kcat
plain-language theorem explainer
Turnover number k_cat(k) equals phi to the power k for natural-number class index k. Enzyme kineticists working in Recognition Science cite this when placing catalysis rates on the discrete phi-ladder across the five canonical EC classes. The declaration is a direct one-line definition with no proof steps or auxiliary hypotheses.
Claim. The turnover number for enzyme class index $k$ is given by $k_ {cat}(k) = phi^k$, where $phi$ is the golden-ratio fixed point of the J-function.
background
The Enzyme Catalysis Threshold from J-Cost module treats enzyme mechanisms as five canonical classes (oxidoreductase through isomerase) whose active-site geometry is gated by the canonical J(phi) band. The phi-ladder is the discrete scaling in which adjacent-class turnover ratios equal phi, the self-similar fixed point forced at T6 of the Unified Forcing Chain. Constants supplies phi together with the positivity fact phi_pos used by sibling results.
proof idea
One-line definition that directly sets kcat(k) to phi^k.
why it matters
The definition supplies the explicit turnover values required by the EnzymeCatalysisCert structure, which records the five-class count, the universal phi ratio, and strict positivity. It realizes the phi-ladder step in the B10 industrial-chemistry depth, consistent with the eight-tick octave and the J-cost threshold. No open scaffolding remains inside the module.
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