- Combinatorial - Pions
- Aether, Gravity, and the Higgs Mechanism
- Quaternionic
- Electron-Planck Mass-E6; and Universe-Neutron Stars-Octonions
- 23!

In the D4-D5-E6-E7 model, a Planck-massblack hole is not a tree-level classical particle such as an electronor a quark, but a quantum entity resulting from the Many-Worldsquantum sum over histories at a single point in spacetime.

Consider an isolated single point, or vertex in the latticepicture of spacetime. In the D4-D5-E6-E7 model, fermions live onvertices, and only first-generation fermions can live on a singlevertex. (The second-generation fermions live on two vertices that actat our energy levels very much like one, and The third-generationfermions live on three vertices that act at our energy levels verymuch like one.)

At a single spacetime vertex, a Planck-mass black hole is theMany-Worlds quantum sum of all possible virtual first-generationparticle-antiparticle fermion pairs permitted by the Pauli exclusionprinciple to live on that vertex.

Once a Planck-mass black hole is formed, it is stable in theD4-D5-E6-E7 model. Less mass would not be gravitationally bound atthe vertex. More mass at the vertex would decay by Hawkingradiation.

In the D4-D5-E6-E7 model, a Planck-mass black hole can be formedin two ways: either as the end product of Hawking radiation decay ofa larger black hole; or

by vacuum fluctuation, creating either a graviton4-pair virtual Planck-mass black hole or as part of a cosmologicalquantum conformal fluctuation that could create a newuniverse.

Since Dirac fermions in 4-dimensional spacetime can be massive(and are massive at low enough energies for the Higgs mechanism toact), the Planck mass in 4-dimensional spacetime is the sum of massesof all possible virtual first-generation particle-antiparticlefermion pairs permitted by the Pauli exclusion principle.

There are 8 fermion particles and 8 fermion antiparticles for atotal of 64 particle-antiparticle pairs. A typical combination shouldhave several quarks, several antiquarks, a few colorlessquark-antiquark pairs that would be equivalent to pions, and someleptons and antileptons.

Due to the Pauli exclusion principle, no fermion lepton or quarkcould be present at the vertex more than twice unless they are in theform of boson pions, colorless first-generation quark-antiquark pairsnot subject to the Pauli exclusion principle. Of the 64particle-antiparticle pairs, 12 are pions.

A typical combination should have about 6 pions.

If all the pions are independent, the typical combination shouldhave a mass of .14x6 GeV = 0.84 GeV. However, just as the pion massof .14 GeV is less than the sum of the masses of a quark and anantiquark, pairs of oppositely charged pions may form a bound stateof less mass than the sum of two pion masses. If such a bound stateof oppositely charged pions has a mass as small as .1 GeV, and if thetypical combination has one such pair and 4 other pions, then thetypical combination should have a mass in the range of 0.66 GeV.

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

Since each fermion particle has a corresponding antiparticle, aPlanck-mass Black Hole is neutral with respect to electric and colorcharges.

The value for the Planck mass given in the ParticleData Group's 1998 review is 1.221 x 10^19 GeV.

Perhaps a Planck-Mass BlackHole could be built using a pion laser.

The combinatorial basis of the Planck mass is related to theI Ching.

The Higgs Mechanism gives mass to theWeak Force gauge bosons, so that

The Weak Force Mass Factor, abour 150 GeV, is determined by theVacuum Expectation Value ofthe Higgs Mechanism, about 250 GeV,

The Mass Me of the smallest charged Elementary Particle, theFirst-Generation Fermion Electron ComptonRadius Vortex Particle, is about 0.5 MeV, so that

If the Higgs VEV gives the linear compressibility of the Aether,**the Gravitational VEV should be given by the 4-volumecompressibility of the Aether**, so that the Gravitational VEV isabout ( 5 x 10^5 )^4 Me = 6 x 10^22 Me = 3 x 10^22 MeV = 3 x 10^19GeV.

Since the Gravitational VEV shouldcorrespond to a pair of Planck Mass Black Holes,

The value for the Planck mass given in the ParticleData Group's 1998 review is 1.221 x 10^19 GeV.

Both physical Spacetime (MinkowskiRP1xS3 and Euclidean S4) and Internal Symmetry Space (CP2)have

From a discrete lattice point of view, the Quaternionic structureis represented by Integral Quaternions.

Integral Quaternions have a D4 lattice structure. Using the squarenorm of distance from the origin, the numberof vertices in each shell of the D4 lattice can be calculated. Ifyou look at shells whose square norm is a power of 2, you see thatall of them have exactly 24 vertices. Here are the number of verticesin a few of the layers of a D4 lattice:

Square Norm of Layer Number of Vertices 1 24 2 24 3 96 4 24 5 144 6 96 7 192 8 24 9 312 ... ... 127 3,072 128 24

The powers of 2 are adjacent to the Mersenne Primes, of the form2^k - 1 for prime k.

There is a sequence of Mersenne Primes of at least length 4:

Power of 2 Mersenne Prime 2^2 = 4 2^2 - 1 = 4-1 = 3 2^3 = 8 2^3 - 1 = 8-1 = 7 2^7 = 128 2^7 - 1 = 128-1 = 1272^127 = about 1.7 x 10^38 2^127 - 1 = about 1.7 x 10^38

(It is not known whether or not 2^(2^127 - 1) - 1 is prime.)

MattiPitkanen has studied relationships between PrimeNumbers and Physics. He has noticed that, since the PrimeQuaternions are the Integral Quaternions whose square norm isrational prime, and since the ordinary distance is the square root ofthe square norm,

if you look at the fourth Mersenne Prime in the sequence, M127 =2^127 - 1 = about 1.7 x 10^38, and the power of 2 to which it isadjacent, 2^127 = also about 1.7 x 10^38,

you see that its ordinary distance from the origin, the squareroot of the square norm,

is about **1.3 x 10^19**, which is a good estimate of the**ratio of the Planck mass to the Proton mass**, and is about 1.22x 10^19 GeV.

