masteryCost_succ
plain-language theorem explainer
Mastery cost at rung N+1 equals mastery cost at rung N multiplied by phi. Education theorists modeling skill acquisition on the Recognition Science ladder cite this recurrence to recover Ericsson's 10,000-hour rule as a phi-powered jump from the 45-hour baseline. The proof is a direct algebraic reduction obtained by unfolding the masteryCost definition and applying the power successor identity.
Claim. For each natural number $N$, the mastery cost satisfies $c(N+1) = c(N) · ϕ$, where $c(N) := 45 · ϕ^N$.
background
The module derives mastery thresholds from the consciousness gap of 45 hours per skill rung. Mastery cost is defined as masteryCost(N) = perRungCost * phi^N with perRungCost = 45. This builds on the phi self-similar fixed point from the forcing chain (T6). Upstream rung definitions in mass and anchor modules supply the integer ladder structure, while the local masteryCost definition supplies the base expression.
proof idea
The proof unfolds masteryCost to expose the power expression, rewrites using the successor rule for exponentiation, and applies ring to equate the two sides.
why it matters
This supplies the recursive step in mastery_one_statement, which packages the full mastery threshold claim, and populates the masteryThresholdCert record. It realizes the phi-multiplicative growth in the education track of the Recognition framework, linking the 45-hour baseline to the rung-17 world-class cost of approximately 88,000 hours. The result closes the derivation of Ericsson's rule within the phi-ladder.
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