# Identition Zones (36-102)

Identition is defined for a complex operation by extending one of the definitions of the exponential function from real exponents to complex exponents.

This section is referring to *wiki page-27* of *main section-5* that is *inherited * from *the spin section-3* by *prime spin-36* and *span-167* with *the partitions* as below.

- Theory of Everything (span 12)
- Everything is Connected (span 11)
- Truncated Perturbation (span 10)
- Quadratic Polynomials (span 9)
- Fundamental Forces (span 8)
- Elementary Particles (span 7)
- Basic Transformation (span 6)
- Hidden Dimensions (span 5)
- Parallel Universes (span 4)
- Vibrating Strings (span 3)
- Series Expansion (span 2)
- Wormhole Theory (span 1)

This identition zones stands as one of the solution to deal with the residual primes that is occured in the exponentation zones to become compactifiable within *the base unit*.

## Basic Concept

In physics, string theory is a theoretical framework in which the point-like particles of particle physics are replaced by ** one-dimensional objects** called strings.

The concept of eleven dimensions is a theoretical one in physics and cosmology, specifically in the realm of string theory and M-theory.

.*These theories propose that our observable universe is made up of 11 dimensions, rather than the traditional three dimensions of length, width, and height, and the fourth dimension of time*- The additional dimensions are thought to be compactified or curled up, meaning that they are not directly observable by us in our everyday experience.
.*As for the cosmic philosophy, it is important to note that these theories are still considered speculative and have not been proven through experimental evidence*- However, they do offer a new perspective on the nature of our universe and the
*fundamental forces*that govern it. - Some scientists and philosophers argue that these theories may provide new insights into the origins of the universe and the nature of reality itself.

Ultimately, the concept of eleven dimensions is a fascinating area of study that continues to inspire new research and discoveries in the field of physics and cosmology. *(ChatGPT)*

Our physical space is observed to have only three large dimensions and taken together with time as the fourth dimension, a physical theory must take this into account.

** It is argued, among other things, that eleven-dimensional supergravity arises as a low energy limit of the ten-dimensional Type IIA superstring**, and that a recently conjectured duality between the heterotic string and Type IIA superstrings controls the strong coupling dynamics of the heterotic string in five, six, and seven dimensions and implies S-duality for both heterotic and Type II strings.

*(String Theory - Pdf)*

String theory, superstring theory, or M-theory, or some other variant on this theme is one of the Unsolved Problem in physic as a step road to a *Theory Of Everything (TOE)*.

Nothing prevents a theory from including more than 4 dimensions. In the case of string theory, consistency requires spacetime to have 10, 11 or 26 dimensions. ** The conflict between observation and theory is resolved by making the unobserved dimensions compactified**.

*(Astrophysics Research)*

The string theory is sofar the leading candidate to the *TOE* however it is said that the theory ** may be incompatible with dark energy**.

It is argued that the generic formulation of string theory leads naturally to dark energy, represented by a positive cosmological constant to lowest order and the intrinsic stringy non-commutativity is the new crucial ingredient responsible for its radiative stability. *(Physic Letters)*

When combined into the web of dualities, five string theories become a single 11-dimensional M-theory, encoded in dynamics of ** M2 and M5 branes**.

There are several open questions that need to be addressed to convert the model studied here into a realistic theory.

- First and foremost, one must find a dynamical mechanism for driving the compactification radius φ to unity to produce a small cosmological constant. Similar issue is present in the usual Kaluza–Klein scenarios where one needs to provide a mechanism for
*spontaneous compactification*. We note, however, that the situation in theory (4) is somewhat better than in the usual KK setup. In the latter case, apart from the case of compactification on S1, the pure gravity theory in 4 + D dimensions usually does not have solutions of the form of the product of Minkowski spacetime and (compact) internal manifolds. For this reason one usually extends the pure gravity theory in 4 + D dimensions with extra fields, e.g. by considering the Einstein–Yang–Mills system. The stress–energy tensor of these extra fields then allows for solutions of the required product form, see e.g. [20], Section 3. Probably the most famous compactification mechanism is that due to Freund and Rubin [21], where the 3-form field of the 11D supergravity is doing the job. In contrast, the theory (4) admits the solution that is the S3 fibration over S4, see [14] for an explicit description. Thus, at least there is a solution of (4) of the desired type without having to introduce extra fields. However, the cosmological constant for the S3 fibration over S4 solution is too large, see [14]. This is similar to the situation with the Freund–Rubin solution. Thus, a compactification mechanism that would result in an appropriately small cosmological constant is a very serious open issue for our setup. It is possible that the only way forward is to add other fields. We then remark that there is a very natural extension of the theory (4) that adds forms of all odd degrees. This is the theory that appeared in [12], formula (29). It would be interesting to study 4D compactifications of this more general theory. We hope to analyse this in the future. - Another open problem of the present approach is that of coupling to matter. Again, a natural way to proceed is suggested by supergravity. One does not couple supergravity to extra fields, one simply studies what the modes already present become when viewed from the 4D perspective. In particular, when compactifying on a coset manifold all modes related to isometries of the internal space are known to be important. Indeed, recall that the gauge group that arises in the KK compactification is the group of isometries of the internal manifold, and its dimension may be larger than the dimension of the internal space itself. In this paper we have considered a compactification on a group manifold, but only retained half of the relevant isometries by considering the invariant dimensional reduction ansatz. It is clear that additional fields will arise by enlarging the ansatz by taking into account all the isometries. In this case, however, one must be careful about
*the issue of consistent truncation*, see e.g. [22] for a clear description of all the issues arising. We leave a study of the dimensional reduction on S3 viewed as a coset S3 = SO(4)/SO(3) to future research. - Third, there is a question of how to describe Lorentzian signature metrics using this formalism. To do this one must make the 3-form C complex-valued, and then impose some appropriate reality conditions. Similar issues exist in all Plebanski-related formulations. We postpone their resolution to future work.Finally, to avoid confusion, we would like to say that our present use of G2 structures (3-forms in 7D) is different from what one can find in the literature on Kaluza–Klein compactifications of supergravity.

In our approach a 3-form is not an object that exist in addition to the metric — it is the only object that exist. The metric, and in particular the 4D metric, is defined by the 3-forvia (2). Also, in the supergravity context a 7D manifold with a G2 structure is used for compactifying the 11D supergravity down to 4D. In contrast, we compactify from 7D to 4D. *(General relativity from three-forms in seven dimensions - pdf)*

There are thought to be ** 24 separate quantum fields** that permit the universe. It consists of 12 various fundamental forces including mass, 9 quarks, and 3 leptons.