The value for the Planck mass given in the ParticleData Group's 1998 review is 1.221 x 10^19 GeV.

- 2^127 = 2^(128 - 1) is the number of subsets of a given 127 elements of a set of 128 elements;
- 2^128 is the square of 2^64;
- 128 is the dimension of the ZeroDivisor Algebra in which, according to of Charles Muses, you first find elements a such that a^2 = 0, and 127 is the number of elements of the algebra other than the identity 1;
- 128 is the dimension of the Cl(7) Clifford Algebra in which you first find spinors that are not pure, and 127 is the number of elements of the algebra other than the identity 1;
- 128 is the dimension of the even subalgebra Cl(7) of the Cl(8) Clifford algebra upon which the D4-D5-E6-E7 physics model is based, and which is the standard building block of Clifford Algebras by the decomposition formula Cl(8N) = Cl(8) x ... (N-fold tensor product) ... x Cl(8); and
- the Quaternionic Internal Symmetry Space CP2 is the characteristic manifold of the SU(3) Color Force, whose most stable bound particle is the Proton, while the Quaternionic Spacetime Euclidean S4 is the Euclidean characteristic manifold of Gravity, whose characteristic particle is the Planck-Mass Black Hole, which in the D4-D5-E6-E7 physics model is stable.

**is about 10^22 times**

**the Mass of the** smallest charged Elementary Particle, theFirst-Generation Fermion **Electron** Particle, about 0.5MeV.

For clarity of discussion, I am considering order-of-magnitudeestimates and ignoring factors like 2 in considering 0.5 MeV to beabout 1 MeV, etc., so I note that

**As to 72**:

In the D4-D5-E6-E7 physics model,the Lie algebra E6 describes the Gauge Bosons, the SpaceTime, and theFermion Particles and Antiparticles. E6is 78-dimensional, and can be constructed from a 72-vertex E6Root Vector Diagram in 6-dimensional Euclidean space.

**As to 24**:

In the D4-D5-E6-E7 physics model,you can see that E6 can be constructedusing the traceless Jordan Algebra J3(O)o. In the notation that Iuse in the 3x3 matrix representation below, O1, O2, and O3 are threeOctonions.

Re(O4) O1 O2 O1* Re(O5) O3 O2* O3* -

The off-diagonal dimensionality of J3(O)o is 24, thedimensionality of the LeechLattice. that is J3(O)o itself is 26-dimensional, as isthe Lorentx Leech Lattice. Suchlattices are useful in building codesand physics models.

**As to 8^8**:

8^8 is the set of all maps from 8 things to 8 things, and so isrelated to the Reflexivity ofOctonions.

**As to (8^8)^3**:

Since 8^8 is related to the Reflexivityof Octonions and three Octionions (off-diagonal matrix elementsof the traceless Jordan AlgebraJ3(O)o) are fundamental to the D4-D5-E6-E7physics model,

which may be its connection to the ratio of the Planck Mass to theElectron Mass.

Saul-Paul Sirag has noted that:

**is about** 10^22 = **(8^8)^3 times**

**the Mass of the smallest Black Hole Star**, about 10 solarmasses, or about 10^34 grams.

Three Octionions ( off-diagonal matrix elements of thetraceless Jordan Algebra J3(O)o ) are fundamental to theD4-D5-E6-E7 physics model.

can be regarded as a 3-dimensional Octonion lattice, with 720 ofits 3x240 + 3x16x240 + 3x16x16x240 = 196,560 units regarded as threesets of 240 vertices of three E8 lattices.

In the D4-D5-E6-E7 physics model,the 3 Octonion dimensions correspond to

- 8-dim spacetime plus
- 8 first generation fermion particles plus
- 8 first generation fermion antiparticles.

Split the 8-dimensional spacetime into 1 time dimension and 7space dimensions.

Then, for any Timeline, you have

- 7 space dimensions plus
- 8 fermion particle dimensions plus
- 8 fermion antiparticle dimensions

for a total of 23 space-fermion dimensions.

The lowest energy state for any Single tTmeline would have a zerotree-level mass neutrino (antineutrino) fermion particle(antiparticle).

Since second-generation and third-generation fermions decay, inthe D4-D5-E6-E7 physics model, intofirst-generation fermions by Standard Model processes; and since, inthe D4-D5-E6-E7 physics model,first-generation quarks decay intoelectrons or positrons plus neutinos or antineutrinos by VirtualBlack Holes; therefore: in the D4-D5-E6-E7physics model, the lowest energy fermion particle (antiparticle)is the electron (positron), which is stable, and the lowest non-zeroenergy state for any Single Timeline would have an electron(positron) fermion particle (antiparticle).

so that nothing could decay from it, then any of the 23space-fermion dimensions could play the role of any spatial dimensionor any fermion particle or antiparticle dimension. Therefore

each such state could have the stable minimum non-zero mass-energyof an electron or positron

so that

where me is the electron mass. Since 23! = 2.585 x 10^22 and me =0.51 x 10^(-3) GeV,

which is roughly consistent with the value for the Planck massgiven in the Particle Data Group's1998 review of 1.221 x 10^19 GeV.

Since each fermion particle has a corresponding antiparticle, sucha Planck-mass Black Hole is neutral with respect to electric andcolor charges because opposite charges would be represented equallyin the superposition.

The idea of looking at the number 23! came to me from studying thetopological model ofMarco Spaans. I do not necessarily agree with his physicalinterpretations, but he did use 23! as the ratio between the Planckmass and a mass related to the electron mass.

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