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 \emph{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)<C_{\rm cell}) cannot serve as the integrand of a universal pointwise lower bound (E_{\rm xc}(\Psi)\ge -!\int F(\mathcal R_\Psi)\rho_\Psi^{4/3}) for arbitrarily large (N). The proof constructs antisymmetric many-cell states whose local data ((\rho_\Psi,T_\Psi)) look like a flat Fermi sea but whose indirect energy is dominated by one-particle self-interaction. Consequently, no universal pointwise bound depending only on ((\rho_\Psi,T_\Psi)) can have (C_D(q)) as its small-(\mathcal R) limit.