Bob commits crime X, which has a statute of limitations of five years. Twenty years later — fifteen years after the statute of limitations has passed — a police officer finds out about the details of Bob's crime.
The officer realizes that Bob cannot be punished for the crime because the statute of limitations has long passed. The officer decides to visit Bob at his home anyway to ask Bob about the crime twenty years earlier. The officer tells Bob what he knows about what Bob did and asks Bob if it is true that he did it. Bob lies and says he didn't do it. Bob is then charged with intentionally lying to a police officer, which in our hypothetical jurisdiction is a felony. The government's proof: twenty earlier, Bob did in fact commit crime X.
Question: Is it constitutional for Bob to be charged and punished for lying to the officer about the crime he committed 20 years earlier?