How do I get through the south exit from Laying The Foundation?

To get inside, you have to get the balls move very fast. This is hardy possible if the resolution of your screen is too high compared to the power of your computer. Try lowering the resolution for this particular puzzle.

Edit: Some people still cannot manage to get through the south exit, so I give a bigger hint. Do not read if you are afraid of spoilers.

It's rather quite tricky to get it down, but the way I did it was... Don't keep moving once you've already broken a lot of the cube. Let it regenerate, and try again. If you keep moving, it'll regenerate too much over any area you want to move through. I found that only one of the sides consistently made a hole big enough to jump through. Move around, try to look for the hole, wait for the cube to regenerate, then move to where you saw the biggest hole. Then move around to break it open again, and usually it breaks open in a similar pattern as before.

Reference: NeoGAF.

One of the walls will regen slower than the others. Thats your best bet for getting in.

Reference: GameFAQs.

There is actually another way to solve this (even with low FPS):

Create some small holes by running, then place some cubes inside. Now, as the wall is about to regenerate, drag a cube outside. If your timing was right you will move the white blocks outside the wall. You can even make the white cubes flow into the destruction field!