The isotropic elastic Calderón problem
From OpenAI Aftermath
The isotropic elastic Calderón problem asks whether the static boundary map from displacements to tractions of an isotropic elastic body determines both Lamé moduli λ and μ inside it. A manuscript in OpenAI's public mathematics collection (github.com/openai/math), dated 24 September 2026, claims global uniqueness for smooth moduli on every bounded connected smooth domain in R3, under μ > 0 and 3λ + 2μ > 0. This page summarizes the published claim and lists publicly available follow-up work. Statements are the respective authors' claims; the publication of a paper or a formal proof is not presented here as independent verification.
Background
- As recounted in the introduction of the manuscript:
- Calderón's scalar conductivity problem and the complex geometric optics method of Sylvester and Uhlmann are the main antecedents; elasticity couples two moduli through vector displacements.
- Nakamura and Uhlmann recovered the boundary Taylor series of the moduli. Their original global uniqueness claim was corrected in an erratum, whose corrected theorem requires the gradients of the shear modulus to be small. Eskin and Ralston developed matrix complex geometric optics with restricted uniqueness results.
- In two dimensions Imanuvilov and Yamamoto proved global uniqueness for smooth coefficients without smallness. In three dimensions, Lin and Nakamura gave local boundary reconstruction, Tan and Liu boundary jets and global recovery under analytic hypotheses, and a preprint of Chen, Jiang, Liu and Tao structured cases.
The original result
- Claim. Theorem 1.1 of Global Uniqueness for the Smooth Isotropic Elasticity Inverse Problem states that on any bounded connected domain in R3 with smooth boundary, if two pairs of smooth real Lamé moduli with μj > 0 and 3λj + 2μj > 0 on the closure have the same full displacement-to-traction map, then they agree throughout the domain. Neither analyticity, closeness to constants nor agreement near the boundary is assumed.
- Method, as described: after boundary determination, the divergence is adjoined as an unknown to obtain a system with Laplacian principal part; exact physical solutions of complex geometric optics type realize local transport frames, the leading transports of the two media are matched, and a bundle over the null conic of complex directions forces the comparison to be trivial, which recovers the coefficients.
- Source (openai/math repository, commit
7ef5c0c): Global Uniqueness for the Smooth Isotropic Elasticity Inverse Problem, dated 24 September 2026, 24 pages. - Formalization, as described by the repository. The scope note lean/docs/372.md states that the formalization proves the uniqueness theorem under the same hypotheses, and that it does not supply a reconstruction algorithm.
Public follow-up work
| Date | Work | Relation | Public claim | Public status (checked 29 October 2026) |
|---|---|---|---|---|
| 2026-10-06 | Boundary recovery of smooth isotropic elastic moduli, an article of the Latent Seminar blog at lowrank.github.io | Reading (exposition) | Explains strain, stress and traction, why the positivity conditions are natural, the claimed inverse theorem, physical solutions and complex exponential tests, and the comparison of transports in all complex directions, with outlines of the key steps. | Blog article; it states that its review covers statements, introductions and selected proof mechanisms and is not an independent verification of the complete proofs. |
| 2026-10-10 | H. Diao, U. Durdiev, Unique Determination of the Lamé Parameters in Three Dimensions, arXiv:2610.14741v1 | Parallel (independent claim) | Proves that the full Dirichlet-to-Neumann map for static isotropic elasticity determines both smooth Lamé parameters in a bounded smooth domain of R3 under strong convexity, with no smallness, symmetry, separability or analyticity hypothesis, through an exact four-component Laplace-type reduction, complex geometrical optics solutions from physical elastic solutions and holomorphically weighted matrix moments. | arXiv preprint, version 1, not refereed. The authors state that a preliminary draft with the uniqueness statement and the main elements of the approach was saved by 4 October 2026, before the public release of the OpenAI manuscript, and that the approach was developed independently of it; the acknowledgments state that ChatGPT was used for exploratory mathematical discussions and editorial assistance. |
| 2026-10-15 | Global uniqueness for isotropic elasticity, a course on Open Mathematics Courses (kokunoyumeto/open-math-courses-public) | Reading (exposition) | Seven lessons on the manuscript's proof: the displacement-to-traction map, boundary jets and the transfer of solutions, an augmented system and a Carleman estimate, planar transport equations, exact solutions with prescribed leading terms, matching the transports, and the null conic with the recovery of the moduli. The course states that boundary determination, interior elliptic regularity, a Carleman estimate and holomorphic frames are used as named results from the literature. | The lessons state that they were written by Claude Opus 5.5 (Anthropic) in October 2026 and self-checked by the writing AI; the course lists its status as author self-checked. |
| 2026-10-16 | H. Diao, U. Durdiev, Logarithmic Stability for the Inverse Boundary Problem in Three-Dimensional Isotropic Elasticity, arXiv:2610.24053v1 | Extension (stability) | Establishes logarithmic stability for the simultaneous recovery of both Lamé parameters from the full Dirichlet-to-Neumann map, uniformly on fixed bounded C22 classes of strongly convex coefficients, and sharp boundary stability exponents for derivatives. The paper states that its quantitative argument yields global uniqueness for C22 Lamé parameters, relaxing the smoothness assumptions of both earlier uniqueness results, which it cites. | arXiv preprint, version 1, not refereed; the acknowledgments state that ChatGPT was used for exploratory mathematical discussions and editorial assistance. |
| 2026-10-20 | AI Proved What?, episode 372 Feel the inside from the surface?, remade version of 24 October 2026 | Reading (exposition) | Video explainer of the claim that on every bounded connected smooth domain in three dimensions, two isotropic materials with smooth moduli satisfying the positivity conditions and the same full static map from surface displacements to surface forces have the same moduli everywhere. The episode states that reconstruction, stability, partial data and anisotropic materials are not claimed, and it mentions the independent proof by Diao and Durdiev posted four days after the release. | Web page and video; the site states that its scripts and animations are produced with AI assistance and that the narration is an AI-generated voice. Status record last checked 27 October 2026: posted as a preprint: OpenAI, 24 pages, dated September 24, 2026; under expert review: not yet; peer reviewed: not yet; checked in Lean: the theorem, re-proving the cited boundary and Carleman inputs. The page also states that the theorem has a Lean proof of about 32,000 lines, not run by the site, which re-proves the boundary and Carleman inputs the paper cites, and that it found no specialist verdict. |
Notes
- The background summarizes the manuscript's own account of earlier work.
- The list covers follow-ups found in public sources up to the date above; it is not a priority or completeness claim.
