ComplexityTier
plain-language theorem explainer
Recognition Science archaeology classifies societies by Z-rung of the recognition substrate into five tiers. Complexity theorists mapping historical data to the phi-ladder would cite this when assigning rung intervals to band through empire levels. The declaration is a direct inductive enumeration that derives decidable equality, representation, boolean equality and finite type structure with no additional lemmas.
Claim. The complexity tiers form an inductive type with five constructors corresponding to Z-rung intervals: band (0-2), tribe (3-5), chiefdom (6-8), state (9-11), empire (12+), equipped with decidable equality, representation, boolean equality and finite type structure.
background
The module treats civilizational complexity C as the Z-rung of the societal recognition substrate. Five tiers follow Bondarenko 2006: band for hunter-gatherer groups under 100 members, tribe for early agriculture (100-2000), chiefdom with monumental architecture (2000-20000), state with writing and cities (20000-1M), empire multi-ethnic over 1M. This fixes configDim D = 5. The upstream band definition supplies arithmetic conjunction on stable states (0/1 multiplication) but is not invoked in the tier enumeration itself.
proof idea
Inductive definition enumerating the five constructors and deriving DecidableEq, Repr, BEq, Fintype. No tactics or upstream lemmas are applied beyond the deriving clauses.
why it matters
The definition supplies the five tiers required by CivilizationCert (which asserts Fintype.card = 5, positive thresholds and phi^2 ratio between adjacent tiers) and by tierCount. It implements the module claim that 5 tiers equal configDim D = 5 and encodes the RS prediction that adjacent thresholds scale by phi^2. It connects archaeology to the phi-ladder and Z-rung arithmetic while leaving open the precise empirical mapping of rung intervals to archaeological records.
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