I don’t know what “power spectrum” is, but the second-to-last graph looks pretty obviously like Brownian motion. This makes sense because the differences between consecutive points in the third graph will be approximately Poisson and independently distributed, so if you renormalize so that the expected value of the difference is zero, then the central limit theorem will give you Brownian motion in the limit.
Anyway regarding the relation of your post to Tegmark’s theory, a random sequence can be a perfectly well-defined mathematical object (well maybe you need to consider pseudo-randomness, but that’s not the point) so you are not getting patterns out of something non-mathematical (whatever that would mean) but out of a particular type of mathematical object.
I don’t know what “power spectrum” is, but the second-to-last graph looks pretty obviously like Brownian motion. This makes sense because the differences between consecutive points in the third graph will be approximately Poisson and independently distributed, so if you renormalize so that the expected value of the difference is zero, then the central limit theorem will give you Brownian motion in the limit.
Anyway regarding the relation of your post to Tegmark’s theory, a random sequence can be a perfectly well-defined mathematical object (well maybe you need to consider pseudo-randomness, but that’s not the point) so you are not getting patterns out of something non-mathematical (whatever that would mean) but out of a particular type of mathematical object.