Geometric limits on magnetic matching of spectral Dirac boundary conditions

Magnetic fields cannot generally make spectral boundary selection coincide with zero-energy Dirac propagation. For a smooth simply connected planar cavity at fixed subcritical flux, we prove that the least squared Hilbert–Schmidt projector discrepancy is a geometric energy, positive for every nondisk. The minimum is attained by fields of one sign. Thus magnetic redistribution cannot compensate boundary shape, even though the tangential spectrum depends only on perimeter and flux. This identifies information about bulk continuation that boundary eigenvalues alone do not contain.