Vel-v5.0: A Scalable Multi-Threaded Swarm Protocol for 3D Volumetric Voxel Space Dominance under Aerodynamic Wind Gust Perturbations

This paper presents the theoretical formalization, parallel computing framework, and massive-scale empirical validation of the Vel-v5.0 swarm protocol. Moving beyond restricted 2D formulations, Vel-v5.0 introduces a decentralized protocol optimized for 3D continuous flight mechanics over non-convex voxel elevations. We model multi-agent coordinate-free navigation using Map-Aware Gradient Repulsion (MAGR) subject to constant gravitational drift and stochastic, high-velocity wind gust perturbations (G). We provide a rigorous continuous-time Lyapunov-like convergence proof under LaSalle’s Invariance Principle to guarantee asymptotic coverage, alongside a stochastic boundedness analysis confirming trajectory containment within the bounded manifold. To validate these analytical guarantees, we built a highly parallelized multi-threaded simulation engine and executed 10,000,000 independent flight trials. The empirical results demonstrate a clear phase transition in convergence behavior. Under the optimized configuration ($N = 150$, $\mu = 0.98$, $r = 8$), the protocol successfully mitigates boundary pile-up instabilities. The cumulative average efficiency converges asymptotically to $\eta = 0.3523$ with an extremely low system rolling variance ($\sigma = 0.0084$). The resulting performance metrics show that MAGR establishes a mathematically robust, scale-invariant foundation for physical 3D multi-rotor deployments.