Multiplication Table (Quadratics Part II)

Abstract: This study develops the mod-4 modular microstructure of the Triangular Multiplication Table (TMT) under the FTC composite–type classification. We show that the multiplication table modulo 4 forms a rigid 4×4 periodic residue block whose intersection with the admissible residue classes of , and  collapses to a universal 3×3 microtile. This residue-restricted microtile repeats across the entire TMT and governs the fine-scale geometric texture seen in the FTC-colored diagram. The present version of Part II focuses exclusively on the modular mechanism leading to this 3×3 structure; additional geometric, hyperbolic, and quadratic phenomena will be developed in future revisions.

Keywords: Triangular Multiplication Table; mod-4 residues; multiplicative tiling’s; composite classifications; 3×3 microtiles; modular structure; OEIS; Fundamental Theorem of Composite Numbers (FTC).

2020 Mathematics Subject Classification: Primary 11A25; Secondary 11B99, 05B45.