masteryCost_pos
plain-language theorem explainer
The positivity theorem for mastery cost asserts that 45 times phi to the power N exceeds zero for every natural number N. Education theorists modeling skill acquisition on the phi-ladder would cite it to anchor baseline positivity in training-cost calculations. The term-mode proof unfolds the definition and applies mul_pos to the positivity of the constant 45 and of phi raised to any power.
Claim. For every natural number $N$, the mastery cost $45 · ϕ^N$ satisfies $45 · ϕ^N > 0$, where 45 is the per-rung baseline and ϕ is the golden-ratio fixed point.
background
The Mastery Threshold from Gap-45 module derives cumulative skill-acquisition costs on the phi-ladder. Per-rung cost is defined as the constant 45, obtained by applying consciousnessGap 3 to skill acquisition. Mastery cost at rung N is then this baseline multiplied by phi to the power N. The construction rests on the general cost functions in the Foundation layer, including the J-cost of recognition events and the derived cost of multiplicative recognizers.
proof idea
The proof unfolds masteryCost to the product of perRungCost and phi raised to N. It then applies mul_pos, supplying the positivity of perRungCost by norm_num (yielding 45 > 0) and the positivity of the power via phi_pos.
why it matters
This result supplies the cost_pos field inside masteryThresholdCert, the certificate that bundles positivity, successor behavior, strict monotonicity, and rung ordering for the mastery thresholds. It anchors the derivation of Ericsson's 10,000-hour rule as a single phi-rung jump atop the 45-hour baseline, consistent with the self-similar fixed point phi (T5-T6) in the UnifiedForcingChain. The module identifies a longitudinal study showing costs outside the band 45 · ϕ^k · [0.5, 2] as the falsifier.
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