A sector-coherent gapless Dirac lower bound

We prove that the Dirac exchange constant is the correct asymptotic lower bound on the indirect Coulomb energy for a broad class of correlated fermionic states in the thermodynamic limit, without invoking a spectral gap at the Fermi surface. The states considered are sector-coherent: the macroscopic Fermi core is filled sector by sector, while the active Fermi shell of width \(\alpha k_{F}\) may carry arbitrary many-body correlations and coherent superpositions. We show that every such state \(\Gamma_{L}\) on a periodic box \(\Lambda_{L}\) satisfies the per-volume bound \(L^{-3}I_{L}(\Gamma_{L})\ge -C_{D}(q)\rho^{4/3}-A_{\rho,q}(\alpha+\alpha^{4/3})-o_{\alpha}(1)\), so that letting \(L\to\infty\) and then \(\alpha\to 0\) recovers the sharp Dirac constant. The proof combines an elementary Riemann-sum calculation for the filled-core exchange, a Pauli-bound estimate on the active-shell density, and a lattice-Coulomb cross-exchange bound. This is the Coulomb side of the gapless kinetic-to-Dirac programme: once the kinetic data have confined excitations to a thin Fermi shell, residual correlations in the shell are harmless at leading order.