The Schrödinger Equation as CirculationDominated Limit of Pre-Coherent Probability Dynamics

The extended Fokker–Planck equation on the pre-coherent space Ms contains three structural terms: diffusion, drift, and circulation. Previous work (Schmieke, 2026m, 2026q) has established that circulation arises from non-commutativity and constitutes time, that drift produces projection, and that diffusion maintains openness. This paper demonstrates that the Schrödinger equation — the fundamental equation of quantum mechanics — is not an independent postulate but the circulation-dominated limit of the pre-coherent dynamics. Specifically, we show that in the regime where the system is near a pointer state (drift subdominant) and weakly coupled to its environment (diffusion subdominant), the Mother Equation reduces to a volume-preserving Hamiltonian flow. This flow is the classical Liouville equation. The transition from the Liouville equation to the Moyal equation — and hence to the Schrödinger equation — requires the additional assumption that the observable algebra has the Weyl structure [x, p] = iħ. We analyze this assumption in detail, showing that it is not derived from the admissibility conditions (A1)–(A3) alone, but that there are structural reasons why the Weyl algebra may be the unique stable realization of (A2). The Schrödinger equation and wavefunction reduction are thereby revealed as two regimes of a single equation: circulation dominance yields unitary evolution; drift dominance yields projection. The measurement problem disappears because there is no contradiction between two regimes of one dynamics. We derive the precise conditions under which the Schrödinger approximation holds and identify the corrections that signal its breakdown.