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 ($\varphi$), 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 ($\kappa_0 \approx 480.091$ 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 ($\approx 327.4 \pm 0.6$ 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.