tierThreshold
plain-language theorem explainer
Archaeologists modeling societal complexity via Z-rung cite tierThreshold to fix population cutoffs between the five canonical tiers. The definition returns 100 times phi raised to twice the rung index k. It is introduced as a direct noncomputable assignment that supplies the inputs for the positivity and ratio properties in CivilizationCert.
Claim. The threshold for complexity tier indexed by natural number $k$ is $100 phi^{2k}$, where $phi$ is the self-similar fixed point of the Recognition Composition Law.
background
The module treats civilizational complexity as the Z-rung of the societal recognition substrate. It adopts five tiers from Bondarenko: band (Z-rung 0-2, <100 members), tribe (3-5, 100-2000), chiefdom (6-8, 2000-20000), state (9-11, 20000-1M), empire (12+, >1M). Adjacent tier thresholds are required to scale by phi squared, with 5 tiers matching configDim D=5. Phi enters from the imported Constants as the fixed point satisfying the Recognition Composition Law.
proof idea
The declaration is a direct definition that sets tierThreshold(k) to 100 multiplied by phi to the power 2k.
why it matters
This definition supplies the explicit thresholds required by CivilizationCert to certify the five-tier structure, the strict positivity of each threshold, and the phi-squared ratio between consecutive tiers. It realizes the RS prediction of geometric scaling by phi^2 along the Z-rung ladder. The construction links the archaeology module to the core phi-ladder fixed by the forcing chain at T5-T6.
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