To expect the primes to be uniform in distribution would seem a just so idealization i.e. atypical. Randomness is just an idealization. One could also consider the clustered primes in 3-dimensions, with still a non-uniform distribution, and compare it to non-uniform distribution of matter in early universe. Evenly distributed matter (DM and LM) for the early universe would seem like a special condition, since non-uniform distribution (voids, proto-voids, and non-void regions) for macro-scale would seem more typical. Thus physical models would seem to have to describe just evolution from such more typical early macro-stage, rather than assuming perfect uniformity (randomness) for such distribution. The latter being just an idealized assumption. One could compare to alternative scenarios (top-down, and bottom-up models – but not enough time). However perhaps one has to accept the early universe distribution of matter as like physical truth; not so unlike the clustered primes as mathematical truth? That is, was there no choice, as the the curtain rises at the end of Dark Age? TMM

see https://sites.google.com/site/zankaon

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