What is the religious ruling regarding the marriage of a man and a woman after the man has uttered the divorce several times in different ways (threat, conditional statement, oath), and in one of these instances, the divorce was carried out by a registrar as a first irrevocable divorce, and the woman claimed that she became completely forbidden to him, while both parties wish to reconcile and mend their relationship for the sake of their children?

The speaker has treated divorce as a jest, while it is a remedy for marital discord, not a means of threat. Your statement, "I have divorced you," if intended for the present, constitutes a divorce. If it is a report about the past, it is an acknowledgment legally, and religiously, she remains your wife unless she denies it. Your statement, "If you leave the door, you will be divorced threefold," is a conditional divorce, and the divorce takes effect upon her departure. She would then be irrevocably divorced from you (major separation), and this is the view of the majority of scholars. Sheikh al-Islam Ibn Taymiyyah disagreed, considering it an oath if the intention was to urge or prevent. If by "you will be" you intended a promise of divorce in the future, then it is not a divorce. As for a divorce issued under the aforementioned threat, it is effective because the threat did not reach the level of coercion. Therefore, we request clarification of your intention with your statement "you will be" and the nature of the latter threat. What appears evident is that the woman is irrevocably divorced from you and is not lawful for you until she marries another husband. It is advisable to consult scholars in your country.