Jonathan, with each essay contest you improve your ideas and they become increasingly interesting. The Mandlebrot set is a great example of how universality works when there are scaling laws, and how symmetry emerges. As you have explained in your own words the Mandlebrot set exhibits a self-similarity symmetry at Misiurewicz points which is approximate at large scales, but it gets more exact as you zoom in. Similar things happen in physics in different contexts. The renormalisation group in quantum field theory and in thermodynamics is an example of a similar phenomena except that the self similarity improves as you zoom out to larger distance scales rather than zooming in. Another difference is that the renomalisation group gives a continuous self-similarity symmetry whereas the Mandelbrot set is self-similar in discrete iterative steps.
One of the most interesting parts of your essay is the description of how octonions arise at some points. I was not aware of this. Emergence of symmetry in systems of universality with scaling is fundamental to my ideas on the emergence of the laws of physics from complex systems of mathematical possibilities. My claim is that the process of quantisation is the iterative step from which symmetry emerges.
In your proposal there is a twist. if I have understood correctly You say that the Mandelbrot set is not merely an example of emergence but is in fact the perfect description of it. You also say that the asymmetry is fundamental. This can happen. If you zoom in on a thermodynamic system the emergent behavior eventually breaks down and you see the interactions between molecules from which thermodynamics emerges. In quantum field theory you may also find that a theory such as QCD is not exact at high energies, then the SU(3) gauge symmetry may be approximate or not. The same goes for the diffeomorphism symmetry of gravity. My long held belief has been that diffeomorphism symmetry is replaced with permutation symmetry over spacetime events at small scales. This fits perfectly with entropic theories of gravity.
Your idea that the Mandelbrot set is somewhere down there at the bottom is not as daft as it sounds. It is the defining characteristic of universal behavior that it turns up in many different circumstances and that is the case for the Mandelbrot set too. Whether Mandelbrot is the whole, or some part of something larger that encompasses all forms of universality is another question.
There is an ongoing debate about whether gauge symmetry becomes an approximation at higher energies, or alternatively that it is revealed as the residual symmetry of a larger one. It may depend on whether you think the limit of the iterative process is taken or alternatively that there is a cutoff scale at which it stops. In particle physics it looks like it stops, but my view is that at the deepest level the limit must be taken so that a huge exact symmetry emerges. This is necessary to hide the irrelevant details of axiomisation and also to protect us from Godel's undecidability. If it turns out that there is a cutoff and the universal symmetry is only approximate then the universe could be a much stranger place. That is certainly something to think about.
