'Ten Martini' Proof Uses Number Theory to Explain Quantum Fractals

> That makes me feel like we live in a simulation perhaps more than anything else I’ve heard

I'm never quite sure what this is meant to mean. Is it comparing to other simulations like computer games or physical simulations where you could change a seed or a data structure and have it manifest in reality? What is expected from a simulation to differ from reality? What does it even mean to make this distinction when we are observing inside the process we are trying to distinguish between real and simulated?

I THINK they are saying that when the parameter alpha (magnetic flux) is an irrational number, the allowed energy levels of the electron form a structure that resembles a Cantor set. The gaps in the energy levels (forbidden energies) correspond to the “missing middle thirds” in the Cantor set

Well, if you consider that the setting of the problem is non-interacting, two-dimensional electrons in an infinite lattice, there's not much real about it. These patterns will likely disappear in a more realistic situation.

https://www.recursivebecoming.info/RBT_v1.0_release.pdf

Also: Church encodings for integers and bootstrapping number systems from the empty set. https://en.m.wikipedia.org/wiki/Church_encoding And Lisp, of course; where the book Gödel, Escher, Bach shows's up how to build a number system from empty lists:

https://wiki.xxiivv.com/site/church_encoding.html

An approximation, I’d imagine. I wouldn’t expect it to manifest in any real physical sense, any more than I’d expect an actual physical “unit circle” object to have a circumference of exactly two Pi.