Geometric Origin of Quantum Waves from Finite Action

Quantum mechanics introduces wave--particle duality as a postulate, yet the geometric origin of wave behavior has never been derived from first principles. This work shows that a finite quantum of action, $\hbar_{\mathrm{geom}}$, compactifies the classical action manifold into a periodic $U(1)$ phase space. Physical observables depend only on modular action $S \bmod 2\pi\hbar_{\mathrm{geom}}$, making interference a direct geometric necessity. We present this as a theorem, proving that any system with finite $\hbar_{\mathrm{geom}}$ must exhibit wave interference, while the classical limit corresponds to decompactification $\hbar_{\mathrm{geom}}\!\to\!0$. Chronon Field Theory (CFT) provides the physical substrate for this geometry: its causal field $\Phi^\mu$ carries quantized symplectic flux $\oint\omega=\hbar_{\mathrm{geom}}$, establishing Planck’s constant as a geometric invariant of causal alignment. This result unifies modular action, quantization, and spacetime geometry, revealing the wave nature of matter as a consequence of finite causal curvature. The framework predicts quantized phase discontinuities in mesoscopic interferometry, offering an avenue for direct experimental validation.