I didn't cover getting "enrolled", but usually that won't be much of a problem: Even the top graduate programs are hungry for good students. If you are ready for their qualifying exams when you show up, then SAY SO. And/or submit a stack of nicely done, hard exercises from a well known, challenging text. Then being enrolled and able to take the exams should be routine.

For an alternative to the Bachelor's, mostly graduate schools will expect a student to have that degree. But, a graduate school doesn't have to care very much about where a student got a Bachelor's. That is, nearly everyone in serious academics knows that a student who shows up ready for the qualifying exams deserves the credit themselves, that is, the school didn't deserve the credit. The US is awash in relatively inexpensive Bachelor's programs, coast to coast, border to border.

There is a hidden point going for a graduate student who can do research: At the top schools, especially Harvard, Princeton, Johns Hopkins, the emphasis in graduate school, for both the profs and the students, is just RESEARCH. In particular, commonly there is no coursework requirement. Princeton has been known to state on their Web site that grad students are expected to prepare for the qualifying exams on their own, that the courses given are for introductions to research, and that there are no courses for preparation for the quals. So, this stuff about independent work to get ready for the quals is mostly what have to do anyway. Some schools publish copies of their old qualifying exams and have lists of references for preparation for the quals. So if you are good with the references and the old exams, then you have a good shot.

All those things said, ugrad school is usually more strict about courses, credits, and grades. Still, there may be some full or partial ways around the time and expense of a full ugrad program: First, it may be possible to impress a grad school just with GRE scores. So, if you are well prepared, then see if you can take the GRE without a Bachelor's. Second, the US is also just awash in junior or community colleges. Since their student quality is usually low, they would be thrilled to have a good student with a shot at getting a Ph.D. These colleges are usually just for two years and give only an Associates degree. These colleges are uaually quite inexpensive. But, then, the last two years of a four year program are mostly just in the major subject, anyway. So, a grad school MIGHT be willing to accept just an Associates degree. Third, there is a fairly strong but rarely written principle in US higher education: Really good students are wanted and go for free or nearly so! If you are ready for Ph.D. quals, then the money for a Bachelor's should be no big problem, even at Harvard. And once you get at a really good school, say, Harvard, you mostly won't be held back. E.g., I knew a bright student who at Harvard as a sophmore took a reading course with A. Gleason, right, who solved one of Hilbert's problems. And Gleason never got his Ph.D.: His work was so good Harvard made him a Harvard Fellow and put him on the faculty. James Simons was an ugrad at MIT and in his senior year took a reading course from Singer, right, as in the Atiyah-Singer index theorem -- watch the video of Simon's talk on YouTube or some such. Simons? Right, he is as in the Chern-Simons result (try Wikipedia), was Chair at Stony Brook, started a hedge fund, for some years paid himself $2 billion a year, and, net, is likely the most successful hedge fund guy.

Yes, the field where this independent study works best is math. In physics, chemistry, or biology, a ugrad program will typically have a lot of lab work can't do independently and some lecture courses with material difficult to get independently. But math remains the most powerful major, maybe even for physics and biology. Even biology? Look up Eric Lander! But, again, for a really good student, ugrad can be for free or nearly so.