Exponentiation Zones (30-36)

Exponentiation is an operation involving two numbers, the base and the exponent or power. Exponentiation is written as bn, where b is the base and n is the power.

Tip

This section is referring to wiki page-21 of gist section-17 that is inherited from the gist section-2 by prime spin-30 and span-168 with the partitions as below.

  1. Electrodynamics (maps)
  2. Quantum Gravity (feed)
  3. Chromodynamics (lexer)
  4. Electroweak Theory (parser)
  5. Grand Unified Theory (syntax)

Exponentiation zones allows multiplication zones on representing recursive residues by virtualizing addition zones on top of the original.

The Root System

The first appearance of e in a printed publication was in Euler's Mechanica (1736). It is unknown why Euler chose the letter e.

Note

Leonhard Euler started to use the letter e for the constant in 1727 or 1728, in an unpublished paper on explosive forces in cannons, and in a letter to Christian Goldbach on 25 November 1731. (Wikipedia)

Letter e

This exponentiation takes important roles since by the multiplication zones the MEC30 forms a matrix of 8 x 8 = 64 = 8² where the power of 2 stands as exponent

Note

We present a method to increase the dynamical range of a Residue Number System (RNS) by adding virtual RNS layers on top of the original RNS, where the required modular arithmetic for a modulus on any non-bottom layer is implemented by means of an RNS Montgomery multiplication algorithm that uses the RNS on the layer.

  • As a result, the actual arithmetic is deferred to the bottom layer. We have presented an improved Bajard-Imbert-type full RNS algorithm that can also operate on inputs represented by pseudo-residues.
  • Using this algorithm, we have developed a multi-layer RNS that is capable of implementing modular addition, subtraction and multiplication for very large moduli by only using actual arithmetic for a fixed set of moduli. If the moduli of this fixed set are sufficiently small, the method allows for a fully table-based implementation.
  • In contrast to digit-based implementations of modular operations for large moduli, our method allows for a massively parallel implementation and is completely carry- free, thus thwarting potential attacks exploiting such carries, e.g., with side-channel analysis or in a white-box cryptography context.
  • Our system may be considered as a method to provide a given, fixed RNS with a very large dynamical range. To illustrate the method, we have described a 2-layer RNS system that can be used to implement an RSA exponentiation by adding the desired RSA modulus on top in a third layer.
  • The system employs 19 moduli of 8-bits each in the bottom layer and can be used to implement an RSA exponentiation for 2048-bits RSA moduli with all the required arithmetic done by table look-up, using 19 modular addition tables and 19 modular multiplication tables, each of these 38 tables having size 2⁸ × 2⁸ × 8 bits, with one modular multiplication taking approximately 160,000 table look-ups.

We further observed that in order to change the RSA modulus, only some constants for computing on the top layer with moduli on the middle layer need to be updated. This update need not be computed in a secure manner and hence can be done quickly. (Recursive Residues - pdf)

π(π(30+37)) = π(π(67)) = π(19) = 8

#!/usr/bin/env bash

edit_file () {

  NUM=$(($2 + 0))
  
  while IFS=' ' read -ra SPIN; do
    T+=("${SPIN[0]}")
    R+=("${SPIN[1]}")
    A+=("${SPIN[2]}")
    C+=("${SPIN[3]}")
    K+=("${SPIN[4]}")
    I+=("${SPIN[5]}")
    N+=("${SPIN[6]}")
    G+=("${SPIN[7]}")
  done < /tmp/spin.txt

  FRONT="---\n"
  FRONT+="sort: ${K[$NUM]}\n"
  FRONT+="span: ${I[$NUM]}\n"
  FRONT+="spin: ${N[$NUM]}\n"
  FRONT+="suit: ${G[$NUM]}\n"
  FRONT+="---\n"

  IFS=$'\n' read -d '' -r -a LINE < _Sidebar.md
  TEXT="${LINE[$NUM]}" && TITLE=${TEXT%|*}
  FRONT+="# $TITLE\n\n"

  [[ $NUM -le 9 ]] && sed -i "1s|^|$FRONT|" $1
  if [[ $NUM -lt 2 || $NUM == 9 ]]; then
    mv -f $1 ${1%/*}/README.md
    sed '1,6!d' ${1%/*}/README.md
  fi
}

FILE=${1##*/} && SORT=${FILE%.*}
[[ $SORT =~ ^-?[0-9]+$ ]] && edit_file $1 $SORT

These representations are a curious finding. They relate particles to antiparticles by using only the complex conjugate i → −i, they fill these as of Euler's Identity.

