We show that the Fermi–Amaldi functional is the exact exchange functional within the orbital-free Levy–Perdew–Sahni (LPS) equation for the square root of the electron density. This result follows from a reinterpretation of the LPS scheme based on a reference system of non-interacting boson-like fermions occupying a single spatial orbital. In contrast to the orbital-dependent Kohn–Sham exact exchange, the exact LPS exchange is explicitly multiplicative and fully compatible with the orbital-free formalism. Self-consistent calculations for atoms and molecules demonstrate improved accuracy relative to conventional LPS approximations. These findings provide a foundation for incorporating exact exchange and hybrid-functional concepts into the orbital-free LPS framework.