cycleDuration
plain-language theorem explainer
Cycle duration for the k-th civilizational stage is defined as phi to the power k. Historical modelers would cite this to obtain the exact phi ratio between successive cycles in the certification structure. The definition is a direct power expression on the phi ladder, mirroring the economics version but without the 4* prefactor or doubled exponent.
Claim. Let $d(k)$ denote the duration of the $k$-th civilizational cycle. Then $d(k) := phi^k$ for each natural number $k$, where $phi$ is the golden ratio.
background
The module treats civilizational cohesion as a recognition coherence field whose timescales follow phi-ladder scaling, with five stages fixed by configDim D = 5. Upstream economics defines an analogous duration as 4 * phi^(2*k) to produce a phi-squared ratio for business cycles. The sociology version simplifies to phi^k to produce the direct phi ratio required by the local certification structure.
proof idea
This is a one-line definition that sets the value directly to phi raised to k.
why it matters
The definition supplies the duration function required by CivilizationCyclesCert, which records both the five-stage count and the phi-ratio property for all k. It realizes the RS prediction of phi-scaled cycle ratios, linking to the self-similar fixed point phi from forcing chain T6. The result remains a scaling definition pending empirical mapping to historical data.
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