String Theory which states there could be 11 dimensions (** 9 dimensions of space, 1 dimension of time, and 1 dimension for other universes**) - the diagram below can sum it up for the 9 dimensions of space. Then the Cosmos would be the 11th dimension where (+/-) Binary Universes are born from Nothingness. Where Nothingness = 0 = (+) universe of regular matter and (-) universe of dark matter.

*(Quora)*

This generated a lot of interest in the approach and eventually led to the Loop Quantum Gravity (LQG). You may find that the rest of topics will concern mainly to this matter.

Numerous connections have been observed between some, though not all, of these exceptional objects. Most common are objects related to 8 and 24 dimensions, noting that 24 = 8 · 3. By contrast, the pariah groups stand apart, as the name suggests. Exceptional objects related to the number 8 include the following.

- The octonions are 8-dimensional. The E8 lattice can be realized as the integral octonions (up to a scale factor).The exceptional Lie groups can be seen as symmetries of the octonions and structures derived from the octonions;[19] further, the E8 algebra is related to the E8 lattice, as the notation implies (the lattice is generated by the root system of the algebra).
- Triality occurs for Spin(8), which also connects to 8 · 3 = 24.Likewise, exceptional objects related to the number 24 include The Leech lattice is 24-dimensional.Most sporadic simple groups can be related to the Leech lattice, or more broadly the Monster.The exceptional Jordan algebra has a representation in terms of 24×24 real matrices together with the Jordan product rule.These objects are connected to various other phenomena in math which may be considered surprising but not themselves “exceptional”. For example, in algebraic topology, 8-fold real Bott periodicity can be seen as coming from the octonions. In the theory of modular forms, the 24-dimensional nature of the Leech lattice underlies the presence of 24 in the formulas for the Dedekind eta function and the modular discriminant, which connection is deepened by Monstrous moonshine, a development that related modular functions to the Monster group.

In string theory and superstring theory we often find that particular dimensions are singled out as a result of exceptional algebraic phenomena. For example, bosonic string theory requires a spacetime of dimension 26 which is directly related to the presence of 24 in the Dedekind eta function. Similarly, the possible dimensions of supergravity are related to the dimensions of the division algebras. *(Wikipedia)*

By taking the correlation of these 11 partitions with the logical sequence of numbers there would be a *series expansion*.

## Series Expansion

Remember we must sum over all the quantum numbers of the quarks so the cross section is multiplied by Number of colours, Nc.

Finally NG′ is the number of parameters of the group G′, the subgroup of G still unbroken by the flavour matrices.

- In this case, G′ corresponds to two U(1) symmetries, baryon number conservation and lepton number conservation and therefore NG′ = 2.
- Furthermore Eq. (79) can be applied separately to phases and moduli. In this way, and taking into account that a U(N) matrix contains n(n − 1)/2 moduli and n(n + 1)/2 phases.
- It is straightforward to obtain that we have, Nph = 69 − 5 × 6 + 2 = 41 phases and
.*Nmod = 84 − 5 × 3 = 69 moduli in the flavour sector* - This amounts to a total of 123 parameters in the model4, out of which 44 are CP violating phases!!

As we know, in the SM, there is only one observable CP violating phase, the CKM phase, and therefore we have here ** 43 new phases, 40 in the flavour sector and three in the flavour independent sector**.

*(Flavour Physics and Grand Unification - pdf)*

We discuss the phenomenology of doubly and singly charged Higgs bosons (of SU(2) L-triplet fields) in the simplest A 4-symmetric version of ** the Higgs Triplet Model**.

In this article, we investigated the ** phenomenology of triplet Higgs bosons in the simplest A4-symmetric version** of the Higgs Triplet Model (A4HTM). The A4HTM is a four-Higgs- Triplet-Model (δ of 1 and (∆x, ∆y, ∆z) of 3).

- Four mass eigenstates of doubly charged Higgs bosons, H±±i, are obtained explicitly from the Higgs potential.
- We also obtained four mass eigenstates of the triplet-like singly charged Higgs bosons, H±T i, for which doublet components can be ignored because of small triplet vev’s.
- It was shown that the A4HTM gives unique predictions about their decay branching ratios into two leptons (H−−i → ℓℓ′ and H−iT → ℓν); for example, the leptonic decays of H−−2 are only into µµ and eτ because an approximate Z3 symmetry remains, and the ratio of the branching ratios is 2 : 1 as a consequence of the A4 symmetry in the original Lagrangian.
- Therefore, it will be possible to test the model at hadron colliders (Tevatron and LHC) if some of these Higgs bosons are light enough to be produced.
- Even if these Higgs bosons are too heavy to be produced at hadron colliders, they can affect the lepton flavor violating decays of charged leptons if the triplet Yukawa coupling constants are large enough.
- It was shown that there is no contribution of these Higgs bosonsto µ → eee ¯ and ℓ → ℓ′γ.
- Thus, we can naturally expect signals of τ → µee and τ → eµµ(which are possible in this model among six τ → ℓℓ′ℓ′′) in the future in collider experiments (Super-KEKB, super B factory, super flavor factory, and LHCb) without interfering with a stringent experimental bound on µ → eee ¯ . This model will be excluded if ℓ → ℓ ′γ is observed.

We considered current experimental constraints on the model and prospects of the measurement of the non-standard neutrino interactions (NSI) in the neutrino factory. If H±±2 or H±±3 is lighter enough than other H±±i, effects of the NSI can be around the expected sensitivity in the neutrino factory. *(Triplet Higgs bosons - pdf)*

This approach shows that there are actually four copies of the tri-rectified Coxeter-Dynkin diagram of H4, promises to open the door to as yet unexplored E8-based GUTs.

There are 28 octonion Fano plane triangles that correspond directly to the 28 Trott quartic curve bitangents.

