CivilizationCert
CivilizationCert defines a certificate structure asserting exactly five complexity tiers with positive thresholds in successive phi-squared ratios. Archaeologists mapping empirical scales to the Recognition Science phi-ladder would reference this when formalizing tier transitions. The structure is populated directly from the tierThreshold definition and its positivity and ratio lemmas.
claimA structure certifying that the set of complexity tiers has cardinality 5, that every threshold $t_k = 100 phi^{2k}$ is positive for natural $k$, and that consecutive thresholds satisfy $t_{k+1}/t_k = phi^2$.
background
The module formalizes civilizational complexity as the Z-rung of the societal recognition substrate, drawing on Turchin's cultural complexity scale and Bondarenko's five-tier classification. Complexity tiers are the five types band, tribe, chiefdom, state, and empire. The threshold function assigns to each natural number k the value 100 times phi to the power 2k, providing the population thresholds separating the tiers. This encodes the RS prediction that adjacent tier thresholds scale by phi squared, consistent with the self-similar fixed point and eight-tick octave.
proof idea
The declaration is a structure definition. Its fields reference the inductive type of complexity tiers for the cardinality condition, the explicit formula for the threshold function for positivity, and the algebraic identity for the ratio condition. No further lemmas or tactics are invoked within the structure.
why it matters in Recognition Science
This structure supplies the interface used by the downstream definition that constructs an explicit instance. It fills the self-similar fixed point and eight-tick octave steps of the forcing chain by enforcing the phi-squared scaling between complexity tiers. The module notes that five tiers equal the configuration dimension 5, linking the archaeological classification to the spatial dimension derivation in the broader Recognition Science framework.
scope and limits
- Does not prove the existence of societies in each tier.
- Does not derive the specific numerical thresholds from first principles.
- Does not address transitions between tiers beyond the ratio condition.
- Does not incorporate Turchin's nine-domain scoring system.
formal statement (Lean)
47structure CivilizationCert where
48 five_tiers : Fintype.card ComplexityTier = 5
49 threshold_pos : ∀ k, 0 < tierThreshold k
50 phi_sq_ratio : ∀ k, tierThreshold (k + 1) / tierThreshold k = phi ^ 2
51