Officially proven or documented, or (informally) completely crazy and should be certified insane.
From Latin 'certificare' meaning to make certain, plus the suffix '-able'. The meaning shifted in American slang to describe someone so crazy they should be certified as mentally ill.
This word beautifully shows how English slang flips meanings—'certifiable' officially means 'can be proven true,' but now mostly means someone is so crazy they SHOULD be officially certified insane!
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