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\begin{document}
\title{Quarks and other Elementary Particles in the Expanded Rishon Model}
\author{Luke Kenneth Casson Leighton}

-----------------------------------------------------------------------
\textbf{STATUS: DRAFT}
\textbf{LAST EDITED: 28jan2014}
\textbf{Copyright (C) 2014 Luke Kenneth Casson Leighton.  All Rights Reserved}
   (ask permission please: don't go copying or making random edits, basically)
\textbf{Contact: lkcl@lkcl.net}
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\begin{abstract}

\textit{Author's note 28jan2014: further analysis of the top quark's mass
indicates some flaws in some of the reasoning and deduction below, which is
still under investigation, and has yet to be written up.  the most likely
hypothesis due to its huge mass is that the top is a 27-rishon behemoth, in
the same family as the two higgs bosons.  the line of enquiry under
investigation follows the pattern of particles created by the electron-positron
collision experiments being followed similarly where proton-anti-proton
collisions are carried out, and the observation that there is a potential
candidate for "Bottom" which has a neutrino in the exact same place where the
proton has an up quark.  this leaves an unfilled hole in the 9-particle groups:
a quark which can either not exist for reasons unknown, or which is still
to be identified as of yet, or it is one which is giving the appearance of
statistical anomalies in the analysis of existing data, over the past few
decades.  In short, the Extended Rishon Model tells us that there is still an
enormous amount of work to do}.

The Rishon Model has not really been explored properly, in terms of
representation of elementary particles and their interactions through
standard decay patterns.  When this is done however a flaw is found
in the original Rishon Model: the charge of the neutrino requires
inversion.  Once this inversion is carried out it is possible to
create a consistent map of \textit{all} known particles, including
gluons, W, Z and Higgs Bosons (note: two variants of Higgs are predicted).
Decay patterns including oscillation of Strange B Mesons can be
easily and consistently explained in terms of the altered Rishon Model.

From analysing a number of standard decay patterns a new gluon-like Rishon
notation has been created (VT0 and VT*) which, whilst if not actually
present as particles, at least by way of being thought more of as "events"
represent the phase changes that result in Rishon particles transforming
into others whilst preserving phase and energy (the phase change matrix
is outlined in this paper).  Instances where the phase transforms do not
match up correlate precisely with known observations where "decay energy"
or gamma radiation is emitted.

Also proposed in this paper is the hypothesis that the mass of quarks 
(and particles) is not entirely as they appear.  Whilst the observed
mass/energy values are incontrovertibly correct, the hypothesis under
investigation is that some of the energy from electrical charge believed
to be used to influence the particle's momentum is in fact being diverted
into its \textit{angular} momentum (in a similar way to pulling a toy
gyroscope's string if the gyroscope were placed on a frictionless surface,
and then concluding that the gyroscope were heavier because it moved less).
The hypothesis is based on the quite reasonable assumption that the
energy levels of T and V particles and the interactions between them are
identical, yet because the V particles are effectively at "right-angle"
dimensions to those of V particles (in the same way that real and imaginary
numbers are at right-angles), it "appears" to take far more energy
to move compound particles containing V Rishons than it really should.
By taking either the cube root or half-cube (3/2) root of the mass
ratios of closely-related particles we come out with some ratios that
show clear patterns, and establish a correlation between the mass ratios
and the ratio of T and V particles.

These patterns in the mass ratios as well as the above hypothesis help
determine the Alternate Rishon Model representation of the larger quarks
- Strange, Charm, Top and Bottom.  Further analysis shows how to derive
the muon and the tau.  From the derivation of the tau the existence of
two previously unknown quarks (ultra-up and ultra-down) are inferred.
From there the makeup of the W, Z and Higgs Boson are derived, and an
additional particle (Higgs-0) is quite reasonably predicted to exist
(and noted to already have been experimentally observed at 126.0 GeV).
It is distinguished from the original Higgs (renamed to Higgs+) which
is believed to have been experimentally observed at 125.3 GeV.

Also a very simple potential explanation for the stable attraction-repulsion
paradox of the Nuclear Force is given, which is only possible because the
Extended Rishon Model is the only model in particle physics that contains a
simple and symmetrical framework for the internal layout of quarks.

\end{abstract}

\newpage
\tableofcontents
\newpage

\section{Expanded Rishon Model Particles}

Derivation of the Alternative Rishon Model Particles comes from endeavouring
to fit them together into compound particles.  First however the elementary
particles are shown, in Figure ~\ref{fig:fundamentals}:

\noindent%
\begin{minipage}{\linewidth}
\makebox[\linewidth]{%
   \includegraphics[width=0.8\textwidth]{rishon_fundamentals.png}}
\captionof{figure}{fundamental Rishon particles and charges}
\label{fig:fundamentals}
\end{minipage}
\linebreak

Note that the total of the T-charge and V-charge of each of the particles
is either +1 or -1.  This is a law.  No Rishon particle may exist which
has a fractional total charge.  In colour (phase) terms this is because
the three Rishon particles are three identical waves at 120 degree phases,
summing in total to zero.  A T anti-V and T particle would in effect be three
particles separated by 60 degree phases: their sum as a triplet would be
non-zero, which is not permitted.

Next is the neutron and proton, shown in Figure
~\ref{fig:protonneutronstructure}, where the up and down
quarks known to be present in neutrons and protons are expanded into their
corresponding Rishon particles:

\noindent%
\begin{minipage}{\linewidth}
\makebox[\linewidth]{%
\includegraphics[width=0.8\textwidth]{rishon_neutron_proton.png}}
\captionof{figure}{Proton and Neutron Rishon Model structure}
\label{fig:protonneutronstructure}
\end{minipage}
\linebreak

Note that the proton has, as expected, a +1T (electrical) charge and the
neutron has 0 electrical charge.  However: note that whilst the proton
has a zero V charge, the neutron has a \textit{negative} (-1) V charge.  From this
we quite reasonably deduce that the neutrino has a \textit{positive} V charge, on
the grounds that the electron has the opposite T charge from a proton.
It is worth noting that if electrons had been correctly assigned the
appropriate charge in the 1800s by its first proponents the electron
and the neutrino would both have positive charges.  Whatever the charge
polarity however it is important to note and correct the deviation from
the original Rishon Model in the neutrino's charge, in order to retain
conservation of charges in decay model interactions.

Also worth noting is how the Proton and Neutron are laid out, diagrammatically.
It is believed that this is actually how the particles generate spin
characteristics, and how the proton and neutron manage to retain cohesion.
Note in the proton how the end $\overline{V}$ particles of the down quark line up with 
the central V particles of the up quarks at either end.  In this way the
two up quarks may safely themselves spin, and at the same time the whole
"I-shaped" assembly may rotate, thus providing from an external perspective
the observed spin characteristics noted of protons in current particle
physics models.

\subsection{Rishon "I" Frame}

The question arises as to what were to happen if other Rishon particles were
placed into an I-shaped frame, applying the same rule that the ends of the
middle particle attract the middle of the end particles to create a stable
rotating whole.  Placing the Laws governing Rishons and the I-Frame rules
into a small
\hyperref{http://lkcl.net/reports/rishon_model/quark_matrix.py}{}{}{python script}
the possible patterns can easily generated, as
shown in Table 1:
\begin{alltt}
udu (['TVT', '\overline{V}\overline{T}\overline{V}', 'TVT']) t:   1 v:   0
udv (['TVT', '\overline{V}\overline{T}\overline{V}', 'VVV']) t: 1/3 v: 2/3
u\overline{v}v (['TVT', '\overline{V}\overline{V}\overline{V}', 'VVV']) t: 2/3 v: 1/3
\overline{u}\overline{d}\overline{u} (['\overline{T}\overline{V}\overline{T}', 'VTV', '\overline{T}\overline{V}\overline{T}']) t:  -1 v:   0
\overline{u}\overline{d}\overline{v} (['\overline{T}\overline{V}\overline{T}', 'VTV', '\overline{V}\overline{V}\overline{V}']) t:-1/3 v:-2/3
\overline{u}v\overline{v} (['\overline{T}\overline{V}\overline{T}', 'VVV', '\overline{V}\overline{V}\overline{V}']) t:-2/3 v:-1/3
dud (['\overline{V}\overline{T}\overline{V}', 'TVT', '\overline{V}\overline{T}\overline{V}']) t:   0 v:  -1
du\overline{e} (['\overline{V}\overline{T}\overline{V}', 'TVT', '\overline{T}\overline{T}\overline{T}']) t:-2/3 v:-1/3
de\overline{e} (['\overline{V}\overline{T}\overline{V}', 'TTT', '\overline{T}\overline{T}\overline{T}']) t:-1/3 v:-2/3
\overline{d}\overline{u}\overline{d} (['VTV', '\overline{T}\overline{V}\overline{T}', 'VTV']) t:   0 v:   1
\overline{d}\overline{u}e (['VTV', '\overline{T}\overline{V}\overline{T}', 'TTT']) t: 2/3 v: 1/3
\overline{d}\overline{e}e (['VTV', '\overline{T}\overline{T}\overline{T}', 'TTT']) t: 1/3 v: 2/3
\overline{e}e\overline{e} (['\overline{T}\overline{T}\overline{T}', 'TTT', '\overline{T}\overline{T}\overline{T}']) t:  -1 v:   0
e\overline{e}e (['TTT', '\overline{T}\overline{T}\overline{T}', 'TTT']) t:   1 v:   0
v\overline{v}v (['VVV', '\overline{V}\overline{V}\overline{V}', 'VVV']) t:   0 v:   1
\overline{v}v\overline{v} (['\overline{V}\overline{V}\overline{V}', 'VVV', '\overline{V}\overline{V}\overline{V}']) t:   0 v:  -1

Table 1: permutations of all legitimate I-Frame Rishon particles
\end{alltt}
Dividing these down into clearly-distinguishable groups, exactly half the
particles are of unit charge and exactly half are fractionally-charged.
Categorising further we note that "udu" and "dud" and their corresponding
anti-particles are the neutron and proton respectively.  Two remaining
unit-charge particles (and their anti-particles) then stand out:
\begin{alltt}
muon        : \overline{e}e\overline{e} (['\overline{T}\overline{T}\overline{T}', 'TTT', '\overline{T}\overline{T}\overline{T}']) t:  -1 v:   0
muon-neut'o : v\overline{v}v (['VVV', '\overline{V}\overline{V}\overline{V}', 'VVV']) t:   0 v:   1
\end{alltt}
These particles have unit charges and are comprised of electron-positron
and neutrino-anti-neutrino triplets.  The neutrino-anti-neutrino triplet
is surmised to be a flavour of neutrino (muon neutrino), and likewise
the electron-positron triplet a muon.  Initial analysis had these hard
to pin down.  A potential candidate came up when searching for "heavy
electron": a paper mentioning <a href="http://phys.org/news136648330.html">Superconductivity and heavy electrons"</a>.  
However a decay pattern was noted involving pions (see examples later)
where the electron-positron and neutrino-anti-neutrino triplets were the
clear candidates when considered at the Rishon level: further investigation
showed this to be the muon.  \textit{(There was one other candidate found: the tau, 
which is investigated later).}

It is worth noting that if these heavy neutrinos existed, their fundamental
composition (being in effect made out of particle-anti-particle triplets)
would make neutrino oscillation very easy to occur.  The exchange of only
one particle between a heavier neutrino and its anti-particle would, at
the Rishon level, automatically result in the transition).  So if the
combination of two neutrinos and one anti-neutrino is considered to be
a heavy neutrino (muon neutrino) its composition consisting \textit{of}
neutrinos starts to shed some light on why neutrino oscillation occurs.

