Enzyme Catalysis as J-Cost Lens Optics: A Zero-Parameter Theory of Catalytic Perfection

We prove that enzyme catalysis is exact saddle-point cancellation in the J-cost landscape. The unique cost function J(x) = ½(x + x⁻¹) − 1, forced by the Recognition Composition Law with zero adjustable parameters, assigns every chemical reaction a transition-state barrier J(x*) > 0. We establish four results: (1) an ideal enzyme satisfies the complementary-cancellation condition J_enz(x*) = −J(x*), reducing the catalyzed barrier to exactly zero (Theorem 3.3); (2) the resulting rate enhancement is k_cat/k_uncat = exp(J(x*)), recovering the full Boltzmann penalty as catalytic speedup (Theorem 3.4); (3) because J is strictly convex and injective on each φ-ladder rung, specificity is forced: an enzyme optimized for one rung cannot cancel the barrier at a different rung (Theorem 3.5); (4) for every reaction coordinate x*, a complementary enzyme exists constructively (Theorem 3.6). The theory makes a falsifiable experimental prediction: the active-site J-cost profile of any highly evolved enzyme is the additive inverse of its substrate's transition-state J-cost profile, testable via transition-state analog crystallography. All four theorems are verified in the Lean 4 proof assistant with zero sorry.