9D-USTE Framework: Unified Formula of Quantum Gravity and Its Geometric Origin

Based on the Nine-Dimensional Universal Unified Evolution Theory of Entropic Topology (9D-USTE $v_\infty$), this paper derives a complete, parameter-free unified formula of quantum gravity from the geometric origin of 9D compactified topological manifolds. Gravity is identified as the geometric curvature of 4D spacetime projected from 9D topological frustration, while quantum fluctuations correspond to dynamic fluctuations of extra-dimensional compactification radii $l_i$ driven by the topological frustration index $K^*$. The unified action principle naturally incorporates general relativity, quantum mechanics, the topological fifth force $\mathcal{F}_T$, dark matter solitons, and dark energy as manifestations of a single topological structure. Spacetime singularities are resolved by topological stabilization without additional assumptions. Three falsifiable observational signatures are proposed: quantum gravitational wave echoes from black hole mergers, topological CMB B-mode polarization with a $\pi/4$ phase shift, and tiny violations of the equivalence principle in high-gradient $K^*$ regions. The framework is self-consistent, parameter-free, and observationally testable, establishing a consistent bridge between Planck-scale physics and cosmic evolution.