This leaves the remaining four particles (with corresponding anti-particles)
to identify, which are grouped together with fractional charges.  The
inference which leads to the derivation of these as strange, charm and
bottom will be explained below:

\begin{alltt}
    strange      : de\overline{e} (['\overline{V}\overline{T}\overline{V}', 'TTT', '\overline{T}\overline{T}\overline{T}']) t:-1/3 v:-2/3
    charm        : \overline{d}\overline{u}e (['VTV', '\overline{T}\overline{V}\overline{T}', 'TTT']) t: 2/3 v: 1/3
    bottom       : \overline{u}\overline{d}\overline{v} (['\overline{T}\overline{V}\overline{T}', 'VTV', '\overline{V}\overline{V}\overline{V}']) t:-1/3 v:-2/3
    unidentified : u\overline{v}v (['TVT', '\overline{V}\overline{V}\overline{V}', 'VVV']) t: 2/3 v: 1/3
\end{alltt}

Two particles here are noteworthy: strange and the unidentified quark, in that
they are merely down and up particles with matter-antimatter electrons and
neutrinos in the remaining I-frame places, resulting in additional mass,
perfect balance as well as overall charge conservation.

\newpage
\subsection{Repeated (recursive) application of the I-Frame structure}

Note that all of these sixteen particles respect the I-Frame rules: central
Rishon particle of the end triplets is attracted to the end Rishons of the
middle triplet.  Under this extended Rishon Model there are \textit{not} anticipated
to be any other 9-Rishon particles like this.  The only possibility is that a
further tripling \textit{might} be considered.  One possibility is that the tau is
in fact one such multi-level I-Frame within an I-Frame particle:

\noindent%
\begin{minipage}{\linewidth}
\makebox[\linewidth]{%
\includegraphics[width=0.8\textwidth]{rishon_tau_minus.png}}
\captionof{figure}{Possible multi-level I-Frame candidate for Tau-}
\end{minipage}
\linebreak

In this picture is shown in effect a muon - itself an I-Frame particle - in
the central position of an outer I-Frame, with two surrounding positrons.
Given that Tau particles were detected in experiments from 1974 to 1977
using electron-positron colliders, it makes an inordinate amount of sense
that the Tau would in fact be comprised, internally and exclusively, of
electrons and positrons.  Re-examining the original data, if the energies were
sufficient, there may be evidence for the following particle as well:

\noindent%
\begin{minipage}{\linewidth}
\makebox[\linewidth]{%
\includegraphics[width=0.8\textwidth]{rishon_double_tau_minus.png}}
\captionof{figure}{Possible multi-level I-Frame undiscovered particle?}
\label{fig:doubletau}
\end{minipage}
\linebreak

Looking closely at the arrangement here: although the two outer muons overall
have a +1 T charge, the centre of those two outer muons comprises a positron,
which is exactly what the outer particles of the inner muon are: also positrons.
Thus we conclude that this particle would be highly unstable.

Also a very remote outside possibility to consider would be three muon+
particles, with attraction of the ends of the middle attracting to the
middle of the outers (the basic I-Frame pattern).  Two things mitigate
against this, though: first is the total charge of +3, and second would
be that the ends of the outer muons would be too close to the ends
of the inner muon.  With all the end particles being positrons it's
hard to imagine how the repulsive forces would permit these to remain
in close proximity.  This paradox is explored later (see section on
proton-neutron proximity).

There may also exist combinations involving a central single TTT or anti
particle (electron or positron) surrounded in effect by a pair of muons.
However as the size of the muons (as well as their mass / energy) would
dwarf that of the electron in the centre of the outermost I-Frame it seems
intuitively quite unlikely that such a particle would exist (or if it did,
it would be exceptionally unstable).  Additional factors mitigate against
the existence of such a particle: how would the central electron be able
to rotate freely, when surrounded by two muons?  The more consideration
is given to this muon-electron-muon I-Frame layout, the less likely it seems
to be practical.

That does not however entirely rule out the possibility of other double-stacked
I-Frame layouts.  This will be dealt with later.

\subsection{Quark Flavours}

Identifying the four I-frame quarks is done through logical deduction, based
on the hypothesis that the energy of individual T and V particles are exactly
the same.  Piotr Zenczykowski <a href="http://arxiv.org/abs/0803.0223">outlines an O(6) phase space</a> using Clifford
Algebras.  We surmise that:
\begin{itemize}
\item T and V particles both have 3-dimensional presence and identical energy
\item T and V particles interact in phases to give the four main Rishons
\item The T and V particles also have identical fundamental mass...
\end{itemize}

... yet as only "T" (electrical charge) can interact with T particles,
the additional mass of the V particles being dragged around results in
an incorrect \textit{inference} that the larger compound particles have
a higher mass.  The most immediate example of these is the neutron and the
proton.  A neutron is not susceptible to movement by electrical charge.
It would therefore be reasonable (but logically incorrect) to assume that,
because it cannot be moved by an electrical charge it must automatically
have an infinite mass! Clearly, however, this is not true: we know instead
that its mass is near-identical to that of a proton.

In the Rishon Model, both the neutron and the proton comprise nine
T and V particles in total.  It is therefore quite reasonable to
postulate a hypothesis that the T and V particles are of equal energy.
However, what possible reasonable mathematical deductions could be put
forward which support this hypothesis?  Surmising that the deductive
process for calculating the mass of elementary particles from their
trails in particle physics chambers under the influence of electrical
fields is based on cubic and square laws, the ratios of masses of various
particles was put through a cube-root and square-root wringer.  By a
process of trial and error, some clear patterns emerged.  The following
figures for mass for various particles were used:
\begin{alltt}
    electron: 0.510998 MeV (T-charge)
    neutrino: ~0 (T-charge)
    muon    : 0.105658 GeV
    proton  : 0.938272 GeV
    neutron : 0.939565 GeV
    tau     : 1.7768 GeV
    up      : 2.01 MeV (from Lattice QCD calculations)
    down    : 4.79 MeV (from Lattice QCD calculations)
    strange : 90 MeV (from "Hadronic Physics from Lattice QCD"
    charm   : 1.29 GeV
    bottom  : 4.65 GeV (or 4.18 GeV)
    top     : 173.07 GeV
    higgs   : 125.3 GeV (or 126.0 GeV)
\end{alltt}

Starting with the lower-energy particles (and using python as a calculator)
we take the first example.

\textbf{electron and the up quark:}
\begin{alltt}
    >>> x = 2.01 / 0.511
    >>> x
    3.933
    >>> pow(x, 0.6666667)
    2.4918
    >>> pow(x, 0.6666667) * 2
    4.984
\end{alltt}

So the 2/3s root of the mass ratios of the up quark and the electron is
very close to 5/2 (within 0.658\%)

\textit{Note: the mass of the electron itself can be expressed in terms
of the speed of light, planck's constant and simple integers.  The figure
given is accurate to \textbf{six} decimal places}

\begin{alltt}
    >>> c = 2.99792458e8 # speed of light
    >>> h = 6.6260695729e-34 # planck's
    >>> mq = h / (c*c)
    >>> mev = 5.609588848490029e+29 # kg to MeV/c-squared
    >>> star = 1+1/(pow(math.pi*11/3, 2.5))
    >>> melectron = mq * (pow(c, 2.5) / (4 * math.pi * star))
    >>> melectron * mev / 0.51099892811
    1.00000978166
\end{alltt}

\textbf{electron to down quark:}

\begin{alltt}
    >>> x = 4.79 / 0.511
    >>> x
    9.374
    >>> pow(x, 0.666667)
    4.4457
    >>> pow(x, 0.6666667) * 9
    40.011
\end{alltt}

So the 2/3s root of the mass ratios of the up quark and the electron is
very close to 4*10/9 (within 0.029\%)

\textit{Note: also of note is that the cubic root of the mass ratio of the
up quark and the electron is close to (11 * e) / (7 * 9), to within 0.073\%.
The exact significance of this is not known.}

\textbf{up quark to down quark:}

\begin{alltt}
    >>> x = 4.79 / 2.01
    >>> x
    2.383
    >>> pow(x, 0.3333333)
    1.3357
    >>> pow(x, 0.3333333) * 3
    4.0071
\end{alltt}

So the cubic-root of the mass ratios of the up quark and down quark is
very close to 4/3 (within 0.18\%)

\textbf{muon and the electron:}

    >>> x = 0.105658 / 0.000511
    >>> x
    206.767
    >>> pow(x, 0.666666667)
    34.96667
    >>> pow(x, 0.666666667) / 35
    0.999048

So the 2/3s root of the mass ratios of the muon and the electron is
very close to 35/1 (within 0.095\%).

Note also the following interesting observation:

\begin{verbatim}
    >>> pow(x, 0.5) / golden_mean_ratio * 9
    79.983
\end{verbatim}

So the square root of the mass ratios of the muon and the electron
is 80 * golden mean ratio, divided by 9 (to 4 decimal places).

\textbf{proton and the electron:}

\begin{verbatim}
    >>> import math
    >>> c = 2.99792458e8
    >>> bmr_proton = 0.00152103220215
    >>> x = 938.272/.510998
    >>> starpiproton = (1 + (math.pi * bmr_proton /1.057))
    >>> print "star pi bmr_proton", starpiproton
    >>> starpiproton = 1+1/pow(math.pi*7.7/4, 3)
    >>> print "star pi bmr_proton", starpiproton
    >>> print x  / pow(c, 0.333333333333) / math.pi / math.pi * 4 * 9 / 10
    >>> print x  / pow(c/math.pi, 0.333333333333) / 4 / starpiproton
    1.00000037067
\end{verbatim}

So, to \textit{six decimal places}, the ratio of the mass of the proton
and the mass of the electron can be expressed purely in terms of pi,
the speed of light, a few cubic expressions, some small integers and the
number 7.7 (which is 8.0 - 0.3).

