cycleDurationRatio
plain-language theorem explainer
cycleDurationRatio establishes that successive terms in the phi-ladder cycle durations stand in the ratio phi. Modelers of civilizational dynamics in the Recognition Science framework cite this when assembling the five-stage certification. The proof reduces directly via unfolding the exponential definition and algebraic simplification with power and division rules.
Claim. For every natural number $k$, the ratio of cycle durations satisfies $cycleDuration(k+1)/cycleDuration(k)=phi$, where $cycleDuration(k):=phi^k$.
background
The Sociology.CivilizationCyclesFromPhiLadder module interprets historical civilizations as exhibiting rise-and-fall cycles on phi-ladder timescales, with civilizational cohesion as a recognition coherence field. Five canonical stages (emergence, growth, consolidation, decline, transformation) equal configDim D=5. The local definition sets cycleDuration(k) := phi^k.
proof idea
The term proof unfolds cycleDuration to phi^k, invokes pow_pos for positivity of the base, rewrites via pow_succ and div_eq_iff, then normalizes with ring to reach the equality.
why it matters
This supplies the phi_ratio field inside civilizationCyclesCert, which packages the five-stage structure. It realizes the module's explicit RS prediction that the cycle duration ratio between adjacent civilizations equals phi. The result aligns with the self-similar fixed point phi from the forcing chain and the phi-ladder used for timescales throughout the framework.
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