Emergence of the Proton-to-Electron Mass Ratio from a Continuous U(1) Topological Graph

The Standard Model mass hierarchy is conventionally treated as a set of fun-damental inputs. This paper presents a dynamical systems model demonstratingthat the proton-to-electron mass ratio (≈ 1836.15) can emerge natively from acontinuous topological graph without hardcoded global scalars. By simulating a5000-node U(1) phase space subjected to gauge-isolated vacuum fluctuations, weestablish an organic Hamiltonian optimization. Much like how the historical adop-tion of continuous activation functions like ReLU resolved training bottlenecks indeep optimization networks, replacing binary interaction thresholds with a continu-ous, shifted probability manifold allows stable topological structures to crystallize.By independently evaluating dynamic binding energy against intrinsic gauge cou-pling, the Monte Carlo execution natively converges on a constituent mass ratio of1809.06 ± 30.42.