Intrinsic Phase Curvature from a C3 Algebra: Multi-Carrier Geometry Beyond U(1) Quantum Mechanics

Abstract Standard quantum mechanics encodes phase degrees of freedom through a single U(1) structure, implicitly assuming a flat phase geometry in which global phase transport is isometric and any curvature must be introduced externally, typically via spacetime metrics or interaction potentials. In this work, we show that replacing the U(1) phase algebra with the minimal cyclic algebra defined by j3=−1 (C₃) leads to an intrinsically curved phase space without invoking any external geometric input. The multi-carrier structure of C₃ generates nontrivial holonomy under closed phase evolution, producing residual phase components that cannot be eliminated by gauge transformations. This holonomy naturally defines an intrinsic connection whose curvature arises solely from the algebraic transition rules between phase carriers. Within this framework, we construct a C₃-generalized Laplace operator and analyze its Green function, demonstrating phase-mediated propagation that departs from flat-space behavior. When embedded into a C₃-generalized Schrödinger equation, curvature effects emerge dynamically through phase gradients, without the explicit introduction of gravitational or geometric potentials. In the weak-field limit, an effective Newtonian potential is recovered from hidden-phase gradients, providing a direct link to classical physics. The results reframe curvature as an emergent property of multi-carrier phase geometry and offer a unified perspective connecting quantum phase structure, measurement, and emergent spacetime features.