Incremental computing promises large speed-ups after small input edits. Yet, most incrementality approaches merely skip unchanged work and recompute the remaining sub-computations, even when the inputs change only slightly. Differential execution avoids this by propagating data changes (i.e., deltas), and prior work has shown how to develop a provably correct differential big-step semantics. Unfortunately, that semantics must still replay the original computation at every step, squandering much of the potential gain of incrementalization. While the semantics clearly needs caching to avoid recomputations, a sound and efficient caching discipline is challenging. First, each execution step must be uniquely identified; second, the identifier must remain stable even when the preceding control flow changes. To this end, we develop lexical tracing, which identifies execution steps through their path in the derivation tree of the big-step semantics. We then extend differential execution with lexical tracing and caching to deliver, for the first time, a formally verified, asymptotically efficient account of differential execution for imperative languages. In particular, we developed a novel mechanized theory of cache stability for lexical traces and their semantic rules, which was essential in proving the differential caching semantics correct and complete in Rocq.