So in other words, relating all this to the op which is the whole point, let's say you got Ernie to generate 1e44 random numbers. I don't know how long that would take, probably a long time haha. But let's say you could do it.

The quantum theory says, that after 1e44 samples your observed statistical distribution will be exactly perfect.

However my claim is, it will be imperfect. You will see some deviations from Pure Randomness, and you will have to account for those. Most likely the accounting will occur on the basis of information and geometry.

So, the testing in the OP which is being done to four or five orders of magnitude, is completely inadequate. It needs to be done to 44 orders of magnitude, not 4.

But I mean, this is a wonderful thing, having machines which can finally do this.

If my thinking is correct, geometry is information to. If you're dividing something into two parts, you're doing work. The work that you're doing ends up as a partition, which is to say geometry. A partition is the same thing when you divide a set, or when you choose a member from a set.

This concept underlies the field of "information geometry". (Google Shun-Ichi Amari).