What is the ruling regarding someone who made a covenant with Allah to abandon a specific act, linking it to fasting a day if they committed it, then forgot and broke their covenant? Are they required to fast the aforementioned day, or offer an expiation for breaking an oath, or is nothing incumbent upon them?

The statement "I vow to God for such-and-such" falls under the ruling of a vow of obstinacy and anger, in which one has the choice between fulfilling it or paying expiation. If the sworn-upon action is done out of forgetfulness, the preponderant opinion is that nothing is binding upon you. However, fasting for a day or paying expiation is more cautious to avoid scholarly disagreement.