### Could the basic parameters of TGD be fixed by a number theoretical miracle?

If the v

_{0}deduced to have value v

_{0}=2

^{-11}appearing in the expression for gravitational Planck constant hbar

_{gr}=GMm/v

_{0}is identified as the rotation velocity of distant stars in galactic plane, it is possible to express it in terms of Kähler coupling strength and string tension as v

_{0}

^{}

^{-2}= 2×α

_{K}K,

α_{K}(p)= a/log(pK) , K= R^{2}/G .

The value of K is fixed to a high degree by the requirement that electron mass scale comes out correctly in p-adic mass calculations. The uncertainties related to second order contributions in p-adic mass calculations however leave the precise value open. Number theoretic arguments suggest that K is expressible as a product of primes p ≤ 23: K= 2×3×5×...×23 .

If one assumes that α_{K} is of order fine structure constant in electron length scale, the value of the parameter a cannot be far from unity. A more precise condition would result by identifying α_{K} with weak U(1) coupling strength
α_{K} = α_{U(1)}=α_{em}/cos^{2}(θ_{W})≈ 1/105.3531 ,

sin^{2}(θ_{W})≈ .23120(15),

α_{em}= 0.00729735253327 .

Here the values refer to electron length scale.
If the formula v_{0}= 2^{-11} is exact, it poses both quantitative and number theoretic conditions on Kähler coupling strength. One must of course remember, that exact expression for v_{0} corresponds to only one particular solution and even smallest deformation of solution can change the number theoretical anatomy completely.
In any case one can make following questions.

- Could one understand why v
_{0}≈ 2^{-11}must hold true. - What number theoretical implications the exact formula v
_{0}= 2^{-11}has in case that it is consistent with the above listed assumptions?

**1. Are the ratios π/log(q) rational?**

The basic condition stating that gravitational coupling constant is renormalization group invariant dictates the dependence of the Kähler coupling strength of p-adic prime exponent of Kähler action for CP_{2} type extremal is rational if K is integer as assumed: this is essential for the algebraic continuation of the rational physics to p-adic number fields. This gives a general formula α_{K}= a π/log(pK). Since K is integer, this means that v_{0}^{2} is of form

v_{0}^{2}= qlog(pK)/π, q rational.

if a is rational.

- Since v
_{0}^{2}should be rational for rational value of a, the minimal conclusion would be that the number log(pK)/π should be rational for some*preferred prime*p=p_{0}in this case. If this miracle occurs, the p-adic coupling constant evolution of Kähler coupling strength, the only coupling constant in TGD, would be completely fixed. Same would also hold true for the ratio of CP_{2}to length characterized by K^{1/2}. - A more general conjecture would be that log(q)/π is rational for q rational: this conjecture turns out to be wrong as discussed in the previous posting. The rationality of π/log(q) for single q is however possible in principle and would imply that exp(π) is an algebraic number. This would indeed look extremely nice since the algebraic character of exp(π) would conform with the algebraic character of the phases exp(iπ/n). Unfortunately this is not the case. Hence one loses the extremely attractive possibility to fix the basic parameters of theory completely from number theory.

The condition for v_{0}=2^{-m}, m=11, allows to deduce the value of a as

a= (log(pK)/π) × (2^{2m}/K).

The condition that α_{K} is of order fine structure constant for p=M_{127}= 2^{127}-1 defining the p-adic length scale of electron indeed implies that m=11 is the only possible value since the value of a is scaled by a factor 4 in m→ m+1.

The value of α_{K} in the length scale L_{p0} in which condition of the first equation holds true is given by

1/α_{K}= 2^{21}/K≈ 106.379 .

** 2. What is the value of the preferred prime p _{0}?**

The condition for v_{0} can hold only for a single p-adic length scale L_{p0}. This correspondence would presumably mean that gravitational interaction is mediated along the space-time sheets characterized by p_{0}, or even that gravitons are characterized by p_{0}.

- If same p
_{0}characterizes all ordinary gauge bosons with their dark variants included, one would have p_{0}=M_{89}=2^{89}-1. - One can however argue that dark gravitons and dark bosons in general can correspond to different Mersenne prime than ordinary gauge bosons. Since Mersenne primes larger than M
_{127}define super-astrophysical length scales, M_{127}is the unique candidate. M_{127}indeed defines a dark length scale in TGD inspired quantum model of living matter. This predicts 1/α_{U(1)}(M_{127})= 106.379 to be compared with the experimental estimate 1/α_{U(1)}(M_{127})= 105.3531 deduced above. The deviation is smaller than one percent, which indeed puts bells ringing!

- The identification of Kähler coupling strength as U(1) coupling strength poses strong conditions on the p-adic length scale evolution of Weinberg angle using the knowledge about the evolution of the electromagnetic coupling constant. The condition
cos

^{2}(θ_{W})(89)= [log(M_{127}K)/log(M_{89}K)] × [α_{em}(M_{127})/α_{em}(M_{89})]× cos^{2}(θ_{W})(127) .Using the experimental value 1/α

_{em}(M_{89})≈ 128 as predicted by standard model one obtains sin^{2}(θ_{W})(89)=.0479. There is a bad conflict with experimental facts unless the experimentally determined value of Weinberg angle corresponds to M_{127}space-time sheet.I will leave leave the implications of this conflict to the future posting.

The reader interested in details is recommended to look previous postings and the new chapter Can TGD Predict the Spectrum of Planck Constants? of the book "TGD: an Overview" and the chapter TGD and Astrophysics of the book "Physics in Many-Sheeted Space-Time".

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