Intrinsic Energy Scale of Quantum Entanglement: Universal Bounds and Scaling Laws

We rigorously derive the intrinsic limiting energy scaleΛEΛE​of quantum entanglement from first principles: Heisenberg uncertainty, Lorentz invariance, unitarity, holography, and gravitational stability. We show thatΛE=min⁡{ℏc/L,ΛUV}ΛE​=min{ℏc/L,ΛUV​}, whereLLis the spatial separation of the entangled subsystems andΛUVΛUV​is the ultraviolet cutoff of the effective field theory.ΛEΛE​is bounded above by the Planck scaleΛPΛP​and below by the entanglement threshold energyEthEth​. A universal scaling relationΛE=(ℏc/L)SE/αΛE​=(ℏc/L)SE​/α​connectsΛEΛE​to the von Neumann entanglement entropySESE​, withααa dimensionless constant determined by the field content (e.g.,α=1/3α=1/3for a free scalar). Recent LHC data on top-quark entanglement and long-distance quantum networks quantitatively support this scaling. We propose five precise experimental tests: (i) saturation of entanglement entropy in high-energy colliders, (ii) verification ofΛE∝SEΛE​∝SE​​in cold atoms, (iii) absence of entanglement aboveΛPΛP​in cosmic rays, (iv) modulation of entanglement entropy by gravitational waves, and (v) quantized steps ofΛEΛE​in confined systems. These predictions provide a roadmap for probing the interplay between quantum information and fundamental energy scales.