masteryDesignCert
The definition constructs a MasteryDesignCert record that packages the gap-45 educational parameters in Recognition Science. Curriculum designers and learning theorists would cite it to fix study-block length at phi hours and mastery at 45 hours per rung. It is assembled by directly supplying five lemmas as the structure fields.
claimA MasteryDesignCert asserts that the number of mastery stages is 5, the optimal study block satisfies $1 < phi < 2$ hours, the recovery ratio is positive, mastery requires exactly 45 hours per rung, and mastery time is strictly positive at every rung level $k$.
background
Recognition Science fixes masteryHoursPerRung at 45 hours via the gap-45 body-plan ceiling. OptimalBlockHours is defined as phi, placing the block between 1 and 2 hours, while recoveryRatio is the reciprocal 1/phi. MasteryAtRung k scales as 45 times phi to the power k, remaining positive for all natural k. The module sets five canonical stages (novice through expert) equal to configDim D = 5, consistent with the phi-ladder conventions imported from Constants.
proof idea
The definition is a direct structure constructor that populates each field of MasteryDesignCert from a dedicated lemma: five_stages from masteryStageCount, block_range from optimalBlock_in_range, recovery_pos from recovery_ratio_pos, mastery_hours from masteryHours_eq_gap45, and mastery_pos from masteryAtRung_pos.
why it matters in Recognition Science
This certificate is consumed by the parallel masteryDesignCert in MasteryDesignFromJCost. It realizes the E5 prediction that five stages, phi-hour blocks, and 45-hour rungs follow from the gap-45 threshold, linking the phi fixed point (T6) and configDim D = 5 to pedagogical constants. The definition closes the educational interface without new axioms.
scope and limits
- Does not derive the numerical value 45 from the Recognition Composition Law.
- Does not prove that phi-hour blocks optimize cognitive performance for all learners.
- Does not model individual variation or exceptions to the 45-hour rung.
- Does not connect these parameters to physical constants such as alpha or G.
Lean usage
import IndisputableMonolith.Education.MasteryDesignFromGap45 def cert : MasteryDesignCert := masteryDesignCertformal statement (Lean)
71noncomputable def masteryDesignCert : MasteryDesignCert where
72 five_stages := masteryStageCount
proof body
Definition body.
73 block_range := optimalBlock_in_range
74 recovery_pos := recovery_ratio_pos
75 mastery_hours := masteryHours_eq_gap45
76 mastery_pos := masteryAtRung_pos
77
78end IndisputableMonolith.Education.MasteryDesignFromGap45