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.

On 4/22/2026 at 8:37 PM, BuddhasDragon23 said:

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.

Whilst I don't have any problem with mathematical handling of infinities so I can't agree with your initial claims about 'blowing up' , I would agree they need special treatment.

Have you studied Gibbs phenomenon ?

These provide a good example, that is easier to study than say Dirac delta functions as they do not require extended methods of integration and measure theory to handle.

Thanks but I am not trying to remove the phenomena, but explore it. The simulation hit a phase‑transition saturation, the classical numerical stack makes it look like the system has developed a literal plateau or jump. That’s a projection, the fixed global scale and standard floating‑point arithmetic collapse all sub‑critical variation into “no change.” If you are thinking in terms of Gibbs overshoot or Dirac deltas you are assuming the feature is real and we’re just struggling to represent a sharp edge or remove the spike.

I am changing the effective coordinates and arithmetic so the “frozen” region unfolds into smooth, structured behaviour. Dynamic renormalisation rescales the variables right at the critical point, and the non‑standard arithmetic preserves the tiny variations that standard numerics flatten. Once you do that, the apparent plateau dissolves and the underlying dynamics reappear. So the point isn’t “better representation of a jump” — it’s “the jump was never physical; it was an artefact of the wrong scale.”

59 minutes ago, BuddhasDragon23 said:

Thanks but I am not trying to remove the phenomena, but explore it. The simulation hit a phase‑transition saturation, the classical numerical stack makes it look like the system has developed a literal plateau or jump. That’s a projection, the fixed global scale and standard floating‑point arithmetic collapse all sub‑critical variation into “no change.” If you are thinking in terms of Gibbs overshoot or Dirac deltas you are assuming the feature is real and we’re just struggling to represent a sharp edge or remove the spike.

I am changing the effective coordinates and arithmetic so the “frozen” region unfolds into smooth, structured behaviour. Dynamic renormalisation rescales the variables right at the critical point, and the non‑standard arithmetic preserves the tiny variations that standard numerics flatten. Once you do that, the apparent plateau dissolves and the underlying dynamics reappear. So the point isn’t “better representation of a jump” — it’s “the jump was never physical; it was an artefact of the wrong scale.”

Perhaps if you drew a few diagrams it might help clarify what you are actually talking about and, dare I say it, put in a few numbers/calculations.

Gibbs phenomena are not 'artifacts' or jumps but can easily be demonstrated in the physical world, as can their development and growth.

Furthermore the calculations accord exactly with observation.

The interesting thing about the Gibbs is that the generating pulse function (which does contain jumps) takes you out of the domain of linear definition for fourier series and generates something which is genuinely emergent in the mathematics.