Software sandboxing or software-based fault isolation (SFI) is a lightweight approach to building secure systems out of untrusted components. Mozilla, for example, uses SFI to harden the Firefox browser by sandboxing third-party libraries, and companies like Fastly and Cloudflare use SFI to safely co-locate untrusted tenants on their edge clouds. While there have been significant efforts to optimize and verify SFI enforcement, context switching in SFI systems remains largely unexplored: almost all SFI systems use \emph{heavyweight transitions} that are not only error-prone but incur significant performance overhead from saving, clearing, and restoring registers when context switching. We identify a set of \emph{zero-cost conditions} that characterize when sandboxed code has sufficient structured to guarantee security via lightweight \emph{zero-cost} transitions (simple function calls). We modify the Lucet Wasm compiler and its runtime to use zero-cost transitions, eliminating the undue performance tax on systems that rely on Lucet for sandboxing (e.g., we speed up image and font rendering in Firefox by up to 29.7\% and 10\% respectively). To remove the Lucet compiler and its correct implementation of the Wasm specification from the trusted computing base, we (1) develop a \emph{static binary verifier}, VeriZero, which (in seconds) checks that binaries produced by Lucet satisfy our zero-cost conditions, and (2) prove the soundness of VeriZero by developing a logical relation that captures when a compiled Wasm function is semantically well-behaved with respect to our zero-cost conditions. Finally, we show that our model is useful beyond Wasm by describing a new, purpose-built SFI system, SegmentZero32, that uses x86 segmentation and LLVM with mostly off-the-shelf passes to enforce our zero-cost conditions; our prototype performs on-par with the state-of-the-art Native Client SFI system.