The sharp kinetic–Dirac interpolation for homogeneous quasi-free fermions

What is the best exchange lower bound that the homogeneous quasi-free framework can deliver, as a function of the kinetic budget? We answer this question with a sharp constant. The bound is expressed through an explicit finite-dimensional variational function $\mathcal G_q(R)$ defined on the simplex of spin populations, and it is attained for every value of the homogeneous kinetic ratio $R\ge 1$ by spin-resolved Fermi balls, with Galilean boosts inserted when the ball configuration alone undershoots the prescribed kinetic excess. We identify the structure of the optimizer: outside the trivial regimes $R=1$ and $R\ge q^{2/3}$, every maximizer has at most two distinct positive spin densities, characterized by a quadratic Lagrange equation. A Taylor expansion of $\mathcal G_q$ at $R=1$ yields a sharp slope of $2/5$ around the unpolarized Fermi ball, refining the looser $\sqrt{R}/2$ slope of the elementary interpolation. The result resolves the quasi-free piece of the Lieb–Oxford problem exactly.