One small patch sees everything?
Three AI preprints claim uniqueness for the smooth anisotropic Calderón problem from one boundary patch, and nonuniqueness for rough conductivities
Covers result 365 in OpenAI’s catalog: all 3 papers. See its connections to other results and the work it cites.
4:04. English captions available. The video loads from YouTube when you press play.
Chapters
The claim, in plain words
Can you tell what is inside a body from voltages and currents measured on its surface? Calderón posed the mathematical version in 1980. On a line, a toy version, the answer is no: our computer builds four different layered conductivities on an interval with exactly the same end-to-end response. And a change of coordinates that fixes the boundary never changes the measurements, so for anisotropic materials the inside can at best be recovered up to such a change.
On a disc the measurements carry one number per mode, but the inside fades fast: our computer finds an inclusion’s effect falling from about 0.3 in the first mode to about 10⁻⁵ by the tenth. For smooth anisotropic metrics in dimension three and up, uniqueness was known only in special cases: real-analytic metrics, by Lee and Uhlmann and by Lassas and Uhlmann, and special geometries or metrics near special ones. Papers this summer still called the smooth problem open, and a 2024 claim to solve it was withdrawn three days later as incorrect.
Result 365 is three OpenAI papers, made public October 6, 2026. The September 24 paper, 43 pages, claims that for smooth metrics in dimension three and up, zero-frequency measurements on any one boundary patch, with inputs and outputs on that same patch, determine the metric up to a change of coordinates fixed on the patch. The October 5 paper, 50 pages, the headline, claims the same while also recovering a unitary connection on a trivial rank-two bundle, up to gauge.
What it doesn’t claim
The September 23 paper gives the contrast: in three dimensions, two different scalar conductivities, bounded above and below but not smooth, with identical full boundary data. Not claimed: stability or a reconstruction method, nonzero frequencies, where smooth counterexamples exist, or inputs and outputs on different patches. Specialists who cite the smooth papers describe them as an announcement.
Where it stands
- Posted as a preprint
- OpenAI, 50 + 43 + 33 pages, dated October 5, September 24 and September 23, 2026
- Peer reviewed
- Not yet; cited by specialists “as an announcement”
- Formalized in Lean
- The rough nonuniqueness theorem only; the smooth results have no Lean proof (scope notes)
Last checked October 19, 2026. All three papers are unrefereed and unchanged since release. Only the nonuniqueness theorem is formalized in Lean, about 75,000 lines, not run by us; the smooth results have no Lean proof. Two groups of specialists cite the smooth results as an announcement, and one notes it is unclear whether it settles the linearized problem. We found no specialist verdict and no reported error. OSVerify has not reviewed it.
How the proof goes
- Growing the matched region. For the smooth result, equal patch data first give equal boundary jets. A small cap glued outside the patch makes the two Green kernels agree there, and harmonic functions from sources in the cap label interior points, defining a matched region. At its frontier, harmonic functions are moved across shrinking spheres; the crux is a uniform bound on this transfer: it makes harmonic differences of order r², so the metric mismatch is of order r^(3/2), better than the assumed order r, and the region can grow. Finally a leftover conformal factor is forced to be one, because in dimension three and up such a factor changes which functions are harmonic; our computer shows the same effect on the energy, which vanishes in dimension two.
- Hiding in rough conductivities. The nonuniqueness paper builds a block from box-shaped tori, one outer and two inner, and nests copies inside each other forever; our computer checks its printed numbers, like two s cubed equals 0.3 seven, below one, so the nested volume shrinks to zero. The block’s flux rule forces every solution to share one surface average, and a hidden potential then changes the conductivity without changing any measurement.
The numbers in the video
- Ours: four different layered conductivities on an interval give exactly the same end-to-end response.
- Ours: on a disc, an inclusion's effect on the boundary map falls from 0.29 in the first mode to about 10⁻⁵ by the tenth.
