  We introduce and study em (or "emergent"), a lambda calculus style rewrite
system inspired from dilations structures in metric geometry. Then we add a new
axiom (convex) and explore its consequences. Although (convex) forces
commutativity of the infinitesimal operations, Theorems 6.2, 8.9 and
Proposition 8.7 appear as a lambda calculus style version of Gleason and
Montgomery-Zippin solution to the Hilbert 5th problem.
