The distance between two points is basically just the Pythagorean theorem: x^2 + y^2 + z^2. If you have extra dimensions, you just add more terms to this. Now, if you know that e.g. x is very large then that puts a lower bound on the distance:

Distance^2 = x^2 + y^2 + z^2 + ... > x^2.

In other words, if x is large there's no way that the two particles are still somehow close together.

Put two iron pellets on a very thin, flexible fabric. Underneath the fabric, put a magnet so each pellet is stuck to one pole of the magnet through the fabric. Tug the magnet "down" in the third dimension, which will fold the fabric and move the pellets so that from their perspective (stuck to the 2D surface of the fabric) they're moving far away from each other. They also maintain a constant and very short distance in the third dimension, but they can't take advantage of it because I forgot to mention the fabric is impervious to them.

This is only true for cartesian spaces. If the low-dimensional manifold is "folded" or "crinkly" inside a higher-dimensional space then you can indeed have particles that look far apart on the manifold but which are actually close together in the higher-dimensional space.

(My terminology here is largely wrong; this is not my field. Perhaps an expert can correct me.)

ER=EPR entanglement could be realized as a non-local distance mapping

Bell's theorem is strong: No local hidden variables can explain quantum entanglement.

Your proposed fifth dimension is exactly such a variable – there is no set of values it can take on that explains the quantum phenomena we can easily and reliably observe.

The fact that you think of it as a distance itself isn't important to Bell's theorem. Such a theory is cannot even obey relativity in our existing 3 dimensions (obeying the speed of light in your extra dimension would be even stricter!)

Superdeterminism asserts what you're implying is incorrect.

Sure. There are assumptions you need for Bell's Theorem to apply. Statistical independence, single-outcome measurements, locality, determinism... All of these feel right to our scientific brains, but something's got to give.

So yes, you can throw out Bell's theorem if you're willing to throw out statistical independence with super-determinism. It's a tough leap to make, though. Super-determinism implies that there aren't really any laws of quantum mechanics. The fact that billions of quantum interactions have been observed in labs to obey consistent probabilities over time is pure coincidence. The fact that two entangled electrons always have opposite spin is pure coincidence. Nothing about the laws of the universe say you couldn't observe the other thing, it's just that the initial conditions of the universe prevented us from making measurements any time we would have measured otherwise. It's a tough pill to swallow.

My comment above is just saying you can't throw out Bell's theorem by saying "maybe there's a fifth dimension". I don't think GP meant a fifth dimension plus superdeterminism.

From what I can gather, in a single world view, it seems that thinking of reality in an information oriented way results in more accuracy. So you might have interactions that aren't dependent on distance, like with entangled particles. Taking on this view is a bit of a mind bender from a normal spatial view of the world though.

Information sharing without physical interaction is "spooky action at a distance"

there are many ways to explain the result and all are extremely strange. particles communicating at FTL speeds? particles actually touching in an unknown dimension?

It is also strange, but I think the many worlds hypothesis makes the most sense to me. Then there is no spooky action at a distance, the observer is just always in a particular version of reality that all states, even distant ones, conform to. I am not a physicist!

But there is still spooky splitting of the universe... Many worlds just replaces one mystery with another

It's not a mystery, it has a testable precise mathematical description - Schrodinger equation.

Yes, but not action at a distance. There's no violation of relativity in MWI.

All the states are already there, they just share the have value before they have split