RMB Mathematical Core: Action Principle and Master Equation of the Space–Matter–Motion Theory

We present an updated mathematical core of the Space–Matter–Motion (RMB) theory in a fully Lagrangian formulation. Starting from a single action principle, we define the RMB potential field, the antisymmetric field tensor, the motion tensor, and the master conservation equation ∇μM^{μν} = 0. In contrast to previous versions, this work introduces a proposed extended field sector in which the transition scale r_c is no longer treated as a fixed parameter, but as an emergent, dynamical quantity associated with an auxiliary relaxation field X. This extension leads naturally to a history-dependent field response and a non-local memory structure, while preserving the minimal core formulation. The framework derives spacetime deformation as a consequence of matter in motion, without assuming an independent background geometry. It serves as a canonical theoretical entry point to the RMB framework, clearly separating foundational structure from phenomenological, cosmological, and scale-dependent applications. The extended sector is formulated at the level of a proposed effective field theory. Detailed derivations of the X-field dynamics, as well as empirical validation against observational data, are deferred to dedicated follow-up studies. This work establishes a consistent mathematical foundation and defines explicit directions for theoretical completion and observational falsifiability.