Calderon's uniqueness question for bounded measurable scalar conductivities in dimension three
Calderon (1980) posed the inverse conductivity problem for real scalar conductivities with : does the Dirichlet-to-Neumann map determine ? In the plane Astala and Paivarinta (2006) proved uniqueness for all bounded measurable conductivities. In dimension uniqueness was proved under derivative regularity only: smooth (Sylvester-Uhlmann), Lipschitz (Caro-Rogers), in dimensions three and four (Haberman). Does determine a bounded measurable, uniformly positive scalar conductivity in dimension ?
- Result
- Disproved(see note)
- Status
- Candidate (review pending)
- AI contribution
- AI-discovered
- Method
- Construction
- Field
- Inverse problems; elliptic PDE
- Posed by
- Alberto P. Calderon, On an inverse boundary value problem (1980)
- Year posed
- 1980
- Years open
- 46y
- Solved
- 2026-09-23
- Model
- Unreleased internal OpenAI model
- Vendor
- OpenAI
- Collaborators
- —
- Verification
- Lean-checked, statement unaudited
- Publication
- Announced
- Collection
- OpenAI math release (October 2026), version adc7f12
- Significance
- 40 / 100
- Disclosed cost
- —
- Wikipedia
- No dedicated article
What was actually shown
Theorem 1.1: there are real with , both equal to 1 near the boundary, differing on a set of positive measure, whose full weak Dirichlet-to-Neumann operators coincide. A corollary places such conductivities in any bounded Lipschitz domain in . The construction nests scalar blocks with toroidal interfaces. This refutes the uniqueness assertion quantified over all . Not shown: counterexamples in each dimension , control of the contrast, or anything about conductivities with some derivative regularity (where uniqueness is known).
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 rests on one manuscript dated September 23, 2026; the family's two smooth-uniqueness manuscripts are a separate entry.
Verification
No independent mathematician has checked this yet. Checked here: the abstract, introduction and Theorem 1.1 of 'Nonuniqueness for bounded measurable scalar conductivities in three dimensions' were read against Calderon's problem. Lean: formalization.yaml lists ComparatorChallenges/Conductivity.json, declaration OAI.ScalarConductivity.main_nonuniqueness in OAI/Analysis/Conductivity/Main.lean. The challenge statement was read: two measurable essentially bounded functions on the ball of radius 3 in R3, between constants 0<c<C, equal to 1 near the boundary, differing on a set of positive measure, with the same weak Dirichlet-to-Neumann operator on H1 modulo H1_0 (weak solutions existing uniquely). That is the headline claim. Not rebuilt here. The paper states it covers dimension three only and does not prescribe the contrast C/c.