rungBarrier
plain-language theorem explainer
The definition sets the J-cost barrier height for a transition state at rung index r to equal phi raised to the power r. Enzyme catalysis researchers in Recognition Science cite this to express the phi-ladder scaling needed for active-site complementarity. It directly supplies the rung-dependent factor in the J-cost lens model. The implementation is a direct one-line assignment with no further reduction.
Claim. The J-cost barrier height at rung index $r$ is given by $phi^r$.
background
Recognition Science treats enzymes as J-cost lenses that cancel the saddle-point cost of a reaction transition state. The J-cost is the functional cost J(x) for configuration x along the reaction coordinate, and the phi-ladder indexes discrete scaling levels for barriers and masses. The module states that the transition-state J-cost scales with a phi-ladder rung, so enzyme complementarity requires matching rungs to produce J_total(x*) = 0. Upstream rung definitions supply the integer indexing convention used across fermion masses, ore classes, and anchor sectors.
proof idea
This is a direct definition assigning phi raised to the input rung. No lemmas are applied and no tactics are used; the body is the primitive scaling assignment.
why it matters
It supplies the phi-scaling factor required by the phi-Scaling of Active Sites result in the enzyme catalysis framework, where active-site topology must occupy the same rung as the transition state. This connects to the Recognition Science forcing chain through the self-similar fixed point phi and the mass formula yardstick times phi to a rung offset. It supports the downstream claim that zero saddle cost yields unit Boltzmann factor and full rate enhancement.
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