What is the authenticity of the Hadith: "He who breaks his fast a day in Ramadan without a valid excuse, a lifetime of fasting will not compensate for it, even if he fasts it"?

The hadith mentioned in the question is weak, and its chain of narration revolves around Abu al-Mutawwis and his father, both of whom are unknown. It has been weakened by imams such as Al-Bukhari, Al-Tirmidhi, Al-Qurtubi, Al-Dhahabi, Ibn Hazm, and Al-Albani, due to reasons related to its instability ( اضطراب) and the ignorance of the narrators' conditions.

Breaking the fast on a day in Ramadan intentionally and without a valid شرعي excuse is forbidden (haram), and it requires making up the fast (qada') and repentance. If the fast was broken by sexual intercourse, then expiation (kaffarah) is obligatory in addition to making up the fast and repentance. The evidence for the obligation of making up the fast is what was narrated in the hadith of Abu Hurayrah concerning the story of the man who had sexual intercourse during Ramadan, where the Prophet, peace and blessings be upon him, commanded him to fast a day in its place.