Praise be to Allah.
It is not essential for divorce to be written down or recorded; divorce takes place as soon as the words are uttered. So if a man said to his wife, “You are divorced” or he says, if she is absent, “She is divorced” or “I have divorced her” and similar phrases, then it counts as a divorce. It is not essential that it be witnessed or that it be done in court or in front of the wife.
Recording the divorce and proving it in court is the correct action, so as to protect people’s rights and prevent tampering with the limits set by Allah, especially if it is an irrevocable divorce, such as if he has divorced her three times. In that case, the sharee‘ah court has to be notified of that and issue documentation, so that the woman who has been divorced can remarry if she wants to, and so that the husband’s rights will be protected and the divorced wife will not ask for maintenance after an irrevocable divorce.
To sum up: divorce takes place once it is uttered or written with the intention of divorce. As for recording it and documenting it, this is to protect rights and is not a condition of divorce taking place.
And Allah knows best.