Uniqueness for bounded measurable scalar conductivities in three dimensions

Does the full weak Dirichlet-to-Neumann map determine a uniformly positive bounded measurable scalar conductivity in R³? The counterexample target uses two conductivities on B(0,3), equal to one near the boundary but differing on a positive-measure set.

Why it matters and what counts as progress

Why it matters: It tests the regularity boundary of scalar Calderón uniqueness in three dimensions.

What counts as progress: Two distinct uniformly positive bounded scalar conductivities with the same full boundary operator, or a uniqueness theorem under weaker regularity.

How the problem developed

1987 · Partial result [1]
John Sylvester and Gunther Uhlmann Proved uniqueness for smooth scalar conductivities in dimensions at least three, leaving lower regularity outside their theorem.
2006 · Partial result [2]
Kari Astala and Lassi Päivärinta Established uniqueness for bounded measurable conductivities in two dimensions; their paper noted that the higher-dimensional measurable case remained open.
23 Sept 2026 · Reported claim [3]
OpenAI Reported two distinct uniformly positive bounded measurable scalar conductivities on a three-dimensional ball, equal to 1 near its boundary, with the same full weak Dirichlet-to-Neumann map.

AI contribution

Nonuniqueness for bounded measurable scalar conductivities in three dimensions

OpenAI

The manuscript constructs two distinct uniformly positive bounded measurable scalar conductivities on the ball B(0,3) in R³. Both equal one near the boundary, differ on a set of positive measure and have the same full weak Dirichlet-to-Neumann operator.

AI involvementAI-led

ImportanceMajor

Technical verificationFormal (Lean 4 theorem with Comparator target)

Human understandingFew