The Law of Geometric Sovereignty: A Single-Parameter Unified Field Theory of Mass and Force

We present a single-parameter scalar field theory, the Law of Geometric Sovereignty (LGS), which models the vacuum as a recursive manifold governed by discrete scale invariance. By introducing a log-periodic restorative potential anchored to the Golden Ratio (), we show that physical particle masses emerge as discrete topological resonances, or oscillons, within the vacuum field. The framework reduces the mass spectrum of the Standard Model to a single dimensional empirical scale (₀ 480. 091 MeV), reproducing the masses of the electron, proton, Higgs, and Z-boson within numerical uncertainty. Furthermore, the model predicts a stable, non-luminous localized state at Node 13 (327. 4 0. 6 GeV), offering a mathematically constrained candidate for dark matter. This upload includes both the manuscript and the Python numerical solver used to extract the resolution-converged oscillon energy plateaus.