It sounds more complicated than it is.
Franklin is a second generation mutate, which means his powers come from the cosmic rays that bombarded his parents. Those cosmic rays changed their genetics, and because of that they gave birth to franklin who carried those genes.
His sister did not.
Franklin from an early age felt a desire to be a part of something. A desire to be special, to feel connected with other individuals who were special.
As such he was drawn to mutant kind, who were a people all connected by being born innately different and special.
Because Franklin's power's are so powerful, he unknowingly altered his own genetics to place an Xgene in his genetic make up. This made it seem as though his powers were the manifestation of his mutant genes.
Realistically his powers were already present. Franklin unknowingly used his powers to alter his physiology to make him seem like a mutant, thoigh he never was.
He never knew he did this. It was all subconscious. So really, he's a second generation mutate, not a mutant.
Hes not the only mutate to do this either. Mister Sinister did something similar by placing an xgene in his own body to be a mutant, though he wasnt himself.
