Geometric Origin of Quantum Waves from Finite Action

Quantum mechanics postulates wave–particle duality and assigns amplitudes of the form $e^{iS/\hslash }$, yet no existing formulation explains why physical observables depend only on the phase of the action. Here we show that if the quantum of action ${\hslash }_{\mathrm{geom}}$ is finite, the classical action manifold $\mathbb{R}$ becomes compact under the identification $S\equiv S+2\pi {\hslash }_{\mathrm{geom}}$, yielding a $U\left(1\right)$ action space on which only modular action is observable. Wave interference then follows as a geometric necessity: a finite action quantum forces physical amplitudes to live on a circle, while the classical limit arises when the modular spacing $2\pi {\hslash }_{\mathrm{geom}}$ becomes negligible compared with macroscopic actions. We formulate this as a compact-action theorem. Chronon Field Theory (ChFT) provides the physical origin of ${\hslash }_{\mathrm{geom}}$: its causal field ${\Phi }^{\mu }$ carries a quantized symplectic flux $\oint \omega ={\hslash }_{\mathrm{geom}}$, making Planck’s constant a geometric topological invariant rather than an imposed parameter. Within this medium, the Real–Now–Front (RNF) supplies a local reconstruction rule that reproduces the structure of the Feynman path integral, the Schrödinger evolution, the Born rule, and macroscopic definiteness as consequences of geometric compatibility rather than supplemental postulates. Phenomenologically, identifying the electron as the minimal chronon soliton—carrying the fundamental unit of symplectic flux—links its spin, charge, and stability to topological properties of the chronon field, yielding concrete experimental signatures. Thus the compact-action/RNF framework provides a unified geometric origin for quantum interference, measurement, and matter, together with falsifiable predictions of ChFT.