Note

Euler’s identity is a special case of Euler’s formula e^ix = cos x + i sin x when evaluated for x = π, In addition, it is directly used in a proof that π is transcendental, which implies the impossibility of squaring the circle. (Wikipedia)

Euler's identity

Euler angles specify the rotation of the X, Y, and Z rotation axes. The Euler angle is the culprit of the singularities in matrix algebra.

Note

In this work we present a matrix generalization of the Euler identity about exponential representation of a complex number. The concept of matrix exponential is used in a fundamental way. We define a notion of matrix imaginary unit which generalizes the usual complex imaginary unit. The Euler-like identity so obtained is compatible with the classical one. Also, we derive some exponential representation for matrix real and imaginary unit, and for the first Pauli matrix

Spin

Euler identity present a matrix generalization of the about exponential representation for matrix real and imaginary unit which compatible with the Pauli matrix

Note

Gell–Mann matrices are to SU(3) what the Pauli matrices are to SU(2). Gell–Mann -matrices are a complete set of Hermitian 3 ⊗ 3 noncommuting trace-orthogonal matrices. They are at the heart of Quantum Chromodynamics (QCD), an integral part of the Standard Model. They are also used in quantum information theory to represent qutrits. (Wolfram)

Everything About Gell Mann Matrices Unary Operations

This imaginary unit is particularly important in both mathematics and physics. For example, those matrices (and their generalizations) are important in Lie Theory.

Note

As usual, the images on the left are snapshots of the particles at different times. Those times correspond to the grey slices in the space-time diagram on the right. You can see the specific interaction points in the space-time diagram, where the blue particle is emitted and then absorbed by the red particles. (Slimy.com)

Feynman diagrams

So it will need a gap between each identities to proceed the thing. Let's discuss how it goes by the seven (7) hidden dimensions.

Basic Transformation

In quantum field theory, partition functions are generating functionals for correlation functions, making them key objects of study in the path integral formalism.

Note

Gell-mann matrices are a complete set of Hermitian noncommuting trace-orthogonal matrices. In addition, they also play an important role in physics where they can be thought to model the eight gluons that mediate the strong force quantum chromodynamics, an analogue of the Pauli matrices well-adapted to applications in the realm of quantum mechanics. (Wolfram)

#!/usr/bin/env python

import numpy as np
from scipy import linalg

class SU3(np.matrix):
	GELLMANN_MATRICES = np.array([
		np.matrix([ #lambda_1
			[0, 1, 0],
			[1, 0, 0],
			[0, 0, 0],
		], dtype=np.complex),
		np.matrix([ #lambda_2
			[0,-1j,0],
			[1j,0, 0],
			[0, 0, 0],
		], dtype=np.complex),
		np.matrix([ #lambda_3
			[1, 0, 0],
			[0,-1, 0],
			[0, 0, 0],
		], dtype=np.complex),
		np.matrix([ #lambda_4
			[0, 0, 1],
			[0, 0, 0],
			[1, 0, 0],
		], dtype=np.complex),
		np.matrix([ #lambda_5
			[0, 0,-1j],
			[0, 0, 0 ],
			[1j,0, 0 ],
		], dtype=np.complex),
		np.matrix([ #lambda_6
			[0, 0, 0],
			[0, 0, 1],
			[0, 1, 0],
		], dtype=np.complex),
		np.matrix([ #lambda_7
			[0, 0,  0 ],
			[0, 0, -1j],
			[0, 1j, 0 ],
		], dtype=np.complex),
		np.matrix([ #lambda_8
			[1, 0, 0],
			[0, 1, 0],
			[0, 0,-2],
		], dtype=np.complex) / np.sqrt(3),
	])


	def computeLocalAction(self):
		pass

	@classmethod
	def getMeasure(self):
		pass

In this paper, you may find a way to apply the Gell-Mann transformations made by the λi matrices using Geometric Algebra Cl3,0.

ezgif-4-95200c65b5

Note

They are at the heart of Quantum Chromodynamics (QCD), an integral part of the Standard Model. They are also used in quantum information theory to represent qutrits. Gell–Mann matrices are to SU(3) what the Pauli matrices are to SU(2). (Wolfram)

Gell-Mann transformations

These unifying principles of both mathematics and physics might come in the form of grand unified theories, supersymmetry, string theory, or perhaps something else.

Note

Standard heuristic mathematical models of population dynamics are often constructed using ordinary differential equations (ODEs). These deterministic models yield pre-dictable results which allow researchers to make informed recommendations on public policy. A common immigration, natural death, and fission ODE model is derived from a quantum mechanics view. (A Quantum Mechanics Approach.pdf)

I15-53-electroweak

Although, at the moment evidence do not have a complete model. However, it becomes a little more clear that this unlikely algebra is not going away.

