1. Various p-adic physics is part of TGD and fused to what I call adelic physics proposed to be a physics including physical correlates of both sensory experience and cognition. p-Adic physics began with p-adic thermodynamics calculations of particle masses assuming conformal invariance leading to p-adic length scale hypothesis: the primes p=about 2^k (but smaller) are of special importance physically. Mass scales are proportional to 2^(k/2) and one has fractality.

Mersenne primes (also Gaussian) are of special importance physically. One can assign to each elementary particle p-adic prime p. Electron corresponds to M_127 = 2^127-1, the largest Mersenne, which does not correspond to super-astronomical p-adic length scale propto sqrt(p)*R, R the size scale of CP_2 about 2^12 Planck lengths. The success of calculations is not trivial since there is an exponental sensitivity to the value of k.

The outcome is naturalization of the mass scale hierarchy: the integers k involved are rather small although mass

scales vary in a huge range. This would solve the naturalness problems related to neutrino mass and intermediate boson masses.

2. Cosmological constant and Planck length (besides CP_2 size) emerge naturally in twistor lift of TGD besides Kahler coupling strength having interpretation as U(1) coupling of standard model. Cosmological constant Lambda obeys p-adic coupling constant evolution and behavs as 1/p so that one can understand both the extremely large value of Lambda during very early cosmology and large value in recent cosmology. p defines a p-adic length scale of order 10^10 ly.

Cosmological constant defines also vacuum energy density 1/L^4 and L corresponds in recent cosmology to p-adic scale of order large neuron size: does fundamental bio-scales have interpretation in terms of fundamental physics?

3. In TGD color gauge fields are at fundamental level geometrized and gauge potentials are proportional to the projections of isometry currents of CP_2 to the space-time surface. At QFT-GRT limit they become color gauge potentials. Color corresponds to orbital quantum number assignable to CP_2 spinorial harmonics rather than being spin-like. Also leptons have color partial waves (triality 0). I do not know whether this could explain the smallness of the theta parameter. A possible hint is that the instanton density for single space-time sheet is non-vanishing only if it has 4-D CP_2 projection. String like objects assignable to hadrons have 2-D CP_2 projection and vanishing instanton density.

4. In TGD CKM mixing and its leptonic counterpart have topological origin. Particles have as space-time correlates what I call partonic 2-surfaces whose topology is characterized by genus g=0,1,2.. The three lowest genera allow always Z_2 conformal symmetry but the higher genera only for hyper-elliptic surfaces. The conjecture is that for higher genera the handles of the partonic 2-surface behave like free particles and one does not have discrete mass spectrum but 2-D system with continuous energy and rest mass.

The relationship between leptonic and quark CKM matrices was new to me. The hierarchy of algebraic extensions of rationals play a key role in TGD. The condition that the elements of CKM matrix belong to an algebraic extension of rationals defining an element of finite subgroup of SU(3) and that these elements are number theoretically universal making sense in the extension of any p-adic number induced by the algebraic extension of rationals gives very strong conditions on the CKM matrix and might also allow insights about the lepton-quark relationship. I have not consider this option explicitly.

]]>I recently started playing around with homeomorphic irreducible trees like you described in your Log#055. I had previously drawn trees up to n=12 and was able to find drawings of those trees on the internet to verify my results. However, I have been unable to find drawn results for any value of n>12. Today, after completing my drawings for n=14, I was able to find the series values for the trees on the On-Line Encyclopedia of Integer Sequences and found that the 78 variations I drew for n=14 is the correct number of results. However, I have been unable to find drawn examples to compare my results against. Do you know to what value of n the homeomorphic irreducible tress have been drawn? If they have been drawn for values greater than 12, as I suspect they have, do you know where I can find a link to the images?

Thank you,

Mike Hofner

peterr211@yahoo.com

Thank you

]]>I get ζ(0) = ζ(1), which should be infinite not, -1/2? ]]>