THE LAW OF ONTOLOGY

We formulate a self-contained law of ontology from the canonical reciprocal cost J(x) = ½(x + x⁻¹) − 1, x > 0. Existence is defined as vanishing defect. We prove that J admits the exact factorization J(x) = (x − 1)²/(2x), and therefore has a unique zero and unique global minimum at x = 1. This yields the scalar law of ontology: x exists ⟺ J(x) = 0 ⟺ x = 1. We then prove the boundary theorem: for every real cost bound C there exists ε > 0 such that 0 < x < ε implies J(x) > C; equivalently, J(x) → +∞ as x → 0⁺. By reciprocal symmetry the same holds as x → ∞. From this we derive explicit exclusion zones for every bounded-cost trajectory. Finally, we extend the scalar theory to recognizer bridges χ_{R,ι,c₀}(c) = ι(R(c))/ι(R(c₀)) and prove that zero cost is equivalent to event-level identity whenever the scale map is injective, and to configuration-level identity whenever the recognizer is complete. The resulting ontology is sharp: being is the unique zero-defect state, non-being is a cost-divergent boundary, and bounded recognition cannot approach it.