A kinetic Dirac interpolation theorem for homogeneous quasi-free fermions

How much exchange energy can a homogeneous fermion gas concede beyond the Dirac value, and what controls the deviation? We answer this question completely within the gauge-invariant quasi-free class. The bound we obtain interpolates between the Dirac constant $C_D(q)$, attained at the unpolarized Fermi ball, and the fully spin-polarized constant $C_D(1)$, attained in the large-kinetic-excess regime. The interpolation parameter is the homogeneous kinetic ratio $R=\tau/(K_D(q)\rho^{5/3})\ge 1$, which the Pauli principle forces above unity. The proof rests on two classical inputs: the Riesz rearrangement inequality, which bounds the exchange integral by the value at indicator-of-ball occupations, and the bathtub principle, which converts the kinetic budget into a spin-polarization constraint. A short Lyapunov-style interpolation between $\ell^1$ and $\ell^{5/3}$ then yields the explicit $\sqrt{R}$ law and the small-excess slope at $R=1$. The result isolates what the quasi-free framework alone can establish about the Lieb–Oxford problem.