Zero-Cost POSIX Compliance: Encoding the Socket State Machine in Lean's Types

The article demonstrates how Lean 4's type system can encode the POSIX socket API state machine (fresh → bound → listening → connected → closed) as phantom type parameters, enforcing correct call ordering at compile time. Invalid operations like sending on an unbound socket or double-closing become type errors rather than runtime bugs, while the proof terms are fully erased at compile time, producing code with the same memory layout and performance as raw C.