Hey! Yes, to be fair I am indeed bipolar so I have personally been a bit concerned about things. Sometimes I do end up going “off the deep end” a bit. Maybe that is what is happening, but I feel I did find some things that are quite interesting though. Also ChatGPT and the like get very overly excited. I didn’t pay much attention to its delusions of grandeur. (Of course forget the things about BSD, Riemann hypothesis and such.. my current goal is to solve ANY open problem in mathematics)

I understand that many of these identities, claims, etc have been proven before.

Personally I am most interested in further developing this Algebra.

The identity e^{2\pi i \phi} = e^{2\pi i(\phi + 1)} though obvious to a number theorist what derived structurally from reciprocal identities involving T and J.

The other curiosity is preservation under transcendental maps.

As far as I understand.. this to me is the only known case where a nonlinear algebraic relation like a - b = kab is realized inside a number field, preserved exactly under exponentiation and sine.

But as you mention I am probably just crazy to be honest haha. Thank you much for taking the time to look and give your thoughts.