An Elementary Proof of the Hambly-Lyons Uniqueness Theorem

We give a self-contained proof, in the bounded variation setting, of the Hambly--Lyons uniqueness theorem, which states that (total) signature identifies the path up to tree-like equivalences. The argument is organized around two key geometric observations. First, tree-like paths have trivial signature because factorization over a loop in a tree $τ:[0,1]\to T$ is preserved under signature lifts, which follows from an elementary property of planar curves. Second, a path with trivial total signature contains a nontrivial subpath with trivial total signature (the sub-interval lemma). This is proven by applying a winding-number argument to a two-dimensional projection of the signature lift. Collapsing all trivial-signature sub-intervals then defines a compact metric tree $T$ through which the original path factors by virtue of the sub-interval lemma.