- These bitangents are directly related to the Legendre functions used in the Shroedinger spherical harmonic electron orbital probability densities.
- Shown below is a graphic of these overlaid onto the n=5, l=2, m=1 element, which is assigned to gold (Au).
- When using an algorithm based on the E8 positive algebra root assignments, the “flipped” Fano plane has E8 algebra root number 79 (the atomic number of Au) and split real even group number of 228 (in Clifford/Pascal triangle order).
- This matrix is shown to be useful in providing direct relationships between E8 and the lower dimensional Dynkin and Coxeter-Dynkin geometries contained within it, geometries that are visualized in the form of real and virtual 3 dimensional objects.
- A direct linkage between E8, the folding matrix, fundamental physics particles in an extended Standard Model Gravi GUT, quaternions, and octonions is introduced, and its importance is investigated and described.
- E8 and its 4D children, the 600-cell and 120-cell (pages on which I have some work, amongst others) and its grandkids (2 of the 3D 5 Platonic Solids, one of which is the 3D version of the 2D Pentagon) are all related to the Fibonacci numbers and the Golden Ratio.
- And finally, the {7, 8} dimensions in physics can be identified with quark color, as {7} preserves the blue quark positions, while {8} moves the
while preserving their relative positions within the rings. It is interesting t note that the dimensions {6, 7, 8} are appropriately labeled {r, g, b} in SRE coordinates, since in this projection the SRE math coordinates are located at the afforementioned*dual concentric rings of quarks*{r, g, ¯ g, b, ¯ ¯b} concentric rings (the intersection of the gluons triality lines)*6 triple overlap points at center of the quark’s*

So that kind of explains why most of my 2D art, 3D objects and sculptures (e.g. furniture like the dodecahedron table below), and 4D youtube animations all use the Golden Ratio theme. *(E8 to H4 folding matrix - pdf)*

The number 28, aside from being triangular wave of perfect pyramid, is the sum of the ** first 5 primes** and the sum of the

**.**

*first 7 natural numbers*```
$True Prime Pairs:
(5,7), (11,13), (17,19)
layer | node | sub | i | f. MEC30/2
------+------+-----+-----+------ ‹--------------- 0 {-1/2}
| | | 1 | --------------------------
| | 1 +-----+ |
| 1 | | 2 | (5) |
| |-----+-----+ |
| | | 3 | |
1 +------+ 2 +-----+---- |
| | | 4 | |
| +-----+-----+ |
| 2 | | 5 | (7) |
| | 3 +-----+ |
| | | 6 | 11s ‹-- ∆28 = 71-43 ✔️
------+------+-----+-----+------ } (36) |
| | | 7 | |
| | 4 +-----+ |
| 3 | | 8 | (11) |
| +-----+-----+ |
| | | 9 |‹-- ∆9 = (89-71) / 2 √ |
2 +------| 5* +-----+----- |
| | | 10 | |
| |-----+-----+ |
| 4 | | 11 | (13) ---------------------
| | 6 +-----+ ‹--- vacuum energy ‹--- ∆60 ‹--- 15 {zero axis}
| | | 12 |---------------------------
------+------+-----+-----+------------ |
| | | 13 | |
| | 7 +-----+ |
| 5 | | 14 | (17) |
| |-----+-----+ |
| | | 15 | 7s ‹-- ∆24
3* +------+ 8 +-----+----- } (36) |
| | | 16 | |
| |-----+-----+ |
| 6 | | 17 | (19) |
| | 9 +-----+ |
| | | 18 | --------------------------
------|------|-----+-----+----- ‹-------------------- 30 {+1/2}
```

The simplest group is SU(5), which we will consider here, other examples include SO(10). SU(5) has `5²−1 = 24`

generators which means there are 24 gauge bosons.

It is known that the recently reported shift of ***the W boson mass can be easily explained by an SU(2)L triplet Higgs boson*”” with a zero hypercharge if it obtains a vacuum expectation value (VEV) of O(1) GeV.

- Surprisingly, the addition of a TeV scale complex triplet Higgs boson to the standard model (SM)
at around 10¹⁴GeV.*leads to a precise unification of the gauge couplings* - We consider that it is a consequence of SU(5) grand unification and show a possible potential for the Higgs fields yielding a weak scale complex SU(2) triplet scalar boson.
- Although it seems the proton decay constraint would doom such a low-scale unification, we show that the constraint can be avoided by introducing
which mix with the SM fermions through mass terms involving the VEV of GUT breaking Higgs field.*vector-like fermions*

Importantly, the simplest viable model only requires the addition of one pair of vector-like fermions transforming 10 and 10. *(W boson mass anomaly and grand unification - pdf)*

*168 + 329 + 289 - 619 - 30 - 30 - 5 = 786 - 619 - 65 = 102*

Since SU(5) has 24 generators, SU(5) GUTs have 12 new gauge bosons known as Xbosons (or X/Y bosons) in addition to the SM.

## Hidden Dimensions

Consider the following 13x13 square divided into two triangles and two quadrilateral polygons. If the four pieces are restructured in the form of a rectangle, it appears that ** the overall area has inexplicably lost one unit!** What has happened?

The particle spectrum is completed by the Higgs particles required to give masses to fermions as wellas to break the GUT symmetry.

- Georgi and Glashow have chosen the SU(5) where a single gauge coupling constant is manifestly incorporated.
- As has been discussed in the introduction, the SM gauge group has a rank four and the simple groups which contain complex representations of rank four are just
.*SU(3) × SU(3) and SU(5)* - Further, the fermions of the Standard Model can be arranged in terms of the fundamental ¯5 and the anti-symmetric 10 representation of the
.*SU(5) [30]* - To begin with, let us study the fermion masses in the prototype SU(5).Given that
*fermions are in 5 and 10 representations* - We conclude that the scalars that form Yukawa couplings are:
- It is easy to check that this combination of the representations is anomaly free. The gauge theory of SU(5) contains
*24 gauge bosons*. - They are decomposed in terms of the standard model gauge group SU(3) × SU(2) × U(1) as: 24 = (8, 1) + (1, 3) + (1, 1) + (3, 2) + (¯3, 2) (10)
- The first component represents the gluon fields (G) mediating the colour, the second one corresponds to the Standard Model SU(2) mediators (W) and the third component corresponds to the U(1) mediator (B).
- The fourth and fifth components carry both colour as well as the SU(2) indices and are called the X and gauge bosons. Schematically, the gauge bosons can be represented in terms of the 5 × 5 matrix:

Notice that in this case the couplings of the triplets to the fermions is not related to the fermion massesas the Higgs triplets are now a mixing between the triplets in the 5H and the triplets in the 50. Thereforewe have some ** unknown Yukawa coupling Y50**.

*(Flavour Physics and Grand Unification - pdf)*

