Similarly, as the number of particles in a system increases, the less you have to worry about Heisenberg uncertainty bounds on your momentum/position measurements.

It’s a misreading of the halting problem to think of it as a ‘barrier’ of some sort. What it really means is that there are some computer programs where the only way to find out what they will do is to actually run them. There’s not a ‘better trick’ you can pull to get the answer. This is kind of intuitive, since we know there are problems we run into where the easiest way we can think of to solve them is to write a program and run it. The halting problem tells us that it’s plausible that that instinct is right - that for some programs, you really can’t do some other analysis of the program that tells you what it’s going to do. You have to just run it.

Like: take a random complex number with magnitude < 2. Keep squaring it and adding the original number. Will you ever get a number with magnitude > 2?

For some numbers you can take a shortcut and say yes or no. But for others the only way to find out is to keep doing it.