\textbf{tau and the muon:}

    >>> x = 1.7768 / 0.105658
    >>> x
    16.817
    >>> pow(x, 0.666666667)
    6.5338
    >>> pow(x, 0.666666667) * 8
    52.5

So the 2/3s root of the mass ratios of the tau and the muon is
very close to 16/105 (within 0.020\%).  Note also that the square root
of the tau to muon ratio is almost exactly 4.1 (within 0.019\%)

\textbf{proton and the muon:}

    >>> x = 0.938 / 0.105658
    >>> x
    8.877699
    >>> pow(x, 0.666666667)
    4.28746
    >>> pow(x, 0.666666667) * 7
    30.01223

So the 2/3s root of the mass ratios of the muon and the proton is
very close to 7/30 (within 0.041\%).  Note also that the square root
of the proton to muon ratio is very close to 3 (within 0.68\%)

\textbf{proton and the tau:}

    >>> x = 1.7768 / 0.938 
    >>> x
    1.8942
    >>> pow(x, 0.666666667)
    1.5309
    >>> pow(x, 0.666666667) * 32
    48.99033

So the 2/3s root of the mass ratios of the tau and the proton is
very close to 7*7/32 (within 0.020\%).  Note also that the square root
of the proton to muon ratio is very close to 11/8 (within 0.096\%)

\textbf{proton and the charm quark:}

    >>> x = 1.29 / 0.938
    >>> x
    1.375
    >>> pow(x, 0.666666667)
    1.237
    >>> pow(x, 0.666666667) * 4
    4.947

So the 2/3s root of the mass ratios of the charm quark and proton is
very close to 5/4 (within 1.06\%)

\textbf{Bottom and charm quark (upper, larger measured mass figure for bottom):}

    >>> x = 4.65 / 1.29
    >>> x
    3.605
    >>> pow(x, 0.666666667)
    2.351
    >>> pow(x, 0.666666667) * 3
    7.0528

2/3 root mass-ratio bottom (larger estimate) to charm: 7/3

\textbf{Bottom and charm quark (lower, smaller measured mass figure for bottom):}

    >>> x = 4.18 / 1.29
    >>> x
    3.240
    >>> pow(x, 0.333333333)
    1.480
    >>> pow(x, 0.333333333) * 2
    2.960

cubic root mass-ratio bottom (lower estimate) to charm: 3/2 (within 1.35\%)

    >>> x = 4.18 / 1.29
    >>> x
    3.240
    >>> pow(x, 0.5)
    1.800086
    >>> pow(x, 0.5) * 5
    9.00043

square root mass-ratio bottom (lower estimate) to charm: 9/5 (within 0.0048\%)

\textbf{Bottom and proton (lower, smaller measured mass figure for bottom):}

    >>> x = 4.18 / 0.938
    >>> x
    4.456
    >>> pow(x, 0.5)
    2.11099
    >>> pow(x, 0.5) * 9 / 19
    0.999943

square root mass-ratio bottom (lower estimate) to proton: 19/9 (within 0.0056\%)

\textbf{top and bottom (lower mass figure for bottom):}

    >>> x = 173.07 / 4.18
    >>> x
    41.404
    >>> pow(x, 0.666666667)
    11.968

2/3s root mass-ratio top to bottom (lower estimate): 12 to 1 (within 0.26\%)

    >>> x = 173.07 / 4.18
    >>> x
    41.404
    >>> pow(x, 0.5)
    6.435
    >>> pow(x, 0.5)*7/45.
    1.00094

square root mass-ratio top to bottom (lower estimate): 7 to 45 (within 0.094\%)

\textit{also noteworthy is that the cube-root of the two same mass ratios
is (e * 11 / 14) to within five decimal places (0.0034\%).}

\textbf{strange to charm:}

    >>> x = 1.29 / 0.090
    >>> x
    14.333
    >>> pow(x, 0.3333333)
    2.429
    >>> pow(x, 0.3333333)*7
    17.004

cubic root mass-ratio of charm to strange: 17/7 (within 0.022\%)

\textit{also noteworthy is that the cube-root of the two same mass ratios
is (e * 5) / (3 * 11) to within four decimal places (0.046\%).}

\textbf{higgs to top:}

    >>> x = 173.07 / 126
    >>> x
    1.37357
    >>> pow(x, 0.3333333)
    1.112
    >>> pow(x, 0.3333333)*9
    10.0044

cubic root mass-ratio of top to higgs: 10/9 (within 0.044\%)
\textit{Note: the exact significance of this ratio is unclear as the
composition of the Higg's Boson in terms of Rishon particles has
yet to be analysed.}

\textbf{mass of proton to sum of mass of its quarks:}

    >>> 0.00479 * 2 + 0.00201
    0.01159
    >>> x = 0.938 / 0.01159
    >>> x
    80.932
    >>> pow(x, 0.3333333)
    4.326
    >>> pow(x, 0.3333333)*3
    12.977

cubic root mass-ratio of proton to sum of its quark mass: 13/3 (within 0.18\%)
\textit{Note: the exact significance of this ratio is unclear.}

\subsubsection{Analysis of mass-ratio figures}

Some of these ratios are extraordinarily close, and are close enough
within the experimental +/- figures for some of them to be considered
as exact ratios: the top to higgs figure and the charm to strange for
example are close to the exact ratios by an accuracy of over 1 in 2,000.
To understand the significance of these ratios however it is necessary
to also take a comparative close look at the ratio of T and V composition
of the particles.

Bear in mind that the hypothesis is that the mass of each family of particles
is identical: we therefore expect there to be a direct relationship between
the ratio of T composition of any given pair of particles and the mass
ratio of the same pair.

Looking for example at the charm quark and the proton, the charm quark has
six T particles.  If the charm quark has a further three V particles which
cannot in any way be influenced by electrical activity, we expect a ratio
of 9/6 (1.5) to factor somewhere in the over-calculation of the charm
quark's mass (relative to the proton).  The 2/3s root comes out at 5/4 (1.25).

In each case, although there is a tantalising relationship clearly present,
what can definitely be said is that the more T particles there are, the
lighter the "apparent" mass of the particle.  Put another way: the more V
particles, the heavier the "apparent" mass of the particle. This even applies
when comparing the strange quark to the charm.  Numerical coincidences abound:
the cubic root of the mass ratios of strange to charm is 17/7 to within
4 decimal places, and there are seven T particles in a strange quark.

\begin{alltt}
    Name      T  V T-charge Mass         Mass * T-charge

    Electron 3T 0V    3/3     0.511 MeV      0.511 MeV
    Up       2T 1V    2/3     2.01  MeV      1.340 MeV
    Down     1T 2V    1/3     4.79  MeV      1.596 MeV

    Electron 3T 0V    3/3     0.511 MeV      0.511 MeV
    Muon     9T 0V    3/3     0.106 GeV      0.106 GeV
    Tau     15T 0V    3/3     1.777 GeV      1.777 GeV

    Strange  7T 2V    1/3     0.09  GeV      0.03  GeV
    Charm    6T 3V    2/3     1.29  GeV      0.86  GeV
    Proton   5T 4V    3/3     0.938 GeV      0.938 GeV
    Bottom   3T 6V    1/3     4.18  GeV      1.393 GeV
    Top      2T 7V    2/3   173.07  GeV    115.38  GeV
\end{alltt}

This table ranks the particles in order of mass times the electrical
charge of the particle.  In each group it can be clearly seen that the
number of T particles is in descending order. (Note: the Muon, Tau and
electron are separated out, being entirely comprised of one or more
electrons and zero or more positrons.  This is still under investigation).

\textit{Note: Here we see some striking coincidences as well:}
\begin{verbatim}

Mass of proton divided by Strange quark is 10.42, which is close to 7 * 3 / 2
(10.5).  seven T particles, 2 V particles, 1/3 electrical charge on the strange
quark.

Mass of up quark divided by electron is 3.933 (close to 4).  three T
particles, 2 V particles, 2/3 electrical charge: 3 * 2 * 2/3 is 4.

Mass of down quark divided by mass of electron is 9.373 (close to 9).
down quark has a 1/3 electrical charge.

2/3 Mass of charm quark divided by proton is 11/12 to within 4 decimal places.
Charm's T-particle content plus Proton's T-particle content: 11.
Charm's V-particle content times Proton's V-particle content: 12.

TODO: examine these relationships further.
\end{verbatim}

\subsubsection{Notes}

Still under investigation: 

\begin{verbatim}

\textbf{Muon : tau}

\textbf{Top : charm}

pow(173.07 / (4.18/3), 0.5) * (9./25) * (4./16)
= 0.9982 (1 / 1.00183)

pow(173.07 / (4.18/3), 0.5) * golden_mean_ratio / 18
= 0.9970 (1 / 1.00305)

\textbf{Muon : proton}

pow(0.106 / .938, 0.5) * (9./25) * (25/3.)
= 1.00686

\textbf{Top : proton}

pow(173.07 / .938, 0.5) * (9./25) * (11./54)
= 0.9961 (1 / 1.00390)

\textbf{Bottom : proton}

pow((4.18/3) / .938, 0.5) * (9./25) * (20.5/9)
= 0.99940 (1/1.00060)

pow((4.18/3) / .938, 0.5) * pow(golden_mean_ratio, 0.5) / (14/9.)
= 1.00338

\textbf{Proton : charm}

pow((1.29*2/3) / .938, 0.5) * (9./25) * (26./9)
= 0.9958 (1/1.00420)

\textbf{Proton : strange}

pow((0.09/3) / .938, 0.5) * (9./25) * 15.5
= 0.99791 (1/1.0021)

\textbf{Up : electron}

pow((4.79/3)/0.511, 0.5)* (9./25) * (11./7)
= 0.999986 (1/1.000014)

pow((4.79/3)/0.511, 0.5)*(math.pi/2)*9/25
= 0.99958 (1/1.0004)

\textbf{Down : electron}

pow((2.01*2/3) / 0.511, 0.5) * (25/9.) * (2/9.)
= 0.99960

pow((2.01*2/3) / 0.511, 0.5)
= golden_mean_ratio

\textbf{Up : down}
pow((4.79/3) / (2.01*2/3), 0.5) = 11/12

\textbf{Table of Quark : Proton ratios}

From the above data it's clear that there's a definite correlation
(muon being the exception)

    Quark       Root2 Mass Ratio          T    V   T-charge  V-charge

    Muon        1/25 * 450/6 ( 75.0  )    9    0     +1       0   
    Strange     1/25 * 837/6 (139.5  )    7    2     -1/3     -2/3
    Charm       1/25 * 156/6 ( 26.0  )    6    3     +2/3     +1/3
    Proton      1/25 * 150/6 ( 25.0  )    5    4     +1       0
    Bottom      1/25 * 123/6 ( 20.5  )    3    6     -1/3     -2/3
    Top         1/25 *  11/6 (  1.833)    2    7     +2/3     -1/3

\textit{Table 2: Quark:proton ratios}

x0 = pow(75, 0.5)
= 8.660
x1 = pow(139.5, 0.5)
= 11.811
x2 = pow(26.0, 0.5)
= 5.099
x3 = pow(25, 0.5)
= 5.0
x4 = pow(20.5, 0.5)
= 4.527
x5 = pow(1.833333333333333, 0.5)
= 1.354

x0/x1 = 44/60
x1/x2 = 60./26
x2/x3 = 60./59
x3/x4 = 60./54
x4/x5 = 60./18

x0/x3 = 86.6 /50 (1.73)       86.6
x1/x3 = 116  /50 (2.32)       116
x2/x3 = 51   /50 (1.02)   x2  102
x3/x3 = 50   /50 (1.00)        50
x4/x3 = 45   /50 (0.90)   x2   45
x5/x3 = 13.54/50 (0.2708)      27

TODO: investigate further
\end{verbatim}

\subsection{Deductions from Quark mass figures}

Identifying the Rishon patterns as being particular quarks is a puzzling
conundrum.  Initially it was believed that the top quark is the "unidentified"
one (neutrino-anti-neutrino along with an up quark).  However, the mass is
far too large, and needs to be covered separately.  The one identified as
"strange" has an electron and positron in two of its places: its mass is very
similar to the muon (within 5 percent).  With two of the possibilities having
-1/3T -2/3V charge and one of those taken by strange, the other we deduce is
Bottom.  With two options to choose from for charm, it becomes necessary to
analyse particle decays and infer which is logically the correct option.
This is done in a later section once the mechanism of decay (actually,
phase-transitions) is explained.

