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Albert Einstein proposed a unification of mechanics and electrodynamics by his special theory of relativity, which describes point-like particles moving through empty space. We show in this paper that mechanics and electrodynamics can be unified in Euclidean space if we relax the idea of point-like particles by describing fermions and bosons as spatially extended wave objects. Based on the process theory of gases presented in Paper II [Kalies et al., AIP Adv. 13(5), 055317 (2023)] of this series, we consider a confined photon gas and show that the momentum work of photons leads to an analogous description of matter and light and to the insight that electromagnetic energy is momentum energy. We present the unified energy equations, which do justice to the wave nature of matter, and it is the momentum work that links physical theories. Without momentum energy, which is larger in real terms than the kinetic energy of mechanics, the theoretical edifice of modern physics, at the fundamental level, is incomplete. [ABSTRACT FROM AUTHOR]