User:Mattatk/sandbox/Least action principle

The stabilization of chip configurations follows the least action principle: each vertex topples no more than necessary. Even if toppling stable vertices is allowed, stabilization via the toppling of exclusively unstable vertices is always at least as efficient. More precisely, if u is a vector such that u(v) is the number of times the vertex v topples during the stabilization (via the toppling of unstable vertices) of a chip configuration z, and n is an integral vector (not necessarily non-negative) such that z-&Delta;′n is stable, then u(v) ≤ n(v) for all vertices v.