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 \(\limsup_{r\downarrow 0}F(r)C_{D}(q)\) 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.