Modular Lattice Structures over Z9: Affine Incidence, Double Domains, and Toroidal Representation

We study a discrete lattice system defined over the modular ring Z9 and governed by a uniform local balance rule imposed on every 2 × 2 block of the grid.A central result is that all admissible configurations of the lattice admit a separable representation as the modular sum of a row function and a column function. This property shows that the global structure of the lattice is determined by two one–dimensional modular components.Within this framework a minimal 3 × 3 configuration naturally emerges as the fundamental structural unit of the system. The nine vertices of this tile (SQV) and the twelve incidence triples (ARG) form a combinatorial structure equivalent to the affine plane AG(2, 3).By considering adjacent tiles, a minimal double domain consisting of eighteen vertices is obtained. The periodic propagation of admissible configurations allows the lattice to be represented on a toroidal surface without altering its local incidence relations.Taken together, these results provide a coherent mathematical description of the modularlattice linking modular arithmetic, finite affine geometry, and discrete lattice structures.