Interior regularity of planar Mumford–Shah minimizers

Formalization

• OpenAI, MumfordShah compact exclusion (OpenAI Math Release, 6 October 2026). The Lean source contains compactPieceExclusion and supporting results for the manuscript's compact-piece exclusion argument, with hypotheses including global absolute minimality, connected complement, linear energy growth and a regular arc in the compact piece; this collection did not build or audit its dependencies. [family-level]

Related prior literature mentioned by others

• Francesco Deangelis, Solution of the Mumford-Shah conjecture (arXiv:2609.26732, v1 22 Sep 2026, v2 5 Oct 2026). The manuscript cites v1 as an independent proof of the same interior structure by a different argument (compact translations and second-variation inequalities); the v2 acknowledgments state that the "initial proof strategy and manuscript draft were generated using ChatGPT Astra (OpenAI)". [AI-generated per source] [family-level]

Possibly relevant literature (found by the agent, not checked)

• Francesco Deangelis, Classification of boundary blow-ups for the Mumford-Shah functional with Dirichlet boundary conditions (arXiv:2610.23513, 16 Oct 2026). It treats blow-ups at the boundary under Dirichlet conditions, which the manuscript explicitly leaves out ("no assertion about finiteness up to $\partial\Omega$"); it does not cite the manuscript.
• Francesco Deangelis, Boundary regularity for the Mumford-Shah functional with Dirichlet boundary conditions (arXiv:2404.07162, Apr 2024). It proves an epsilon-regularity theorem at the boundary, the region the manuscript's Theorem 1.1 excludes.
• Matteo Focardi, Fine regularity results for Mumford-Shah minimizers: porosity, higher integrability and the Mumford-Shah conjecture (arXiv:1610.03784, Oct 2016). It surveys links between the conjecture and higher integrability of the gradient, including an energetic characterization, which relates to the manuscript's weak-$L^4$ corollary.
• Yi Ru-Ya Zhang, A growth estimate for the planar Mumford--Shah minimizers at a tip point: An alternative proof of David--Léger (arXiv:2502.15605, Feb 2025). It proves an $r^{1/2}$ growth estimate at a crack tip, the behaviour the manuscript cites as the reason for the endpoint exponent 4 in $\nabla u\in L^{4,\infty}_{\mathrm{loc}}$.

Digests and expositions

• Latent Seminar, Interior regularity of planar Mumford–Shah minimizers (6 October 2026). This exposition explains the claimed compact-piece exclusion mechanism and the crack-tip scaling behind weak-L⁴ integrability, explicitly limiting its review to statements, introductions and selected proof mechanisms. [family-level]
• Gradient Damage, Boltzmann Equation, Hot Spots Conjecture and 7 More in Partial Differential Equations (15 October 2026). The video compilation includes a chapter beginning at 0:22 on the planar Mumford–Shah claim, linked to this manuscript and presented as a visual illustration rather than a proof. [family-level]