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6.5. Estimate of the Planck mass.

The Planck mass is the minimal nontrivial instanton mass for

the F4 model.

It is based on the last Hopf fibration S7 -> S15 -> S8 = OP1.

Physically, the minimal nontrivial F4 instanton corresponds to a black hole in 8-dimensional spacetime consisting of one spacelike point. From the point of view of sum-over-histories path integral quantum structure of the F4 model, all future paths from the one spacelike point must lead to the same spacelike point, so that the minimal Planck mass black hole consists of the sum of all possible virtual 8-dimensional particle- antiparticle fermion pairs permitted by the Pauli exclusion principle.

When 8-dimensional the F4 model is reduced to the 4-dimensional formulation of the F4 model, the 8-dimensional black hole goes to a 4-dimensional Spin(5) gravitational instanton minimal Planck mass black holes corresponding to the Hopf fibration

S3 -> S7 -> S4 = HP1.

Since Dirac fermions in 4-dimensional spacetime are massive, the Planck mass in 4-dimensional spacetime is the sum of masses of all possible virtual 8-dimensional particle-antiparticle fermion pairs permitted by the Pauli exclusion principle.

There are 8 fermion particles and 8 fermion antiparticles for a total of 64 particle-antiparticle pairs. A typical combination should have several quarks, several antiquarks, a few colorless quark-antiquark pairs that would be equivalent to pions, and some leptons and antileptons.

Due to the Pauli exclusion principle, no fermion lepton or quark could be present at the vertex more than twice unless they are in the form of boson pions, colorless first-generation quark-antiquark pairs not subject to the Pauli exclusion principle. Of the 64 particle-antiparticle pairs, 12 are pions.

A typical combination should have about 6 pions.

If all the pions are independent, the typical combination should have a mass of .14&endash;6 GeV = 0.84 GeV. However, just as the pion mass

of .14 GeV is less than the sum of the masses of a quark and an antiquark, pairs of oppositely charged pions may form a bound state of less mass than the sum of two pion masses. If such a bound state of oppositely charged pions has a mass as small as .1 GeV, and if the typical combination has one such pair and 4 other pions, then the typical combination should have a mass in the range of 0.66 GeV.

Summing over all 2^64 combinations, the total mass of a one-vertex universe should give mPlanck Å 1.217-1.550 x 10^19 GeV.

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