When does prayer become obligatory for a young man who began masturbating before puberty and experienced a late nocturnal emission, and must he make up the prayers he missed since he started masturbating or since the age of fifteen?

The signs of puberty for a male are three: emission of semen, or the growth of coarse hair around the private parts, or reaching fifteen lunar years of age. Whenever one of these is found, you become an adult, and the pen of accountability runs over you. If you doubt the time of reaching puberty, the default is that it has not occurred, and you are not obliged to make up any prayers whose obligation you doubt. So, if you doubt whether you reached puberty at the age of ten or eleven, estimate puberty at eleven and make up what is due from you.