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

Let $\rho_\Psi(x)$ and $T_\Psi(x)$ denote the one-particle density and the local kinetic energy density of a normalized antisymmetric $N$-fermion wavefunction $\Psi$, and let $E_{\mathrm{xc}}(\Psi)$ be its indirect Coulomb (exchange-correlation) energy. Set $C_{\mathrm{cell}} := \tfrac{3}{5}(4\pi/3)^{1/3} \approx 0.96720$ and let $C_D(q) := \tfrac{3}{4}(6/\pi)^{1/3} q^{-1/3}$ be the Dirac exchange 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 pointwise lower bound

$$ E_{\mathrm{xc}}(\Psi)\ \ge\ -\!\int_{\mathbb{R}^3} F\bigl(\mathcal{R}_\Psi(x)\bigr)\,\rho_\Psi(x)^{4/3}\,dx,$$

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 the pair $(\rho_\Psi, T_\Psi)$ can converge to the Dirac constant in the low-$\mathcal{R}$ limit.

As a positive companion we establish a Gaussian scale-window identity that rewrites the indirect energy of the admissible family $w_{\alpha,\beta}$ exactly in terms of local number variances $\mathrm{Var}_\Psi(N_g)$, and combine it with a canonical-commutator uncertainty inequality to bound it from below by a conjugate-current variance $\mathrm{Var}_\Psi(J_g)$. Together the two results identify the local data that any pointwise replacement for the Dirac interpolation must include: number — or, equivalently, conjugate-current — fluctuations on the test scale.