From a handle-theoretic perspective, the simplest  contractible 4-manifolds, other than the 4-ball, are Mazur manifolds. We produce the first pairs of Mazur manifolds that are homeomorphic but not diffeomorphic.  Our diffeomorphism obstruction comes from our proof that the knot Floer homology concordance invariant \(\nu\) is an invariant of the trace of a knot \(K \subset S^3\), i.e. the smooth 4-manifold obtained by attaching a 2-handle to \(B^4\) along \(K\). This provides a computable, integer-valued diffeomorphism invariant that is effective at distinguishing exotic smooth structures on knot traces and other simple 4-manifolds, including when other adjunction-type obstructions are ineffective.  We also show (modulo forthcoming work of Ozsváth and Szabó) that the concordance invariants \(\tau\) and \(\epsilon\) are not knot trace invariants. As a corollary to the existence of exotic Mazur manifolds, we produce integer homology 3-spheres admitting two distinct \(S^1\times S^2\) surgeries, resolving a question from Problem 1.16 in Kirby's list.