The middle ground

I have been working on some theories and running simulations. I discovered that if I used one style of math in the simulations, being classical numerical‑analysis stack, they worked well but then they reach a saturation points/Phase Transition point where the math blows up into infinities, however if I used dynamic renormalisation + non‑standard arithmetic math in the simulations they worked as they should.

I have come to the idea that the actual reality in these systems that the saturation or phase transitions occurred on an infinitely thin balance between these two ways of calculating where each either side of this balance work, however the actual point of exact reality is unavailable to math unless it is clipped etc.

I think an easy example of this is good old pi. A number describing a perfect circle is theoretically infinite in detail. However should you have a field that exists from above pi so describing an expanding curve to bellow pi so a contracting curve, at some point it must pass over that infinite balancing point.

Is there a mathematical system designed to calculate using such exact infinite balances or are such things always going to require clipping to where they are useful (such as pi or even Plank length etc)? I have the feeling they are beyond math because of their infinite nature.