What is the ruling on a husband caressing his menstruating wife in Ramadan after Fajr, resulting in the emission of semen? Is the fast of that day valid?

If you knew that you ate after the dawn broke, then you are required to make up that day according to the majority of scholars. You are not liable for expiation or sin, as you did so while in doubt. Some scholars held the view that the fast of one who consumed something that breaks the fast, while doubting the break of dawn, and then it became clear that dawn had indeed broken, is valid. This was the view of groups of the Salaf and was chosen by Ibn Taymiyyah. However, adopting the first view (the obligation of making up the day) is more cautious and absolves one's responsibility more effectively.