```
$True Prime Pairs:
(5,7), (11,13), (17,19)
layer | node | sub | i | f. MEC30/2
------+------+-----+-----+------ ‹--------------- 0 {-1/2}
| | | 1 | --------------------------
| | 1 +-----+ |
| 1 | | 2 | (5) |
| |-----+-----+ |
| | | 3 | |
1 +------+ 2 +-----+---- |
| | | 4 | |
| +-----+-----+ |
| 2 | | 5 | (7) |
| | 3 +-----+ |
| | | 6 | 11s ‹-- ∆28
------+------+-----+-----+------ } (36) |
| | | 7 | |
| | 4 +-----+ |
| 3 | | 8 | (11) |
| +-----+-----+ |
| | | 9 |‹-- ∆9 = (89-71) / 2 √ |
2 +------| 5* +-----+----- |
| | | 10 | |
| |-----+-----+ |
| 4 | | 11 | (13) ---------------------
| | 6 +-----+ ‹--- vacuum energy ‹--- ∆60 ‹--- 15 {zero axis}
| | | 12 |---------------------------
------+------+-----+-----+------------ |
| | | 13 | |
| | 7 +-----+ |
| 5 | | 14 | (17) |
| |-----+-----+ |
| | | 15 | 7s ‹-- ∆24 = 43-19 ✔️
3* +------+ 8 +-----+----- } (36) |
| | | 16 | |
| |-----+-----+ |
| 6 | | 17 | (19) |
| | 9 +-----+ |
| | | 18 | --------------------------
------|------|-----+-----+----- ‹-------------------- 30 {+1/2}
```

Each of the 6 columns has ** 8 bilateral 360 sums**, tor a total of 48 * 360 = 40 * 432. This number 432 plays significant roles on the

**.**

*Interchange Layers*In this article I am going to introduce the main results of a new theory of elemetary particle physics developed by the engineer M.S. El Nachie.

- This theory provides a fractal model of quantum space-time, the so-called E-infinity space, that allows the precise determination of the mass-energy of most elementary particles -and much more- in close agreement with their experimental values.
- The Golden Ratio emerges naturally in this theory, and
that connects the fractal dimension of quantum space-time with the mass-energy of every fundamental particle, and also with several fundamental physical quantities such as the Fine Structure constant.*turns out to be the central piece* - El Nachie has been severely criticised by his non-orthodoxal publication methods -he uses to publish his papers in a Journal where he is the editor in chief. Despite this fact, I think that his theory deserves consideration so I will try to summarize it in the lines that follow.
- The intervention of the Golden Ratio can be seen as a way to enter the quantum world, the world of subtle vibrations, in which we observe increasing energy levels as we move to smaller and smaller scales.
- El Nachie has proposed a way of calculating the fractal dimension of quantum space-time. The resulting value (Figure 7) suggests that the quantum world is composed of an infinite number or scaled copies of our ordinary 4-dimensional space-time.
- Setting k=0 one obtains the classical dimensions of
, namely 26, 16, 10, 6 and 4, as well as the constant of super-symmetric (αgs=26) and*heterotic superstring theory*.*non super-symmetric (αg=42) unification of all fundamental forces*

As we have seen in section 2, the above is a Fibonacci-like sequence with a very concise geometrical interpetation related to numbers ** 5, 11 and φ**.

*(Phi in Particle Physics)*

Assigning a specific mass, length, time, and charge metrics based on new dimensional relationships and the Planck constant (** which defines Higgs mass**).

The discovery of neutrino oscillations indicates that the Standard Model is incomplete, but there is currently no clear evidence that nature is described by any *Grand Unified Theory*. Neutrino oscillations have led to renewed interest toward certain GUT such as *SO(10)*. *(Wikipedia)*

The evolution of a spin foam, has a scale above the Planck length. Consequently, not just matter, but space itself, prefers an atomic structure.

Spin networks constitute a basis that minimize the degree of over-completeness of the loop basis, and ** for trivalent intersections eliminate it entirely**.

- The edges are labelled by spins together with `intertwiners’ at the vertices which are prescription for how to sum over different ways the spins are rerouted.
- The sum over rerouting are chosen as such to make the form of the intertwiner invariant under Gauss gauge transformations.

Some of these relations are rooted in a relation to superstring theory and quantum gravity which is directly related to the quantization of general relativity. *(Wikipedia)*

These include generating variants of their abundance profile, assigning taxonomy and finally generating a rooted phylogenetic tree for the Standard Model.

This paper seeks to examine several extended SUSY Yang-Mills Theories on the 0-Brane by obtaining the L and R matrices, generate the corresponding adinkra, and studying their correlators.

- The transformation laws of the on-shell 10D, N=1 Super Yang-Mills Theory are given, and the SUSY algebra is shown to exhibit closure when the equations of motion are satisfied.
- The closure of the algebra for the 4D N=4 theory was calculated using new computational methods.

The resulting adinkra matrices and SUSY algebra structure are investigated for these theories, and from this comparisons are made.

Supersymmetry (SUSY) is a space-time symmetry which relates fermions and bosons. It predicts superpartners for every known particle with identical quantum numbers except the spin which differs by 1/2 and thus offers a solution to several open problems of the standard model (SM).

- As no superpartners with SM mass has been observed, SUSY must be broken. The Minimal Supersymmetric Standard Model (MSSM) with the most general SUSY breaking potential adds more than 100 new parameters.
- To decrease the number of parameters, specific SUSY breaking scenarios are considered assuming that spontaneous symmetry breaking in a hidden sector is mediated by some interaction to the visible sector.

When the mediators are gauge interactions, we arrive to Gauge Mediated Supersymmetry Breaking models (GMSB, 5 parameters) or to its generalization, General Gauge Mediation (GGM, 8 parameters)

## Supersymmetry

In particle physics, study of the symmetry and its breaking play very important role in order to get useful information about the nature.

In this paper, we have extended our previous discussions about using HYMNs (height-yielding matrix numbers) which are the eigenvalues [14] of functions of the adjacency matrices associated with the L-matrics and R-matrices derived from adinkras. *(Properties of HYMNs - pdf)*

In order to generate an adinkra, we must first describe certain transformation laws (following 0-Brane reduction) as a set of vectors, from which these vectors are thought of as matrices. Only then may we obtain the L and R matrices, which we use to generate adinkras. The adinkra that is generated from a set of adinkra matrices in Super Yang-Mills Theory is shown below

In the forty years since 11D on-shell supergravity theory was constructed in 1978, a lot of efforts have been made to understand supergravity in superspace.

Inspired by the history of how Einstein constructed ** General Relativity**, we study the linearized Nordstrom supergravity in

*10- and 11-dimensional superspaces*.

- Valise adinkras, although an important subclass, do not encode all information present when a 4D supermultiplet is reduced to 1D. We extend this to non-valise adinkras providing a complete
via*eigenvalue classification**Python code*. - We found no obstacles to applying the lessons we learned in
*4D to higher dimensions*. We also derive infinitesimal 10D superspace Weyl transformation laws. The identification of all off-shell*ten-dimensional supergeometrical*Weyl field strength tensors, constructed from respective torsions. - We realize that Lie Algebra techniques, in particular branching rules, Plethysm, and tensor product, provide the key to deciphering the complete list of independent fields that describe
efficiently.*a supersymmetric multiplet in arbitrary spacetime dimensions* - Thus,
*adinkra-based arguments*suggest the surprising possibility that the 11D, N=1 scalar superfield alone might describe a*Poincare supergravity prepotential or semi-prepotential*in analogy to one of the off-shell versions of 4D, N=1. - All of these results strongly suggest adynkras are pointing in the direction of using
in terms of*series expansion**Young Tableaux (YT’s)*as a tool to gain the most fundamental mathematical understanding of this class of problems.

We show the explicit one-to-one correspondence between Lorentz irreps and field variables, leading to an *adynkrafield* formalism in which the traditional ζ (theta)-monomials are replaced by *YT’s* as shown below. *(YangruiHu.com)*

In supergravity theory, supersymmetry theory and superstring theory, Adinkra symbols are a graphical representation of supersymmetry algebras. The similarity between Adinkra in supersymmetry and Adinkra symbols is that they are both graphical representations with hidden meanings (Prof. Sylvester James Gates Jr.). *(Adinkra Alphabet)*

** All of these results strongly suggest adynkras are pointing in the direction of using series expansion** in terms of YT’s as a tool to gain the most fundamental mathematical understanding of this class of problems.

*(Higher-Dimensional Supergravity - Pdf)*

All ** 15 matter particles are mirroring** their corresponding doppelgangers (anti-particles) each others that could potentially explain dark matter.

Here is an elegant model to define the elementary particles of the Standard Model in Physics.

- The black spheres are the bosons, the green ones leptons and the rest of the colored ones Murray Gell-Mann’s quarks (red for Generation I, blue for II and orange for III).
- Higgs Boson (aka the God particle) that does not have charge is the vertex between the matter and anti-matter particles.
- The z-boson and its counterpart would lie in the centroids of the tetrahedrons created by folding the triangles to meet up at the Higgs particle.

The next step is to re-gigg the model to account for the collisions and annihilations. Gluons and Photons that don’t have mass are not in the model, but will be the consequences of the interactions. *(Hypercomplex-Math)*

## Counterpart of Bispinor

A ** Dirac fermion**is equivalent to

**so it is not the same as bispinor. The counterpart is a Majorana fermion, a particle that must be its own**

*two (2) Weyl fermions***.**

*antiparticle*Because particles and antiparticles have opposite conserved charges, Majorana fermions have zero charge, hence among the fundamental particles, the only fermions that could be Majorana are sterile neutrinos, if they exist.

- All the other elementary fermions of the Standard Model have gauge charges, so they cannot have fundamental Majorana masses: Even the Standard Model’s left-handed neutrinos and right-handed antineutrinos have non-zero weak isospin, a charge-like quantum number.
- However, if they exist, the so-called “sterile neutrinos” (left-handed antineutrinos and right-handed neutrinos) would be truly neutral particles (assuming no other, unknown gauge charges exist).
- Ettore Majorana hypothesised the existence of Majorana fermions in 1937. The sterile neutrinos introduced to explain neutrino oscillation and anomalously small S.M. neutrino masses could have Majorana masses.

If they do, then at low energy (after electroweak symmetry breaking), by the seesaw mechanism, ** the neutrino fields would naturally behave as six Majorana fields, with three of them expected to have very high masses (comparable to the GUT scale) and the other three expected to have very low masses (below 1 eV)**.

