Mathematical and Physical Elevation of the LQG-ER=EPR Unified Axiomatic System: Coherent Projection, Gauge Invariance, and the Emergence of the Seonggil Field Equations

This paper presents a fully elevated axiomatic system that unifies the discrete spacetime structure of Loop Quantum Gravity (LQG) with the ER=EPR hypothesis, which posits that quantum entanglement forms wormholes. Grounded in differential geometric consistency, we mathematically formulate how the boundary entanglement of Spin Networks emerges as the geometric area of an Einstein-Rosen (ER) bridge. Furthermore, we rigorously prove SU(2) gauge invariance and develop a strict four-step semiclassical projection utilizing Hall-Thiemann coherent states to demonstrate how the microscopic quantum geometry overcomes classical limitations, culminating in the emergence of the Seonggil Field Equations via the entanglement stress-energy tensor. Finally, we integrate a numerical verification algorithm that simulates topological network entanglement entropy, elevating the theoretical completeness of this fusion to an extreme level.