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The Lee-Uhlmann conjecture: smooth anisotropic Calderon uniqueness in dimension at least three, from one boundary patch

Let (M,g)(M,g) be a compact connected smooth Riemannian manifold with boundary, n=dim⁡M≥3n=\dim M\ge3, and Λg\Lambda_g its Dirichlet-to-Neumann map for Δg\Delta_g. Diffeomorphisms fixing the boundary preserve Λg\Lambda_g. Calderon (1980) posed the conductivity problem; Lee and Uhlmann (1989) proved that Λg\Lambda_g determines gg up to such a diffeomorphism for real-analytic metrics, and the smooth case became the anisotropic Calderon problem, solved before only for analytic or Einstein metrics, near special backgrounds, or on conformally transversally anisotropic manifolds. The partial-data version, with input and output on the same open boundary patch, was open even for smooth scalar conductivities. For smooth metrics and n≥3n\ge3, does Λg\Lambda_g (or its restriction to one boundary patch) determine gg up to a diffeomorphism fixing the measured boundary?

Result
Proved(see note)
Status
Candidate (review pending)
AI contribution
AI-discovered
Method
Argument
Field
Inverse problems; elliptic PDE on manifolds
Posed by
John M. Lee and Gunther Uhlmann (Comm. Pure Appl. Math., 1989), extending Alberto Calderon's 1980 inverse conductivity problem to anisotropic metrics
Year posed
1989
Years open
37y
Solved
2026-09-24
Model
Unreleased internal OpenAI model
Vendor
OpenAI
Collaborators
—
Verification
Unreviewed
Publication
Announced
Collection
OpenAI math release (October 2026), version adc7f12
Significance
55 / 100
Disclosed cost
—
Wikipedia
No dedicated article

What was actually shown

Theorem 1.1: for n≥3n\ge3, compact connected smooth MM with smooth boundary and any nonempty proper relatively open Γ⊂∂M\Gamma\subset\partial M, if smooth metrics satisfy Λg1,Γ=Λg2,Γ\Lambda_{g_1,\Gamma}=\Lambda_{g_2,\Gamma} (energy form with input and output on Γ\Gamma), then g2=Φ∗g1g_2=\Phi^*g_1 for a smooth diffeomorphism with Φ∣Γ=Id\Phi|_\Gamma=\mathrm{Id}. Corollaries: full-boundary uniqueness, and uniqueness of smooth positive scalar conductivities on smooth Euclidean domains from one same-patch measurement. The October 5 companion adds a smooth unitary connection on a trivial Hermitian rank-two bundle, determined up to gauge fixed on the patch. Not shown: dimension two, nonsmooth metrics, stability estimates, or reconstruction.

What the AI did

The release README says every result was produced by an unreleased internal OpenAI model with a fixed procedure of roughly three hours of ChatGPT Pro thinking compute per result. This family is not among the README's exceptions (the Hodge conjecture for CM abelian varieties, and the Re(s) > 11/12 zero-free region, whose write-up was human-edited). The manuscripts are authored 'OpenAI' and name no human author. This entry draws on two manuscripts: the smooth anisotropic uniqueness theorem (September 24, 2026, principal) and a stronger October 5 version that also recovers a unitary connection on a trivial rank-two bundle. The family's September 23 nonuniqueness paper for bounded measurable conductivities is a separate entry.

Verification

No independent mathematician has checked this yet. Checked here: the introduction, Theorem 1.1 and Corollaries 1.2-1.3 of 'Smooth anisotropic uniqueness in the Calderon problem from one boundary patch' were read against the problem as the manuscript and its references describe it. The theorem covers arbitrary smooth metrics, any nonempty proper open patch for both input and observation, nonorientable manifolds and several boundary components. The release's only Lean formalization in this family (Conductivity) concerns the separate bounded-measurable nonuniqueness entry, so this entry is unreviewed. The proof's central step is a uniform estimate for transfer functionals on shrinking spheres; it was not refereed here. Given its standing in inverse problems, this claim warrants expert review before anything else.

Sources

Changelog1 change

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