The Functional That Is Not Given: Kohn and Sham's Exchange–Correlation Functional, Priced

In 1965 Walter Kohn and Lu Jeu Sham gave density-functional theory its working equations: the interacting electron system is mapped onto a fictitious non-interacting system with the same ground-state density, the kinetic energy is computed through that system's orbitals, and everything the mapping does not compute is swept into one object, the exchange–correlation functional E_xc[n]. I credit that paper at full strength: it turned an existence theorem into the equations essentially all of density-functional practice still solves, and band structures, cohesive energies, geometries and reaction energetics across chemistry and materials science are computed inside its framework daily. Nothing in this paper falsifies a single density-functional number or disputes the mapping's fruitfulness. What I evaluate is the standing of the framework's two load-bearing layers in 1965. The first is an assumed representability: the interacting system's exact ground-state density is assumed to be the ground-state density of a non-interacting system — non-interacting v-representability, assumed in 1965, not proved. The second is a named, not given, functional: no explicit form for E_xc[n] for arbitrary density is given in 1965 or derived since, and the 1965 working form uses the homogeneous electron gas's energy per electron point-by-point at the local density — an import, not a derivation. My bench evaluates what that intake carries, on declared inputs only. On a declared grid in the Wigner–Seitz radius, r_s = [1, 2, 3, 4, 5, 10, 20], the local-density exchange energy per electron runs from ε_x = −0.458165 hartree at r_s = 1 to ε_x = −0.022908 hartree at r_s = 20, and its scaling verifies exactly: ε_x · r_s = −0.458165 hartree at every grid point, maximum deviation 1.665e-16 hartree, every doubling ratio 0.500000, every consecutive log-log elasticity −1.000000 — exactness of shape belonging to the uniform gas, not to the functional for arbitrary density, of which 1965 derives 0 terms. The open face is wide: sized — not closed — by one declared, imported correlation parametrisation, the correlation remainder the 1965 form leaves uncomputed is 0.164462 of the local exchange–correlation energy per electron at r_s = 2, rising to 0.334172 at r_s = 20. Grade for this census entry, printed at full size: **REASON FOUND / STILL TURNING** — explicitly not a kill and not a falsification. The program headline stands unmoved: **vacuum catastrophe NOT SOLVED / PARTIAL, producing-count 0.**