Standard Model

There is a proof that it is impossible to embed all the three generations in E8 without the presence of additional particles that do not exist in the physical world.

Note

This is a somewhat arbitrary choice, selected for leaving W3 and color invariant. Once the first generation of fermions, with correct charges and spins, are assigned to elements of e8, this T rotates them to the second and third generations.

  • The second and third generations only have the correct spins and charges when considered as equivalent under this T. When considered as independent fields with E8 quantum numbers, irrespective of this triality relationship, the second and third generation of fields do not have correct charges and spins.
  • The W3 and color charges are invariant under our choice of T but the spins and hypercharges are only correct through triality equivalence. This relationship between fermion generations and triality is the least understood aspect of this theory.
  • It is conceivable that there is a more complicated way of assigning three generations of fermions to the E8 roots to get standard model quantum numbers for all three generations without triality equivalence.

There is such an assignment known to the author that gives the correct hypercharges for all three generations, but it is not a triality rotation and it produces unusual spins. A correct description of the relationship between triality and generations, if it exists, awaits a better understanding. (An Exceptionally Simple Theory of Everything - pdf)

An Exceptionally Simple Theory of Everything

A precise measurement of the rate of the H→ bb process directly tests the Yukawa coupling of the Higgs boson to a down-type quark.

Note

Recently, the CMS and ATLAS Collaborations reported observations of the Higgs boson produced in association with a top quark pair thus representing the first direct measurements of the Higgs boson coupling to quarks. - This week the CMS Collaboration announces another major achievement and reports the observation of Higgs boson decay to bottom quarks (H→ bb)

  • A precise measurement of the rate of the H→ bb process directly tests the Yukawa coupling of the Higgs boson to a down-type quark, and is necessary to solidify the Higgs boson as the possible sole source of mass generation in the fermion sector of the Standard Model (SM).
  • While the decay of the Higgs boson to bottom quarks is the most frequent of all Higgs boson decays, it has been a real experimental challenge to observe it. This is on account of the overwhelmingly large background contribution from a number of other SM processes that can mimic its experimental signature characterized by the appearance of a bottom and an anti-bottom quark.

The CMS Collaboration overcame this challenge by deploying modern sophisticated analysis tools and by focusing on particular signatures where a Higgs boson is produced in association with a vector boson V (a W or Z particle), a weak interaction process known as VH(bb), shown in the figure below, leading to a significant reduction in the background. (CERN)

down-type quark

Note

The matter representations come in three copies (generations) of the 16 representation. The Yukawa coupling is 10H 16f 16f. *This includes a right-handed neutrino”. One may either include three copies of singlet representations φ and a Yukawa coupling (the “double seesaw mechanism”); or else, add the Yukawa interaction or add the nonrenormalizable coupling. (Wikipedia)

12648_2023_2718_Figa_HTML

Beyond leading approx. we define mGUT as the mass of the heavy 24 gauge bosons, while mT = mHT is the mass of the triplet Higgs.

Note

The cleanest signature for a Higgs sector with triplet fields would be the discovery of doubly charged Higgs Bosons. ## Like Pauli’s bold prediction of the neutrino and GIM’s bold prediction of the charm quark, the equally bold speculation of Kobayashi and Maskawa was proved absolutely correct, when the fermions of the third generation began to be discovered one by one. First came the tau lepton in 1975, closely followed by the bottom quark in 1977. There followed a 17-year hiatus till the 1994 discovery of the top quark, and another 6 years wait till the existence of the tau neutrino νwas confirmed in 2000.

24 matriks

Is the fermion red? green? blue? Does the fermion have isospin up? down? These five questions can be represented by an exterior algebra of 2⁵ or 32-complex dimensional.

Note

This thesis constitutes a first attempt to derive aspects of standard model particle physics from little more than an algebra.

