Papers 1 and 2 of this series derived the single-particle and two-particle Schrödinger equations from classical stochastic electrodynamics (SED), resting on the foundational assumption that the coupled electron-vacuum system is ergodic. The present paper converts that assumption into a theorem. We write the SED Hamiltonian explicitly and show that it is isomorphic to the Caldeira—Leggett model of a particle coupled to a harmonic oscillator bath, with spectral density J(ω) ∝ ω^3 (super-Ohmic) and a physical ultraviolet cutoff at the Compton frequency ωc = mc^2/ℏ. With the cutoff, the system is finite-dimensional. We apply the Ford—Kac—Mazur theorem to establish that the velocity autocorrelation function of the electron decays to zero, proving that the system is mixing. Since mixing implies ergodicity, the stationary measure is unique and time averages equal phase-space averages almost everywhere (Birkhoff—von Neumann). As a corollary, the zero-point energy per field mode is rigorously derived as ε(ω) = ℏω/2, completing the foundation of Papers 1 and 2 without additional postulates. The remaining unproved assumptions of the three-paper programme—Nelson’s stochastic mechanics, the fermionic antisymmetry condition, and the identification of the initial cross-correlation with the vacuum two-point function—are identified and discussed.