Wireless Echo: From Continuous Laplacian Potential Transport to Non‑Linear Resonant Field Solitons in Disordered Topologies

We propose a unified, multi‑scale framework for long‑range wireless power transfer (WPT) and non‑dissipative information routing in complex, non‑homogeneous media. Moving away from conventional radiative coupling prone to Anderson localization, we model energy transport across two distinct ontological scales: discrete structured topologies (Stochastic Block Models) and continuous non‑linear field manifolds. On the network scale, continuous Laplacian potential transport triggers a macroscopic phase transition at a critical threshold βc = 0.8, unlocking a near‑perfect transmission efficiency of 98.72%. On the continuous field scale, we extend this mechanism into a non‑linear Schrödinger framework with dynamic topological charge feedback q(t), revealing self‑stabilizing tunneling envelopes. Finally, we demonstrate that a 2D multi‑dimensional Sine‑Gordon kink soliton routes localised phase curvature past dense spatial obstacles (media defects) with zero wave dissipation. This combined framework establishes Topological Resonance as a universal physical foundation for the Roudy Protocol across discrete and continuous manifolds.