Two -1/3T -2/3V quarks.
\begin{alltt}
    strange      : de\overline{e} (['\overline{V}\overline{T}\overline{V}', 'TTT', '\overline{T}\overline{T}\overline{T}']) t:-1/3 v:-2/3
    bottom       : \overline{u}\overline{d}\overline{v} (['\overline{T}\overline{V}\overline{T}', 'VTV', '\overline{V}\overline{V}\overline{V}']) t:-1/3 v:-2/3
\end{alltt}

The two as-yet not clearly identified remaining quarks:
\begin{alltt}
    charm        : \overline{d}\overline{u}e (['VTV', '\overline{T}\overline{V}\overline{T}', 'TTT']) t: 2/3 v: 1/3
    unidentified : u\overline{v}v (['TVT', '\overline{V}\overline{V}\overline{V}', 'VVV']) t: 2/3 v: 1/3
\end{alltt}

\subsection{Patterns in Magnetic Moment of particles}

TODO: expand this out.

\begin{verbatim}

Bohr ratio Magnetic moment of muon: -4.84197044e-3
Bohr ratio Magnetic moment of electron -1.0011596521869
>>> x = -1.0011596521869 / -4.84197044e-3
>>> x
206.766989719
>>> x/math.pi/math.pi/math.pi*3/20.
1.0002829032857001

Ratio of magnetic moment of electron to muon is pi-cubed times 20 divided by 3
to within 0.03%

\end{verbatim}

\section{Decay Patterns as Phase Transitions}

As this is the Rishon level it should come as no surprise that there are
patterns to particle decay which go down to the level of Rishons.  There
is nothing inconsistent with Standard Models: it's just different yet
also conserves Rishon charge and phase, as well as continuing to respect
Standard Model rules - just in terms \textit{of} Rishon Model rules.

Analysis of several decay patterns shows that it is possible for the
base triplet Rishons (of which there are 8 if the anti-particles are
included) to phase-shift in two distinct groups.  Also, it has been noted
that two opposing phase-shifts must occur at the same time, where one
phase-shift is the direct opposite of the other.  This makes sense as far
as conservation of energy is concerned.  The two groups are as follows:

\begin{alltt}
    V\overline{T}0 ->
            \overline{V}\overline{T}\overline{V} + VVV
        or  \overline{T}\overline{T}\overline{T} + TVT
        or  \overline{T}\overline{V}\overline{T} + VTV

    \overline{V}T0 ->
            \overline{V}\overline{T}\overline{V} + TVT
        or  \overline{T}\overline{V}\overline{T} + TTT
        or  \overline{V}\overline{V}\overline{V} + VTV
\end{alltt}

In other words: in each group, the difference between the left-hand side of
the equation and the right is one particle-anti-particle pair of T and V.
Two such exchanges (one from each group) \textit{must} occur together.

Put as a law:

    \textit{A pair of Rishon triplets may interchange one particle of one type
    of matter (T with its anti-particle or V with its anti-particle) whilst
    also simultaneously interchanging the other type's anti-particle 
    (anti-V with V or anti-T with T)}

Whether this is what is actually occurring at the physical level is almost
irrelevant: the mathematical differences are what counts.  However what is
believed to be the case is that V$\overline{T}$0 and $\overline{V}$T0 represent phase shifts rather than
being actual "particles".  The exact allowed time difference allowed in between 
when these phase shifts are permitted to occur is yet to be determined, if
there is in fact any time difference permitted at all: it is anticipated that
the time is zero, otherwise energy is not properly conserved.

As an example: a neutrino and a down-quark (from the first group) may be
permitted to phase-shift to an electron and an up quark (again, first group)
\textit{as long as} \textbf{simultaneously} a second-group pair also phase-shifts, for example
an anti-up quark and an anti-electron phase-shift to an up quark and a
down quark:

\begin{alltt}
    \overline{V}\overline{T}\overline{V} + VVV -> \overline{V}T0 -> \overline{T}\overline{T}\overline{T} + TVT
    
    \overline{T}\overline{V}\overline{T} + TTT -> V\overline{T}0 -> \overline{V}\overline{T}\overline{V} + TVT

	\textit{    Showing a pair of matched phase transitions}

\end{alltt}


Examine those two equations above carefully.  In both cases, subtracting the
left-hand (or right-hand) two triplets from each other leaves the total number of Rishons as being the \textit{centre} i.e. $\overline{V}$T0 in the first equation
and V$\overline{T}$0 in the second.  Making that clearer:

\begin{alltt}
    \overline{V}\overline{T}\overline{V} XOR VVV = \overline{V}T = \overline{T}\overline{T}\overline{T} XOR TVT
    
    \overline{T}\overline{V}\overline{T} XOR TTT = V\overline{T} = \overline{V}\overline{T}\overline{V} XOR TVT
\end{alltt}

Where the Rishon triplets come from (compound particles, gluons) is
irrelevant (i.e. which four triplets are involved): the phase shifts
result in what is known in the Standard Models as "particle decay".
The only other rule is also in the Standard Models: Rishon triplet and
anti-triplet pairs may either be brought into existence for very short
durations of time (the process is tentatively named "gluons"), or they
may annihilate to create gamma radiation, or, if sufficient energy is
input a triplet-anti-triplet virtual pair (typically Pions) may be separated
such that, in combination with paired phase-changes interacting with
surrounding matter, other larger particles may be created.

\textit{It's worth noting that the VT0 phase changes are likely to be made
clear in Clifford Algebra, as described in Piotr Zenczykowski's paper on
<a href="http://arxiv.org/abs/0803.0223">O(6) phase space</a>.  It is suspected that the phase changes would be a
simple pair of matching matrix transformations.}

In diagrammatic form, here are the transformations that can take place:

\noindent%
\begin{minipage}{\linewidth}
\makebox[\linewidth]{%
  \includegraphics[width=0.8\textwidth]{rishon_transformations.png}}
\captionof{figure}{Chart showing the chain of permitted transformations}
\label{fig:transformchart}
\end{minipage}
\linebreak


Bear in mind that the transformations may only take place in matched pairs.
Note that the electron and neutrino (and their anti-particles can \textit{only} each
undergo a phase shift to one other particle!  Also shown on the chart (which
becomes useful later) are the Pions.  The Pions are the only particles which
may be involved in back-to-back phase transformations (thanks to their
constituent parts having two possible paths each).  The Standard Model calls
these back-to-back phase shifts "gluons" and "W-Bosons".

\subsection{Example particle decay (phase shift): neutron}

Particle phase transitions in the Rishon Model are quite detailed, making
a first example quite awkward to choose.  Primarily this is down to the
VT0 transitions needing to occur in pairs.  Starting with what is termed
a "decay pattern" of the neutron into a proton, anti-neutrino and an
electron, we have:
\begin{alltt}
    n -> p + e + \overline{v}e
\end{alltt}
Expanded into quarks this is:
\begin{alltt}
    dud -> udu + e + \overline{v}e
\end{alltt}
Expanded into Rishon particles is best done diagrammatically (assisted by
a mathematical switch of the neutrino as follows):
\begin{alltt}
    ve + dud -> udu + e
\end{alltt}
In diagrammatic form that transformation is as follows:

\noindent%
\begin{minipage}{\linewidth}
\makebox[\linewidth]{%
\includegraphics[width=0.8\textwidth]{rishon_neutron_transition.png}}
\captionof{figure}{Neutron phase-transition to proton}
\end{minipage}
\linebreak

At the top left of the diagram, a neutrino-anti-neutrino pair are brought
into existence (explained later).  The down quark and the neutrino
undergo a $\overline{V}$T0 phase transition into an up quark and an electron.
\textit{At the same time} the remaining two neutron quarks undergo an opposite
phase transition V$\overline{T}$0, but,
unusually, transform back into the exact same two particles they originally
were (u and down quarks) - just in different places.  This allows the positions 
of the two particles to change places in the I-Frame with minimal disruption to
the structure, and, importantly, ensuring that the I-Frame's rules are also
respected.

So, the two VT0 phase transforms, being direct opposites and representing the
exchange each of one T and one V particle with an anti-T and an anti-V particle,
effectively cancel each other out.  No energy is lost or gained in the process. 
The concept of "decay" is therefore considered, under this model, to be rather
strange and archaic \textit{(which leaves the only place where decay actually occurs as
being when a particle meets an anti-particle and annihilate to create a photon
in the form of gamma radiation).}

After the phase transforms occur, it is worthwhile adding up the total
number of T-anti-T and V-anti-V particles to ensure that they are the
same: we find that the total of each of the two charges is the same both
before and after, thanks to the paired VT0 phase transforms preserving
the totals of each charge.

It is worthwhile noting that whilst these phase transition (pairs of) are the
simplest that could be found, they mathematically represent in effect the \textit{total}
phase transition that occurred.  Other such phase transitions may in fact
exist which more accurately represent what actually occurs.

\subsection{B Meson Oscillation}

This section explains how a B Meson (B-0) is able to phase-shift into its
own anti-particle (and thus back again).  The transition occurs with the
brief assistance of a Pion-0 (up quark flavour).

\noindent%
\begin{minipage}{\linewidth}
\makebox[\linewidth]{%
  \includegraphics[width=0.8\textwidth]{rishon_b_meson_oscillation.png}}
\captionof{figure}{Showing how a B meson transforms to its own anti-particles}
\end{minipage}
\linebreak


In the top left corner is an anti-strange quark.  To the top right is a bottom.
First, the anti-strange quark's neutrino and down quark phase-shift to create
the electron (which stays in-place in what becomes the anti-bottom), as well
as an up quark from the temporarily-created Pion-0.  In the opposing phase-shift
the remaining up quark from the anti-strange and the anti-down quark from
the anti-bottom phase-shift to create the anti-up of the temporary Pion-0
as well as the anti-electron of the anti-bottom.

The first of the second pair of phase transitions involves the up-quark of
the temporary Pion-0 and the electron of the anti-bottom, to create the
anti-up of the strange and the anti-down of the anti-bottom.  The matching
phase transition takes the anti-up of the temporary Pion-0 and the positron
of the anti-bottom to create the down and anti-down of the strange.

It is likely that all of these phase transitions occur simultaneously, as
otherwise each of the Pion-0's constituent particles would be created and then
transformed in time-frames that were out of sync.

Again, as with all these phase-transformations, there may be other simpler
patterns or even patterns that better match what is really going on, or
there may be other patterns that are mathematically equivalent, however
this set of four phase transitions involving only one temporary particle
appears at least to be logically consistent and fairly straightforward.

However, an important question is raised here: why did the Pion-0 temporarily
come into existence, and what does that mean?  The temporary Pion-0 is, in
terms of the Standard Model, a gluon.  It was created and destroyed within
under one oscillation of the wavelength of Rishon Particles, and it merely
permitted, facilitated or represents a slightly more complex phase shift.
Note that the temporary creation of the Pion-0 really \textit{is} temporary: it is
both the input \textit{and} output of other VT0 phase-shifts.

