Hmm. I would love to follow this course of action. Can you point to evidence that schools will allow you to take their qualifying exams and let you get a PhD based on your independent research without being formally enrolled?
Also, as far as I can tell, most of the article is complaining about the bachelor's level, which you assume as a given. For those who haven't done that, do you have an alternative for that level?
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.
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.
You mentioned the GRE, and I assume you mean the famous GRE general test. But another option to look at to prove mastery of material is the GRE subject test. While the GRE General is mostly an IQ test, GRE subject tests have a knowledge component to them. And, they are quite challenging: for the GRE CS test less than 1% of the test takers in the past 3 years achieved score above 900 (maximum score possible is 990).
EDIT: anyone interested can download the test booklet here which has an example test:
I meant both tests but definitely the subject tests.
Usually the CEEB, SAT, and GRE tests are designed to have Gaussian distribution with mean 500 and standard deviation 100. So, can get a table of the cumulative Gaussian distribution and see what percentile 4 standard deviations above the mean is; I would guess less than 1%.
I got 800 on the GRE subject math test, the only 800 I got on any of those tests, and that 800 always intimidated my wife, MUCH smarter than I am, PBK, etc.
Why especially the subject tests? Because the question was how to skip a Bachelor's degree and do not pass GO, do not collect $100, and go directly to grad school. There the grad school may still want a Bachelor's but for anything less really good scores on the relevant GRE subject tests may be the difference. Show up with 750 or better on GRE subject math, physics, and computer science, and offer to take the Ph.D. quals right away, and may, just MAY, be permitted to 'enroll'. Blow away the quals, publish a paper or two, even in a conference, maybe knock out some code just to prove are not all theory, and may be regarded as a good student. Then will be closer to the front of the line for various kinds of financial aid.
How to skip a Bachelor's is chancy. For the importance and potential of good independent work, at the best research universities that's rock solid. Read the story of the guy who gave the name a 'good' algorithm, Jack Edmonds. Read what Feynman did at Princeton. Read what Gleason did at Harvard. Independent work was just crucial in how I got my Ph.D.: It helped that I did the research for my dissertation independently in my first summer and worte a 50 page manuscript. Then it helped that took a 'reading course', selected a long outstanding problem, and found a solution which also solved a problem in a famous paper by Arrow, Hurwicz, and Uzawa (poor Uzawa was left out of the prize). It helped that the department chair taught a flunk out course, an advanced, second, course in linear algebra and I took it as my first course in linear algebra and blew everyone else away. How? I'd done a LOT in linear algebra independently and in my career in 'scientific programming' before grad school. It even helped that I was the only student that year who showed that there are no countably infinite sigma algebras. That's the kind of stuff that can help one get through grad school.
Skipping the Bachelor entirely might be one use case for the GRE subject test, but I think a more common use case is where the applicant to grad school has a Bachelor is a different field (Math grad going to CS school). I imagine one wouldn't need blow out grades in that case.
Also, I don't think what you're saying about the Gaussian distribution is true for GRE Subject tests as they have significantly varying distributions. Check out this table:
I had a 'qualification': "Usually the CEEB, SAT, and GRE", and with that we both can be right!
For those tests, at the level of detail of the distribution of the scores, it can be tough to get solid data.
But it is easy and common in educational testing to scale 'raw' scores so that the 'scaled scores' are Gaussian.
Also in educational testing, it is easy and common to have enough data on individual questions so that the distribution will be known fairly accurately for a test made of such questions -- my father did that for years as the main 'educational architect' at one of the world's largest and most important technical schools, with 40,000 students there at any one time.
Having those tests be accurately Gaussian more than
2.5 standard deviations away from the mean is likely challenging. I've often suspected that on some of the SAT tests they didn't give any 800 scores.
There have been suggestions that those testing companies have not always been very open about just what they were doing in their details!
Also, as far as I can tell, most of the article is complaining about the bachelor's level, which you assume as a given. For those who haven't done that, do you have an alternative for that level?