Hyper-Relational Quantum-Continuum Field Theory (HR-QCFT): A Unification Framework via Rough Operator Algebra and Non-identity Calculus

This paper proposes the Hyper-Relational Quantum-Continuum Field Theory (HR-QCFT) to overcome the fundamental topological conflict between quantum discontinuity and classical spacetime continuity. Existing frameworks, such as Loop Quantum Gravity (LQG) and Quantum Hydrodynamics (QHD), encounter fragmentation of topological foam near singularities and divergence issues due to UV/IR mixing. To resolve these challenges, this study introduces Non-identity Calculus and Hypertensor bases anchored in Rough Operator Algebra (ROA), which unify macroscopic spacetime and microscopic quantum states into a single continuous algebraic structure. This framework mathematically proves the topological phase transition of the Barbero-Immirzi parameter (γ_Immirzi) and rigorously deduces the trajectory of the Big Bounce alongside the origin of large-scale cosmic structures through a multidimensional quantum Bohm potential.