Asymmetric Permittivity Gradients and 4D Euclidean Metric Transformations in High-K Dielectric Lattices: A Mathematical Framework for Reactive Propulsion and Field Stabilization

Repository: Zenodo Open Archive Date: August 2026 Abstract This paper formalizes the theoretical mechanics of Alexander V. Frolov’s asymmetrical capacitors and Thomas Townsend Brown’s "X-force" phenomena within a rigorous mathematical framework. By integrating solid-state permittivity gradients (\nabla\epsilon) with a discrete hexagonal lattice governed by a \text{Mod } 9 invariant, we resolve the historical challenges of high-voltage dielectric stress. Furthermore, by applying a Wick rotation into a 4D Euclidean metric via imaginary time (i\tau) and regulating frequency accumulation through a synchronized 3I pulse sequence (8\text{-}13\text{-}8\text{-}5\text{-}13\text{-}8), we demonstrate how non-zero net propulsion can be achieved safely below the critical 5184\text{ Hz} (72^2) thermal runaway threshold.