The planar Mumford–Shah regularity conjecture and local weak-L4 gradient bounds

What OpenAI claims

Resolves the interior regularity conjecture for reduced absolute planar Mumford–Shah minimizers with bounded fidelity data. Locally, the closed discontinuity set is a C^1,α arc, a regular crack tip, or three arcs meeting at 120°; only finitely many global connected components meet any compact interior region.

Paper

1. Interior regularity of planar Mumford–Shah minimizers

   24 September 2026 · Preprint, not peer reviewed · PDF

   OpenAI's abstract We prove the interior regularity assertion of the planar Mumford–Shah conjecture for reduced absolute minimizers with bounded fidelity data. Every interior point of the closed discontinuity set has a neighborhood consisting of a C^1,α arc, an arc ending at that point, or three such arcs meeting at 120 degrees. Only finitely many global connected components meet any relatively compact open set.

Status

Preprint. The paper has not been independently reviewed. Treat the result as OpenAI's claim until other mathematicians have checked it. If a video exists, it illustrates the statement; it is not the proof.