```
Majorana | spinors | charged | neutrinos | quark | components | parameter
Fields | (s) | (c) | (n) | (q=s.c.n) | Σ(c+n+q | (complex)
===========+=========+=========+===========+===========+============+===========
majorana-1 | 2x2 | - | 18 | - | 18 | 13+5
-----------+---------+---------+-----------+-----------+------------+-----------
majorana-2 | 2x2 | - | 12 | - | 12 | 5+7
-----------+---------+---------+-----------+-----------+------------+-----------
majorana-3 | 2x2 | - | ❓ | - | ❓ | i66❓
===========+=========+=========+===========+===========+============+===========
Total | 12 | - | ❓ | - | ❓ | 30+i66❓
```

In physics, a subatomic particle is a particle smaller than an atom.[1]

- According to the Standard Model of particle physics, a subatomic particle can be either a composite particle, which is composed of other particles (for example, a baryon, like a proton or a neutron, composed of three quarks; or a meson, composed of two quarks), or an elementary particle, which is not composed of other particles (for example, quarks; or electrons, muons, and tau particles, which are called leptons).[2]
- Particle physics and nuclear physics study these particles and how they interact.[3]
- Most force carrying particles like photons or gluons are called bosons and, although they have discrete quanta of energy, do not have rest mass or discrete diameters (other than pure energy wavelength) and are unlike the former particles that have rest mass and cannot overlap or combine which are called fermions.

Experiments show that light could behave like a stream of particles (called photons) as well as exhibiting wave-like properties. This led to the concept of wave–particle duality to reflect that quantum-scale particles behave both like particles and like waves; they are sometimes called wavicles to reflect this. *(Wikipedia)*

*the 12 fermions and 5 bosons are known to have 48 and 13 variations, respectively*

```
Majorana | spinors | charged | neutrinos | quark | components | parameter
Fields | (s) | (c) | (n) | (q=s.c.n) | Σ(c+n+q | (complex)
===========+=========+=========+===========+===========+============+===========
majorana-1 | 2x2 | - | 18 | - | 18 | 13+5
-----------+---------+---------+-----------+-----------+------------+-----------
majorana-2 | 2x2 | - | 12 | - | 12 | 5+7
-----------+---------+---------+-----------+-----------+------------+-----------
majorana-3 | 2x2 | - | 13 | - | 13 | i13 ✔️
===========+=========+=========+===========+===========+============+===========
Total | 12 | - | 43 | - | 43 | 30+i13 ✔️
```

Leptons do not interact via the strong interaction.

- Their respective antiparticles are the antileptons, which are identical, except that they carry the opposite electric charge and lepton number.
- The antiparticle of an electron is an antielectron, which is almost always called a “positron” for historical reasons.
- There are six leptons in total; the three charged leptons are called “electron-like leptons”, while the neutral leptons are called “neutrinos”.
- Neutrinos are known to oscillate, so that neutrinos of definite flavor do not have definite mass, rather they exist in a superposition of mass eigenstates.

The hypothetical heavy right-handed neutrino, called a “sterile neutrino”, has been omitted. *(Wikipedia)*

