Why would primes near powers of two (or small primes) be important?

The earlier posting What could be the role of complexity theory in TGD? was an abstract of an article about how complexity theory based thinking might help in attempts to understand the emergence of complexity in TGD. The key idea is that evolution corresponds to an increasing complexity for Galois group for the extension of rationals inducing also the extension used at space-time and Hilbert space level. This leads to rather concrete vision about what happens and the basic notions of complexity theory helps to articulate this vision more concretely.

Also new insights about how preferred p-adic primes identified as ramified primes of extension emerge. The picture suggests strong resemblance with the evolution of genetic code with conserved genes having ramified primes as their analogs. Category theoretic thinking in turn suggests that the positions of fermions at partonic 2-surfaces correspond to singularities of the Galois covering so that the number of sheets of covering is not maximal and that the singularities has as their analogs what happens for ramified primes.

p-Adic length scale hypothesis states that physically preferred p-adic primes come as primes near prime powers of two and possibly also other small primes. Does this have some analog to complexity theory, period doubling, and with the super-stability associated with period doublings?

Also ramified primes characterize the extension of rationals and would define naturally preferred primes for a given extension.

1. Any rational prime p can be decomposes to a product of powers P^k_i of primes P_i of extension given by p= ∏_i P_i^k_i, ∑ k_i=n. If one has k_i≠ 1 for some i, one has ramified prime. Prime p is Galois invariant but ramified prime decomposes to lower-dimensional orbits of Galois group formed by a subset of P_i^k_i with the same index k_i . One might say that ramified primes are more structured and informative than un-ramified ones. This could mean also representative capacity.
   
   
2. Ramification has as its analog criticality leading to the degenerate roots of a polynomial or the lowering of the rank of the matrix defined by the second derivatives of potential function depending on parameters. The graph of potential function in the space defined by its arguments and parameters if n-sheeted singular covering of this space since the potential has several extrema for given parameters. At boundaries of the n-sheeted structure some sheets degenerate and the dimension is reduced locally . Cusp catastrophe with 3-sheets in catastrophe region is standard example about this.
   

   Ramification also brings in mind super-stability of n-cycle for the iteration of functions meaning that the derivative of n:th iterate f(f(...)(x)== fⁿ⁾(x) vanishes. Superstability occurs for the iteration of function f= ax+bx² for a=0.
   
   

3. I have considered the possibility that that the n-sheeted coverings of the space-time surface are singular in that the sheet co-incide at the ends of space-time surface or at some partonic 2-surfaces. One can also consider the possibility that only some sheets or partonic 2-surfaces co-incide.
   

   The extreme option is that the singularities occur only at the points representing fermions at partonic 2-surfaces. Fermions could in this case correspond to different ramified primes. The graph of w=z^1/2 defining 2-fold covering of complex plane with singularity at origin gives an idea about what would be involved. This option looks the most attractive one and conforms with the idea that singularities of the coverings in general correspond to isolated points. It also conforms with the hypothesis that fermions are labelled by p-adic primes and the connection between ramifications and Galois singularities could justify this hypothesis.
   
   

4. Category theorists love structural similarities and might ask whether there might be a morphism mapping these singularities of the space-time surfaces as Galois coverings to the ramified primes so that sheets would correspond to primes of extension appearing in the decomposition of prime to primes of extension.
   

   Could the singularities of the covering correspond to the ramification of primes of extension? Could this degeneracy for given extension be coded by a ramified prime? Could quantum criticality of TGD favour ramified primes and singularities at the locations of fermions at partonic 2-surfaces?
   

   Could the fundamental fermions at the partonic surfaces be quite generally localize at the singularities of the covering space serving as markings for them? This also conforms with the assumption that fermions with standard value of Planck constants corresponds to 2-sheeted coverings.
   
   

5. What could the ramification for a point of cognitive representation mean algebraically? The covering orbit of point is obtained as orbit of Galois group. For maximal singularity the Galois orbit reduces to single point so that the point is rational. Maximally ramified fermions would be located at rational points of extension. For non-maximal ramifications the number of sheets would be reduced but there would be several of them and one can ask whether only maximally ramified primes are realized. Could this relate at the deeper level to the fact that only rational numbers can be represented in computers exactly.
   
   
6. Can one imagine a physical correlate for the singular points of the space-time sheets at the ends of the space-time surface? Quantum criticality as analogy of criticality associated with super-stable cycles in chaos theory could be in question. Could the fusion of the space-time sheets correspond to a phenomenon analogous to Bose-Einstein condensation? Most naturally the condensate would correspond to a fractionization of fermion number allowing to put n fermions to point with same M⁴ projection? The largest condensate would correspond to a maximal ramification p= P_iⁿ.
   
   

Why ramified primes would tend to be primes near powers of two or of small prime? The attempt to answer this question forces to ask what it means to be a survivor in number theoretical evolution. One can imagine two kinds of explanations.

1. Some extensions are winners in the number theoretic evolution, and also the ramified primes assignable to them.
   These extensions would be especially stable against further evolution representing analogs of evolutionary fossils.
   As proposed earlier, they could also allow exceptionally large cognitive representations that is large number of points of real space-time surface in extension.
   
   
2. Certain primes as ramified primes are winners in the sense the further extensions conserve the property of being ramified.
   
   
   1. The first possibility is that further evolution could preserve these ramified primes and only add new ramified primes. The preferred primes would be like genes, which are conserved during biological evolution. What kind of extensions of existing extension preserve the already existing ramified primes. One could naively think that extension of an extension replaces P_i in the extension for P_i= Q_ik^{k_i so that the ramified primes would remain ramified primes.}
      
      
   2. Surviving ramified primes could be associated with a exceptionally large number of extensions and thus with their Galois groups. In other words, some primes would have strong tendency to ramify. They would be at criticality with respect to ramification. They would be critical in the sense that multiple roots appear.
      

      Can one find any support for this purely TGD inspired conjecture from literature? I am not a number theorist so that I can only go to web and search and try to understand what I found. Web search led to a thesis (see this) studying Galois group with prescribed ramified primes.
      

      The thesis contained the statement that not every finite group can appear as Galois group with prescribed ramification. The second statement was that as the number and size of ramified primes increases more Galois groups are possible for given pre-determined ramified primes. This would conform with the conjecture. The number and size of ramified primes would be a measure for complexity of the system, and both would increase with the size of the system.
      
      

   3. Of course, both mechanisms could be involved.
      
      

Why ramified primes near powers of 2 would be winners? Do they correspond to ramified primes associated with especially many extension and are they conserved in evolution by subsequent extensions of Galois group. But why? This brings in mind the fact that n=2^k-cycles becomes super-stable and thus critical at certain critical value of the control parameter. Note also that ramified primes are analogous to prime cycles in iteration. Analogy with the evolution of genome is also strongly suggestive.

For details see the chapter Unified Number Theoretic Vision or the article What could be the role of complexity theory in TGD?.

For a summary of earlier postings see Latest progress in TGD.

Articles and other material related to TGD.