In the D4-D5-E6 and HyperDiamond Feynman Checkerboard Models: 0. Below the Planck length, or above the Planck energy, the HyperDiamond structure of spacetime breaks down. 1. Either discrete or continuous structures can be used above Planck length. Discrete are used at Planck length. 2. Either compact or noncompact structures can be used. 3. Antisymmetric representations of the D4 Weyl group give the Lie algebra Spin(0,8) and its Clifford algebra. 4. 28 generators of Spin(0,8) are gauge bosons. 5. 8-dim vector representation space is spacetime. 6. 8-dim +/-halfspinor row left and right ideals give Dirac operator. 7. 8-dim +halfspinor column left ideal gives fermion spinor particles. 8. 8-dim -halfspinor column right ideal gives antiparticles. 9. 1-dim scalar representation gives Higgs scalar. 11. classical Lagrangian in 8-dim with first generation fermions is constructed from global E6 symmetry and octonion structure of the Spin(0,8) representations. 12. Second and third generation fermions come from global E7 and E8 symmetry and produce CP symmetry violation. 13. 8-dim E8 HyperDiamond lattice spacetime splits into an associative 4-dim HyperDiamond lattice physical spacetime plus a coassociative 4-dim HyperDiamond lattice internal symmetry space. 14. quantum 4-dim Lagrangian is defined by Many-Worlds sum over Feynman Checkerboard path histories. 15. Dimensional reduction converts Spin(0,8) gauge bosons into gauge bosons for the standard model (with Higgs) plus gravity. 16. Symmetric representations of Weyl groups give Casimir operators, Cohomology, and BRST transformations. 17. quantum theory can be done using BRST transformations.

The breakdown of the 28 Spin(0,8) gauge bosons can be seen by looking at reduction of the Clifford Algebra Cl(0,8) of Spin(0,8) to get SU(3)xSU(2)xU(1) plus Higgs and Gravity

Click on this Picture to read about the Spin(0,8) Clifford Algebra:

Here is

Kobayshi-Maskawa parameters come from quark constituent masses and lepton masses.

d s b u 0.975 0.222 0.00249 - 0.00388 i c -0.222 0.974 0.0423 -0.000161 i -0.0000365 i t 0.00698 -0.0418 0.999 -0.00378 i -0.001086 i

The Kobayashi-Maskawa parameters were calculated from the following

tree-level lepton masses and quark constituent masses:

Me = 0.5110 MeV (assumed);

Me-neutrino = Mmu-neutrino = Mtau-neutrino = 0;

Md = Mu = 312.8 MeV (constituent quark mass);

Ms = 625 MeV (constituent quark mass);

Mc = 2.09 GeV (constituent quark mass);

Mb = 5.63 GeV (constituent quark mass);

Mt = 129.5155 (rounded to 130) GeV (constituent quark mass).

according to the following formulas:

phase angle d13 = 1 radian

sin(a) = [Me+3Mu+3Md] /Sqrt( [Me^2+3Mu^2+3Md^2] + [Mmu^2+3Mc^2+3Ms^2] )

sin(b) = [Me+3Mu+3Md] /Sqrt( [Me^2+3Mu^2+3Md^2] + [Mtau^2+3Mt^2+3Mb^2] )

sin(g') = [Mmu+3Mc+3Ms] /Sqrt( [Mmu^2+3Mc^2+3Ms^2] + [Mtau^2+3Mt^2+3Mb^2] )

sin(g) = sin(g') Sqrt( [Mmu+3Mc+3Ms] / [Me+3Mu+3Md] )

As calculated in the HyperDiamond Feynman Checkerboard model:

tree-level weak boson masses are

Mw+ = Mw- = 80.326 GeV and Mz0 = 91.862 GeV;

Higgs field Vacuum Expectation Value = 252.514 GeV;

electromagnetic fine structure constant = 1/137.03608;

weak force Gfermi = (Gweak)(Mproton)^2 = 1.045 x 10^(-5);

color force strength = 0.6286 (at 0.245 GeV);

color force strength = 0.167 (at 5.3 GeV);

color force strength = 0.121 (at 34 GeV); and

color force strength = 0.106 (at 91 GeV)

Color force strength calculation used Perturbative QCD.

If Nonperturbative QCD is taken into account, then

color force strength = 0.123 (at 91 GeV)

Pions and Protons are 4-D versions of: Sine-Gordon Breathers made up of quark-antiquark pairs; and Nontopological Solitons similar to tHooft-Polyakov monopoles made up of rgb quark triples.

In this model, the quark constituent masses are fundamental.

The Planck mass is the mass of a condensate of pions at one point.

Some thoughts about cohomology, quadrics, and dimensional reduction.

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