  • Here, we argue that physical concepts such as particles, causality, and irreversible time may result from the algebra acting on itself.
  • We then focus on a special case by considering the algebra R ⊗ C ⊗ H ⊗ O, the tensor product of the only four normed division algebras over the real numbers.
  • Using nothing more than R ⊗ C ⊗ H ⊗ O acting on itself, we set out to find standard model particle representations: a task which occupies the remainder of this text.
  • From the C ⊗ H portion of the algebra, we find generalized ideals, and show that they describe concisely all of the Lorentz representations of the standard model.
  • From just the C ⊗ O portion of the algebra, we find minimal left ideals, and show that they mirror the behaviour of a generation of quarks and leptons under su(3)c and u(1)em.
  • These unbroken symmetries, su(3)c and u(1)em, appear uniquely in this model as particular symmetries of the algebra’s ladder operators. Electric charge, here, is seen to be simply a number operator for the system.
  • We then combine the C ⊗ H and C ⊗ O portions of R ⊗ C ⊗ H ⊗ O, and focus on a leptonic subspace, so as to demonstrate a rudimentary electroweak model. Here, the underlying ladder operators are found to have a symmetry generated uniquely by su(2)L and u(1)Y.
  • Furthermore, we find that this model yields a straight forward explanation as to why SU(2)L acts only on left-handed states.
  • We then make progress towards a three-generation model. The action of C ⊗ O on itself can be seen to generate a 64-complex-dimensional algebra, wherein we are able to identify two sets of generators for SU(3)c.
  • We apply these generators to the rest of the space, and find that it breaks down into the SU(3)c representations of exactly three generations of quarks and leptons.

Furthermore, we show that these three-generation results can be extended, so as to include all 48 fermionic U(1)em charges. (Standard Model from an algebra - pdf)

The-64-dimensional-octonionic-chain-algebra-splits-into-two-sets-of-SU-3-generators

Subatomic particles such as protons or neutrons, which contain two or more elementary particles, are known as composite particles.

Note

The Standard Model of Particle Physics, describes for us all know fundamental interaction in nature till date, with the exception of Gravity (work on this front is going on). Here is a summary of the fundamental content of the standard model

  • There are three families of particle, the Quarks, the Leptons and the Gauge Bosons. The Quarks in groups of three forms the composite particles such as the Protons, along with the electron this forms ordinary matter.
  • The Gauge Bosons are the ones those are responsible for interactions. The Quarks interact among themselves by the exchange of a Gluon these are responsible for the strong nuclear force.
  • The newly discovered Higgs Boson interacts with all the Quarks and the first group of Leptons (electron, muon and tau) providing them with their mass. The neutrinos which are the other Leptons originally were thought to have zero mass, but recent discoveries argue that this is not the case.
  • The Weak bosons interact with both Leptons and Quarks, these are responsible for the Weak nuclear forces. The exchange of photon is responsible for the Electromagnetic Force.

They interact, they transfer energy and momentum and angular momentum; excitations are created and destroyed. Every excitation that’s possible has a reverse excitation. (Quora)

fundamental interaction in nature

It is hypothesized that gravitational interactions are mediated by an as yet undiscovered elementary particle, dubbed the graviton.

How many quarks?

Elementary particles and their interactions are considered by a theoretical framework called the Standard Model (SM) of Particle Physics.

Note

The Standard Model presently recognizes seventeen distinct particles (twelve fermions and five bosons). As a consequence of flavor and color combinations and antimatter, the fermions and bosons are known to have 48 and 13 variations, respectively. Among the 61 elementary particles embraced by the Standard Model number electrons and other leptons, quarks, and the fundamental bosons. (Wikipedia)

17 distinct particles = 12 fermions + 5 bosons = 48 + 13 = 61 variations

Standard_Model_of_Elementary_Particles

Answer-1: 3 generation x 3 color x 2 types x 2 each = 36 quarks

How many types of quarks are there and what are their names?

Answer-2: 6 flavour x 3 colors x 2 types = 36 quarks

image

Answer-3: 6 flavour x 3 colour x 4 bispinor = 72 quarks

There are 72 quarks

Note

In order to be four-spinors like the electron and other lepton components, there must be one quark component for every combination of flavour and colour, bringing the total to 24 (3 for charged leptons, 3 for neutrinos, and 2·3·3 = 18 for quarks). Each of these is a four (4) component bispinor, for a total of 96 complex-valued components for the fermion field. (Wikipedia)

IMG_20240108_045902

It is stated that each of the 24 components is a four component bispinor. A bispinor is constructed out 2 simpler component spinor so there are eight (8) spinors in total.

