Finite exterior-power stability and the Dirac exchange constant

A spectral gap above the closed-shell Fermi sea would, if it persisted in the thermodynamic limit, immediately give the Dirac exchange constant. It does not persist: the one-particle gap at the Fermi surface collapses. This paper develops a thin-shell substitute and uses it to bound the indirect Coulomb energy of arbitrary states. In a finite cell, small kinetic excess above the closed-shell determinant forces overlap with the Fermi sea and closeness of any bounded observable to its Fermi-sea expectation. In the thermodynamic limit, the same mechanism with a thin Fermi shell forces all gapless fluctuations into a shell of $O(\eta L^3)$ orbitals