Extension of 9D-USTE Non-Commutative Geometry: Topological Coupling Law of Quantum Entanglement and Remote Regulation

Based on the non-commutative geometry framework of the Nine-Dimensional Universal Unified Evolution Theory of Entropic Topology (9D-USTE), this paper reconstructs quantum entanglement as a non-commutative topological coupling effect on a 9-dimensional topological manifold. By introducing three core non-commutative observables—the non-commutative topological coupling tensor Θ_ij, entanglement entropy control parameter Σ, and topological causal connectivity κ* (the local operator form of the global κ* in previous cosmological works)—we derive the fundamental topological coupling conservation law: Θ_ij · Σ · κ* = Ĉ, where the non-commutative topological constant Ĉ ∝ θ / ℓ_P² (θ: non-commutative parameter, ℓ_P: Planck length). This law reveals that the nonlocal nature of quantum entanglement is a projection effect of 9D topological connectivity onto 4D spacetime. We further propose the critical conditions for remote regulation and a unique observable signature: a topological phase shift Δφ = π/2 · Θ_ij. This work provides a testable, geometrically complete foundation for quantum entanglement and topological quantum control, unifying quantum information with 9D-USTE cosmology.