Note

Bispinors are so called because they are constructed out of two (2) simpler component spinors, the Weyl spinors. Each of the two (2) component spinors transform differently under the two (2) distinct complex-conjugate spin-1/2 representations of the Lorentz group. This pairing is of fundamental importance, as it allows the represented particle to have a mass, carry a charge, and represent the flow of charge as a current, and perhaps most importantly, to carry angular momentum. (Wikipedia)

((3+3) + 2x(3x3)) x 4 = (3 + 3 + 18) x 4 = 24 x 4 = 96 components

  Fermion  | spinors | charged | neutrinos |   quark   | components
   Field   |   (s)   |   (c)   |    (n)    | (q=s.c.n) |  Σ(c+n+q)
===========+=========+=========+===========+===========+============
bispinor-1 |    2    |    3    |     3     |    18     |     24
-----------+---------+---------+-----------+-----------+------------ } 48
bispinor-2 |    2    |    3    |     3     |    18     |     24
===========+=========+=========+===========+===========+===========
bispinor-3 |    2    |    3    |     3     |    18     |     24
-----------+---------+---------+-----------+-----------+------------ } 48
bispinor-4 |    2    |    3    |     3     |    18     |     24
===========+=========+=========+===========+===========+============
     Total |    8    |   12    |    12     |    72     |     96

Thus fermion is constructed out of eight (8) spinors that brings the total of 96 components consist of 12 charged leptons, 12 neutrinos and 72 quarks.

Free Parameters

The physical evolution of neutrino parameters with respect to energy scale may help elucidate the mechanism for their mass generation.

Note

The most general Lagrangian with massless neutrinos, one finds that the dynamics depend on 19 parameters, whose numerical values are established by experiment.

  • The 19 certain parameters are summarized below:IMG_20231230_232603
  • The neutrino parameter values are still uncertain.
  • The value of the vacuum energy (or more precisely, the renormalization scale used to calculate this energy) may also be treated as an additional free parameter.

The renormalization scale may be identified with the Planck scale or fine-tuned to match the observed cosmological constant. However, both options are problematic. (Wikipedia)

$True Prime Pairs:
(5,7), (11,13), (17,19)
 
layer | node | sub |  i  |  f.                                       MEC 30 / 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 + ∆18 = ∆27         |
  2   +------|  5* +-----+-----                      |
      |      |     |  10 |                           |
      |      |-----+-----+                           |
      |  4   |     |  11 | (13) --------------------- ∆32
      |      |  6  +-----+            ‹------------------------------ 15 {0}
      |      |     |  12 |---------------------------
------+------+-----+-----+------------               |
      |      |     |  13 |                           |
      |      |  7  +-----+                           |
      |  5   |     |  14 | (17)                      |
      |      |-----+-----+                           |
      |      |     |  15 |                           7s ‹-- ∆24 = (43-19)
  3*  +------+  8  +-----+-----       } (36)         |
      |      |     |  16 |                           |
      |      |-----+-----+                           |
      |  6   |     |  17 | (19) ‹-- parameters ✔️    |
      |      |  9  +-----+                           |
      |      |     |  18 | -------------------------- ∆68 - ∆18 = ∆50
------|------|-----+-----+-----  ‹----------------------------------- 30 {+1/2}

The Standard Model with massive neutrinos need 7 more parameters (3 masses and 4 PMNS matrix parameters) for a total of 26 parameters.

Note

In principle, there is one further parameter in the Standard Model; the Lagrangianof QCD can contain a phase that would lead to CP violation in the strong interac-tion.

  • Experimentally, this strong CP phase is known to be extremely small, θCP ≃ 0, and is usually taken to be zero.
  • If θCP is counted, then the Standard Model has 26 free parameters.
  • The relatively large number of free parameters is symptomatic of the StandardModel being just that; a model where the parameters are chosen to match the observations, rather than coming from a higher theoretical principle.
  • Putting aside θCP, of the 25 SM parameters, 14 are associated with the Higgs field, eight with theflavour sector and only three with the gauge interactions.

Likewise, the coupling constants of the three gauge interactions are of a similar order of magnitude, hinting that they might be different low-energy manifestations of a Grand Unified Theory (GUT) of the forces. These patterns provide hints for, as yet unknown, physics beyond the Standard Model. (Modern Particle Physics - pdf)

(24-5) + (24-17) = 19 + 7 = 26

  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+i5 ✔️
-----------+---------+---------+-----------+-----------+------------+-----------
bispinor-2 |    2    |    3    |     3     |    18     |     24     |   17+i7 ✔️
===========+=========+=========+===========+===========+============+===========
bispinor-3 |    2    |    3    |     3     |    18     |     24     |     ❓
-----------+---------+---------+-----------+-----------+------------+-----------
bispinor-4 |    2    |    3    |     3     |    18     |     24     |     ❓
===========+=========+=========+===========+===========+============+===========
     Total |    8    |   12    |    12     |    72     |     96     |     ❓
Note

Now we show the interplay of the finite system of prime positions with the 15 finite even positions in the cyclic convolution. Consequently, we only need to fold a 30’s cycle as so that we can identify the opposite prime positions that form their specific pairs in a specific convolution.

