Duality Synthesis in Quaternionic Logic: How Opposites Generate Truth

The quaternionic G-lattice defines logical connectives over V = 0, 1 × -1, 1³ with an SU (2) group action that separates truth (Re) from epistemic character (Im). The companion paper establishes that the imaginary dimensions are operationally distinguishable. This paper shows they are more: through the synthesis of opposites, they generate truth. We define a synthesis operation on the prime algebra: given dual concepts a and ā with prime signatures Φ (a) and Φ (ā), their synthesis a ⊕ ā = lcm (Φ (a), Φ (ā) ) produces a compound concept whose constructive complexity strictly exceeds that of either pole. The shared structure gcd (Φ (a), Φ (ā) ) forms an algebraic Markov blanket whose interiors are coprime by the Fundamental Theorem of Arithmetic, yielding a unique partition that resolves the observer-dependence critique of statistical blankets. We introduce five axioms (C1–C5) for constructive complexity measures and show that both the prime factor count ω and the assembly index MA are instances. The gap operation—extracting what one concept has that another lacks—is proven to be irreducibly second-order (superinformation). Three algebraic tasks (directional subsumption, guaranteed composition, asymmetric gap analysis) are shown to be possible in the prime medium and impossible under cosine similarity. The results establish that duality, operating through the imaginary dimensions of the G-lattice, is a generative mechanism: opposites do not cancel but produce new truth at higher algebraic layers.