settlementLevelCount
Five settlement levels are fixed in the phi-ladder urban model, matching the configuration dimension of five. Researchers modeling rank-size distributions from hamlet to metropolis in archaeology or central place theory would cite this constant when deriving population ratios. The definition consists of a single constant assignment with no computation.
claimThe number of canonical settlement levels forced by the configuration dimension is $5$.
background
The module derives urban settlement scaling from the phi-ladder, where hierarchies follow power laws with adjacent-level population ratios equal to phi squared or phi cubed. These ratios fall inside Christaller's empirical band of 3 to 7. The five levels (hamlet through metropolis) are set by fixing the configuration dimension to 5. No upstream lemmas are required for this definition.
proof idea
Direct constant definition that assigns the natural number 5.
why it matters in Recognition Science
This supplies the fixed count required by the equality theorem settlementLevelCount_eq and the UrbanDensityCert structure, which certifies that the population ratio lies between 3 and 8. It instantiates the RS prediction that settlement hierarchies scale as phi to the k per level, consistent with the self-similar fixed point phi. The definition closes the structural theorem for this archaeology application.
scope and limits
- Does not derive the value five from the forcing chain or Recognition Composition Law.
- Does not compute rung spacing or population ratios between levels.
- Does not address spatial dimensions or constants such as G or alpha.
- Does not validate the count against specific archaeological site data.
Lean usage
theorem settlementLevelCount_eq : settlementLevelCount = 5 := rflformal statement (Lean)
32def settlementLevelCount : ℕ := 5
proof body
Definition body.
33