adoptionTime
plain-language theorem explainer
The definition adoptionTime(k) assigns phi to the power k as the recognition threshold for the k-th adopter category on the phi-ladder. Economists extending Rogers' diffusion model would cite it when deriving time ratios between successive categories. It is introduced as a direct power expression in the reals that downstream results unfold to obtain the constant ratio phi.
Claim. Let $k$ be a natural number and let $phi$ be the self-similar fixed point of the Recognition Science forcing chain. The adoption time for category index $k$ is $phi^k$.
background
The module maps Rogers' five adopter categories (innovators 2.5 percent, early adopters 13.5 percent, early majority 34 percent, late majority 34 percent, laggards 16 percent) onto a phi-ladder of social recognition costs, with configDim equal to 5. The phi-ladder is the sequence of thresholds generated by successive powers of the fixed point phi that appears at step T6 of the UnifiedForcingChain. Adoption transition times are therefore predicted to scale by the constant factor phi, consistent with the Recognition Composition Law.
proof idea
The declaration is a direct definition that sets adoptionTime k equal to the real number phi raised to the power k. No lemmas are applied; the expression serves as the base case that downstream theorems unfold.
why it matters
This definition supplies the explicit time scale required by the InnovationDiffusionCert structure, which records both the five-category partition and the universal ratio property. It instantiates the phi fixed point (T6) inside an economic model and thereby links the eight-tick octave and D=3 spatial structure of the monolith to diffusion predictions. The module reports zero sorry or axiom, closing the scaffolding for this fragment of the Recognition Science economics tier.
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