OpenAI · 365-03 · 2026-09-23 · Partial differential equations
Nonuniqueness for bounded measurable scalar conductivities in three dimensions
preprints/Nonuniqueness-for-Bounded-Measurable-Scalar-Conductivities-in-Three-Dimensions-September-23-2026
Family 365. Joint metric and connection recovery from one boundary patch.
The authors are OpenAI. This companion does not verify the proof and does not adopt it as a theorem of Ngo Cheung’s series. It is not a premise. O.1 stays NOT_CLOSED. Anti-de Sitter results are not a de Sitter state, not Em02-E’s relational lapse, not XVI-P’s QdS scheme, and not XVI-Q’s sigma-image record. No quantum coefficient is admitted. No compilation is asserted.
Read against the standing problems
Standing problems. No standing problem is named in this text. It was still read against the nine problems and does not close any of them. The authors are OpenAI. The companion does not verify the proof. This is not a premise of a manuscript by Ngo Cheung. O.1 stays NOT_CLOSED. No quantum coefficient is admitted. No compilation is asserted. XVI-P · XVI-N
Zero-frequency measurements on any nonempty open boundary patch determine a smooth metric and smooth unitary connection on a trivial Hermitian rank-two bundle over a compact connected manifold of dimension at least three, up to diffeomorphism and gauge fixed on that patch. Inputs and observations use the same patch. In contrast, distinct uniformly positive bounded measurable scalar conductivities on a three-dimensional ball can have identical full-boundary data.
We construct two distinct uniformly positive bounded measurable scalar conductivities on a ball in ℝ3 with the same full Dirichlet-to-Neumann operator. Both conductivities equal one near the boundary. This gives nonuniqueness in the scalar Calderón problem at bounded measurable regularity.
Preprint source and Lean scope
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This preprint is named in the Lean scope note for 365: Joint metric and connection recovery from one boundary patch. The authors are OpenAI. This companion does not verify the formalization. It is not a premise. The OAI library is not imported. No compilation is asserted. O.1 stays NOT_CLOSED.