The planar Mumford-Shah conjecture (interior regularity of the discontinuity set)

Theorem 1.1: for a reduced absolute minimizer $(u,K)$ with $g\in L^\infty$ , every $x\in K$ has a neighborhood where $K$ is one $C^{1,\alpha}$ arc through $x$ , one arc ending at $x$ , or three arcs meeting at $x$ at $120^\circ$ ; only finitely many connected components of $K$ meet any $U\Subset\Omega$ . Corollary: $\nabla u\in L^{4,\infty}_{\mathrm{loc}}$ , sharp because of crack tips. The key step excludes bounded components in blow-up limits. It does not treat behaviour at $\partial\Omega$ , higher dimensions, or minimizers in weaker senses than absolute ones.

Exact FrontierDelta

Prior state unknown→proved

Scope and record

Occurred: Sep 24, 2026

Delta type: SOURCE CLAIM

Assumptions: VibeMathed verification: unreviewed. Publication: announcement. AI contribution: ai-discovered. VibeMathed editorial classifications, scores, notes, relations, and dataset structure are CC BY 4.0. Source statements and linked content retain their own rights.

Canonical aliases: The planar Mumford-Shah conjecture (interior regularity of the discontinuity set) · Mumford-Shah conjecture (planar, interior)

Confidence: Not scored

Registry verification: unreviewed · announcement · candidate

Open the source record ↗

Attribution

VibeMathed
registry · event recorded by

Unreleased internal OpenAI model
model · ai model contributor · OpenAI

Evidence vector

SourceReplayReproducedFormal proofStatement auditExternal checkExpert reviewPeer review

Named evidence

VibeMathed record: The planar Mumford-Shah conjecture (interior regularity of the discontinuity set)

specialist registry · passed

No independent mathematician has checked this yet. Checked here: Theorem 1.1 was read against the planar Mumford-Shah conjecture. It is stated for reduced absolute minimizers on bounded Lipschitz domains with bounded measurable data and gives, near every point of $K$ , a $C^{1,\alpha}$ arc, an arc ending at the point, or a $120^\circ$ triple junction, plus local finiteness of the global components of $K$ . The manuscript states it makes no assertion about finiteness up to $\partial\Omega$ . The proof was not refereed. No Lean formalization of this manuscript is in the release (no lean/docs/366.md at the pinned commit). The manuscript itself cites a human preprint posted two days earlier, F. Deangelis, 'Solution of the Mumford-Shah conjecture', arXiv:2609.26732v1 (22 September 2026), which it credits with a proof of the global classification and the same interior structure by a different argument.

Source ↗

The planar Mumford-Shah conjecture (interior regularity of the discontinuity set)

witness · pending

Primary source named by the VibeMathed record. Presence in this import is not an independent mathematical verification.

Source ↗

Concurrent human proof: Deangelis, Solution of the Mumford-Shah conjecture (arXiv:2609.26732)

paper · pending

paper link supplied by the VibeMathed registry.

Source ↗

Lineage and corrections

Concurrent human proof: Deangelis, Solution of the Mumford-Shah conjecture (arXiv:2609.26732) evidence for this event

This event attributed to Unreleased internal OpenAI model

The planar Mumford-Shah conjecture (interior regularity of the discontinuity set) evidence for this event

The planar Mumford-Shah conjecture (interior regularity of the discontinuity set) parent of this event

VibeMathed record: The planar Mumford-Shah conjecture (interior regularity of the discontinuity set) evidence for this event

Act on this frontier

Verify, challenge, or extend the result.

Help verify Open canonical problem Explore evidence graph