The exact structure-factor reduction of the Dirac problem

What exactly stands between kinetic control of a homogeneous fermion gas and the Dirac lower bound on its indirect Coulomb energy? We give an explicit, exact answer in the form of a one-line obstruction. Working in the periodic box with spin degeneracy $q$, we rewrite the indirect Coulomb energy through the static structure factor and compare it term by term to the closed-shell Fermi sea. The difference is bounded below by a single nonnegative quantity, the Coulomb-weighted Fermi-structure deficit $\mathfrak D_L(\Gamma_L)$. The Dirac lower bound $-C_D(q)\rho^{4/3}$ holds for any homogeneous sequence along which $\mathfrak D_L=o(L^3)$, and conversely any putative violation of the Dirac bound forces a macroscopic deficit along a subsequence. The formulation isolates the mathematical breakthrough still needed: a kinetic proof that small kinetic excess prevents many-body cancellations of density modes below the Fermi-sea structure factor.