settlementCost
plain-language theorem explainer
settlementCost applies the J-cost to the ratio of actual over expected population in a settlement hierarchy. Archaeologists and physicists modeling Zipf scaling or central-place hierarchies in Recognition Science cite this when deriving cost from the phi-ladder. The definition is a direct one-line wrapper around Jcost.
Claim. settlementCost(actual_pop, expected_pop) := J(actual_pop / expected_pop), where J(x) = (x + x^{-1})/2 - 1.
background
J-cost quantifies multiplicative deviation from unity via J(x) = (x + x^{-1})/2 - 1, the T5 uniqueness function in the forcing chain. This module treats settlement hierarchies as phi-ladder structures whose adjacent-level population ratios are powers of phi, matching the empirical 3-7 range of Christaller's central-place theory. The module states that five canonical levels are forced by configDim D = 5 and that rank-size relations follow power laws from the Recognition Composition Law.
proof idea
This is a one-line wrapper that applies Jcost to the ratio actual_pop / expected_pop.
why it matters
The definition supplies the cost function used by settlementCost_at_fit to prove zero cost on exact match. It completes the structural theorem for urban density from the phi-ladder, connecting settlement scaling to the eight-tick octave and the D = 3 spatial dimensions of the forcing chain. No open scaffolding remains in the module.
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