Spontaneous Symmetry Breaking and Emergent Scale Separation: A Lagrangian Analog for the Proton-to-Electron Mass Hierarchy

The origin of dimensionless constants, such as the proton-to-electron mass ratio (mp/me ≈1836.15), remains a fundamental open question in theoretical physics. This paper presents an algorithmic proxy that simulates the evolution of a randomized state vector expanding through orthogonal topological bottlenecks. Just as historical algorithmic shifts like Batch Normalization (Ioffe & Szegedy, 2015) resolved gradient collapse in deep neural networks, balancing a linear pseudo-gravity against a quadratic vacuum penalty here drives the network into a stable thermodynamic equilibrium, organically yielding an emergent mass hierarchy of approximately 1842.82.