Complex Phase Extensions of Deterministic Beables in Quantum Gravity: Unifying Geometric Holonomy and Internal Gauge Structure

This article presents a complex phase extension of the deterministic beable framework for quantum gravity, originally formulated using discrete exterior calculus (DEC) on Lorentzian simplicial complexes. By introducing an imaginary component to the connection, the model incorporates internal gauge-like phase degrees of freedom alongside causal–geometric holonomy. The resulting complexified connection enables deterministic modulation of interference patterns, entanglement structures, and Bell-type correlations. The framework unifies geometric and internal phase evolution in a gauge-invariant, topologically robust setting, with applications ranging from quantum gravity to neuromorphic computation. Experimental implications include fringe shifts in zero-flux Aharonov–Bohm setups, torsion-controlled phase gates, and phase-coded memory in analog neural circuits. This work builds on the real-valued beable formulation and proposes a testable, computationally expressive model for phase evolution in quantum systems.