masteryAtRung
plain-language theorem explainer
masteryAtRung scales the fixed base of 45 hours by phi to the power k to give total mastery hours at rung k. Educational modelers in Recognition Science cite the scaling when constructing phi-ladder curricula from the gap-45 threshold. The definition is a direct real multiplication of the upstream constant by the power term.
Claim. The mastery hours required at rung $k$ equal $45 phi^k$, where $phi$ is the golden ratio.
background
The module develops pedagogical design from the gap-45 mastery threshold. Upstream result masteryHoursPerRung states: Mastery hours per rung = 45. Scaling by phi^k embeds the self-similar fixed point from the forcing chain into education, with five stages matching configDim D = 5. Module documentation notes optimal blocks of phi hours and recovery ratio 1/phi, aligning with observed cycles near 100 minutes.
proof idea
One-line definition that casts the natural-number constant to real and multiplies by phi raised to k.
why it matters
The definition supplies the rung scaling used by masteryAtRung_pos and MasteryDesignCert to certify positivity, block range, and five stages. It realizes the phi-ladder prediction for expertise growth from the gap-45 ceiling, approximating the 10,000-hour rule as 45 times phi^5. No open scaffolding questions are closed here.
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