Final Turn is an expression used by many prominent Vanguard players, either occasionally or consistently. It was first used by Toshiki Kai , and has since been used by multiple other characters. So far in the series, the first person to have used this phrase is Kai although it is unknown whether the phrase originated from him or not. Though Kai is the one that made this phrase famous, he does not use it in every game that he wins. Kai's Final Turn call is usually accompanied by his theme.
Final Turn is a declaration of assured victory, claiming that the user will win the match in the current turn. It is called either before (Kai and most others) or after (Ren) the Stand and Draw Phase. With Aichi Sendou, it has become synonymous with his use of PSY Qualia and later his use of the Shadow Paladin deck.
It should be noted that near the end of the anime's first season, Final Turn acquired rather negative connotations. Aichi used it in a cruel, derogatory manner to demean his opponents before finishing them off while under the influence of PSY Qualia, and even Kai was implied to have begun using the phrase after Ren used it in a similar state. Kai mostly stopped using the phrase after he became less cold and single-minded about Vanguard, most importantly after Aichi fell under the influence of PSY Qualia. It was used again sparsely during the second season, most importantly by Leon Soryu, who also possessed PSY Qualia. In season 3, it is repeatedly used mostly by Reverse fighters.
It must also be noted that using the phrase in a real life environment is frowned upon, and is even a punishable offense during some tournaments.