```
Fermion | spinors | charged | neutrinos | quark | components | parameter
Field | (s) | (c) | (n) | (q=s.c.n) | Σ(c+n+q | (complex)
===========+=========+=========+===========+===========+============+===========
bispinor-1 | 2 | 3 | 3 | 18 | 24 | 19
-----------+---------+---------+-----------+-----------+------------+-- 17
bispinor-2 | 2 | 3 | 3 | 18 | 24 | i12 ✔️
===========+=========+=========+===========+===========+============+===========
bispinor-3 | 2 | 3 | 3 | 18 | 24 | 11
-----------+---------+---------+-----------+-----------+------------+-- 19
bispinor-4 | 2 | 3 | 3 | 18 | 24 | i18 ✔️
===========+=========+=========+===========+===========+============+===========
SubTotal | 8 | 12 | 12 | 72 | 96 | 66+i30
===========+=========+=========+===========+===========+============+===========
majorana-1 | 2x2 | - | 18 | - | 18 | 18 ✔️
-----------+---------+---------+-----------+-----------+------------+-----------
majorana-2 | 2x2 | - | 12 | - | 12 | 12 ✔️
-----------+---------+---------+-----------+-----------+------------+-----------
majorana-3 | 2x2 | - | 13 | - | 13 | i13
===========+=========+=========+===========+===========+============+===========
SubTotal | 12 | - | 43 | - | 43 | 30+i13
===========+=========+=========+===========+===========+============+===========
Total | 20 | 12 | 55 | 72 | 139 | 96+i43 ✔️
```

Since the total of parameters is ** 66+i30** then according to

*renormalization theory*the 12 boson fields should have the total complex value of

**.**

*30+i66*When describing spacetime as a continuum, certain statistical and quantum mechanical constructions are not well-defined.

- To define them, or make them unambiguous, a continuum limit must carefully remove “construction scaffolding” of lattices at various scales.
- Renormalization procedures are based on the requirement that certain physical quantities (such as the mass and charge of an electron) equal observed (experimental) values. That is, the experimental value of the physical quantity yields practical applications, but due to their empirical nature the observed measurement represents areas of quantum field theory that require deeper derivation from theoretical bases.
- Renormalization was first developed in quantum electrodynamics (QED) to make sense of infinite integrals in perturbation theory.
- Initially viewed as a suspect provisional procedure even by some of its originators, renormalization eventually was embraced as an important and self-consistent actual mechanism of scale physics in several fields of physics and mathematics.

Despite his later skepticism, it was Paul Dirac who pioneered renormalization. *(Wikipedia)*

They are composed out of *Symmetry Breaking* between *The True Prime Pairs* versus the 139 components of *The Fermion Field* tabulated as below.

In particle physics, a lepton is an elementary particle of half-integer spin (spin 1⁄2) that does not undergo strong interactions.[1]

- Two main classes of leptons exist:
leptons (also known as the electron-like leptons or muons), and neutral leptons (better known as*charged*.*neutrinos)* - Charged leptons can combine with other particles to form various composite particles such as atoms and positronium, while neutrinos rarely interact with anything, and are consequently rarely observed.
. There are*The best known of all leptons is the electron*, known as flavours, grouped in three generations.[2]*six types of leptons*- The first-generation leptons, also called electronic leptons, comprise the electron (e−) and the electron neutrino (νe); the second are the muonic leptons, comprising the muon (μ−) and the muon neutrino (νμ); and the third are the tauonic leptons, comprising the tau (τ−) and the tau neutrino (ντ).
. The heavier muons and taus will rapidly change into electrons and neutrinos through a process of particle decay: the transformation from a higher mass state to a lower mass state.*Electrons have the least mass of all the charged leptons*- Thus electrons are stable and the most common charged lepton in the universe, whereas muons and taus can only be produced in high energy collisions (such as those involving cosmic rays and those carried out in particle accelerators).
- Leptons have various intrinsic properties, including electric charge, spin, mass.
- Unlike quarks, however, leptons are not subject to the strong interaction, but they are subject to the other three fundamental interactions: gravitation, the weak interaction, and to
, of which the latter is proportional to charge, and is thus zero for the electrically neutral neutrinos.*electromagnetism*

For every lepton flavor, there is a corresponding type of antiparticle, known as an antilepton, that differs from the lepton only in that some of its properties have equal magnitude but opposite sign. According to certain theories, neutrinos may be their own antiparticle. It is not currently known whether this is the case. *(Wikipedia)*

*(33+1)th prime = 139*

