The Cost Curve: Discrete Hierarchy on a Continuous Landscape

Minimum edge cost Cedge(n) = (π/n)² across the gain-group hierarchy. Continuous gold curve; integer groups as dots. Hover any dot for the full calculation.

Continuous curve (π/n)²
Extremal (above threshold)
Generic (below threshold)
Threshold C(Sc) = 0.20267
Slilaty ↔ Rhodes — Term Mapping
(F−B)/(F+B) Minty ratio → S, the balance score on [−1, 1]
Colors on [0, k) identified → S¹, forced by the axiom
"How hard is coloring?" → Vertical axis — edge cost C_edge(n)
"What group level?" → Horizontal axis — gain-group order n
"0 is special" → Ground state S = 0, unique fixed point, zero cost
"The wall" (no Z₃ in ℤ) → Steep section of a smooth hill (55% drop, Z₂→Z₃)
"Where does it get easier?" → Phase transition at n* = 6.9785
"Murky road ahead" → A continuous curve with a known threshold

n* = π/θ* = 6.9785… where θ* solves cos θ = 2θ. Irrational, zero free parameters. The circle doesn't know about integers.