Hi Dr. Cosmic Ray
Thank you for reading and for asking very pertinent questions.
For your first question, a very important one: how comes E8 from D4, as lattices ?
The answer is illustrated by Figure 4b
You see here a fermion (in Lisi's base) from one family, encoded by two triplets of bits:
One gives a number in 0-7 to select one of 8 out-coming short links (of length 2),
The other selects one of the 8 out-coming long links (of length 2*square(2)). A long links is (1+i)*the short link, when each of 24 external leaves is connected through a short and a long links to external leaves of neighbor modules. So we encode here 64 fermions by 64 E8 roots.
Just by rotating the diagram (the three central bits) by 1/3 and 2/3 of circle you get two other families. (thats what I call 'elegance')
Triality reduces to 1/3 circle rotation, and 1/2(-1+i+j+k) quaternionic product.
Now you have 192 of 240 roots, corresponding to fermions of standard model.
So E8 lattice is twice projected to D4, one of the projection being "twisted" as you said.
This twisting is the (1+i) quaternionic multiplication, and geometrically is two 45° rotations in two perpendicular planes and one square(2) scaling
One D4 is the gluonic part of the fermion
The other is the higgsonic. (in [math]\left\{ \frac{1}{4}\omega _L^3, \frac{1}{4}\omega _R^3,\frac{1}{2} W^3,\frac{1}{2} B_{1 }^3\right\}[/math] root system)
Positive weak force boson W+ has quantum coordinates {0,0,+2,0}, represented by quaternion 2j , and its opposite, -2j, for its anti-boson W-.
Left chiral (self-dual) part of the spin connection is represented by 2. Right chiral (anti self-dual) part is represented by 2i.
The 16 hypercubic vertices of the radius 2 24-cell, with coordinates ,each +1 or -1 are 16 frame-higgs bosons, therefore we name this sector "higgsonic".
On this basis, fermions coordinates are permutations of {2a,2b,0,0} where a and b are 1 or -1.
Due to the fact that the sixth element, the icositetrachoron (or 24-cell) is self dual, both D4 merges in the same lattice.
For your second question, my sixth element is not a tachyon because it is only the geometric structure of the proto-space as a 4D crystal.
The structure (as a trivalent subgraph) of a cell of empty space at Planck scale, illustrated in Fig 4a, is the cell of this crystal, and hold symmetries and geometry of the sixth element, the 24-cell.
As you explain in your paper how fermion are vertices of a direct lattice and bosons are struts of the dual lattice, my hyperdiamond lattice represent F4 lattice and its dual (itseff) in same lattice, because of self duality of 24-cell (with a square(2) factor between both). Then the figure has one more step of complexity. Two half-fermions are symmetrically encoded with bosonic part in the dual. So we have bosonic higgsonic sector, and bosonic gluonic sector in its dua. And fermionic has two halves (of the E8 representation), one in dual of bosonic higgsonic sector, and the second half in dual of bosonic gluonic sector.
By this innovative way of thinking, my theory escapes to mathematical issue which were found by Jacques Distler in Lisi's theory.
My model allow an explicit construction of the LQG gravitationnal field (the tetrad), as a field of four quaternions which values are implicitly encoded by the local topology.
The connection and the metric tensor both derives from the tetrad (and its derivatives, that can be defined along the supporting spin network, whom nodes are getting a 4D position field also implicitly defined)
In this context, E8 may be enough, and no need to go to E8xE8...
Its difficult to explain all my theory in a 10 pages paper, and I m working on a more extensive demonstration.
All the best,
Ray