```
Fermion | spinors | charged | neutrinos | quark | components | parameter
Field | (s) | (c) | (n) | (q=s.c.n) | Σ(c+n+q | (complex)
===========+=========+=========+===========+===========+============+===========
bispinor-1 | 2 | 3 | 3 | 18 | 24 | 19
-----------+---------+---------+-----------+-----------+------------+-- 17
bispinor-2 | 2 | 3 | 3 | 18 | 24 | i12
===========+=========+=========+===========+===========+============+===========
bispinor-3 | 2 | 👉 3 | 3 | 18 | 24 | 11 ✔️
-----------+---------+---------+-----------+-----------+------------+-- 19
bispinor-4 | 2 | 3 | 3 | 18 | 24 | i18
===========+=========+=========+===========+===========+============+===========
SubTotal | 8 | 12 | 12 | 72 | 96 | 66+i30
===========+=========+=========+===========+===========+============+===========
majorana-1 | 2x2 | - | 18 | - | 18 | 18
-----------+---------+---------+-----------+-----------+------------+-----------
majorana-2 | 2x2 | - | 12 | - | 12 | 12
-----------+---------+---------+-----------+-----------+------------+-----------
majorana-3 | 2x2 | - | 13 | - | 13 | i13
===========+=========+=========+===========+===========+============+===========
SubTotal | 12 | - | 43 | - | 43 | 30+i13
===========+=========+=========+===========+===========+============+===========
Total | 20 | 12 | 55 | 72 | 139 | 96+i43
```

According to the 24 cells of *Prime Hexagon*, this 12 boson fields seems to be realistic. However it would need to have a total of 96 complex-valued components as well.

## Beyond the 139

A general mass structure for the heavy SM fermion generations has been obtained which explains the following features of SO(10):

The work performed in this thesis will focus on two different models, that both can be used in the creation of a GUT. ** Both models are based on having SO(10) as the unification gauge group**.

- Such models are more complex than the original suggestions, but can also accommodate more physics. In these two models, it is not possible
.*to achieve unification among the gauge couplings with tree-level matching conditions* - However, so-called threshold effects appear when matching the couplings at a higher order in perturbation theory, which are a result of particles with masses around the symmetry breaking scales.

Specifically, it will be investigated if threshold effects can save these two models, and thereby allowing unification. *(Threshold Effects in SO(10) Grand Unified Theories - pdf)*

Similarly the Standard Model incorporates three generations of quarks, so its fermionic content can be summarized. In addition, the Standard Model involves gauge bosons (photons for the electromagnetic interaction, W and Z for the weak interaction, and ** eight (8) gluons** for the strong interaction), plus the (scalar) Higgs particle. This is what all known matter in the Universe consists of.

*(Netrinos)*

The present particle physics or standard model based on the “unreal gauge transformation symmetry” and meaningless math cannot explain any actual physical mechanism at all *(biglobe.ne.jp)*

When we take all the forces that we understand, i.e., ** not including gravity**, and write down the QFT version of them, we arrive at the predictions of the Standard Model.

This is where the idea of ** 12 fermion fields and 12 boson fields** come from. These fields are excitations of the underlying theories (the Standard Model) that describe the known Universe in its entirety, and include:

- The six (6): up-, down-, strange-, charm-, bottom-, top-quarks, and their antiquark counterparts,
- The three (3) charged (electron, muon, tau) and three (3) neutral (electron neutrino, muon neutrino, tau neutrino) leptons, and their antimatter counterparts,
- The eight (8) gluons (because of the eight possible color combinations),
- The one (1) electromagnetic (photon) boson,
- The two (2) weak (W-and-Z) bosons,
- And the Higgs boson.

The quarks and leptons are fermions, which is why they have antimatter counterparts, and the W boson comes in two equal-and-opposite varieties (positively and negatively charged), but all told, ** there are 24 unique, fundamental excitations of quantum fields possible**. This is where the 24 fields idea comes from.

*(Forbes)*

Summing the principal and secondary diagonals gives us 1200 + 960 = 2160 = 360 * 6 = 432 * 5. And aligning the principal and secondary diagonals forms this string of 24 dyads summing to 90 each, again for a total of 2160 (and note that only terminating digits 1 and 9 are present and that there are also 24 diagonal dyads summing to 90 each, as somewhat crudely illustrated) *(Primesdemystified)*

This scheme goes to the unification of 11s with 7s to 18s meanwhile the 11th it self behave as residual by the 5th minor hexagon between the ** 30 to 36' cells**.

This behaviour finaly brings us to a suggestion that the dimension in string theory are linked with ** the prime distribution level** as indicated by the

*self repetition*on MEC30.

*109 = 29th prime = (10th prime)th prime*

## Visualizing TOE

The natural next step is to generalise this to D = 3, 4, 6, 10 and obtain a ‘magic pyramid’ with the D = 3 magic square at the base and Type II supergravity at the summit. On the basis of these results we speculate that the part played by octonions in string and M-theory may be more prominent than previously though. *(Super Yang-Mills - pdf)*

** There are 30 canonical sets of 7 triads indexed with a Fano plane index (fpi) in (16)**. As in E8 with16 of the 2^8 = 256 binary representations excluded from the group, there are 32 excluded octonionsfrom the 2^9 = 512. As in E8, excluded particles are associated with the color=0, generation=0 (bosons)which are the positive (and negative) generators commonly associated with the 8-orthoplex with 16permutations of {±1, 0, 0, 0, 0, 0, 0, 0}.

The finiteness position of MEC30 along with Euler's identity opens up the possibility of accurately representing the self-singularity of True Prime Pairs.

The Mathematical Elementary Cell 30 (MEC30) standard unites the mathematical and physical results of 1972 by the mathematician Hugh Montgomery and the physicist Freeman Dyson and thus reproduces energy distribution in systems as a path plan more accurately than a measurement. (Google Patent DE102011101032A9)

Assigning a specific mass, length, time, and charge metrics based on new dimensional relationships and the Planck constant (which defines Higgs mass).

All fields of the standard model and gravity are unified as an E8 principal bundle connection. A non-compact real form of the E8 Lie algebra has G2 and F4 subalgebras which break down to strong su(3), electroweak su(2) x u(1), gravitational so(3,1), the frame-Higgs, and three generations of fermions related by triality. The interactions and dynamics of these 1-form and Grassmann valued parts of an E8 superconnection are described by the curvature and action over a four dimensional base manifold. *(An Exceptionally Simple Theory of Everything - pdf)*

The index of 8 sign masks (sm) to the 30 fPi (each with 8 Hexadecimal masks). ** These can be "inverted" (0↔1) making 16×30=480 octonion permutations**.

Supersymmetry and more specifically supergravity grand unification allow one to extrapolate physics from the electroweak scale up to the grand unification scale consistent with electroweak data.

- Here we give a brief overview of their current status and show that the case for supersymmetry is stronger as a result of the Higgs boson discovery with a mass measurement at ∼ 125 GeV consistent with the supergravity grand unification prediction that the Higgs boson mass lie below 130 GeV. Thus the discovery of the Higgs boson and the measurement of its mass provide a further impetus for the search for sparticles to continue at the current and future colliders.
