Pattana-Relational Dynamics (PRD) v1.2 : Emergent Spacetime from SU(5) Relational Algebra and the Einstein Limit

This work presents a formal mathematical development of Pattana-Relational Dynamics (PRD), a framework in which spacetime geometry and gravitational dynamics emerge from a network of fundamental relational operators. The theory maps the twenty-four causal relations (Paccaya conditions) to the generators of the SU(5) Lie algebra, forming a relational operator structure acting on a complex Hilbert space.A relational metric tensor is constructed from correlations of operator derivatives, allowing curvature tensors and a relational Ricci scalar to be defined within the emergent geometric structure. The dynamics of the relational system are governed by a PRD action principle combining a Yang–Mills–like operator field term with a curvature term weighted by a scale-dependent function.By applying the variational principle and analyzing the macroscopic limit where the relational scale greatly exceeds the fundamental scale, the resulting field equations reduce to the Einstein field equations with an effective cosmological constant. This demonstrates that classical general relativity arises naturally as the low-energy limit of the relational operator dynamics.The PRD framework therefore provides a mathematically consistent bridge between relational causal structure, operator algebra, and emergent spacetime geometry, offering a new approach to the unification of quantum mechanics and gravity.