>The claim here is actual "continuous" objects likely exist. How can you demonstrate this?

You cannot demonstrate either continuous or discrete beyond what a model shows. All of knowledge in science is from models, nothing more.

For example, all objects that you perceive as discrete are modeled as waves in QM (more specifically TQFT). Given that it seems everything behaves as a wave, how can you prove discrete items exist to the level of proof you seem to require? I doubt it's possible to do so. Your perception of discrete is wrong, according to QM. What was once wave/particle duality in the early 1900s got replaced with larger and larger systems and TQFT until now everything in QM is a field, including you, including marbles, including particles, including everything. And this view allows the theory to accurately predict all relevant experiments. Discrete you, marbles, electrons, fails experimentally.

>All continuous models can be reproduced in discrete models up to the resolution of "all current observations"

This is not true. If I recall my QCD correctly, some calculations spit out infinities if the underlying space is not continuous. I suspect there are other places that add divergences if you discretize things.

Roughly - if you perform calculations in QCD (or QED), you basically add up an infinite number of terms, each with smaller and smaller probability, to get a finite answer (which, for QED, is astoundingly accurate). If you force this infinite sum to not be allowed to go to zero spatially, then you get infinite terms that cannot shrink, and you get an infinity. If you try to fix this by only including a finite number of terms, you get other issues, such as not getting the right answer when tested experimentally.

I also seem to recall that in QED if you do not have continuous space, then the self-interaction of an electron again spits out infinities, which of course is one major impetus for developing QED to begin with.

So no, you cannot simply discretize math and get all calculations to still given the same answers you see experimentally. It's not that simple.

For example, you mentioned LQG - researchers have tried for a long, long time, with much effort, to get it to reproduce continuous GR with a discrete model, and they so far have failed. It is not as simple as simply making the discreteness smaller. There are fundamental issues in getting such things to match the continuous theories.

So, despite it being a really useful thing to do, and despite there being lots of work trying to get a discrete model to match GR or SM, I am unaware of anyone ever making it work.

Interestingly, trying to find a better way to explain it to you, I found this [1] question about the inability of anyone to make a discrete model match current continuous ones (which backs my view that it has not been done, and perhaps will be found impossible). He makes the interesting point that your view of LQG making space(time) discrete is not correct. The reason is that this would violate Lorenz invariance (which is also the thing those experiments listed) are looking at, and so far Lorenz has never been broken experimentally, to the point of pushing it below the plank limit. So the "discreteness" of LQG is much more subtle than a naïve view.

I asked for an example of a discrete theory that matches some significant piece of physics - do you have one or not? Your claim that it can be done is not matched by any evidence you have put forth, nor any that I have ever encountered.

[1] https://math.stackexchange.com/questions/186076/is-there-suc...