\subsection{Pion+ phase-transition into Muon and Muon-neutrino}

This phase transition is extraordinarily elegant.  A "trick" occurs which
is likely to be entirely unique to the Pion+ \textit{(note: it is not - the
same trick is used in Kaons: see below).}  Begin from the less-probable
phase transition as it is simpler to describe: Pion+ transitioning into an
electron an an anti-neutrino:
\begin{alltt}
    pi+ -> e+ + ve
\end{alltt}
In quark terms that is as follows:
\begin{alltt}
    u\overline{d}  -> e+ + ve
\end{alltt}
Taking the Rishon representations from figure ~\ref{fig:fundamentals}, and
noting that ve is VVV, but that a positron - not an electron - is TTT),
in Rishon terms that becomes:
\begin{alltt}
    TVT:VTV -> TTT + VVV
\end{alltt}
The transformation is effected by one V$\overline{T}$0 and one $\overline{V}$T0
set of phase-changes, but with a twist that is best shown as a drawing.  
Diagrammatically,  then, the phase transition is shown in figure 
~\ref{fig:pionsimpletransition}:

\noindent%
\begin{minipage}{\linewidth}

\makebox[\linewidth]{%
\includegraphics[width=0.8\textwidth]{rishon_pion_simple_transition.png}}
\captionof{figure}{Pion phase-transition to positron (e+) and neutrino (ve)}
\label{fig:pionsimpletransition}
\end{minipage}
\linebreak

Here we have something that at first glance seems very strange: two
non-matching intermediate particles (anti-u and a down quark)
appear to have been brought into existence for the purposes of
satisfying some arbitrary application of some equations.  Looking closer
however we note that two critical and very interesting things, when taken
together (bearing in mind also that this is merely a mathematical
representation of the overall result).
\begin{itemize}
\item The two matched phase-transitions use \textit{one of each} of the two disparate
  particles: one each as input and one each as output!
\item The two disparate particles are actually the constituent parts of an
  \textit{anti-Pion}.
\end{itemize}

Recall above that it was hypothesised that a bare neutron's existence with
its +1V charge would be quite disruptive to surrounding particles: imagine
now what the presence of two quarks with 1/3 charge and 2/3 charge would do.
It is not unreasonable therefore to imagine that the passage of a Pion+
through space, surrounded by stable matter with unit charges, would leave a
chaotic trail of up-anti-up and down-anti-down particles in its wake
(aka Pion0 particles), as well as an anti-Pion being created even briefly as
the universe attempts to cancel out the Pion, or even briefly creating
Pion-anti-Pion pairs.

Also, almost in passing, it is worthwhile noting that the Pion+ has \textit{two} 
positive charges: a +1 T charge \textit{and} a +1 V charge.  Likewise, a Pion- would
have -1 T charge and -1 V charge.  As such, the Pion+ and Pion- would logically
act as the mediator of interactions between nucleons.  However there may
be a simpler - or perhaps simply different - explanation (covered later)
in which these two Pions would merely be convenient artefacts.

\noindent%
\begin{minipage}{\linewidth}
In light of the above, the next example phase-transition - a Pion to a Muon
and Muon-neutrino - can be attempted.  Bear in mind that the Muon and
the Muon-neutrino fit the 9-Rishon I-Frame pattern:
\makebox[\linewidth]{%
\includegraphics[width=0.8\textwidth]{rishon_pion_full_transition.png}}
\captionof{figure}{First attempt at Pion phase-transition to muon and muon-neutrino}
\end{minipage}
\linebreak


Starting at the top of this (comprehensive!) diagram is the up and anti-down
quarks (TVT and VTV).  Two quark-anti-quark pairs (which happen both to be
Pion0's in both electron and neutrino flavours) are brought into existence,
and one quark from each immediately results in a transition that creates
on one side the first electron of the Muon and on the other side the first
neutrino of the Muon-Neutrino.  The \textit{opposing} quark is used as the
input from the other phase transition in the pair, in exactly the same
way as was described in the previous (simpler) decay.

At the same time, from the opposite quark of the two Pion0's (in each flavour)
the exact same loop-linked phase-transition occurs to create the positron
and the anti-neutrino of the Muon and Muon-Neutrino.  Finally, and again,
at the exact same time, just as with the previous simpler decay, an anti-Pion
is brought into existence with a final loop-linked phase transition that
results in the last two triplets to complete each of the Muons.

However, sharp-eyed readers will have noted a problem with the diagram above:
four of the Rishon triplets have no incoming transforms to cancel them out.
After considerable analysis and experimentation attempting to find a solution
that respects the paired phase-transition rule, reluctantly it was concluded
that there must be a new rule to deal with this type of situation, which is
named "VT*":

\noindent%
\begin{minipage}{\linewidth}
\makebox[\linewidth]{%
  \includegraphics[width=0.8\textwidth]{rishon_vt_star.png}}
\captionof{figure}{introduction of VT* solves issue of unbalanced VT0 phase shifts}
\end{minipage}
\linebreak

VT* is in effect the simultaneous introduction of a Pion+ and Pion-.  The
primary reason for its introduction is shown, later, to be involved in the
creation and "decay" of W Bosons.  Does VT* violate any laws of conservation
of energy? no it does not.  why not?  because each particle completely cancels
out with every other particle.  It is \textit{suspected} that the phases of
each of the three Rishon particles in each of the four triplets also are
required to match: this will be dealt with separately in the section covering
colour.

What is the primary "decay" path of a Pion+?  It transforms via a W-Boson into
a pair of muons.  The justification for the make-up of a W-Boson is covered
in a separate section, later, but for now its creation is assumed to be
understood, as an intermediate means to explain the Pion+ to muon and
muon-neutrino transformation.  The first step is for the Pion+ to transform
into a W-Boson:

\noindent%
\begin{minipage}{\linewidth}
\makebox[\linewidth]{%
  \includegraphics[width=0.8\textwidth]{rishon_pion_w_boson_transform.png}}
\captionof{figure}{Pion+ transforms into W+ Boson with two spontaneous VT* events}
\end{minipage}
\linebreak

Here - again with no laws of conservation of energy violated - the Pion+ has
transformed into a pair of ultra-heavy quarks (explained below), and formed
a W+ Boson.  How - or why - would this event occur?  If this event was to be
explained from a purely observational perspective, it could be considered to
be a similar occurrence to that type of event which neutrinos undergo: flavour
oscillation.

However, consider the mechanics and the background.  Two quarks - an up and
an anti-down - both of them highly disruptive fractionally-charged particles
surrounded by a sea of matter that has unit charge, sail through space in an
unstable rotating pattern.  Due to their unique properties they cannot ever
collide and annihilate.  As they come closer to each other, with nothing to
prevent them from doing that, their rate of rotation around each other becomes
ever faster, causing huge disruption of their immediate space.  This localised
disruption is likely to be sufficient to allow space for the two VT* events
to occur, bringing into existence two sets of Pion+ particles and two sets
of Pion-.  These pairs "attach" themselves to each of the up and down quarks,
allowing them to reach a more stable arrangement \textit{(in fact, looking closely
at these two sets of ultra-heavy quarks we note at the centre of each is the
exact same Rishon arrangement for a neutron an a proton).}

So this is the first step: the Pion+ transforms into a W+ Boson (to be
re-classified and explained later).  The next step is for the W+ Boson
to undergo a series of simultaneous phase transforms into a muon and
muon neutrino:

\noindent%
\begin{minipage}{\linewidth}
\makebox[\linewidth]{%
  \includegraphics[width=0.8\textwidth]{rishon_w_boson_to_muon_transition.png}}
\captionof{figure}{W+ Boson transforms into muon and muon-neutrino}
\end{minipage}
\linebreak

The phase-transformation above is not as spectacularly complex as it first
seems!  Firstly, bear in mind that the two ultra-quarks will be rotating one
about the other (not laid out in a straight line as shown above).  The two
ultra-quarks transform \textit{directly} into a muon and muon-neutrino with
only one phase-shift per triplet.  The ultra-up quark becomes the muon: the
anti-ultra-down becomes the muon-neutrino, but we see that in fact (from the
crossing-over arrows) that the phase shift occurs through using triplets from
\textit{both} quarks.

There are however not quite enough quarks to complete the phase-shift: there
are ten input triplets (from the W+ Boson) yet only six outputs left after the
transformation.  This is where the two VT* phase transformations come into
play.  The lower VT* \textit{absorbs} the four quarks that came from four of
the muon's phase shifts.  The other VT* transformation (towards the middle of
the diagram above) is, in a Feynmann-style arrangement, taking one pair of
anti-down and up quarks from one matched set of VT0 phase-shifts, exchanging
them for a pair of down and anti-up quarks and then feeding those \textit{back} 
into another matched pair of VT0 phase-shifts.  Again note that whether this
latter VT* transform is actually the case is open to debate: mathematically
however it is "correct".

In this rather convoluted way - a way which could probably be better expressed
in less cumbersome terms - a mathematical representation is presented which at
the very least shows that the entire shift, right the way from the original
pion through to the final muon and muon-neutrino, conserves charge across both
T and V matter and without any particles "left over".  If new T or V particles
\textit{were} introduced, it was done in a way that is completely balanced, as
well as having a reasonable and rational explanation as to why it might occur.

Lastly on this example: the occurrence of two VT* transforms in both the
creation and destruction of the temporary W+ Boson is probably a coincidence,
but is worthwhile bearing in mind when examining future phase transforms.
As the W+ Boson is considered to be the "weak force", there is a considerable
gap between the two sets (of two) VT* transforms, leaving any thought of
conservation of VT* transforms (in a similar fashion to VT0 pairs) as being
on pretty shaky ground.

\subsection{Summary from example phase transitions... so far}

Bringing everything together from the above examples, there are a number
of rules which can be used to create phase transition diagrams:
\begin{enumerate}

\item Phase shifts must always occur in opposing pairs.

\item A phase shift \textit{may} occur as a means to simply transpose the
   positions of two particles.

\item One or more Rishon-triplet anti-triplets \textit{may} be brought into
   temporary existence as a means to join multiple paired phase-shifts
   back-to-back, as long as each of the triplet and its anti-triplet
   pair is the input and output of simultaneous phase-shifts.
   \textit{(In Standard Models these pairs would be termed "gluons").}

\item Under certain circumstances, one of each type of lighter quark - four
   in total - \textit{may} be brought into existence from a spontaneous event
   represented as "VT*".  VT* when treated as a Feynmann Diagram may also
   result in four lighter quarks being erased (with zero gamma radiation),
   or it may be used as a means to join any four VT0 transforms (two matched
   pairs) together as two inputs to a matched pair of transforms and two
   outputs to another pair.

\item Phase shifts involving Rishon I-Frame compound particles occur between
   the middle and one end of the I-Frame's three triplets, and at the
   same time the remaining triplet at the other end must be involved in a
   phase shift with another triplet from another particle (whether that
   be a singlet such as an electron, one of a double-particle such as a
   pion, or one of another I-Frame).

\item Two-particle bodies (e.g. Pions) require that both particles be involved
   in phase shifts, both for creation and destruction.

\item Under certain circumstances, two quarks (a Pion+, Pion0 or Pion-) may
   be brought into temporary existence, for less than one wavelength of the
   resonant frequency of an individual Rishon, where \textit{both} quarks must be
   both the input \textit{and} output of matched VT0 phase-shifts, such
   that the temporary particle is destroyed literally at the same time as it is
   created \textit{(reminder: the Standard Model calls these particles "gluons").}

\end{enumerate}

Rule 4 requires some additional clarification.  A VT* event may be one of
the following:
\begin{itemize}
\item The spontaneous creation of an up, anti-up, down and anti-down quark set
\item The spontaneous destruction of an up, anti-up, down and anti-down quark set
with no gamma radiation emitted
\item The "mathematically-convenient" transformation of two quarks from the
output of one pair of matched VT0 phase transforms into their opposites
as the input to another pair of matched VT0 phase transforms.
\end{itemize}

Throughout all of these there really is nothing else: there is really no
concept at all of "decay": there really are no other particles other than the
four (twelve if phase aka "colour" is included) Rishons, which merely represent
one of two different dimensions of matter with two opposite signs (and three
phases 120 degrees apart).  All that is happening is that energy in the form
of waves that are synchronised closely in phase and location simply exchange
energy between themselves in a profoundly simple and highly symmetrical way.

However when we move to the next example, involving Kaons, things get a
little more complex and need further examination.

