The Law of Existence: Proven Uniqueness of J-Cost Minimization Across Mathematics, Physics, Biology, and Consciousness

Recognition Science (RS) treats cost as primitive and uses it to define ontology before any physics is introduced. This paper isolates a single meta-structure: a uniquely forced cost functional J on positive ratios, its interpretation as a defect (existence cost), the resulting Law of Existence (a unique defect-zero element), and a derived meta-principle: lim_{x→0^+} J(x) = ∞, i.e. “nothing cannot exist” because “nothing” carries infinite cost. On top of this cost foundation we define RS-native predicates for exists, true, and real: existence is defect-zero, truth is stability under recognition iteration, and reality is existence plus membership in a discrete RS skeleton (here represented by a φ-generated ladder).