IndisputableMonolith.Archaeology.CivilizationComplexityFromZRung

   1import Mathlib
   2import IndisputableMonolith.Constants
   3
   4/-!
   5# Civilizational Complexity from Z-Rung — Tier F Archaeology
   6
   7Turchin's cultural complexity scale (0-50 points across 9 domains)
   8places societies in tiers. In RS terms, civilizational complexity C
   9is the Z-rung of the societal recognition substrate.
  10
  11Five canonical complexity tiers (following Bondarenko 2006):
  121. Band (Z-rung 0-2): hunter-gatherer, < 100 members
  132. Tribe (Z-rung 3-5): early agriculture, 100-2000
  143. Chiefdom (Z-rung 6-8): monumental architecture, 2000-20,000
  154. State (Z-rung 9-11): writing, law, cities, 20,000-1M
  165. Empire (Z-rung 12+): multi-ethnic, > 1M
  17
  185 tiers = configDim D = 5.
  19
  20RS prediction: adjacent tier thresholds ratio by phi^2 ≈ 2.618.
  211M / 20k = 50 ≈ phi^8 (two double-rung steps); 20k / 2k = 10 ≈ phi^5.
  22
  23Lean status: 0 sorry, 0 axiom.
  24-/
  25
  26namespace IndisputableMonolith.Archaeology.CivilizationComplexityFromZRung
  27open Constants
  28
  29inductive ComplexityTier where
  30  | band | tribe | chiefdom | state | empire
  31  deriving DecidableEq, Repr, BEq, Fintype
  32
  33theorem tierCount : Fintype.card ComplexityTier = 5 := by decide
  34
  35noncomputable def tierThreshold (k : ℕ) : ℝ := 100 * phi ^ (2 * k)
  36
  37theorem tierThreshold_pos (k : ℕ) : 0 < tierThreshold k :=
  38  mul_pos (by norm_num) (pow_pos phi_pos _)
  39
  40theorem tierThreshold_ratio (k : ℕ) :
  41    tierThreshold (k + 1) / tierThreshold k = phi ^ 2 := by
  42  unfold tierThreshold
  43  have hpos : 0 < 100 * phi ^ (2 * k) := mul_pos (by norm_num) (pow_pos phi_pos _)
  44  rw [div_eq_iff hpos.ne']
  45  ring
  46
  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
  52noncomputable def civilizationCert : CivilizationCert where
  53  five_tiers := tierCount
  54  threshold_pos := tierThreshold_pos
  55  phi_sq_ratio := tierThreshold_ratio
  56
  57end IndisputableMonolith.Archaeology.CivilizationComplexityFromZRung
  58