What is the ruling on someone who finds a fluid resembling semen or prostatic fluid (madhy) after waking up, without having had a wet dream (ihtilam) or feeling the fluid flow with lust, and is their ghusl valid if they begin with the intention and tasmiyah, then wash their palms and hands up to the elbows three times?

Nocturnal emission is the discharge of semen during sleep. If one finds semen on their garment, ghusl (ritual bath) becomes obligatory, even if they do not recall having a nocturnal emission. If you are unsure about the nature of what you saw, whether it is semen or something else, you have a choice: either consider it semen, perform ghusl, and repeat the prayers you performed, or do not consider it semen, in which case your prayer is valid if you have washed away the impurity and prayed with a pure garment. This is a detailed explanation according to the Shafi'i school, which is the relied-upon opinion. If you prayed with impurity out of ignorance or forgetfulness, you do not have to repeat the prayer. The essential elements of ghusl for janabah (ritual impurity) are the circulation of water over the entire outer surface of the body with the intention (niyyah), and ensuring water reaches the roots of the hair.