I'll tell you right now... I feel like there is a way to model the 3n+1 system of equations (or really any such generalized system) using Godel numbering as a representation of each operation, as a prime number based programming language of nature, and then try to glean something from the output primes to see if there is something that predicts the single 4->2->1 outcome we always see. e.g. if it is a certain form of Fermat prime or something.
It would require me to put my computers to work because these numbers get very big, but the real limitation is my time because I have three kids and cannot afford to quit my job.
As an example of just how arbitrary Gödel numbers are, consider the expressions `((3*n)+1)` and `((n*3)+1)`. Those expressions are clearly equivalent, yet the Gödel numbers for those two expressions are wildly different, with a relation that is rather complex.
IIUC Gödel himself never "used" or analyzed Gödel numbers as such, only the idea that they could be constructed.
...a better example is just "x 1". You could add that ad infinitum to any equation and it cannot change the meaning of the outputted Godel number. ...but remember that getting a different result does not mean the interpretation of the Godel number is different. For example, multiples of ten are also even. 64 and 9 are different numbers, but they are both perfect squares. ...and the goal is to find something descriptive of the resultant number - not the number itself.
...but that's one of the main areas to explore. There are different types of primes to experiment with to see if anything meaningful can be discovered.
Your idea smells like wishful thinking.
https://www.youtube.com/watch?v=VPQx1q9cfkU
...so maybe there's something to this line of research. I don't think this is a bad idea at all.
Write down every quantitative expression in Mathematical Physics. Assign a unique numerical ID to each expression. Identify the relations between each of the expressions to create a graph. Does that graph feature any patterns? Is there a path from any expression to every other expression?
The resulting number's geometric properties might provide some insight into the properties of the original system.
The trick is figuring out which primes to use initially, and how to interpret the resulting number.
....but my gut tells me there is something there.
In any case, the geometric (or perhaps you meant topological) properties of the objects are independent of the specific primes used, so "figuring out which primes to use initially" is a bit of a non sequitur.