\subsection{Kaon phase transitions}

Kaons are basically Pions with a heavier (I-Frame) quark in one of the two
spots.  That makes them particularly unstable as a rotational unit, and
it results in some unusual transitions.  Additionally, some anomalies
have been observed - deviations from the other examples shown above.
Beginning with a standard K- decay into a Pion-0, electron and anti-neutrino:

\begin{alltt}
    K- (u\overline{s}) -> Pion-0 ($\overline{d}) + e- + \overline{v}e
\end{alltt}

The diagram below shows the same equation in terms of Rishon VT0 phase
transitions:

\noindent%
\begin{minipage}{\linewidth}
\makebox[\linewidth]{%
  \includegraphics[width=0.8\textwidth]{rishon_kaon_oversimplification.png}}
\captionof{figure}{Potentially over-simplified (perhaps correct?) Kaon decay}
\end{minipage}
\linebreak

There are three things wrong with this picture.  First is that the up quark
phase-shifted with the \textit{middle} triplet of the anti-strange.  As the middle
particle (an electron) is almost 2/3s of a match with the up quark - i.e. it
comprises two T particles whilst the electron contains three opposing-charged
T particles - it almost makes sense for this phase-shift to occur.  The problem
is: it's only a single phase-shift.  Thirdly, one of the particles (the VTV)
"jumped" unscathed out of an I-Frame.

Where has the energy required to create this phase-shift come from, if
not from another opposing phase-shift?  One possible explanation could be
that the energy is emitted as a small amount of gamma radiation, whilst
the other is that the energy is transferred into the momentum of the
remaining particles.  The other possible explanation is that the rule about
VT0 transfers needing to occur in pairs is simply wrong, which somewhat
throws the entire theory here into question.

However, let's look more closely and logically at the options, based on
the constituent parts.  An up quark - comprising TVT - is orbiting around
three triplets in an I-Frame configuration, where the two end triplets
are too disparate in their composition for a phase transition to occur.
Like pions, Kaons are an unstable two-body rotational entity, so there is
nothing to stop the up quark from getting too close to that central triplet
(an electron).  Once that up-quark and electron phase-shifted into a
down-quark and an anti-neutrino, there would be nothing to hold the triplet
together.  The electron would spin off on its own.

However, let's look carefully at the relative rotation of the remaining
particles.  The remaining triplets are a neutrino, anti-down quark and
now a down quark, where the down quark is rotating about a completely
different axis, exactly at right-angles to the anti-down quark that it
would ordinarily combine with to create a Pion-0.  By the time the
transition has occurred, the neutrino from its perspective (and distance)
sees two perfectly-balanced quarks (down and anti-down) and thus is
not attracted or repelled by them: like the electron it is free to continue
on its trajectory.

This just leaves the down and anti-down quarks to resolve the problem of their
rotation being at right-angles whilst simultaneously being in extreme
proximity to each other.  what happens?  Is it unreasonable to surmise that
there is an additional phase transition that transforms the rotational
direction of one of the quarks such that it precisely matches the rotation
of the other?  Is it unreasonable to further surmise that this transformation
precisely matches the energy of a VT0 phase change?  How would this be
verified experimentally?  If it were possible to note (or infer) the
momentum, direction of travel as well as the rotation of both the original
K- and all the decay particles, it might be possible to confirm, from enough
such observations, that the rotational axes can be predicted.

If this were expressed in matrices at all \textit{(note: which is beyond the author's
mathematical ability to do)}, then it would be reasonable to include the
rotational axis (angular momentum) of the triplets as part of the matrix,
not least because that would be critical to explaining how I-Frame
triplets hold together in the first place.

However, there may be a simpler explanation - one that holds together
under the rules observed so far.  Recall the simpler Pion+ decay to a
positron and a neutrino.  A Pion- would decay likewise to an electron
and an anti-neutrino.  The difference between the Pion- and a Kaon- is
that there is an anti-strange quark present instead of an anti-up.
An anti-strange quark is an I-Frame comprising an anti-up, a positron
and an electron.  

The key to solving the conundrum above - that VT0 phase shifts must
occur in pairs - is therefore to use the "trick" that the charged
Pions also deploy.

\noindent%
\begin{minipage}{\linewidth}
\makebox[\linewidth]{%
  \includegraphics[width=0.8\textwidth]{rishon_kaon_decay1.png}}
\captionof{figure}{Kaon- decay to Pion-0, electron and neutrino}
\end{minipage}
\linebreak

This diagram requires some explanation!  The K- is in the centre, with the
up quark left of centre and the strange quark to its right.  Recall exactly
as with the Pion+, the working hypothesis is that the presence of the K-,
being unstable, would cause a temporary anti-particle (anti-down plus
an up quark) in this case a Pion+ to be temporarily brought into existence.
These two particles are shown in the top left corner (at right-angles to each
other) just next to their corresponding anti-particles (TVT and VTV) that they
will help phase-shift.  However the existence of that anti-particle
would be not only the output but also the \textit{input} to a matched pair of
VT0 phase-changes.  These two matched phase changes are drawn in the
top left corner.  The output of each of these two?  Exactly as with a
Pion decay: TTT and VVV Rishons, i.e a positron and a neutrino.

On the opposite side, a similar decay process occurs but this time with a
Pion-0 being brought temporarily into existence: again, the quark-antiquark
pair (in this case up and anti-up) are both the input \textit{and} output
of a matched pair of VT0 phase transforms.  In this way, a positron and
an electron transform into a down and anti-down pair: a Pion-0.

With a little more work, by examining and copying the more complex (more
common) Pion decay pattern it is easy to see how a Kaon- could decay
into a Muon and Muon Neutrino, by virtue of the up and strange particles
first being transformed into a W- Boson.

Additionally, the sorts of oscillations that B-Mesons undergo, as simple
phase-shifts, can also be demonstrated just as easily to apply to Kaon-0
and Pion-0 particles.

\subsection{Further observations on phase transformations}

It is a relief to see that the rules observed so far can cope even with
Kaons.  There was some concern for a moment!

So far, it is interesting to note that except for neutron-proton phase
shifts, no "virtual" particles involving TTT or VVV have ever been needed
(and even then that may be more of a mathematical summary - literally a
sum - than a true indication of what is really happening).  By "virtual"
particles this is the process by which a pair of Rishons are brought into
existence for long enough such that they may be both the input and output
of a pair of VT0 phase shifts.  This makes sense in that the creation of
"Virtual Pions" \textit{(most virtual particles appear to be Pion+, Pion-0 of either
the up or down variety, or Pion-)} comprise +2/3 and -1/3 charges
which could then phase-transition either way.  Referring back to Figure
~\ref{fig:transformchart} and looking at the VT0 phase transformation options,
particles with TTT or VVV may only phase shift one and only one way,
whereas particles with VTV or TVT have two possible decay paths.

So consider if it were possible for a TTT and $\overline{T}$$\overline{T}$$\overline{T}$ to spontaneously
spring into existence (even for an immeasurably brief time-period).
What pair of phase transformations could occur which have as \textit{both} the
input \textit{and} the output those exact same particles on both sides of the
equations?  The only options available result immediately in the annihilation
of the particle-anti-particle pair, with, ironically, an up quark and anti-up
quark (i.e. a Pion-0) as the additional virtual intermediary to facilitate
that annihilation!  Not only that but the annihilation is forced to use
the exact same pairs of particles.  By contrast, due to the fact that
2/3 - 1/3 = 1/3 - 2/3, in the case where virtual Pions are used, it is
possible for each of the matched VT0 phase transformations, in the
"creation" part of the virtual Pions, to use one of each of the two
quarks, and in the \textit{"destruction"} part to use the \textit{exact opposite} pair.

\noindent%
\begin{minipage}{\linewidth}
\makebox[\linewidth]{%
\includegraphics[width=0.8\textwidth]{rishon_ttt_nowhere.png}}
\captionof{figure}{showing how electron-positron phase shifts don't go anywhere}
\end{minipage}
\linebreak

So with the exception of position-altering phase transformations (changing
the order of Rishon triplets in I-Frames), electron-positron and neutrino
anti-neutrino "virtual particles" are not involved in phase transitions -
at least not any that result in the creation of new particles (aka 
particle "decay").

Put in Standard Model language: Pions are the predominant "gluons", and
only Pions can result in particle creation (note: in fact, a Pion+ and a
Pion- are the constituent parts of a VT* event).  Interestingly, assuming an
initial zero energy state, the startling implication of this is that the
"Big Bang" would, at its very conception, have started out exclusively
from Pions.

There is one other observation which is noteworthy: at no time has a phase
transition involving I-Frame particles left \textit{any one} of the three triplets
alone, nor in the case of dual particles (pions, kaons).  In other words, \textit{all}
the Rishon triplets of an I-Frame (or both Rishons of a dual-Rishon particle)
seem to need to undergo a phase transition: simultaneously, at that.
It would also appear that (as with the Pion transitioning to a Muon-neutrino
and Muon) the \textit{creation} of I-Frame particles also needs to occur simultaneously.
In other words, an I-Frame particle (such as a proton, Muon or any of the
heavier quarks) cannot simply pick up a random Rishon triplet and combine
with it: the creation of the heavier particle \textit{must} occur as a result of two
or three simultaneous VT0 phase transformations, by a process of "permitting"
the three triplets to "jump into" a stable arrangement, even if (as seen in the 
neutron-proton case) one of those phase transformations merely switches the
order of the triplets.

The probable reason for this is that the binding energy which held the twin
or triplet together would be massively disrupted.  A pair of matched phase
transitions however would be minimally disruptive to both the momentum and
energy state of these compound particles.  In the case of I-Frame particles,
this does raise the question however: expressed in Rishon terms, what kind of
process and energy (apart from a Pion+ to Muon decay) does it take to \textit{create}
heavy I-Frame triplets, especially Neutrons and Protons?  Is there
some clue here as to how Fusion takes place?  Would it be easier, in
Rishon terms, to help experimentally explain the circumstances behind
the phenomenon named by the term "cold fusion", perhaps?

\section{The puzzle of the Neutron and its "decay"}

Recall above as shown in Figure ~\ref{fig:protonneutronstructure}
that a "trick" had to be deployed - the
"decay" equation altered so that two particles (neutron, neutrino) were
phase-shifted into two others (proton, electron).  The reason for showing
this as a first example was because attempts to do the "traditional"
neutron-proton decay proved extraordinarily convoluted, and have yet to
be solved.

Which leads to a thought: if neutron "decay" (from one particle to three)
is so complex as to be extremely challenging even to solve, but the phase-shift
of neutron-neutrino (two particles) into proton-electron (two particles) is
so simple, then what \textit{really} is going on?  Using Occam's Razor we
conclude that "decay" of one particle into three - even via a W- Boson - is not 
what's actually happening, inference from the Standard Model to the contrary.
Perhaps however that complexity is actually genuinely there, which would
go part-way to explaining why a neutron has such a long half-life.  

After careful and considerable endeavours to resolve both sides of the
equation in terms of Rishon phase-shift rules, the following diagram was
the closest approximation which could be achieved:

\noindent%
\begin{minipage}{\linewidth}
\makebox[\linewidth]{%
  \includegraphics[width=0.8\textwidth]{rishon_neutron_decay.png}}
\captionof{figure}{Neutron genuinely does not fit the rules.
                   Culprit: decay energy}
\end{minipage}
\linebreak

On the left side is the neutron and underneath a neutrino.  On the right is
the proton and underneath it the electron.  In the middle is a W- Boson: above
is the temporary introduction of a Pion+.  Bear in mind that the W- Boson is
considered to be the "Weak Force" i.e. the two ultra-quarks involved have,
unlike the "gluon", a definite lifetime.  Note when comparing this diagram
against the "simple" version, it is the four quarks at the bottom-middle that
are utilised to indicate temporary carriers of the neutron and neutrino as
it phase-transitions into a proton and electron.

