The Mercedes-star partition of a disk

Among triples of pairwise disjoint nonempty connected open subsets of the unit disk, the three sectors of opening $2\pi/3$ minimize the largest first Dirichlet eigenvalue. The minimum is $j_{3/2,1}^2$, where $j_{3/2,1}$ is the first positive zero of the Bessel function $J_{3/2}$. The proof transfers any triple with smaller energy to a triple on the unit sphere with energy below its known sharp minimum $15/4$. A common radial change of variables matches the angular terms after division by positive reference profiles. A Wronskian comparison shows that the remaining radial term has the required sign. The argument imposes no condition on the cells' boundary regularity, topology, symmetry, or coverage of the disk.