Combinatorial Hierarchy: Two Decades Later

Abstract

Combinatorial Hierarchy (CH) is a hierarchy consisting of Mersenne integers M(n)= _MM(n-1) =2^M(n-1)-1and starting from M₁=2.  The first members of the hierarchy are given by 2, 3, 7, 127, M₁₂₇ =2^127-1and are primes. The conjecture of Catalan is that the hierarchy continues to some finite prime.  It was proposed by Peter Noyes and Ted Bastin that the first levels of hierarchy up to M₁₂₇ are important physically and correspond to various interactions.  I have proposed the levels of CH define a hierarchy of codes containing genetic code corresponding to M₇ and also memetic code assignable to M₁₂₇. In this article I consider the argument that the hierarchy ends at M₁₂₇ and find that it should end already at M₇ for which the condition used saturates and which corresponds to genetic code in TGD interpretation. The failure of condition at M₁₂₇ level has interesting “Godelian interpretation".  I find also that in TGD Universe genetic code and its memetic counterpart are realized at the level of fundamental particles. Already earlier I have ended up with alternative realizations at the level of dark nucleons and sequences of 3 dark nucleons.