Modals shade meaning. To say something is logically impossible — 'He ___ be at home; his car is gone' — which fits?

"Can't" is correct here because it expresses logical impossibility or deduction based on evidence — the missing car is proof that he can't be home, even though nobody directly witnessed him leaving. English modal verbs like "can't," "must," and "might" let speakers signal exactly how certain they are about a conclusion, not just describe rules or permissions.

"Mustn't" expresses prohibition, meaning something is forbidden, which doesn't fit a deduction about someone's location. "Shouldn't" expresses advice or expectation about what ought to happen, which also doesn't match the logical, evidence-based reasoning the sentence is making.

This same modal system lets speakers distinguish "he must be home" (strong deduction he is) from "he can't be home" (strong deduction he isn't) using nearly identical grammar, a subtlety that native speakers navigate instinctively but that takes real study for learners to master.