A gapless kinetic-to-Dirac theorem for homogeneous occupation states

We prove that the Dirac exchange constant is the correct asymptotic indirect-energy lower bound, in the thermodynamic limit, for a broad class of homogeneous fermionic occupation states with controlled kinetic excess and without any spectral gap assumption at the Fermi surface. The states considered are occupation-diagonal density matrices \(\Gamma_L\) in the plane-wave basis of a periodic box, with arbitrarily correlated occupation probabilities. Writing \(\varepsilon_L\) for the per-volume kinetic excess above the finite-volume Fermi sea, normalized by \(k_F^{2}\rho L^3\), we show that for every small \(\alpha>0\) the per-volume indirect energy is bounded below by \(-C_D(q)\rho^{4/3}-A_{q,\rho}(\alpha+\varepsilon_L/\alpha)\rho^{4/3}-o_\alpha(1)\). Optimizing in \(\alpha\) gives the interpolation \(\liminf_{L\to\infty}L^{-3}I_L(\Gamma_L)\ge -(C_D(q)+2A_{q,\rho}\sqrt{\varepsilon})\rho^{4/3}\), where \(\varepsilon=\limsup\varepsilon_L\). In particular, sub-extensive kinetic excess forces the sharp Dirac bound. The proof factors into an abstract finite shell-exchange principle and two elementary lattice-Coulomb marginal estimates