An Ant on the Clock: Exponential Escape Bounds for Langton's Ant

The classical two-colour Langton ant exits every square of side n within (13*3^(n-1)-2*n^2-4*n-5)/2 updates, uniformly over all initial colourings and internal poses. An explicit integer clock increases by at least one on every update, giving an exponential bound with leading term (13/6)*3^n. An anisotropic rational potential gives order N^(floor(H/2)+1) for N-column rectangles of fixed height H. The clock also yields decidable complete-state reachability, two-sided orbit equivalence and a computable orbit normal form. Exact algebraic checks and finite-state audits accompany the proofs; related supercritical local clock lemmas are formalized in Lean.