The Rotor Field Equation—A Unified Feedback Law for Matter, Fields, and Spacetime

This paper introduces the Rotor Field Equation, a unified mathematical framework describing the evolution of curvature energy within the vacuum manifold through a hierarchy of geometric feedback processes. The equation organizes physical phenomena according to the rank of spatial coupling in the curvature field, producing distinct dynamical regimes that correspond to familiar laws of physics. Scalar feedback governs local relaxation, vector feedback produces directed transport, and second-order coupling generates propagating wave modes that correspond to electromagnetic radiation. When curvature disturbances acquire complex phase structure, the feedback dynamics lead naturally to equations governing quantum evolution, while closed circulation of curvature produces stable rotor structures corresponding to particles. In structured environments, synchronized curvature feedback across many rotor sites can generate macroscopic coherent states such as superconductivity. At larger scales, higher-rank feedback processes influence the geometry of the surrounding manifold, providing a geometric pathway toward gravitational and cosmological dynamics. The Rotor Field Equation therefore provides a single curvature-feedback hierarchy capable of describing physical processes across a wide range of scales, suggesting that many established laws of physics may represent specialized limits of a common geometric field governing the vacuum.