A sharp obstruction to local $(\rho,T)$ Dirac interpolation and an exact fluctuation–current bridge

A long-standing question in the Lieb–Oxford programme asks whether the universal lower bound on the indirect Coulomb energy of fermions admits a pointwise local refinement whose low-kinetic-ratio limit reproduces the Dirac exchange constant. We show that the answer is no, in a sharp form. Let (\rho_\Psi(x)) and (T_\Psi(x)) denote the one-particle density and local kinetic energy density of a normalized antisymmetric (N)-fermion wavefunction~(\Psi), and let (E_{\xc}(\Psi)) be its indirect Coulomb (exchange-correlation) energy. Write (C_{\mathrm{cell}}:=\tfrac{3}{5}(4\pi/3)^{1/3}\approx 0.96720) and let (C_{D}(q):=\tfrac{3}{4}(6/\pi)^{1/3}q^{-1/3}) be the Dirac constant for spin multiplicity~(q). We prove that any bounded (F:[0,\infty)\to[0,\infty)) with (\limsup_{r\downarrow 0}F(r)<C_{\mathrm{cell}}) fails the natural local lower bound for some antisymmetric (\Psi) with~(N) arbitrarily large. Since (C_{\mathrm{cell}}>C_{D}(q)) for every (q\ge 1), no universal local lower bound depending only on ((\rho_\Psi,T_\Psi)) can converge to (C_{D}(q)) in the low-(\mathcal R) regime. As a positive companion we establish a Gaussian scale-window identity that rewrites the indirect energy of an admissible kernel family exactly in terms of local number variances, and combine it with a canonical-commutator uncertainty inequality to bound it from below by a conjugate-current variance.