero-frequency boundary measurements on any nonempty open boundary patch determine both a smooth Riemannian metric and a smooth unitary connection on the trivial Hermitian rank-two bundle over a compact connected smooth manifold of dimension at least three with smooth boundary. Both inputs and observations are restricted to the same patch. The metric and connection are determined up to a diffeomorphism and a unitary gauge that restrict to the identity on the measured patch.

PDF LaTeX source
Cite (BibTeX) 
@misc{OAI:Determination-of-a-metric-and-a-unitary-connection-from-one-boundary-patch-October-5-2026,
  author = {{OpenAI}},
  title = {{Determination of a metric and a unitary connection from one boundary patch}},
  howpublished = {OpenAI Math Release preprint
                  \href{https://github.com/openai/math/blob/main/preprints/Determination-of-a-metric-and-a-unitary-connection-from-one-boundary-patch-October-5-2026/paper.pdf}{OAI:Determination-of-a-metric-and-a-unitary-connection-from-one-boundary-patch-October-5-2026}},
  year = {2026}
}

Smooth anisotropic uniqueness in the Calderón problem from one boundary patch

September 24, 2026 43 pages

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.

PDF LaTeX source
Cite (BibTeX) 
@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}
}

Nonuniqueness for bounded measurable scalar conductivities in three dimensions

September 23, 2026 33 pages Main result formalized in Lean

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.

PDF LaTeX source
Cite (BibTeX) 
@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's note on what the formalization covers, from lean/docs/365.md.

Joint metric and connection recovery from one boundary patch

The following describes the scope of the Lean formalization related to the following accompanying paper(s):

Scope

The Calderón inverse problem asks whether boundary measurements determine an interior conductivity. The formalized result gives nonuniqueness for bounded measurable scalar conductivities on the ball B(0,3)⊂R3B(0,3)\subset\mathbb R^3: two uniformly positive conductivities differ on a set of positive volume, equal 11 near the boundary, and have the same full weak Dirichlet-to-Neumann operator. Weak solutions exist uniquely for every boundary trace.

Comparator links

Result Comparator statement
Scalar conductivity nonuniqueness Conductivity.lean

Data from github.com/openai/math at commit adc7f12, committed October 6, 2026 at 21:58 UTC, last checked for changes about 8 hours ago. Titles, subjects, summaries, abstracts and Lean notes are OpenAI's; page counts are read from the PDFs. The map, related results, search, kinds of results and the named-problem index are Emergent Mind's, built with text embeddings and an LLM, and may contain errors.

An Emergent Mind Labs project. Emergent Mind is not affiliated with OpenAI. None of these results has been peer reviewed. Cite the manuscripts themselves, using the BibTeX on each result's page.