CivilizationCyclesCert
plain-language theorem explainer
CivilizationCyclesCert packages the two RS predictions for historical cycles: the civilizational stages form a finite type of cardinality exactly 5 and successive cycle durations stand in the exact golden-ratio relation. Historians or complexity theorists applying the phi-ladder model to Spengler-style rise-and-fall dynamics would cite the certificate when checking consistency with Recognition Science timescales. The structure is assembled directly from the Fintype cardinality of the five-stage enumeration and the ratio property of the module's
Claim. A certificate structure asserting that the set of civilizational stages has cardinality 5 and that the ratio of successive cycle durations equals the golden ratio $phi$ for every natural number $k$, where cycle duration at rung $k$ is defined to be $phi^k$.
background
In the Recognition Science sociology tier, civilizational cohesion is treated as a recognition coherence field whose temporal evolution follows the phi-ladder. The module enumerates five canonical stages (emergence, growth, consolidation, decline, transformation) whose count equals the configuration dimension 5. Cycle durations are supplied by the local definition mapping each integer rung $k$ to $phi^k$ in RS-native units. This structure records the cardinality fact together with the exact ratio property. The upstream cycleDuration definition in the economics sibling module uses the scaled form $4 phi^{2k}$; the sociology version isolates the pure ratio by adopting the simpler exponential.
proof idea
The declaration is a structure definition whose fields are populated by sibling definitions in the same module. The five_stages field is supplied by the Fintype.card instance on the inductive CivilizationalStage type. The phi_ratio field is supplied by the algebraic reduction in the sibling cycleDurationRatio lemma, which cancels the exponential terms to obtain exactly $phi$.
why it matters
The certificate closes the formalization of the phi-ladder prediction for historical cycles inside the sociology tier and is instantiated by the downstream definition civilizationCyclesCert. It realizes the module claim that cycle-duration ratios equal $phi$ and that the stage count matches configuration dimension 5. The construction sits downstream of the phi-forcing steps T5-T6 and the eight-tick octave T7; it leaves open whether the five-stage taxonomy can be derived from the octave rather than postulated.
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