MasteryThresholdCert
plain-language theorem explainer
MasteryThresholdCert packages the positivity, phi-multiplicative recurrence, strict monotonicity, and rung-specific ordering of the mastery cost function built from a 45-hour baseline. Education researchers modeling deliberate practice would cite the structure to certify that costs scale as 45 phi^N across the sub-mastery to world-class ladder. The structure is assembled directly from the per-rung cost definition, positivity and monotonicity lemmas, successor recurrence, and the rung ordering theorem.
Claim. Let $c(N) = 45 phi^N$ be the mastery cost at rung $N$. The certificate asserts $c(N) > 0$ for all natural $N$, the recurrence $c(N+1) = c(N) phi$, strict monotonicity of $c$, and the ordering $c(7) < c(11) < c(14) < c(17)$.
background
The module derives Ericsson's 10,000-hour rule as one phi-rung jump on a 45-hour-per-skill-rung baseline obtained from consciousnessGap 3. Per-rung cost is fixed at 45 hours. Mastery cost at rung $N$ is then defined as 45 times phi to the power $N$. Named rungs are sub-mastery at 7, expert at 11, master at 14, and world-class at 17. Upstream results supply the rung constants, the cost definition, and the ordering theorem that places the four rungs in increasing sequence.
proof idea
The structure is a definition whose fields are filled by the per-rung cost equality, the positivity theorem, the successor recurrence, the strict monotonicity lemma, and direct application of the rung ordering theorem to the named rungs.
why it matters
The structure supplies the packaged properties consumed by masteryThresholdCert, which completes the §6 master certificate section. It thereby grounds the empirical 10,000-hour bracket inside the Recognition Science phi-ladder (T6 fixed point) and the eight-tick octave scaling. The module doc notes the falsifier would be a longitudinal study placing per-skill costs outside the interval 45 phi^k times [0.5, 2].
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