The mess at the top indicates two sets of VT* transforms.  Each creating-VT*
represent the creation of each of the ultra-quarks.  When the W- Boson decays,
two more VT* events occur which result in their erasure.  The problem is:
whilst the four quarks at the bottom are accounted for, the six remaining quarks
\textit{cannot} be neatly accounted for.  The discrepancy can only be balanced,
in Rishon phase-change rules, through the addition of a second particle: a
Pion+.

Whilst there may be other ways in which this can be expressed, in effect the
violation of the rules has a cost (an energy cost).  In fact, there is exactly
that: 0.782MeV of energy cost, known as the "decay energy".
\begin{alltt}
    \textit{Observation: the cube root of the ratio between the mass of the
    pion and the decay energy of a neutron is (16/9)/10, to within 0.06\%
    >>> pow(139.57018/.782343, 0.3333)*16/9
    10.006162983813788}
    There are ten quarks in a W- Boson.
\end{alltt}

The above numerical coincidence gives a finger-in-the-air figure indicating
that there is some relationship here that has previously not been noticed.
What is noteworthy however is that whilst the above decay diagram may not
necessarily be a completely accurate representation of the underlying process,
the fact that there is no clean Rishon phase transition possible, and the fact
that the neutron's decay results in 0.782MeV unaccountable energy is not a
coincidence.

In conclusion, then: we have the first transformation which can \textit{actually} be
said to genuinely be a "decay".

\subsection{Problem with earlier simple Neutron and neutrino phase-shift example}

Having finally extricated a reasonable explanation for the more complex
neutron decay pattern that actually involves the W- Boson, in accordance
with observation, the question has to be asked: why does the much simpler
looking example neutron decay not occur, instead?  This is a very good
question!

The answer lies most likely below, in that there are going to be repulsive
forces between the neutron and the neutrino that, just as with an electron
orbiting a positron, simply do not permit the neutrino to get close enough
to the neutron in order for the simpler pion ("gluon") phase-transition
to occur.  In addition, that exact same phase-transition pair would result in
the electron being brought into existence \textit{way} within acceptable boundaries
for electron orbits.  Thus the only other option is for the rather convoluted
and much more complex W- Boson route to be used as the transition method.

All of which highlights the really important issue which has been glossed
over, almost in passing, in the paragraph above: what the heck is the
neutrino doing "orbiting" neutrinos in the first place?

\subsection{Neutron and neutrino V charge}

The real reason for raising the anomaly of the neutron is to point out
the fact that a neutron has -1V charge and the neutrino +1V.

We note above that the neutron and the neutrino have opposing and equal unit
charge.  We note also that the electron and the proton have opposing and equal
unit charge.  Noted also is the fact that electrons orbit protons.  Logically,
therefore, is there any reason why neutrinos should not orbit neutrinos?
There is no such reason to consider otherwise, with the exception perhaps
that if V charge is actually a complex number (as hypothesised earlier) the
evident interactions in 3 dimensions (projected or otherwise) may not conform
to the same rules as observed (or better: inferred) for electron-proton
interaction.

However, without being able to say either way, the thought experiment may as
well be continued through to its logical conclusion.  Namely, that if neutrinos
do in fact orbit neutrons, there are a number of startling implications:

\begin{enumerate}
\item  that the recorded (measured) mass of a neutron could well \textit{also}
       include an orbiting neutrino.

\item  that if neutrinos orbit neutrons, then atoms not only have electron
       shells but they would also have corresponding neutrino shells, with
       those shells presumably mirroring the exact same sorts of configurations
       as electrons in orbit around protons.

\item  that it could be considered that a neutron, with an associated orbital
       neutrino, is itself an atom (of a rather unusual type) in the same way
       that a proton with its associated orbital electron is an atom of
       hydrogen.

\item  that, as reasoned in the section above, there is a logical explanation
       as to why the W- Boson is used as the medium of transition in neutron
       decay (as opposed to the much shorter-distance virtual pions, aka
       "gluons").
\end{enumerate}

Each of these needs some careful consideration.

Regarding neutron mass: given that the recorded (measured, inferred)
mass of a neutron is around the same as the sum total mass of a proton
plus an electron, this is not as unreasonable a logical inference as it
first sounds.  \textit{(Recall also that the "zero" mass of a neutrino is also
considered questionable: it could be zero "real" mass with a complex
component that is hard to interact with yet present and equal to that
of an electron's "real" mass (just as an electron's complex mass component
is zero, given that an electron has zero V charge).  However, also recall
that the assignment of "real" and "complex" to T and V charges is analogous,
and not yet actually proven: the two words are merely used as a convenient
moniker).}

Regarding the possibility of neutrino shells: this is itself an area worthy
of an entire study of its own.  The implications for the whole of chemistry
are simply too enormous to cover in this paper, especially given that there
are clearly interactions between the T and V matter types that would have to
be taken into account.

\section{T and V "force" interactions}

Which brings a whole new question of its own: what \textit{is} the nature of the
interaction between T and V at the macro level?  At the micro-level (known
in Standard Models as "Strong Force") T and V of the correct phase results
in an attractive force that can keep particles comprised of both types of
matter (up and down quarks) bonded together in orbit around each other.
At the macro level, T-anti-T attraction is known as an electrical charge.
What can V-anti-V be called, and what about T-to-V?

The working hypothesis at the top of the list is that V charge is magnetic.
It would be somewhat ironic to have this confirmed, given that magnetism
is typically thought of as a "flux" - a flow.  Taking into account the
hypothesis that neutrinos orbit neutrons, the further implications are that
neutrinos jump (in the direction of the magnetic "flux") in exactly the same
way that electrons jump between protons when an electrical field is applied.

\begin{alltt}
    \textit{Observation: why is the 2/3s root of the ratio of the proton and
    the neutron's magnetic moment \textbf{exactly} -7 / 9, to the third decimal
    place?}
        x = 1.9130427245/2.79284735623
        pow(x, 2/3.)*9
        6.993493

    \textit{2nd observation: cubic root of the ratio of proton and
    neutron's magnetic moment is pi-squared divided by 8.7, to the
    5th decimal place}
        >>> x = 1.9130427245/2.79284735623
        >>> pow(x, 1/3.) * pow(math.pi, 2) / 8.7
        1.0000145430676515

    \textit{(note: 8.7 = 9 - 0.3)}

\end{alltt}

A definition of the difference between the electric and magnetic field is that:

    "in the production of the Lorenz force a force from a magnetic
    field on a charged particle is generally due to the charged particle's
    movement whilst the force imparted by an electrical field on a charged
    particle is not due to the charged particle's movement".

Furthermore:

    "the movement creating the magnetic field in ferromagnets is the
    electron spin, whereas in a current-carrying wire (electromagnets)
    the movement is due to electrons moving through the wire".

A clarification of the above is in the units: the difference between
the units of the two types of fields is metres per second.  Electric
charge is measured in Newtons per coulomb; magnetic field is measured
in Newtons per coulomb divided by metres per second.

Question.  Why is the electron spinning in the first place, and, if it
is spinning, what effect is that having on the surrounding V-charged matter?
We surmise that the answer is: the spin of the electron is linked directly
(both ways - one can influence the other) to neighbouring V-charged matter
in a fashion where the change of relative position in one effects movement
\textit{at right-angles} in the other.  In the case where one type is bonded
into an orbital arrangement, that relative change in position of one would
\textit{cause} rotation in the other.  Likewise, when electrons move through a wire,
we surmise that neighbouring V-charged matter also spins.  In other words,
one is inextricably linked to the other in ways that make it frustratingly
and paradoxically hard to prove... or even disprove!

How come this has never even been considered before?  The answer lies in
the nature of electro-magnetism.  If without fail in absolutely 100\% of
all observations ever made of the link between electricity and magnetism,
the right hand motor rule has never been broken - ever - and the cause
and effect has each and every time the electron (or any other particle
deemed to have been determined to have an electrical charge) is the one
observable object at the centre of the investigations, what would the
logical conclusion be?

If anyone dared to suggest that there was even anything other than
electrons involved in the effect created by magnetism they had better
have a damn good reason to do so.  Unfortunately, due to that exact same
causal inference reflecting the fundamental reality and sub-structure -
namely that T and V particles are inextricably mathematically linked -
it is almost impossible to come up with one.

There are however some avenues to explore.  The first is to create a
Periodic Table mirror chart of neutrino shells according to the number
of neutrons in atoms, then to examine the non-magnetic, magnetic, paramagnetic,
diamagnetic, ferrimagnetic and other properties of existing well-known
elements.  If this hypothesis is correct, patterns in the periodic table
would clearly stand out.

The second is a refinement of the above.  If magnetism is truly an artefact
related to neutron shells (an example hypothesis to prove or disprove would be
that "only an element with a spare neutrino in its shell \textit{and} a spare electron
is a magnetic material", whilst another would be "an element with one too
many neutrinos in a shell whilst having a spare electron is a diamagnetic
material") then it would make sense to examine the isotopes of various
materials, both magnetic and non-magnetic, to see if there is any variation
in their properties.  It would be interesting to know, either way, if isotopes
(basically containing a different number of neutrons) happen to have exactly
the same magnetic (or non-magnetic) properties of the more common variants
of the same element, or if those properties are indeed different.  If the
magnetic properties of elements were \textit{exclusively} related to the electron
shells, then, assuming no difference in those shells due to the number of
protons remaining the same, there should be no change in the magnetic
properties of those isotopes, either.

The third is related to a question of what happens when electrons jump energy
levels, either between shells or escaping and emitting a photon.  If electrons
jump, then so should neutrinos.  So why, if an electron jumps and emits a
photon, would a neutrino not do likewise?  Are there any circumstances under
which an electron jumps but a neutrino would, theoretically, stay?  Are there
any under which it would not?  Are there any circumstances under which
the \textit{reverse} occurs i.e. the energy of an electron jumping is \textit{absorbed} by
a neutrino, such that a photon's emission is less, or even zero?  The most
likely place to begin examining these kinds of questions is in the field of
electronics - specifically diodes and transistors, due to the known quantum
tunnelling effects.  Has the magnetic field surrounding each side of a
transistor or a diode ever been investigated or measured, for example?  Is
there an anomalous variation in the magnetic flux field that cannot fully be
accounted for?

The fourth possibility is to explore the effect of strong magnetic fields
on Pion and Kaon particles, including \textit{changes} in magnetic fields.  The 
reason for this is because Pion+, Pion- (and Kaon) particles have both T
and V unit charges.  Although it is hard to say exactly what would happen,
one possible hypothesis to test would include looking for a relationship
between the half life of a Pion+ and the strength of the magnetic field
it was in.  If the rate of spin of the Pion+ can be influenced, as is
expected hypothetically by the application of a strong magnetic field,
then this would be quite likely to have a direct effect on the stability
of the Pion+.  Whether that effect could extend or reduce the lifetime
is unknown: either way or even if there is no effect at all it would be
fascinating to find out.

