First Calculus Supplementary Material, Book III: Dyadology — The Universal Geometric Product integrating ontological gender arithmetic (M/F) and geometric algebra in one operation

Supplementary Material to the First Calculus — Book III: Dyadology This monograph develops the universal geometric product, the algebraic foundation of the qualitative framework introduced in The First Calculus. It reconstructs geometric algebra from a concrete dispositional starting point: the ontological gender opposition (M/F), interpreted as the primitive dyadic distinction between toBe and toHave. From this foundation, a four-term quadratic structure generates four irreducible dispositional units {j, ε, i, η}, which together form the complete basis of a two-dimensional Clifford algebra Cl(1,1). In contrast to standard expositions of Geometric Algebra, which assume a background scalar line and scalar-valued quadratic form, this work derives both metric structure and compositional structure from First-Classness (FC) alone. The geometric product is defined directly in terms of gendered dyadic composition, and from it two complementary operational readings emerge: • a symmetric metric product with Minkowskian signature (measurement mode), and• a non-commutative associative composition (transformation mode), identified with the Clifford product of Cl(1,1). These two readings are not independent constructions but complementary articulations of a single universal product. The scalar identity element η appears not as an external numerical primitive but as an emergent dispositional unit within the fourfold structure. The lightlike dyad ε is null under measurement yet active under composition, expressing the complementary opposition between metric determination and oriented transformation. The resulting algebra provides the operational backbone for the broader First Calculus project. The four dispositional units reappear in subsequent volumes as the generators of Colour Geometry, the structural basis of the genetic code, and the algebraic articulation of physical force structure. This volume therefore establishes the formal dyadic arithmetic upon which the biological and physical applications depend.