Non-Equilibrium Hydrodynamics on the Heegner Lattice: Discrete Exterior Calculus, Exact Conservation Laws, Singular Reconnection Dissipation, and Anisotropic Turbulent Transport

Discretizing space using the maximal class-number-one Heegner lattice establishes an exact topological foundation for solenoidal vortex dynamics and anisotropic turbulence. Exact Topological Conservation: Discrete Exterior Calculus guarantees that vorticity remains strictly divergence-free and gauge-invariant across all grid spacings without continuous-limit approximations or unphysical lattice dislocations. Inviscid Singular Dissipation: Colliding vortex filaments break smooth velocity regularity at the contact point, driving anomalous energy dissipation that remains finite and non-zero even as molecular viscosity approaches zero. Acoustic Radiation and Wave Cascades: Reconnection events emit quadrupole sound waves featuring a characteristic flat low-frequency plateau, while shedding residual linking helicity into a nonlinear four-wave cascade of helical waves along the filaments. Directional Transport Anisotropy: Turbulent collision rates, fourth-rank eddy viscosity tensors, and wall boundary-layer profiles exhibit pronounced directional stiffness ratios dictated by the lattice's algebraic generator, providing clear experimental signatures for superfluid helium, optical lattice condensates, and acoustic metamaterials.