DE BROGLIE RELATION IN THE TVM FRAMEWORK Derivation of the Wave Nature of Particles from First Principles and the Initial State of the Universe

Abstract

This paper rigorously establishes the De Broglie relation from the first principles of the Theory of Time Modulation (TVM), without introducing subsequent hypotheses, postulated quantum conditions, or empirical parameters. In standard quantum mechanics, the De Broglie relation was introduced in 1924 as a hypothesis – an assumption that particles of matter possess wave properties. Although it was later experimentally confirmed, its fundamental nature was never explained: why do particles have a wavelength? Why is the relation exactly ? Where does wave‑particle duality originate? Starting from the initial state defined within the framework of the main TVM paper [1] – a state of pure energy in a minimal volume – and the only possible mechanism that leads to the evolution of the initial state into today's universe, this paper demonstrates that wave‑particle duality is not a quantum mystery, but a direct and geometrically inherited property of the primary state of the universe. By analyzing transformation and stationary states codified through equations (20) and (20a), it is shown how the action boundaries ( and ) dictate the transfer of circular (rotational) motion of pure energy into localized particle momentum. Through mathematical induction and analysis of the internal formalism, the strict invariance of Planck's constant and the universality of the relation across all transformation levels are proven.