What is the Sharʿī ruling concerning a husband who swore by triple divorce to his wife that she uttered a specific word, repeating the oath, while both are certain of their positions (he that she said it, and she that she did not)?

If you broke your oath, your wife would be divorced by triple Talaq, and this is the opinion of most scholars. However, Sheikh al-Islam Ibn Taymiyyah holds that the ruling on swearing by divorce, when the intention is not to effect divorce but rather to threaten or emphasize a matter, is the same as swearing by Allah. Thus, if the oath is broken, the swearer must offer expiation for an oath (kaffarat yameen), and no divorce occurs. When divorce is intended, he believes that a triple divorce pronounced at once counts as a single divorce. The more preponderant opinion is that you do not break your oath as long as you swore believing in the truth of what you swore upon, even if reality turns out to be contrary to what you swore. Consequently, neither divorce nor expiation is binding upon you. It should be noted that the permissible oath is swearing by Allah Almighty, whereas swearing by divorce is among the oaths of the wicked.