Nonuniform-flux saturation of the matching-point bound

Determine whether any nonuniform flux configuration at the matching point can simultaneously satisfy both equality conditions in the trace-norm and Lieb-theorem energy bounds.

Background

At the matching point, the authors derive a global lower bound on the unprojected Majorana energy by decomposing the hopping matrix into two uniform-magnitude honeycomb hopping matrices and applying the trace-norm triangle inequality together with Lieb’s theorem. The uniform flux-free sector is shown to saturate this bound, but the analysis does not determine whether another, nonuniform flux sector can also saturate both inequalities.