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

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