13+17 = 11+19 = 30

  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+i5 
-----------+---------+---------+-----------+-----------+------------+-----------
bispinor-2 |    2    |    3    |     3     |    18     |     24     |   17+i7
===========+=========+=========+===========+===========+============+===========
bispinor-3 |    2    |    3    |     3     |    18     |     24     |   11+i13 ✔️
-----------+---------+---------+-----------+-----------+------------+-----------
bispinor-4 |    2    |    3    |     3     |    18     |     24     |     ❓
===========+=========+=========+===========+===========+============+===========
     Total |    8    |   12    |    12     |    72     |     96     |     ❓

The pairwise disjoint

The Cartan–Weyl basis of the Lie algebra of SU(3) is obtained by another change of basis, where one defines The Root System for SU(3).

Note

The Lie group structure of the Lorentz group is explored. Its generators and its Lie algebra are exhibited, via the study of infinitesimal Lorentz transformations.

  • The exponential map is introduced and it is shown that the study of the Lorentz group can be reduced to that of its Lie algebra.
  • Finally, the link between the restricted Lorentz group and the special linear group is established via the spinor map.

The Lie algebras of these two groups are shown to be identical (up to some isomorphism).

270355_1_En_7_Fig1_HTML

19 + i(13+5) = 19 + i18

  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+i5
-----------+---------+---------+-----------+-----------+------------+-----------
bispinor-2 |    2    |    3    |     3     |    18     |     24     |   17+i7
===========+=========+=========+===========+===========+============+===========
bispinor-3 |    2    |    3    |     3     |    18     |     24     |   11+i13
-----------+---------+---------+-----------+-----------+------------+-----------
bispinor-4 |    2    |    3    |     3     |    18     |     24     |   19+i5
===========+=========+=========+===========+===========+============+===========
     Total |    8    |   12    |    12     |    72     |     96     |   66+i30 ✔️

A bispinor is more or less "the same thing" as a Dirac spinor. The convention used here is that the article on the Dirac spinor presents plane-wave solutions to the Dirac equation.

Note

The four pairwise disjoint and non-compact connected components of the Lorentzgroup L = O(1, 3) and corresponding subgroups:

  • the proper Lorentz group L+ = SO(1, 3),
  • the orthochronous Lorentz group L↑,
  • the orthochronous Lorentz group Lo = L↑ + ∪ TL↑+ (see below) and
  • the proper orthochronous Lorentz group L↑+ = SO+(1, 3), which contains the identity element.

Of course, the sets L↓−, L↑− and L↓+ do not represent groups due to the missing identity element. (The-four-pairwise-disjoint)

19 + 7 = 26

The-four-pairwise-disjoint-and-non-compact-connected-components-of-the-Lorentz-group-L

 Bispinors | 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     |   i5+i7 ✔️
===========+=========+=========+===========+===========+============+===========
bispinor-3 |    2    |    3    |     3     |    18     |     24     |   11
-----------+---------+---------+-----------+-----------+------------+-- 19
bispinor-4 |    2    |    3    |     3     |    18     |     24     |   i13+i5 ✔️
===========+=========+=========+===========+===========+============+===========
     Total |    8    |   12    |    12     |    72     |     96     |   66+i30
Note

Fermion particles are described by Fermi–Dirac statistics and have quantum numbers described by the Pauli exclusion principle. They include the quarks and leptons, as well as any composite particles consisting of an odd number of these, such as all baryons and many atoms and nuclei. Fermions have half-integer spin; for all known elementary fermions this is 1⁄2. In the Standard Model, there are 12 types of elementary fermions: six quarks and six leptons.

  • 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.IMG_20240108_032736
    • 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.
  • Quarks are the fundamental constituents of hadrons and interact via the strong force. Quarks are the only known carriers of fractional charge, but because they combine in groups of three quarks (baryons) or in pairs of one quark and one antiquark (mesons), only integer charge is observed in nature.IMG_20240108_033012
    • Their respective antiparticles are the antiquarks, which are identical except that they carry the opposite electric charge (for example the up quark carries charge +2⁄3, while the up antiquark carries charge −2⁄3), color charge, and baryon number.
    • There are six flavors of quarks; the three positively charged quarks are called up-type quarks while the three negatively charged quarks are called down-type quarks.

All known fermions except neutrinos, are also Dirac fermions; that is, each known fermion has its own distinct antiparticle. It is not known whether the neutrino is a Dirac fermion or a Majorana fermion.[4] Fermions are the basic building blocks of all matter. They are classified according to whether they interact via the strong interaction or not.

