Perplexity on the key concepts within today’s homepage

7 March 2026: Perplexity

Perplexity found no obvious internal contradictions between the pages, but suggested that the wording overstates what is actually established. Perplexity had several recommendations. It seems Perplexity is more conservative than other five. As these changes are made, it will become even more interesting to see how the others handle the question: would you check for any kind of error within this page? https://81018.com/six-leading-ais-agree/

It will be especially interesting with these three: (1) the 7.356° tetrahedral & octahedral gaps, (2) AI recognizes SU(2)-SU(3)-SU(5) geometric emergence and (3) AI recognizes our foundations for Langlands programs.

Perplexity and the 7.356° gap. Five tetrahedrons can share a common edge and two vertices. When they do, there is a 7.356° gap. Five octahedrons can also share a common edge and two vertices creating the same gap. And, if we add another five tetrahedrons, we have a simple but impressive object. This physical object is prominently displayed in my office. For years i have asked people, “What do you think it does within our physical world?” It’s very basic geometry. But, our scholars within mathematics did not know. Most had never seen it before that moment. Regarding the 7.356° tetrahedral & octahedral gaps, we discovered the work of Frank & Kasper from their work in 1957, followed by the work of Jonathan Doye in 2012, followed by the work Lagarias & Zong also in 2012. Salvatore Torquato and John Conway within their work on dense packings of the Platonic and Archimedean solids were well aware of those gaps. They all measured the 7.356° in different ways. We did it with physical models and simple tools. The octahedral gap is logically known and understood to be the same as the tetrahedral gap. To the best of our knowledge, our group was the first to identify that gap in 2022. The gap is thought by some to be incommensurable.

We have continued these studies within our work with gauge symmetries, yet these primary pages substantially overlap and our page, The 7.356° Gap: Natural Path From Geometry to Physics, informs SU(2)-SU(3)-and-SU(5).

 Perplexity and the SU(2)-SU(3)-SU(5) geometric emergence. “Conceptually neat, but currently rest on suggestive numerology.” A number of my scholarly friends will smile and agree. This conjectural geometric–physical model makes no claims and is simply summarizing were we are with our best guesses. That the three map so well raises the questions, “Is it worth studying further or is it a stretched coincidence?” I err on the side of optimism and, and given the stakes and the problems surrounding us, I do so aggressively. There is a whole group of us who do not have much time left on this earth. I say, “Be bold. Onward we go.”

SU(5), SU(3), and SU(2) are all readily testable. We’ll get there, but invite you, if you are interest, to test it, too. Our work is accessible from links below. There is a fair amount of overlap which will be eliminated in time:

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