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 fundamental inputs. This paper presents a dynamical systems model demonstrating that the proton-to-electron mass ratio (≈ 1836.15) can emerge natively from a continuous topological graph without hardcoded global scalars. By simulating a 5000-node U(1) phase space subjected to gauge-isolated vacuum fluctuations, we establish an organic Hamiltonian optimization. Much like how the historical adoption of continuous activation functions like ReLU resolved training bottlenecks in deep optimization networks, replacing binary interaction thresholds with a continuous, shifted probability manifold allows stable topological structures to crystallize. By independently evaluating dynamic binding energy against intrinsic gauge coupling, the Monte Carlo execution natively converges on a constituent mass ratio of 1809.06 ± 30.42.