A no-go theorem for purely local $(\rho,T)$ Dirac interpolation

We prove a sharp impossibility result for local Lieb–Oxford-type lower bounds on the indirect Coulomb energy of antisymmetric fermionic states. For a normalized \(N\)-fermion wavefunction \(\Psi\) write \(\rho_\Psi\) for its one-particle density, \(T_\Psi\) for its local kinetic energy density, and \(E_{\rm xc}(\Psi)\) for its indirect (exchange-correlation) energy. Let \(\mathcal R_\Psi\) be the dimensionless local kinetic ratio. Two constants control the discussion: the Dirac exchange constant \(C_D(q)=\tfrac{3}{4}(6/\pi)^{1/3}q^{-1/3}\) and the one-cell self-interaction constant \(C_{\rm cell}=\tfrac{3}{5}(4\pi/3)^{1/3}\), which is strictly larger than \(C_D(q)\) for every spin multiplicity \(q\ge 1\). We show that any bounded function \(F\) with \(\limsup_{r\downarrow 0}F(r)