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

For a bounded domain $\Omega\subset\mathbb R^2$ and $g\in L^\infty(\Omega)$ , the Mumford-Shah functional is $\int_{\Omega\setminus K}|\nabla u|^2+\mathcal H^1(K)+\int_\Omega|u-g|^2$ , minimized over closed sets $K$ and functions $u$ smooth off $K$ . Minimizers exist (De Giorgi, Carriero, Leaci; Ambrosio), $K$ is regular off a closed $\mathcal H^1$ -null set (David; Ambrosio-Fusco-Pallara), and Bonnet reduced the question to classifying global minimizers. Mumford and Shah (1989) conjectured that $K$ has only the geometry suggested by image segmentation. Is the discontinuity set of every minimizer, inside $\Omega$ , locally a finite union of $C^{1}$ arcs that are regular, end at crack tips, or meet in threes at $120^\circ$ ?

Result: Proved(see note)
Status: Candidate (review pending)
AI contribution: AI-discovered
Method: Argument
Field: Calculus of variations; free-discontinuity problems
Posed by: David Mumford and Jayant Shah (Comm. Pure Appl. Math. 42, 1989)
Year posed: 1989
Years open: 37y
Solved: 2026-09-24
Model: Unreleased internal OpenAI model
Vendor: OpenAI
Collaborators: —
Verification: Unreviewed
Publication: Announced
Collection: OpenAI math release (October 2026), version adc7f12
Significance: 52 / 100
Disclosed cost: —
Wikipedia: No dedicated article

What was actually shown

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.

What the AI did

The release README says every result in openai/math was produced by an unreleased internal OpenAI model with a fixed procedure, on average about three hours of ChatGPT Pro thinking compute per result. This result is not one of the README's two exceptions (the Hodge conjecture for CM abelian varieties and the Re(s) > 11/12 zero-free region). The manuscript is authored 'OpenAI' and names no human author. For the endpoint and compactness theory the manuscript invokes the De Lellis-Focardi monograph; its new step excludes bounded components of blow-up limits by a translation and harmonic-interaction argument.

Verification

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.

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

What was actually shown

What the AI did

Verification

Sources

Discussion