Does a divorce occur with the statement, "If you leave the house without my knowledge, consider yourself divorced," in the event that the wife leaves out of concern for her son and forgets the threat, even though the husband's intention was to threaten, not to divorce, and he did not utter the words "By Allah, I divorce you" (ʿalayya al-ṭalāq)?

The phrase mentioned in the question is an idiom that could imply divorce or something else. If the husband did not intend divorce by it, then it does not occur. Since the husband did not intend divorce but rather intended a threat, divorce does not occur by the wife leaving without his permission.

If the husband explicitly conditioned divorce upon his wife leaving without his permission, and she left forgetfully of the divorce condition, then divorce does not occur, and the husband does not break his oath.