Einstein Equations from M3(C): Algebraic Derivation within Cognitional Mechanics

This work provides the concrete algebraic realization for the gravitational sector of the Cognitional Mechanics Grand Unification (CM-GUT) framework. In this formulation, spacetime geometry and the Einstein field equations are derived not as fundamental physical postulates, but as necessary stability conditions of a non-commutative computational manifold. The central claim is that gravity represents a geometric constraint required for the persistence of logical computation under finite-depth conditions. By employing the M3(C) algebra—3x3 complex matrices—as the minimal realization of CM axioms, we demonstrate how the spatial dimensionality, the Lorentz signature, and the gravitational field equations emerge entirely through deterministic operator evolution. While the primary focus of this paper is the establishment of the gravitational sector, the choice of M3(C) as the foundational substrate naturally implies a unified origin for both spacetime geometry and the gauge-theoretic structures associated with the Standard Model. This approach suggests that what has historically been called fundamental physics is a stable projection of a deeper, logic-constrained computational process. By deriving the Einstein equations from algebraic stability requirements, this work establishes general relativity as an effective description of CM algebraic dynamics at macroscopic scales, providing a structural account of its necessity.