Integer Clifford Algebra Cl(3, 0; Z) as a Unified Semantic Operator

We reformulate the quaternionic semantic operator of Ornelas Brand (2026) as a single primitive operation—the geometric product—in the integer Clifford algebra Cl(3,0; Z). The Hamilton product, Hodge duality, and Boolean lattice operations (meet, join, complement) each arise as a derivation of this single primitive. A further proposition establishes that every dual pair (a, a*) synthesises to the pseudoscalar I up to a grade-dependent sign, formalising the dialectical triad thesis/antithesis/synthesis inside the algebra. The entire framework is closed under integer arithmetic, eliminating floating-point arithmetic from the semantic pipeline and enabling formal verification in proof assistants operating over Z. All lemmas have been computationally verified over 340,638 random and exhaustive test cases with zero failures.