Universal Scale Equality Between the Intrinsic Entanglement Energy and the Topological Quantum Gravity

We prove a parameter-free, universal identity that unifies quantum information and quantum gravity: the intrinsic energy scale of quantum entanglement \ (₄\) is exactly equal to the topological quantum gravity scale \ (c²\) in SI units. Starting from the holographic area law, topological entanglement entropy, Ryu–Takayanagi holography, and the arithmetic rigidity of a K3 surface with \ (Z₃\) symmetry, we fix the dimensionless area-law coefficient \ (₀\) purely from topology, removing all free parameters. All length, entropy, and field dependence cancels exactly, yielding \ (₄ = c²\). This result shows that the maximum energysustainable by nonlocal quantum entanglement is identical to the fundamental ultraviolet scale of quantum gravity, providing a sharp boundary between quantum information and microscopic spacetime geometry.