Imagine choosing the shape of a container for a quantum state. Making the state vary sharply costs energy. But its density also repels itself, which can favour spreading out. If the container must keep the same volume, which shape wins?

This Evidence Press candidate studies that competition in a three-dimensional Hartree model. Its claim is precise: when the repulsion is sufficiently weak, a ball is the unique minimising shape, apart from moving it and ignoring the appropriate negligible sets.

The word unique matters. Earlier work already showed that minimisers exist and become close to balls as the charge becomes small. Close to round is not the same as exactly round. A tiny but genuine deformation could still have survived.

The new argument closes that gap in two stages. It first uses the earlier global theorem to bring every minimiser into one fixed neighbourhood of a ball. It then studies the boundary condition for an optimal shape. After removing translations and enforcing the volume constraint, the linearised boundary equation is invertible. Nearby optimal shapes must therefore be balls too.

Several technical links are essential. The state selected by the local argument must really minimise the energy. The neighbourhood must use the same regularity throughout. And the equality statement must respect the distinction between sets of zero volume and sets invisible to the Sobolev energy.

This is a written analytic proof candidate, not a numerical simulation. The accompanying automated checks test elementary identities and file integrity, not the theorem. There is no explicit largest allowed charge, no claim about strong repulsion, and no external peer review or formal verification. A recent conference abstract mentions related rigidity work, so historical priority remains uncertain.

The manuscript, review response and source comparison are linked from the release page. This briefing uses an AI-generated voice and is not additional mathematical evidence.