The Generative Arithmetic Fabric

The Generative Arithmetic Fabric (GAF) is a geometric arithmetic workbench built from cumulative Cayley growth on the root lattice D3, a distinguished shell edge, and exact readout operators. Its graded edge projects to an elementary affine semigroup on which primitive reduction, Pick triangulations, Farey mediants, roots of unity, divisor jumps, and Dirichlet inversion provide a common dictionary of classical arithmetic constructions.The full FCC graph contributes a separate metric layer. For the cumulative symmetric generator, the proposal-visit ledger is reduced to local word-metric data and expressed through the graph Laplacian of the birth-distance field. Exact shell-stratum classifications yield a shell-median crossover, a boundary crossover, and an all-cutoff proof that the largest-jump selector recovers six 14-vertex, 36-edge square-face components for every G >= 12. Analogous mechanisms are proved for the cubic and BCC lattices.The contribution is structural: a common graded arithmetic register, an exact local-metric layer, and controlled generator deformation. No faster factoring bound or priority claim for the classical identities is asserted.