VibeMathedMath problems solved with AI
Sign in

The elastic Calderon problem in three dimensions: do boundary measurements determine smooth isotropic Lame moduli?

Let Ω⊂R3\Omega\subset\mathbb R^3 be a bounded connected smooth domain and λ,μ∈C∞(Ω‾)\lambda,\mu\in C^\infty(\overline\Omega) Lame moduli with μ>0\mu>0, 3λ+2μ>03\lambda+2\mu>0. The static displacement-to-traction map Λλ,μ\Lambda_{\lambda,\mu} sends boundary displacements ff to the tractions σλ,μ(uf)n\sigma_{\lambda,\mu}(u_f)n of the solution of div⁡(λ(div⁡u)I+2μe(u))=0\operatorname{div}(\lambda(\operatorname{div}u)I+2\mu e(u))=0 with trace ff. Nakamura and Uhlmann claimed global uniqueness in 1994; their 2003 erratum proved it only when ∥∇μj∥Cm\|\nabla\mu_j\|_{C^m} is small. Does Λλ1,μ1=Λλ2,μ2\Lambda_{\lambda_1,\mu_1}=\Lambda_{\lambda_2,\mu_2} imply λ1=λ2\lambda_1=\lambda_2 and μ1=μ2\mu_1=\mu_2 for all such smooth moduli?

Result
Proved(see note)
Status
Candidate (review pending)
AI contribution
AI-discovered
Method
Argument
Field
Inverse problems; linear elasticity
Posed by
Elastic analogue of Calderon's problem; global uniqueness claimed by G. Nakamura and G. Uhlmann (1994), reduced to a small-gradient result in their 2003 erratum
Year posed
2003
Years open
23y
Solved
2026-09-24
Model
Unreleased internal OpenAI model
Vendor
OpenAI
Collaborators
—
Verification
Lean-checked, statement unaudited
Publication
Announced
Collection
OpenAI math release (October 2026), version adc7f12
Significance
30 / 100
Disclosed cost
—
Wikipedia
No dedicated article

What was actually shown

Theorem 1.1: on any bounded connected domain Ω⊂R3\Omega\subset\mathbb R^3 with C∞C^\infty boundary, if λj,μj∈C∞(Ω‾)\lambda_j,\mu_j\in C^\infty(\overline\Omega) satisfy μj>0\mu_j>0, 3λj+2μj>03\lambda_j+2\mu_j>0 and Λλ1,μ1=Λλ2,μ2\Lambda_{\lambda_1,\mu_1}=\Lambda_{\lambda_2,\mu_2}, then λ1=λ2\lambda_1=\lambda_2 and μ1=μ2\mu_1=\mu_2. No analyticity, smallness or prior boundary agreement is assumed. It is a uniqueness theorem for smooth coefficients and full static data only: no reconstruction algorithm, no stability estimate, no partial-data or low-regularity version, and nothing about anisotropic elasticity.

What the AI did

The release README says the results were produced by an unreleased internal OpenAI model with one fixed procedure, on average about three hours of ChatGPT Pro thinking compute per result. This result 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 manuscript is authored 'OpenAI' and names no human author.

Verification

No independent mathematician has checked this yet. Checked here: the introduction and Theorem 1.1 were read against the Nakamura-Uhlmann problem. lean/docs/372.md points to ComparatorChallenges/ElasticityUniqueness.json (theorem OAI.Elasticity.global_uniqueness, solution module OAI.MathematicalPhysics.Elasticity.Uniqueness); the solution file exists at the pinned commit and the challenge is not in formalization.yaml. The statement was read here: for every bounded connected open set in R3\mathbb R^3 with smooth boundary charts and moduli smooth up to the boundary with μ>0\mu>0, 3λ+2μ>03\lambda+2\mu>0, equality of the variationally defined Dirichlet-to-Neumann pairings forces both moduli to agree on Ω\Omega. That is the headline claim. Not rebuilt here. The DN map is defined on a trace quotient rather than H1/2H^{1/2}, as the file notes. No reconstruction or stability estimate is claimed.

Sources

Changelog1 change

Discussion