Combinatorial Hierarchy: two decades later
Combinatorial Hierarchy (CH) proposed by Noyes and Bastin is a hierarchy consisting of Mersenne integers M(n)= MM(n-1)=2M(n-1)-1 and starting from M1=2. The first members of the hierarchy are given by 2,3,7,127,M127=2127-1 and 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 (see this). 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 (see this).
Pierre Noyes and Ted Bastin proposed also an argument why CH contains only the levels mentioned above. This has not been part of TGD view about CH: instead of this argument I have considered the possibility that CH does not extend beyond M127. With the inspiration coming from email discussion I tried to understand the argument stating that CH contains M127 as the highest level and ended up with a possible interpretation of the condition. Zero energy ontology (ZEO) and the representation of quantum Boolean statements A→ B as fermionic parts of positive and negative energy parts of zero energy states is essential. This led to several interesting new results.
- To my best understanding the original argument of Noyes and Bastin does not allow M127 level whereas prime property allows. States at M127 level cannot be mapped to zero energy states at M7 level. Allowing a wild association with Gödel's theorem, one could say that that there is hube number of truths at M127 level not realizable as theorems at M7 level.
A possible interpretation is that M127 level corresponds to next level in the abstraction hierarchy defined by CH and to the transition from imbedding space level to the level of "world of classical worlds (WCW) in TGD. The possible non-existence of higher levels (perhaps implied if MM127 is not prime) could be perhaps interpreted by saying that there is no "world of WCWs"!
- Rather remarkably, for M7, which corresponds to genetic code (see this), the inequality serving as consistency condition is saturated. One can say that any set of 64 mutually consistent statements at M7 level can be represented in terms of 64 Boolean maps at M3 level representable in terms of zero energy states. One obtains an explicit identification for the Boolean algebras involved in terms of spin and isospin states of fermions in TGD framework at level M7 so that genetic code seems to be realized at the fundamental elementary particle level thanks to the dimension D=8 of imbedding space. Even more, the level M127 corresponding to memetic code emerges in the second quantization of fermions at M7 level. Here color triplet property of quarks and color singletness of leptons and the identification of elementary particles as pairs of wormhole contacts are in essential role.
For a summary of earlier postings see Latest progress in TGD.