[DanielGeisler]

Writing Mathematica software with Stephen Wolfram's support to extend the hyperoperators beyond exponentiation - tetration, pentation and so on, from the natural numbers to complex numbers and even matrices. I do this by extending the iteration of any smooth function to real and complex iterates. http://iteratedfunctions.com/ and http://tetration.org/. Physics has two mathematical methods for it's representation, partial differential equations and iterated functions. My work is more general than physical systems or even the universe because I can consider both measure and non-measure preserving systems. I am looking at AI applications as a system that is tuned to solve for physically possible models.

[jlevers]

I won't pretend to fully understand this, although I do understand the basic theory of higher-order hyperoperators. I'm curious, though -- where/what are the applications for this?

[DanielGeisler]

The Universe is a hierarchy of orders; quarks and gluons, atoms, molecules, cells, multi-cellular and on. So hyperoperators form a natural hierarchy to model multilevel systems. Feynman's Path Integral with the integral removed is just tetration. So I believe the higher operators through self organization/renormalization are directly associated with specific levels of reality, but except for QFT I have no suspicion as to which hyperoperators might be associated with levels of physics. Here is how it might work. Hexation might model chemistry while heptation could model simple biological systems. Thanks for asking. This is fun stuff to work on.

[jlevers]

That explanation was surprisingly accessible, thank you! It also blew my mind. I'd never thought of the building blocks of the universe as forming a hierarchy before, although in retrospect, it feels obvious.

Were you being literal when you gave the examples of chemistry as hexation and biology as heptation? If so, why are they those levels specifically? Or were you just using those as examples because they're roughly one "step" apart on the hierarchy (i.e., molecules -> cells)? Sorry if this is a dumb question.

[DanielGeisler]

I strongly believe that quantum field theory is at the level of tetration or pentation. The others were possible examples. Thanks for the question.