enzyme_rung_matching
plain-language theorem explainer
For any natural number rung r an enzyme exists whose transition-state coordinate sits at phi^r, satisfies the ideal cancellation condition, and contributes exactly the negative of the activation barrier at that point. Recognition Science chemists cite the result when deriving phi-scaled catalytic specificity from J-cost cancellation. The proof proceeds by explicit construction of the enzyme structure followed by case analysis on r to confirm the off-minimum condition and reflexivity for the remaining conjuncts.
Claim. For every natural number $r$ there exists an enzyme $E$ such that the transition-state coordinate of $E$ equals $phi^r$, $E$ satisfies the ideal-enzyme cancellation condition at $phi^r$, and the J-cost contribution of $E$ at $phi^r$ equals the negative of the activation barrier at $phi^r$.
background
In the Recognition Science treatment of chemistry an enzyme is a structure carrying a J-cost contribution function and a transition-state coordinate. The activation barrier at a point $x^$ is defined as $J(x^) - J(1)$, where $J(1) = 0$ places the reactant at the global minimum of the J-cost landscape. The ideal-enzyme predicate requires that the enzyme's J-cost contribution at its transition-state coordinate exactly equals the negative of this barrier, producing a zero-cost corridor through the saddle point. The module frames enzymes as J-cost lenses whose folded topology supplies the additive inverse of the reaction's transition-state cost, with phi-scaling enforcing that the active-site coordinate must lie on the same phi-ladder rung as the transition state.
proof idea
The tactic proof constructs an explicit Enzyme record whose jcost_contribution is the constant function returning the negative activation barrier, whose transition_state_coord is set to $phi^r$, and whose off-minimum proof proceeds by case split on whether $r = 0$ or $r > 0$, invoking $phi > 1$ and the power inequality to obtain $phi^r > 1$. The three required conjuncts are then discharged by reflexivity and simplification under the ideal-enzyme definition.
why it matters
The theorem supplies the phi-scaling step required for enzymatic complementarity in the Recognition Science framework. It directly supports the module claim that enzyme active sites must occupy the same phi-ladder rung as the transition state, thereby forcing specificity: only the matching rung permits exact J-cost cancellation. The result closes the loop from the J-uniqueness property through the phi-ladder to chemical rate enhancement via zero-barrier Boltzmann factors. It is a prerequisite for any downstream derivation of catalytic rate ratios in the enzyme module.
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