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 \emph{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