Mr. Langan developed the Cognitive-Theoretic Model of the Universe (CTMU), but it’s not regarded as a valid scientific theory. Any mention of it is typically met with derision.
From the post “Chris Langan, the CTMU, and the process and pitfalls of developing research”:
Like Mr. Langan, I have also claimed to have one of the highest IQs. The goal of the math challenges is to create what is effectively a self-administered IQ test to support this assertion. In avoiding the above pitfalls, the concept does add to and builds on the existing literature, is expressed entirely in the language of mathematics, and the results are objectively true in a mathematically rigorous sense.
So this is why I embarked on the math challenge to create an IQ test of effectivity an unlimited ceiling. My objective was to come up with an original result(s) that ‘contribute to the literature’ for my particular topic, which I eventually succeeded at doing.
My paper proves three conjectured results and provides a new proof of a difficult result, which is used to find new results, among other findings.
I had to:
1. Develop the concept of using the 6th degree integral. Although there is some literature on the cubic, the 6th degree analogue is unknown.
2. Find a way to factor it so it could be integrated. An author of a paper involving a cubic integral used a certain method, which failed on the 6th-degree version, so I had to invent a way to make it work. The author had used a double integral which was converted to the integral of an inverse tangent and polynomial. This did not work when applied to the 6th degree one, as after computing the first of the double integrals, the second integral was undoable. Converting it to a logarithm via integration by parts and other substitutions made it doable, combined with the factorization.
3. Come up with the integral ratio method and the series-shifting method. This is a consequence of the observation that the ratio of two subtly different integrals is a rational value, which can be proven by converting said integrals to infinite series and observing that a constant factors out. Steps 1-3 meant overhauling how prior papers had approached this topic.
4. Prove difficult conjecture using a new method of my own creation. My proof is the first successful attempt using an infinite series approach (using #3) after earlier attempts by other authors from the early ’80s up until as recently as 2019 had failed. A second (to mine) analytic/algebraic proof of the conjecture, published in ’95, is difficult and involves many steps and ad hoc substitutions, whereas mine is three steps and minimal or no prior assumptions. I am still surprised my proof went unnoticed for so long, but it happens. Sometimes it takes having a new set of eyes look at a problem.
5. Prove three conjectures from separate papers using methods I had developed. Like #4, these conjectures eluded authors despite being fairly strait-forward to prove, usually no more than four or five steps. One mysterious conjecture involves a 6th-degree equation again. Knowing the root structure can shed some insight as to how it was derived, but this was otherwise not possible.
6. Derive new results. So, this established the legitimacy of my method, being that I used to come up proofs of two new identities in the case where it works, which is quite restricted as it has to be amenable to #3 & #4. Not all related integrals have this shifting property, or can be factored in such a way that #4 works.
7. Literature review. This meant going through every published text to check for originality. The literature on almost any math topic is vast and it’s not uncommon for people to accidently re-derive stuff. One paper concludes with “…this result is not found in standard bibliographic sources [8, 9], though it resembles (source)”. I’m like, no, it does not merely resemble–it is literally the same formula as the source you cited. Sure, his derivation is different–and there is value in that–but it’s still the same result.
This gives some idea what is involved. But this is what was necessary to find something new.
Moreover;
8. Answer questions on Mathoverflow/Mathexchange (complete), some involving fairly detailed/technical proofs.
9. Obtain the required endorsement. This meant sending emails to get a qualified arXiv author to sign off on my paper, as a form of decentralized quality control. Not all authors are qualified to endorse. This is necessary but insufficient to publish.
As addendum, did AI help? Chat GPT was invaluable for rendering LaTeX, but near useless for computation. It made mistakes when trying to simplify certain expressions.
The free version of Cheat GPT is also extremely limited in terms of computing power and will rate-limit users even after performing as few as five ‘4×4’ (four equations, four unknowns) matrix computations within a 5-minute window. This is a joke. Even free JavaScript programs online do not have such limitations, as the computations are run client-side.