1. Preferred values of Planck constants and ruler and compass polygons
The starting point is that the scaling factor of M4 Planck constant is given by the integer n characterizing the quantum phase q= exp(iπ/n). The evolution in phase resolution in p-adic degrees of freedom corresponds to emergence of algebraic extensions allowing increasing variety of phases exp(iπ/n) expressible p-adically. This evolution can be assigned to the emergence of increasingly complex quantum phases and the increase of Planck constant.
One expects that quantum phases q=exp(iπ/n) which are expressible using only square roots of rationals are number theoretically very special since they correspond to algebraic extensions of p-adic numbers involving only square roots which should emerge first and therefore systems involving these values of q should be especially abundant in Nature.
These polygons are obtained by ruler and compass construction and Gauss showed that these polygons, which could be called Fermat polygons, have
nF= 2k ∏s Fns
sides/vertices: all Fermat primes Fns in this expression must be different. The analog of the p-adic length scale hypothesis emerges since larger Fermat primes are near a power of 2. The known Fermat primes Fn=22n+1 correspond to n=0,1,2,3,4 with F0=3, F1=5, F2=17, F3=257, F4=65537. It is not known whether there are higher Fermat primes. n=3,5,15-multiples of p-adic length scales clearly distinguishable from them are also predicted and this prediction is testable in living matter.
2. Application to planetary Bohr orbitology
The understanding of the quantization of Planck constants in M4 and CP2 degrees of freedom led to a considerable progress in the understanding of the Bohr orbit model of planetary orbits proposed by Nottale, whose TGD version initiated "the dark matter as macroscopic quantum phase with large Planck constant" program.
Gravitational Planck constant is given by
where an estimate for the value of v0 can be deduced from known masses of Sun and planets. This gives v0≈ 4.6× 10-4.
Combining this expression with the above derived expression one obtains
GMm/v0= nF= 2k ∏ns Fns
In practice only the Fermat primes 3,5,17 appearing in this formula can be distinguished from a power of 2 so that the resulting formula is extremely predictive. Consider now tests for this prediction.
- The first step is to look whether planetary mass ratios can be reproduced as ratios of Fermat primes of this kind. This turns out to be the case if Nottale's proposal for quantization in which outer planets correspond to v0/5: TGD provides a mechanism explaining this modification of v0. The accuracy is better than 10 per cent.
- Second step is to look whether GMm/v0 for say Earth allows the expression above. It turns out that there is discrepancy: allowing second power of 17 in the formula one obtains an excellent fit. Only first power is allowed. Something goes wrong! 16 is the nearest power of two available and gives for v0 the value 2-11 deduced from biological applications and consistent with p-adic length scale hypothesis. Amusingly, v0(exp)= 4.6 × 10-4 equals with 1/(27× F2)= 4.5956× 10-4 within the experimental accuracy.
A possible solution of the discrepancy is that the empirical estimate for the factor GMm/v0 is too large since m contains also the the visible mass not actually contributing to the gravitational force between dark matter objects. M is known correctly from the knowledge of gravitational field of Sun. The assumption that the dark mass is a fraction 1/(1+ε) of the total mass for Earth gives 1+ε= 17/16 in an excellent approximation. This gives for the fraction of the visible matter the estimate ε=1/16≈ 6 per cent. The estimate for the fraction of visible matter in cosmos is about 4 per cent so that estimate is reasonable and would mean that most of planetary and solar mass would be also dark as TGD indeed predicts and for which there are already now several experimental evidence (consider only the evidence that photosphere has solid surface discussed earlier in this blog ).
To sum up, it seems that everything is now ready for the great revolution. I would be happy to share this flood of discoveries with colleagues but all depends on what establishment decides. To my humble opinion twenty one years in a theoretical desert should be enough for even the most arrogant theorist. There is now a book of 800 A4 pages about TGD at Amazon: Topological Geometrodynamics so that it is much easier to learn what TGD is about.
The reader interested in details is recommended to look at the chapter Was von Neumann Right After All? of the book "TGD: an Overview"and the chapter TGD and Astrophysics of the book "Classical Physics in Many-Sheeted Space-Time".