enzyme_jcost_lens_summary
plain-language theorem explainer
For any transition-state coordinate x*, an ideal enzyme exists whose J-cost contribution exactly cancels the bare activation barrier, forcing the catalyzed barrier to zero. Researchers modeling recognition-based catalysis would cite this to establish zero-cost corridors through transition states. The proof is a direct term composition of the existence theorem for ideal enzymes with the zero-barrier lemma under the ideal condition.
Claim. $For all real numbers x^*, there exists an enzyme E with transition-state coordinate x^* such that the J-cost contribution of E at x^* equals the negative of the activation barrier at x^*, and the catalyzed barrier at x^* is zero.$
background
In the Recognition Science treatment of enzyme catalysis, an enzyme is a structure carrying a J-cost contribution function (a map from reaction coordinate to added J-cost) and a designated transition-state coordinate off the minimum. The ideal enzyme condition requires that this contribution at the transition coordinate exactly negates the bare activation barrier. The catalyzed barrier is defined as the sum of the activation barrier and the enzyme's J-cost contribution at that point.
proof idea
This is a term-mode one-line wrapper. It invokes the ideal_enzyme_exists theorem to obtain an enzyme satisfying the transition-state coordinate match and the ideal condition, then applies the ideal_enzyme_zero_barrier lemma to conclude that the catalyzed barrier vanishes.
why it matters
This result completes the existence half of the core claim that enzymes act as exact J-cost saddle-point lenses, zeroing the transition barrier. It supports the module's results on complementary cancellation and catalytic rate enhancement, where zero saddle cost recovers the full activation energy as rate. In the broader framework it aligns with J-uniqueness and phi-scaling of barriers on the ladder.
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