Warrant and guarantee are the same word split by French dialects — Norman French kept the Germanic "w," Parisian French changed it to "gu."
A document issued by a legal authority authorizing an action, especially the arrest of a person or search of premises. Also a justification or guarantee.
From Anglo-Norman warant, from Old Northern French warant ('protector, guard, guarantee'), from Frankish *warand ('guarantor'), from Proto-Germanic *war- ('to be aware, guard'). Related to guarantee, which is a doublet from the same source via a different French dialect. Key roots: *wer- (Proto-Indo-European: "to perceive, watch out for"), *war- (Proto-Germanic: "to be aware, guard"), *warand (Frankish: "guarantor").
"Warrant" and "guarantee" are the same word — both come from Frankish *warand, but "warrant" entered English through Norman French (which kept the Germanic 'w'), while "guarantee" came through Central/Parisian French (which changed 'w' to 'gu'). This w/gu split is a telltale marker of Norman vs. Parisian French influence, visible in many English doublets: war/guerre, warden/guardian, William/Guillaume. Every time you see this pattern, you're seeing the linguistic fault line