rung_ordering
plain-language theorem explainer
The sub-mastery, expert, master-craftsman, and world-class rungs satisfy the strict numerical order 7 < 11 < 14 < 17. Education researchers modeling skill-acquisition costs on the phi-ladder cite this ordering to justify monotonic growth of mastery costs. The proof is a term-mode reduction that unfolds the four constant definitions and discharges each inequality by direct numerical comparison.
Claim. Let $s=7$, $e=11$, $m=14$, and $w=17$ denote the sub-mastery, expert, master-craftsman, and world-class rungs. Then $s < e$, $e < m$, and $m < w$.
background
The module derives Ericsson's 10,000-hour rule as successive phi-rung jumps above a 45-hour baseline per skill rung, with total cost $45 · phi^N$ for crossing N rungs. Sub-masteryRung is defined as 7, expertRung as 11, masterRung as 14, and worldClassRung as 17 (the human cognitive ceiling taken from the upstream AnimalZComplexityBound ladder). The local setting is Track I12 of Plan v5, where per-rung cost equals consciousnessGap 3 and the full ladder up to rung 17 brackets the empirical mastery interval.
proof idea
The term proof opens with refine to split the conjunction into three subgoals. Each subgoal unfolds the relevant pair of rung constants (subMasteryRung with expertRung, expertRung with masterRung, masterRung with worldClassRung) and applies norm_num to verify the numerical inequality.
why it matters
The ordering is used by masteryCost_rung_ordering to transfer strict monotonicity of masteryCost across the named rungs, and it populates the rung_ordering field in both AnimalZComplexityBoundCert and BHEchoCatalogCert. It supplies the concrete rung sequence required by the education track of the Recognition framework, aligning with the phi-ladder and the eight-tick octave that fix D=3 and the alpha band.
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