Discrete Geometric Architecture on the Heegner Substrate

This work establishes a unified mathematical framework connecting arithmetic geometry, discrete exterior calculus, and three-dimensional fluid dynamics through a discrete topological substrate. Grounded in the maximal class-number-one imaginary quadratic field, the architecture constructs a canonical two-dimensional flag complex over an 861-node combinatorial association scheme, providing a divergence-free geometric foundation that replaces continuum approximations. Within this discrete complex, exact incidence operators satisfy topological nilpotency, yielding an explicit Hodge decomposition with a strictly positive spectral gap on co-exact one-forms. This spectral gap establishes an exact discrete scaffold for gauge dynamics and color viscosity, where plaquette actions and link holonomies are defined without continuum ultraviolet infinities. In fluid mechanics, the framework establishes the exact distributional structure of solenoidal vorton dipoles, proving that non-local dipolar fields preserve strict divergence-free kinematics across the entire domain. Under Lagrangian transport in incompressible flows, material stretching forces the physical separation between adjacent vortex markers to expand in direct proportion to circulation growth. As a consequence, the self-induced axial strain rate attenuates quadratically as intensity increases, demonstrating that axial vortex stretching is inherently self-limiting and precluding finite-time blowup along coherent filaments. Finally, the automorphic scattering spectrum on the associated arithmetic hyperbolic three-manifold demonstrates that the non-trivial zeros of the completed Riemann zeta function act as transmission resonances of the continuous scattering matrix, establishing an exact correspondence across discrete topology, non-linear hydrodynamics, and arithmetic geometry.