ER = EPR in Loop Quantum Gravity: the Immirzi Parameter and the Continuum Limit

Abstract: We recast the finite-region analysis of Einstein's equations that underpins the ER=EPR program into the loop quantum gravity (LQG) framework. By translating curvature-energy uncertainty relations into holonomy-flux kinematics, and by identifying Planckian Einstein-Rosen throats with single-puncture cuts through spin networks, we obtain a precise dictionary between entanglement and quantum geometry. Within this dictionary we derive the Barbero-Immirzi parameter directly from the entanglement/area increment of a minimal bridge, and show that a boundary edge-mode construction renders the Bekenstein - Hawking entropy coefficient universal and independent of $\gamma$ under a natural complex polarization. We further establish a refinement renormalization flow for spin-foam amplitudes driven by the finite-region curvature energy bound, which suppresses bubble divergences and yields a regulator-independent continuum limit under explicit conditions. Finally, we indicate observational consequences that follow from an $N$-party generalized uncertainty relation.