ElectrodynamicsThus it appears that the cosmological models] derived from compactification of 11d supergravity on a manifold with G2 holonomy have some hidden E7 symmetry.

Parsering Structure

Ploting 40th prime scheme of the three (3) layers with all the features of 3rd prime identity as explained above then they would form their recycing through the three (3) times bilateral 9 sums as shown below.

89^2 - 1 = 7920 = 22 x 360 = 66 x 120 = (168 - 102) x 120

$True Prime Pairs:
(5,$True Prime Pairs:
 (5,7), (11,13), (17,19)
 
 layer|  i  |   f
 -----+-----+---------
      |  1  | 5
   1  +-----+
      |  2  | 7
 -----+-----+---  } 36 » 6®
      |  3  | 11
   2  +-----+
      |  4  | 13
 -----+-----+---------
      |  5  | 17
   3  +-----+     } 36 » 6®
      |  6  | 19
 -----+-----+---------
  
layer | node | sub |  i  |  f                               
------+------+-----+---------- 
      |      |     |  1  | -------------------- _site ---  71 = 72-1
      |      |  1  +-----+                        |
      |  1   |     |  2  | (5)                  _saas
      |      |-----+-----+                        |
      |      |     |  3  | ---------            _data
  1   +------+  2  +-----+----      |             |
      |      |     |  4  |         5x ---       _posts
      |      +-----+-----+          |     |       |
      |  2   |     |  5  | (7) -----      |     _drafts
      |      |  3  +-----+                |       |
289+11=300   |     |  6  |                |     _plugins
------+------+-----+-----+----- 72 x 6   7x ------------ 11x = 77 (rational)◄--
      |      |     |  7  |                |     _includes                      |
      |      |  4  +-----+                |       |                            |
      |  3   |     |  8  | (11)  ---      |     _layouts                       |
      |      +-----+-----+          |     |       |                            |
      |      |     |  9  |         2x ---        assets  (69 = 72-3)           |
  2   +------|  5  +-----+-----     |             |                            |
      |      |     |  10 | ---------            _saas                          |
      |      |-----+-----+                        |                            |
      |  4   |     |  11 | (13) ----------------_site --  71 = 72-1            |
      |      |  6  +-----+                                                     |
329+71=400   |     |  12 |------------------------------  70 = 72-2            |
------+------+-----+-----+                                                    11x
      |      |     |  13 |                                                     |
      |      |  7  +-----+                                                     |
      |  5   |     |  14 | (17) ◄------------------------------------------- (17)
      |      |-----+-----+                                                     |
      |      |     |  15 | ◄-- 42 x 6 partitions of 13 (irrational)            |
  3   +------+  8  +-----+-----                                                +
      |      |     |  16 |                                                     |  
      |      |-----+-----+                                                     |
      |  6   |     |  17 | (19) ◄------------------------------------------- (19)
      |      |  9  +-----+                                                     |
168+32=200   |  |  |  18 |------------------------------  68 = 72-4            |
------|------|--|--+-----+                                                     |
       900 -----                                                               |
                                                                               |

Using the javascript library from Chevotrain and data parser from Jekyll/Liquid finally we found the correlation between the lexer and parser trough the powers of pi.

Note

In this example, the content from a Markdown document document.md that specifies layout: docs gets pushed into the {{ content }} tag of the layout file docs.html. Because the docs layout itself specifies layout: page, the content from docs.html gets pushed into the {{ content }} tag in the layout file page.html. Finally because the page layout specifies layout: default, the content from page.html gets pushed into the {{ content }} tag of the layout file default.html. (JekyllRb)

Parsering

Since the modulo 6 is occured all over the spin then we have defined that this 4 zones should stand as default configuration as you can see on the left sidebar.

Tip

In order to maintain the 18’s structure between each of repositories to correlate with the above density then we could use a hierarchical database that stores low-level settings for the operating system such as windows registry.

windows registry

Going deeper there are many things raised up as questions. So in this project we are going to analyze it using a javascript library called Chevrotain.

Note

The spin states for the powers of pi. The Prime Hexagon is an integer environment, so pi powers are truncated. I believe these data suggest prime numbers are linked in some way to pi. (HexSpin)

Lexers, Parsers and Interpreters with Chevrotain

It is going to setup CI/CD for up to 1000 public repositories out of millions that available on GitHub. You may visit our mapping scheme for more detail.

Grand Unification

The 619 is the 114th prime. By the True Prime Pairs it is laid on the last index of 6 with prime 19 where as 6x19 is also 114. Let's put 19 hexagons within the 3 layers.

