Theory
Banach Fixed Point: if α < 1 and σ=0, the map is a contraction — opinions converge to a unique fixed point regardless of initial conditions.
Random walk: α=1 removes the restoring force. Noise drives opinions on an unbounded random walk — no social law emerges.
Noisy norm: the middle ground. Variance stabilizes at σ²/(1−α²), a finite steady state — a statistical social law.