Pattana-Relational Dynamics (PRD) v1.3 : Full 24-Operator SU(5) Framework and Emergent Einstein Tensor

Pattana-Relational Dynamics (PRD) proposes a relational foundation for physics in which spacetime geometry is not fundamental but emerges from correlations among quantum relational operators.In this work the framework is extended to a complete 24-operator SU(5) algebra, representing the full set of causal relations (Paccaya conditions). These operators act on a relational Hilbert space, and their quantum correlations generate an emergent classical spacetime metric.A relational metric operator is constructed from covariant derivatives of the SU(5) generators. Taking expectation values with respect to physical states yields an effective classical metric tensor. From this metric the full geometric structure—Christoffel symbols, Riemann curvature tensor, Ricci tensor, and Ricci scalar—can be derived using standard differential-geometric methods.The resulting relational curvature defines an Einstein-like tensor satisfying the contracted Bianchi identity. Field equations obtained from a variational principle reproduce the standard Einstein field equations in the macroscopic limit. A scale-dependent weighting function connects Planck-scale quantum relational dynamics to classical spacetime behavior.This formulation provides a mathematically consistent pathway from quantum relational operators to emergent general relativity, offering a background-independent framework for quantum gravity. The theory also suggests testable implications for cosmology and black hole thermodynamics.