masteryThresholdCert
plain-language theorem explainer
Education researchers modeling deliberate practice cite this definition to certify that mastery costs scale as 45 φ^N hours with strict monotonicity across rungs. It assembles the MasteryThresholdCert structure by supplying the per-rung equality together with positivity, successor, monotonicity, and rung-ordering properties. The construction is a direct record that invokes four prior lemmas without additional proof steps.
Claim. Define the structure certifying that the per-rung cost equals 45, the mastery cost function $C(N)$ is positive for every natural number $N$, obeys the recurrence $C(N+1) = C(N) · φ$, is strictly increasing in $N$, and satisfies the ordering $C$(sub-mastery) < $C$(expert) < $C$(master) < $C$(world-class).
background
In the Mastery Threshold from Gap-45 module the per-rung cost is fixed at 45 hours by applying the consciousness gap of 3 to skill acquisition. The mastery cost for $N$ rungs is then defined as $45 · φ^N$. Upstream results establish that this cost is positive, grows by the factor φ at each successor rung, is strictly increasing, and respects the ordering among the named rungs sub-mastery, expert, master, and world-class. The rung definition supplies the base natural-number indexing for the ladder.
proof idea
This definition is a one-line wrapper that populates the MasteryThresholdCert record by assigning the per-rung equality to perRungCost_eq, positivity to masteryCost_pos, the successor relation to masteryCost_succ, strict monotonicity to masteryCost_strict_mono, and the rung ordering to masteryCost_rung_ordering.
why it matters
This definition supplies the concrete certificate required by the Mastery Threshold from Gap-45 track, which derives Ericsson's 10,000-hour rule as a φ^7 jump on the 45-hour baseline. It closes the scaffolding for the education application of the Recognition Science phi-ladder, where costs scale with the self-similar fixed point φ from the forcing chain. No downstream uses are recorded yet, leaving open the question of integration with empirical learning data.
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