masteryHours_eq_gap45
plain-language theorem explainer
The theorem equates mastery hours per rung to the constant 45 in the Recognition Science educational model. Pedagogical designers cite it when anchoring study durations to the gap-45 threshold. The proof is a direct reflexivity step on the constant definition of masteryHoursPerRung.
Claim. Let $M$ be the number of hours required to reach mastery at each rung on the phi-ladder. Then $M = 45$.
background
The module Educational Design from Mastery Threshold sets 45 hours per rung to achieve mastery, identified as the gap-45 body-plan ceiling. Optimal blocks are phi hours (approximately 97.6 minutes) with recovery ratio 1/phi, and the 10,000-hour rule approximates gap-45 times phi^5. Five canonical stages (novice to expert) equal configDim D = 5. The local setting treats masteryHoursPerRung as the base constant before rung-specific scaling. Upstream results supply the definition masteryHoursPerRung : Nat := 45 together with rung maps from fermion masses and asteroid ore classes.
proof idea
The proof is a term-mode reflexivity application. It matches the defining equation of masteryHoursPerRung directly via rfl without invoking any lemmas from the rung definitions or other upstream results.
why it matters
The result supplies the mastery_hours field inside MasteryDesignCert, which bundles five-stage count, block-range validity, recovery positivity, and positive mastery at rung. It fills the gap-45 prediction in the RS educational framework, consistent with the phi-ladder and the eight-tick octave structure. No open scaffolding remains; the equality closes the constant anchor for downstream certification.
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