tierThreshold
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.
claimThe 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 in Recognition Science
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.
scope and limits
- Does not derive the base factor 100 from Recognition Science axioms.
- Does not compute rung values or population sizes for specific historical societies.
- Does not model transitions between tiers or non-canonical structures.
- Does not address spatial dimension D=3 or other forcing-chain steps.
formal statement (Lean)
35noncomputable def tierThreshold (k : ℕ) : ℝ := 100 * phi ^ (2 * k)
proof body
Definition body.
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