What do you mean by "the scientific process"? This is a subject that is up for debate and is philosophical in nature. If you cannot get a universally agreed upon definition for this then "science" can never converge on a 100% accurate model.

You then go into comparing a non-physical entity as an example of a physical system. Universal Turing Machines are an idealised entity. It cannot match, in any way, any physical system in a 100% accurate way. So saying that you cannot simulate an idealised model of computation as the means of saying that that we have a 100% accurate model that cannot be simulated is logically circular.

It does not make the distinction you are trying to make. Simulations are models and all models are idealised. So simulations cannot never be 100% accurate as no model can be 100% accurate of physical systems.

As far as "reality is mathematical" and empiricism, you are the one claiming that empiricism is mathematical in nature when it is a philosophical viewpoint and you are claiming that the OP is claiming empiricism.

The problem you are forgetting is that every theory and every model are idealised representations of the universe about us. Whether or not we are able to successfully use mathematics to describe all that we find is another matter entirely.

When scientists, both experimental or theoretical, say that their theories or models on any the subject are correct, they are misleading themselves and everyone else. Every model and theory is inaccurate. All make assumptions that ignore the reality that they are investigating.

I have said this before, just because a theory or model gives some predictive ability to what one may see, it doesn't mean that that the theory or model is correct. Too many people, including the scientists that promote a theory or model, get so caught up in being right that they forget that the theory and model is only something that we can use to try and understand what is going on about us in the universe.

I used to have no problems with biological evolutionary theory and models. However, after reviewing the results of experiments in the field, it became obvious that something was really wrong with the model (models) being used. The results did not support what was being promoted.

I used to have no problems with various things like the "big bang" or "black holes" or "neutron stars" or even general relativity, until I started to look at what was being seen in actual experiments and the problems with the underlying assumptions to the theories being used. Anomalous results that were being found and ignored. The ad hominem attacks of those who did not ascribe to the "consensus" theories and models really lead me to investigate the basic assumptions being used.

All that I see is a growing problem in science where it is no longer about developing an understanding of our universe but "follow the party line" or be anathema and be excommunicated. To those who know their history, the Inquisition is alive and well today.

Science is a tool we can use to gain some understanding. Philosophy is a tool we can use to gain understanding. We should be capable of putting these tools in our toolbox and using them accordingly. But to those who only have a hammer, then everything is a nail.