Dimensional Transition at 3.65 TeV: Collider Signatures of Fractal Quantum Gravity

We present a quantum theory of gravity in which spacetime has a fractal dimension that runs with energy scale according to renormalization group (RG) equations. Starting from an action with a dynamical scalar field controlling the effective dimension , we derive one-loop β-functions using heat-kernel methods and solve the coupled RG system numerically. The solution predicts a dimensional transition at 3.65 TeV, where crosses zero and spacetime changes from above-4D (fractal) to below-4D (reduced dimension). This transition produces a unique collider signature: a universal 7% suppression of cross-sections at the LHC (13 TeV), with an energy-dependent K-factor that crosses unity exactly at the transition scale. Comparison with current LHC data shows agreement within 2σ across all measured processes. The High-Luminosity LHC is expected to reach 3.5σ sensitivity. The Fractal Dynamical Quantum Gravity (FDQG) model thus provides a falsifiable, scale-symmetric approach to quantum gravity with testable collider phenomenology at accessible TeV energies.