Is an expiation for a broken oath (kaffarat al-yamin) incumbent upon someone who swore by Allah, "By Allah, I will not dismount unless you are with me, or if you leave me, I will dismount alone, and you are content," then dismounted, forgetting the oath, remembered it halfway down, and completed the descent?

If the reason for the oath (such as a child not crying) no longer exists, then expiation is not obligatory upon you. This is similar to someone who swore not to buy something due to a crowd, and then the crowd dispersed, so he bought it. However, if the reason for the oath remains, then it is obligatory to perform expiation because the issue continues after remembering the oath. As for someone who did the sworn-against act forgetfully, there is no expiation upon him.