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If I am correct that there is a simple underlying program for the universe, then this means that theoretical physics must at some level have only a very small amount of true physical input—and the rest must in a sense all just be mathematics.
The same is true of the right-hand edge in rule 30—though the left-hand edge in this case expands only about 0.2428 cells on average per step. … Nothing as simple is true for the left-hand edge, and indeed this seems to execute an essentially random walk—with an average motion of about 0.2428 cells per step.
As emphasized by Benoit Mandelbrot in connection with a variety of systems in nature, the same is also true for random walks whose step lengths follow a power-law distribution, but are unbounded.
Identical snowflakes
The widespread claim that no two snowflakes are alike is not in practice true.
It is still true, however, that at a small scale the boundaries consist of discrete cells.
And in many cases these functions end up trying to prove theorems; so for example FullSimplify[(a + b)/2 ≥ Sqrt[a b], a > 0 && b > 0] must in effect prove a theorem to get the result True .
What is usually said is that prices are in fact determined not by true value, but rather by the best estimates of that value that can be obtained at any given time.
In my experience the same is eventually true with computer languages.
But normally such a statement cannot be proved true or false within the system itself.
Rule 90R has the property that of the diamond of cells at relative positions {{-n,0},{0,-n},{n,0},{0,n}} it is always true for any n that an even number are black.