As a math professor: why would you not formally enroll?
We have formal admissions procedures and the like, but this is not to screen out by arbitrary criteria. Grad applications are screened by math professors, not some stuffed suits somewhere who can't do trigonometry, and if you are well prepared for grad study then it will show and you will be admitted. And, typically, funded with a stipend (usually there are 10-20 hours of teaching a week you have to do, depending on institution).
I think most professors would be happy to let you sit in on an advanced class without enrolling. But doing a whole Ph.D. that way? Perhaps it is possible, but I can't imagine why anyone would, and I don't know of anyone that has.
The question was, could a student without a Bachelor's just show up at a grad school, offering to take the Ph.D. qualifying exams and believing that they are well prepared, AND, without a Bachelor's degree, be permitted to do so? In particular, to take the qualifying exams, would they have to be 'enrolled' and would that be possible without a Bachelor's degree? And, although not said, maybe the student needs financial aid and hopes to get it based on their good qualifying exam performance.
So, they are ready to take the exams. But they have no Bachelor's, are not enrolled, without a Bachelor's would likely not be accepted or enrolled, and need financial aid. So, can they take the quals? If so, then how? That was the question. I suggested maybe an Associates degree, some really good GRE scores, and offering to take the quals BEFORE applying for admission.
We have formal admissions procedures and the like, but this is not to screen out by arbitrary criteria. Grad applications are screened by math professors, not some stuffed suits somewhere who can't do trigonometry, and if you are well prepared for grad study then it will show and you will be admitted. And, typically, funded with a stipend (usually there are 10-20 hours of teaching a week you have to do, depending on institution).
I think most professors would be happy to let you sit in on an advanced class without enrolling. But doing a whole Ph.D. that way? Perhaps it is possible, but I can't imagine why anyone would, and I don't know of anyone that has.