The foundational architecture of Recognition Science has historically developed along two parallel tracks. Recognition Geometry establishes that spatial structure -- neighborhoods, locality, and pseudometrics -- emerges as a quotient of the act of observation, eliminating the need for a pre-existing continuous manifold. Concurrently, the Law of Logic theorems demonstrate that the physical J-cost metric and the Peano arithmetic tower are forced consequences of any comparison operator satisfying the classical Aristotelian conditions. This paper unifies the two programs. We show that any valid recognizer mapping a configuration space to an observable event space naturally induces a comparison operator on those events. This induced operator structurally satisfies identity, non-contradiction, excluded middle, and composition consistency. A recognizer therefore automatically generates a Law-of-Logic realization on its event space. This identification closes the foundational gap and establishes a single forcing chain: the act of recognition generates distinguishability; distinguishability induces the Law of Logic; the Law of Logic forces the physical metric and the arithmetic iteration object; and the finite resolution of that recognition yields the topological exhaust we call space. Space, number, and physics are thus shown to be strictly downstream consequences of a single primitive act.