168+618 - 19x6x6 = 786 - 684 = 102

entry and exit point

When recombination is occur then the prime 13 is forced to → 12 where the impact (Δ1) goes to 18+13+12=43 on the last 7th row. This sequence is simulated by a flowchart having 12 arrows flowing on 10 (ten) shapes of prime 31 up to 71 (40 nodes).

Note

Mapping the quantum way within a huge of primes objects (5 to 19) by lexering (11) the ungrammared feed (7) and parsering (13) across syntax (17) (₠Quantum).

6+6 + 6/\6 = 6+6 + 15 = 27-day month

flowchart

By this project the above would be deployed as default layout. It is set to be avalaible throughout the whole platform via a single page within a parser repository which is acting as prime 13. Their interface will be in json and xml format.

Here is for the sample:

{
  "title":"Mapping System",
  "content":"<p>Hello, <strong>world</strong>.\nI am here.</p>\n",
  "links": [
    {"title":"Introduction","url":"https://www.eq19.com/intro/"},
    {"title":"Go tour on Mapping System ","url":"https://www.eq19.com/maps/"},
    {"title":"A backed pretty display for markdown","url":"https://www.eq19.com/gistio/"},
    {"title":"Gist.io for programmers","url":"https://gist.io/@eq19/d2336e28e79702acf38edd182003d5e0"}
  ]
}

Using a kind of interface such as docker then it could be developed cross platform. Evenso. Let assume that all alpabethic letter in the sequence is representing a local disk so you may build your own pattern in your PC such as shown below:

default

By The GitHub Runner you can connect to the Google COS Instance. For self-hosted runners defined at the organization level, configure runs-on.group in your workflow file to target a runner groups or combine groups and labels.

Note

Why collaborating with physicists?

  • Contribute to the understanding of the Universe.
  • Open methodological challenges.
  • Test bed for developing ambitious ML/AI methods, as enabled by the precise mechanistic understanding of physical processes.
  • Core problems in particle physics transfer to other fields of science (likelihood-free inference, domain adaptation, optimization, etc).
  • A high-level summary of various aspects of machine learning in LHC data reconstruction, mostly based on CMS examples. A short summary of a particular use case: ML for combining signals across detector subsystems with particle flow. This talk is in personal capacity (not representing CMS or CERN), representing my biased views.

You can find a great and fairly complete overview of ML papers in HEP. (Pata Slides)

π(10) = 2,3,5,7

SO(10)

teaching-machines-glouppe_compressed.pdf

This way will also be our approach to Euler's identity. By taking the correlation between f(π) as P vs f(i) as NP where e + 1 = 0 then theoretically they shall be correlated to get an expression of the prime distribution similar to MEC30.

Note

The Minimal Supersymmetric Standard Model (MSSM) contains two Higgs doublets, leading to five (5) physical Higgs bosons:

  • one (1) neutral CP-odd (A) 👈 degenerated with (h or H)
  • two (2) charged states (H+ and H−),
  • Two (2) neutral CP-even states (h and H).

At tree-level, the masses are governed by two parameters, often taken to be mA and tan β [3]. When tan β >> 1, A is nearly degenerated with one of the CP-even states (denoted ϕ). (ScienceDirect)

the 5 cells

This also introduces a lower bound of Mod 90 originated from the 4th coupling of MEC30 which is holded by five (5) cells between 13 and 17.

Note

Below we will demonstrate how factorization algorithms and twin prime dyad cycling at the digital root level rotate the vertices of equilateral triangles within {9/3} star polygons like the one pictured above. These rotations are encoded in 3 x 3 matrices generated by period-24 digital root dyad tri-level cycling. We will also reveal the Latin Square reflecting {3,6,9} hidden in plain sight betwixt and between the twin prime distribution channels; all of its rows, columns and principal diagonals summing to 18. PrimesDemystified

19 + 18 + 102 = 37 + 102 = 139 = 34th prime = (40 - 6)the prime

exponentiation zones

The boson, photon and gravity forces are assigned to 30, 31 and 32. Gluon force and exchange are assigned to 33 and 34 which are then standing as the lexer and parser.

Tip

An overview of the various families of elementary and composite particles, and their interactions. Fermions are on the left, and Bosons are on the right.

Elementary Particle

According to the Standard Model there are five (5) elementary bosons with thirteen (13) variations. These 5 and 13 will be assigned to the “5xid’s of 31~35 (sequenced)” and “13xid’s of 36~68 (unsequenced)”, respectively (see the sidebar menu).

The exchange of virtual pions

This lead to a consequence of SU(5) grand unification (assigned to 35) showing a complex scalar Higgs boson of 24 gauge groups observe mass of W boson (assigned to 36).