Pattana-Relational Dynamics (PRD) v1.1 : Minimal 2-Operator Prototype and Emergent Einstein-Like Tensor

This work introduces a minimal proof-of-concept formulation of Pattana-Relational Dynamics (PRD), a theoretical framework in which spacetime geometry and gravitational dynamics emerge from relational quantum operators acting on a relational Hilbert space. Using a simplified two-operator model, the paper constructs an effective metric from expectation values of relational operators and derives the associated curvature tensors. From these structures, an Einstein-like tensor naturally emerges and satisfies the contracted Bianchi identity.A variational principle is formulated through a relational action functional, leading to field equations analogous to the Einstein field equations. By introducing a scale-dependent weighting function, the framework demonstrates that the relational dynamics reduce to classical general relativity in the macroscopic limit.This prototype establishes a mathematically consistent pathway from relational quantum operators to emergent spacetime geometry, providing a potential foundation for unifying quantum mechanics and general relativity within a relational causal framework.