Exelegcho is the strengthened form of elegcho (to rebuke/convict), with ex intensifying to 'thoroughly' or 'completely.' It appears only in Jude 15: 'to execute judgment on all and to convict [exelegxai] all the ungodly of all their deeds of ungodliness.' This is the judgment of the Lord coming with ten thousands of His holy ones — a complete and final accounting.
The exelegcho of Jude's prophecy (quoting 1 Enoch) is the eschatological verdict: every ungodly deed and every harsh word spoken against God will be fully exposed and judged. This thorough conviction is not cruelty — it is completion. Nothing hidden will remain hidden; every concealed thing will be revealed (Luke 12:2). For those in Christ, this conviction has already been borne by their substitute. For those outside Christ, the exelegcho of the last day will leave no defense. The cross is the only shelter from ultimate conviction.