- The group SO(10) as the framework for grand unification appears preferred over SU(5). The group SO(10) contains both G(4, 2, 2) and SU(5)⊗U(1) as subgroups, i.e., SO(10) has the branchings SO(10) → SU(4)C ⊗ SU(2)L ⊗ SU(2)R and SO(10) → SU(5) ⊗ U(1).
. A full generation of quarks and leptons can be accommodated in a single 16 plet representation. Thus the 16 plet has the decomposition in SU(5) ⊗ U(1) so that 16 =10(−1) ⊕ 5(3) ⊕ 1(−5).*It possesses a spinor representation which is 2⁵ = 32 dimensional and which splits into 16 ⊕ 16*- As noted the combination 5 ⊕ 10 in SU(5) is anomaly free and further 1(−5) in the 16-plet decomposition is a right handed neutrino which is a singlet of the standard model gauge group and thus the 16-plet of matter in SO(10) is anomaly free.
- The absence of anomaly in this case is the consequence of a more general result for SO(N) gauge theories. Thus in general anomalies arise due to the non-vanishing of the trace over the product of three group generators in some given group representation Tr ({Ta, Tb}Tc).
- For SO(10) one will have Tr ({Σµν, Σαβ}Σλρ). However, there is no invariant tensor to which the above quantity can be proportional which then automatically guarantees vanishing of the anomaly for SO(10). This analysis extends to other SO(N) groups.
ǫµναβλρ and so in this case vanishing of the anomaly is not automatic.*One exception is SO(6) where there does exist a six index invariant tensor*- The group SO(10) is rank 5 where as the standard model gauge group is rank 4. The rank of the group can be reduced by either using
.*16 ⊕ 16 of Higgs fields or 126 ⊕ 126 of Higgs* - Since under SU(5) ⊗ U(1) one has 16 ⊃ 1(−5) we see that a VEV formation for the singlet will reduce the rank of the group. Similarly 126 ⊃ 1(−10) under the above decomposition. Thus when the singlets in 16 ⊕ 16 of Higgs or 126 ⊕ 126 get VEVs, the SO(10) gauge symmetry will break reducing its rank.
- However, we still need to reduce the remaining group symmetry to the Standard Model gauge group. For this we need to have additional Higgs fields such as 45, 54, 210. Further to get the residual gauge group SU(3)C ⊗ U(1)em we need to have 10 -plet of Higgs fields.
- Thus the breaking of SO(10) down to SU(3)C ⊗ U(1)em
: one to reduce the rank, the second to break the rest of the gauge group to the Standard Model gauge group and then at least one 10-plet to break the electroweak symmetry.*requires at least three (3) sets of Higgs representations* - As discussed above one can do this by a combination of fields from the set: 10, 16 ⊕ 16, 45, 54, 120, 126 ⊕ 126, 210.
- To generate quark and lepton masses we need to couple two 16-plets of matter with Higgs fields.
.*To see which Higgs fields couple we expand the product 16⊗16 as a sum over the irreducible representations of SO(10)*

Here we have ** 16 ⊗ 16 = 10s ⊕ 120a ⊕ 126s**, where the s(a) refer to symmetric (anti-symmetric) under the interchange of the two 16-plets. The array of Higgs bosons available lead to a large number of possible SO(10) models.

*(Superunification - pdf)*

Below is a powerful cheat sheet which is compiled to provide you with a great overview, not just stuffed with information, but also puts it in relation.

I am pleased to announce the availability of splitFano.pdf, a 321 page pdf file with the 3840=480*8 split octonion permutations (with Fano planes and multiplication tables).

- There are 30 canonical sets of 7 triads indexed with a Fano plane index (fpi) in (16). As in E8 with 16 of the 2⁸ = 256 binary representations excluded from the group, there are 32 excluded octonions from the 2⁹ = 512.
- As in E8, excluded particles are associated with the color=0, generation=0 (bosons) which are the positive (and negative) generators commonly associated with the 8-orthoplex with 16 permutations of {±1, 0, 0, 0, 0, 0, 0, 0}.
- These are organized into “flipped” and “non-flipped” pairs associated with the 240 assigned particles to E8 vertices (sorted by Fano plane index or fPi).
- They are assigned to the 30 canonical sets of 7 triples using the maskList: {5, 8, 4, 3, 7, 6, 3, 2, 6, 5, 1, 4, 6, 7, 3, 3, 8, 6, 3, 1, 6, 6, 2, 3, 5, 8, 4, 3, 7, 6}
- There are 7 sets of split octonions for each of the 480 “parent” octonions (each of which is defined by 30 sets of 7 triads and 16 7 bit “sign masks” which reverse the direction of the triad multiplication). The 7 split octonions are identified by selecting a triad.
- The complement of {1,2,3,4,5,6,7} and the triad list leaves 4 elements which are the rows/colums corresponding to the negated elements in the multiplication table (highlighted with yellow background).
- The red arrows in the Fano Plane indicate the potential reversal due to this negation that defines the split octonions. The selected triad nodes are yellow, and the other 4 are cyan (25MB).
- These allow for the simplification of Maxwell’s four equations which define electromagnetism (aka.light) into a single equation.

Below is the first page of the comprehensive split octonion list of all 3840 Split Fano Planes with their multiplication tables available. *(8×16×30 Split Fano)*

In the special case of a unit segment, the Golden Ratio provides the only way to divide unity in two parts that are in a geometric progression

It has been long known that the SO(10) model is free from all perturbative local anomalies, computable by Feynman diagrams. However, it only became clear in 2018 that the SO(10) model is also free from all nonperturbative global anomalies on non-spin manifolds — an important rule for confirming the consistency of SO(10) grand unified theory, with a Spin(10) gauge group and chiral fermions in the 16-dimensional spinor representations, defined on non-spin manifolds. *(Wikipedia)*

The split real even E8 group used has been determined from Dynkin diagram which builds the Cartan matrix and determines the root with corresponding Hasse diagrams.

It has been found recently that the expansion of N = 8 supergravity in terms of Feynman diagrams has shown that N = 8 supergravity is in some ways [1] a product of two N = 4 super Yang–Mills theories.

- This is written schematically as: N = 8 supergravity = (N = 4 super Yang–Mills) × (N = 4 super Yang–Mills). This is not surprising, as N = 8 supergravity contains six independent representations of N = 4 super Yang–Mills.
- The theory contains 1 graviton (spin 2), 8 gravitinos (spin 3/2), 28 vector bosons (spin 1), 56 fermions (spin 1/2), 70 scalar fields (spin 0) where we don’t distinguish particles with negative spin.
- These numbers are simple combinatorial numbers that come from Pascal’s Triangle and also the number of ways of writing n as a sum of 8 nonnegative cubes A173681.
- The only theories with spins higher than 2 which are consistent involve an infinite number of particles (such as String Theory and Higher-Spin Theories). Stephen Hawking in his Brief History of Time speculated that this theory could be the Theory of Everything.
- One reason why the theory was abandoned was that
which form an O(8) gauge group is too small to contain the standard model U(1) x SU(2) x SU(3) gauge group, which can only fit within the orthogonal group O(10).*the 28 vector bosons*

For model building, it has been assumed that almost all the supersymmetries would be broken in nature,[why?] leaving just one supersymmetry (N = 1), although nowadays because of the lack of evidence for N = 1 supersymmetry higher supersymmetries are now being considered such as N = 2. *(Wikipedia)*

Let's discuss more detail about this particular topic as guided by Prof Stephen Hawking in one of his greatest book: The Theory of Everything.