Topology Kills the Ghost: Brane-Cluster UV Completion of Quantum Gravity

We propose a brane-cluster formalism for ultraviolet quantum gravity in which the higher-curvature operators that soften graviton propagation are not inserted phenomenologically, but arise from the intersection topology of a finite network of transversely embedded branes. The brane union ℬ determines an intersection chain complex (C•, ∂), and each nontrivial homology class [Cₖ] ∈ Hₖ(ℬ) is assigned a massive cluster mode Φₖ coupled to curvature through λₖΦₖR. Eliminating the cluster sector produces an effective gravitational kernel containing R□ⁿR corrections and a momentum-space propagator whose ultraviolet behaviour scales as Gᵉᶠᶠ_μν,ρσ(p) ∼ p⁻⁶, while the low-energy limit returns to the Einstein propagator. The extra higher-derivative poles are treated through the fakeon prescription, so that ghost excitations are excluded from the physical asymptotic spectrum without removing their virtual contribution to ultraviolet damping. The construction is developed through homological charge conservation, BRST quantization, Slavnov–Taylor consistency, one-loop counterterm closure, and explicit curvature-sector renormalization. The resulting framework links brane topology, ghost control, and UV-safe gravitational dynamics in a single effective field-theoretic mechanism, with testable consequences for short-distance gravity, gravitational-wave dispersion, regular black-hole interiors, and topology-driven quantum-gravity signatures.