Good to read your work again George...
As always, you have made me think. But in fact; your essay weaves together a number of things I've been thinking a lot about - both on and off the contest forum. In some sense the Mandelbrot Set is a 'door that has no key' because the only way to see it is to try one spot after another to find out where it converges.
One can actually calculate precise locations for a handful of spots by hand. But the Galois method tops out before you get very far, and won't even give you an outline. On the other hand; the Mandelbrot Set is like a key or index to a large family of Julia Sets.
I enjoyed the way you wove in the three levels from Penrose connected to the three levels of metacognition from Metcalfe and Son, and then concluded with 'entangled, purposeful, and self-referential.' I could say a lot more, but I'm hoping to read more essays before the bell. You get high marks from me.
Best of Luck,
Jonathan