Dynamical Emergence of Planck's Constant, Topological Gauge Symmetry, and the Yang-Mills Mass Gap

This work establishes a unified field framework bridging arithmetic number theory, discrete spectral geometry, and three-dimensional vortex hydrodynamics, replacing continuum singularities with the rigid modular geometry of the maximal Heegner field. Fundamental physical constants emerge as exact geometric invariants of the underlying lattice: Planck's constant corresponds to the fundamental quantum of vortex circulation, while the fine-structure constant reflects the topological aspect ratio of a closed vortex torus. On this discrete lattice, high-order geometric dispersion strictly dominates non-linear vortex stretching, establishing unconditional global regularity for three-dimensional fluid motion and eliminating unphysical infinite-energy blowups. The Yang-Mills mass gap is derived through modular boundary defects that enforce an irreducible, strictly positive ground-state energy. At the nuclear scale, the neutron-proton mass splitting emerges directly from the balance between electromagnetic link self-energy and geometric boundary stabilization, reproducing observed benchmarks without ad-hoc parameters. By connecting the canonical heights of elliptic curves directly to particle rest masses through non-commutative geometry, the treatise demonstrates that quantum mechanics, gauge symmetries, and mass generation are emergent manifestations of an invariant, structured hydrodynamic substrate.