A Covariant Action for the Referential Entropy Cycle Theory: Emergence of the Einstein Field Equations and the Schrödinger Equation

We present a covariant action for the Referential Entropy Cycle Theory (RECT), a monistic framework based on a single axiom: the global maximization of referential entropy drives continuous phase selection and cyclic mass-energy conversion under strict conservation of total mass-energy. The fundamental dynamical variable is a phase (energy-potential-difference) field Φ. From the proposed action we derive the equations of motion and verify energy-momentum conservation both in flat spacetime (via Noether’s theorem) and in curved spacetime (via the Bianchi identities). In the macroscopic high-density continuum limit the action yields the Einstein field equations, with the effective stress-energy tensor sourced by the phase-field dynamics. In the microscopic quantum regime a non-relativistic reduction produces the time-dependent Schrödinger equation, together with a natural nonlinear extension arising from the self-interaction potential. These results demonstrate that both general relativity and quantum mechanics emerge as effective descriptions from a single underlying covariant action, thereby establishing the mathematical consistency of the core structure of RECT. Further development of the relational origin of force and the full cosmic cycle is left to subsequent work.