- Ours: scaling lengths by 2 multiplies the Dirichlet energy by 2^(n−2) — no change in two dimensions.
- Lean: 595 files and 75,520 lines in the import closure of the rough-conductivity comparator (counted by us; not run by us).
Read the source
- Determination of a metric and a unitary connection from one boundary patch (PDF), 50 pages.
- Smooth Anisotropic Uniqueness in the Calderón Problem from One Boundary Patch (PDF), 43 pages.
- Nonuniqueness for Bounded Measurable Scalar Conductivities in Three Dimensions (PDF), 33 pages.
- The Lean formalization notes.
- The collection’s catalog, where this is result 365.
The 3 papers in this result
OpenAI groups related papers, such as alternative proofs, supporting results and consequences, into one result; we label the first paper it lists the main paper and the others companions.
Transcript
Read the full script
0:00The question
Can you tell what is inside a body from voltages and currents measured on its surface? Calderón posed the mathematical version in 1980. On a line, a toy version, the answer is no: our computer builds four different layered conductivities on an interval with exactly the same end-to-end response. And a change of coordinates that fixes the boundary never changes the measurements, so for anisotropic materials the inside can at best be recovered up to such a change. On a disc the measurements carry one number per mode, but the inside fades fast: our computer finds an inclusion’s effect falling from about 0.3 in the first mode to about 10⁻⁵ by the tenth. For smooth anisotropic metrics in dimension three and up, uniqueness was known only in special cases: real-analytic metrics, by Lee and Uhlmann and by Lassas and Uhlmann, and special geometries or metrics near special ones. Papers this summer still called the smooth problem open, and a 2024 claim to solve it was withdrawn three days later as incorrect.
1:07The claim
Result 365 is three OpenAI papers, made public October 6, 2026. The September 24 paper, 43 pages, claims that for smooth metrics in dimension three and up, zero-frequency measurements on any one boundary patch, with inputs and outputs on that same patch, determine the metric up to a change of coordinates fixed on the patch. The October 5 paper, 50 pages, the headline, claims the same while also recovering a unitary connection on a trivial rank-two bundle, up to gauge. The September 23 paper gives the contrast: in three dimensions, two different scalar conductivities, bounded above and below but not smooth, with identical full boundary data. Not claimed: stability or a reconstruction method, nonzero frequencies, where smooth counterexamples exist, or inputs and outputs on different patches. Specialists who cite the smooth papers describe them as an announcement.
2:07How the proof goes
For the smooth result, equal patch data first give equal boundary jets. A small cap glued outside the patch makes the two Green kernels agree there, and harmonic functions from sources in the cap label interior points, defining a matched region. At its frontier, harmonic functions are moved across shrinking spheres; the crux is a uniform bound on this transfer: it makes harmonic differences of order r², so the metric mismatch is of order r^(3/2), better than the assumed order r, and the region can grow. Finally a leftover conformal factor is forced to be one, because in dimension three and up such a factor changes which functions are harmonic; our computer shows the same effect on the energy, which vanishes in dimension two. The nonuniqueness paper builds a block from box-shaped tori, one outer and two inner, and nests copies inside each other forever; our computer checks its printed numbers, like two s cubed equals 0.3 seven, below one, so the nested volume shrinks to zero. The block’s flux rule forces every solution to share one surface average, and a hidden potential then changes the conductivity without changing any measurement.
3:21Where it stands
All three papers are unrefereed and unchanged since release. Only the nonuniqueness theorem is formalized in Lean, about 75,000 lines, not run by us; the smooth results have no Lean proof. Two groups of specialists cite the smooth results as an announcement, and one notes it is unclear whether it settles the linearized problem. We found no specialist verdict and no reported error. If they hold, steady measurements on any small patch of the boundary determine a smooth body’s interior geometry in three or more dimensions, while rough enough conductivities can hide, as the Lean proof confirms in three dimensions.
Spot a mistake?
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