Distance

Mechanism

A function d(x, y) on a space that assigns a non-negative real number to each pair of points, satisfying the metric axioms: (1) non-negativity, d ≥ 0; (2) identity of indiscernibles, d(x, y) = 0 iff x = y; (3) symmetry, d(x, y) = d(y, x); (4) triangle inequality, d(x, z) ≤ d(x, y) + d(y, z). The substrate of 'how close' in any space — geographic, vector-embedding, cognitive-similarity, network-hop, graph-theoretic. Every metric space has a distance function; agents cannot redefine what metric-ness requires. Pseudo-metrics (which relax identity of indiscernibles) and quasi-metrics (which relax symmetry) are distinct descendants, not specializations of Distance itself.