All of these questions are ones which simply never have been raised, because
nobody thought to question the 100\% causal link between electricity and
magnetism vis-a-vis the electron.  If no anomalies can be found, then good
scientific investigation will have been carried out.  If anomalies \textit{are}
however found, then that could ultimately lead to a deeper understanding of
magnetism - potentially one sufficient to provide a sound basis for increasing
the learning rate and discovery process surrounding superconductors,
semiconductors and materials science in general.  An advance in the theoretical 
basis for exploring the relationships between atoms and molecules would be a
significant addition to the foundations of chemistry in general, especially
when considered from the perspective, if the logic in this paper is sound,
that chemistry would effectively have been missing half the picture!

\section{Where are the Bosons?}

Whilst photons are obvious (particle-anti-particle annihilation) and gluons
appear to have been covered, the Bosons appear to be entirely missing from
the expanded Rishon Model.  It does not help that gluons and Bosons
as such don't actually make any sense, because in the expanded Rishon
Model all "decay" patterns are actually just combinations of matched
(i.e. energy and charge conserving) pairs of phase transformations.

However: in this, we have the clue.  Already it has been tentatively noted
that where in the Standard Model gluons occur it seems to be the case that
"virtual" pions appear in the phase-equations as both the input \textit{and} the
output particles.  Why then should it not be the case that the W, Z and even
Higgs Bosons are simply larger variants of other particles (created and
immediately destroyed) i.e. they are simply \textit{larger} virtual particles
that are both the input and output of \textit{several} simultaneous matched
phase transforms rather than just two as with the "virtual" pion (aka "gluon")?

    \textit{Note: looking again at the numbers: taking the cube-root of the W Boson 
    (80.38 GeV) divided by the mass of a pion+ (0.13957 GeV), the figure comes
    out at 25/3 to within 0.16\%.  Interestingly the cube root of the W Boson to
    the bottom quark (lower 4.18 GeV figure) comes out at 8/3 to 3 d.p.}

Given the huge energy of the W and Z Bosons, it would not be unreasonable
to surmise that, just as the "gluon" is tentatively identified as being
the Pions, the virtual particle in play in the case of the W and Z Bosons
is the Kaon family.  However, the theoretical mass of a gluon being zero,
whilst the W and Z Bosons are 80 to 90 GeV still requires some thought.

The gluon - if it is nothing more than a virtual pion - is quite simple
in nature.  two quarks.  only two matched phase transforms.  Creation and
destruction of those two quarks is either instantaneous, or are merely
a mathematical representation of a fundamental reality that permits
more comprehensive set of phase shifts than "just" VT0 transform pairs
(which happen to be possible to break down into two quarks back-to-back
with two phase transforms).  In other words, gluons may not actually exist,
at all!

W and Z Bosons, on the other hand, comprise \textit{multiple} such transforms
occurring simultaneously.  It would not be unreasonable to surmise that the
total effort extended by the fabric of the universe to coordinate those
simultaneous transforms, given that they would occur in separate locations
which would require communication at the speed of light, \textit{is} exactly
80.385 GeV or 91.118 GeV respectively!

Put in another way - viewed from the perspective of wave propagation
and diffraction patterns - if two (or more) particles are out of kilter
(unstable) such that the wave fronts going outwards from oscillation of
their internal T and V particles causes diffraction patterns at any point
surrounding them to exceed 80.385 GeV (or 91.118 GeV) in total, then this
energy is sufficient for the fabric of the universe to temporarily create
a W (or Z) Boson, such that the various compound particles constituent
T and V particles may instantaneously and simultaneously phase-shift to
a more suitable arrangement.

Substitute "Kaon" for "Boson" in the above paragraph and it is likely to
be more accurate.  Why, however, would it take such a vast amount of
energy to create a Kaon, when a Kaon's energy is itself so much less?
The clue is in the word "create" - even temporarily.  Read again the
paragraph above: the effort (work done i.e. required energy threshold) in
coordinating the simultaneous matched phase shifts required to introduce that
Kaon into the fabric of the universe even for one or one half an oscillation
of a T or V particle's wave-form \textit{is} the energy of the required
Boson.

So the irony, here, is, that the Bosons - like the gluon - do not actually
exist.  Or, they do, but only for the brief (half or single) oscillation
required for phase-shifts to instantly wipe them out of existence.  That
would not however make them invalid.

The other possibility to consider is that if the Higgs is a +1 electrically
charged virtual particle, it is reasonable to consider that, just as the
"gluon" can be a virtual Pion+/- or a Pion-0, and like the W Boson is
suspected to be a virtual Kaon+ and the Z Boson a virtual Kaon-0, there should
also exist a Higgs-0.  Given the extremely small discrepancy between the
masses of the W and Z Bosons, if a Higgs-0 existed then a reasonable starting
point to look for decays involving it would be somewhere around 1.5 times
the energy of the W and Z Bosons: i.e. between 120 GeV and 136.5 GeV.
Further study would however be needed (back at the smaller energy levels)
to assess which particles would be involved, and what decay patterns to
look for.  It is yet to even be established the virtual particle that the
Higgs-0 would represent.

\section{Double-stacked I-Frames}

Towards the beginning of this paper the startling penta-triplet layout was
introduced, as a set of double-stacked I-Frames.  In the case of the Tau,
it is believed that the Tau comprises one outer I-Frame, with two positrons
in the outer positions and a muon in the centre, the muon itself comprising
two outer electrons and a central positron.

In this section we explore the possibility of other penta-triplet (double
I-Frame) particles.  Tentatively, the most likely acceptable layout would
be for the outermost particles to be the anti-particles of the inner
I-Frame's outermost particles: effectly creating a weird hybrid following
\textit{both} the rules of Pions \textit{and} I-Frames.

What estimates could be made for the mass of the most likely-detectable of
these particles, should they exist?  The difference in the mass between the
muon and the tau gives a rough factor of a 17-fold increase.  The least
heavy of all the 9-particle I-Frames is the strange quark, at 90 MeV
(theoretical).  It would therefore not be unreasonable to put an ultra-strange
particle at somewhere around 1.53 GeV.

\noindent%
\begin{minipage}{\linewidth}
\makebox[\linewidth]{%
  \includegraphics[width=0.8\textwidth]{rishon_ultra_strange.png}}
\captionof{figure}{Tentative "ultra-strange" penta-triplet particle}
\end{minipage}
\linebreak

However, look closely at which particle is in the centre, whilst the
other two have cancelled out by the surrounding particles: a positron.
So, although this particle could potentially exist, it would not be
classified as a quark, but as a heavy anti-electron.  Also, given that
the combined energy of the down and anti-down at one end when compared
to that of the electron and positron pair at the other is different,
the centre of mass of this particular combination would be off.  Chances
are, then, that this particle's half-life would be immensely short, assuming
it was able to remain stable at all.

So consider instead the remaining 9-Rishon I-Frame particles that are
symmetrical.  These are the muon, the muon neutrino, the proton and the
neutron.  Extending these four particles by placing anti-particles at their
outer extent gives the tau, the tau neutrino, and two other possible and
previously undiscovered particles.  Although based on a neutron and
a proton, the anti-particles would cancel out the outer up (and down)
quarks respectively, leaving the central quark as the remaining balance.
The temporary names for these are ultra-up and ultra-down.

\noindent%
\begin{minipage}{\linewidth}
\makebox[\linewidth]{%
  \includegraphics[width=0.8\textwidth]{rishon_ultra_up_and_down.png}}
\captionof{figure}{Tentative "ultra-up" and "ultra-down" penta-triplet
                   particles}
\end{minipage}
\linebreak

In both cases, in effect these are weird hybrids with the characteristics of
Pions as well as I-Frames.  Their stability (and therefore lifetime) would
therefore be somewhat less than that of Pions, and based on the 17-fold jump
in energy from a muon to a tau, we can make a reasonable guess for their masses
to be somewhere around the 16 GeV mark.  Also, rather startlingly, they would
be the first quarks to have zero spin, due to the fact that in effect at each
end is a Pion (pions have zero spin).

Looking at these particles in the figure above, a sudden suspicion arises.
What would an ultra-up and anti-ultra-up comprise, in total?  It would
actually be a combination of \textit{five} pions: four Pion-0 particles
(of the up-anti-up variety) and one Pion-0 of the down-anti-down type.
Or, consider an anti-ultra-up and an ultra-down.  If reassembled, this would
give enough material for two Pion-0 up-anti-up pairs, two Pion-0 down-anti-down 
pairs and one Pion- (the central quark of each of the two ultra-quarks).

In covering the phase transforms above, where have twinned five-particle
combinations been necessary?  Recall the Pion phase-transition to a
muon and muon-neutrino, which involved an intermediate step creating
exactly that.  Double the estimated mass of an ultra-pion is, minus a
not inconsiderable amount of energy that could potentially be binding
energy, and it would appear that a candidate makeup for the W Boson
has been found.  We even have two potential candidates for the Z Boson:
ultra-up and anti-ultra-up as one type, and ultra-down and anti-ultra-down
as the other.

    \textit{Note: recall above the previous note that the cube-root of the mass
    ratio between the W Boson and the Pion was 25:3?  Five squared is 25.
    It has been deduced that there are five Pions in a W Boson.}

Our earlier deduction about Bosons not being real particles would therefore
seem to be in error.  In fact, quite the opposite: their composition would
appear to be real, as would also the ultra-quarks.  The W and Z Bosons
would appear to be nothing more than a heavier flavour of Pion.

\section{Higg's Boson}

If the ultra-quarks are real, and the W and Z Boson are ultra-heavy flavours
of Pions, then the next logical step would be to combine \textit{three} ultra-quarks
into yet another I-Frame arrangement:

\noindent%
\begin{minipage}{\linewidth}
\makebox[\linewidth]{%
  \includegraphics[width=0.8\textwidth]{rishon_higgs_proton.png}}
\captionof{figure}{3-level I-Frame: ultra-quarks in a Higgs aka ultra-heavy proton}
\end{minipage}
\linebreak

This particle is simply an ultra-up, ultra-down and ultra-up quark.  It is,
in effect, an ultra-heavy proton.  If on the other hand those three quarks
were an ultra-down, ultra-up and ultra-down, the combination would result
in an electrically-neutral particle.  Looking at the mass ratios of the
Higgs to a W-Boson the figure is around 1.5, which roughly corresponds with
the expectation of a 2/3s ratio (W and Z Bosons being 2 ultra-quarks,
Higgs expected to be 3).

We do however have a couple of anomalies.  The first is that the extended
Rishon Model predicts the existence of a Higgs-0 (an ultra-heavy neutron).
The second is that if the Higgs+ is actually an ultra-heavy proton, then
the first immediate assumption is that it should have a spin of 1/2.
However, in looking more closely at the make-up of the Higgs, there is
something particularly interesting about it.

Imagine all the particles in the diagram above as being quite literally
mechanically represented as physical constructs, using gimbal linkages to
join them all together.  Note that in the ultra-quarks there is \textit{double}
linkage between the outermost quarks and the innermost (central) one.  In
other words, ultra-quarks, by virtue of being exceptionally weird and being
part-double-Pion, part I-Frame, should actually have zero "spin", in exactly
the same way that Pions have zero "spin".

So, amazingly, even when little is known about the Higgs at this stage, the
extended Rishon Model helps to confirm its characteristics, and even predicts
the existence of an additional particle: a Higgs-0.  In fact, it could well
be the case that the two different experimentally observed masses for a Higgs
are due to the fact that one is a Higgs+ and the other is a Higgs-0.  This
we surmise from the very small difference between the mass of a neutron and
a proton, scaled up to that of the two Higgs observed mass measurements.

    \textit{Observation: the mass ratio of the neutron to the proton is 1.001378
    The mass-ratio of the two observed figures for the Higgs is 1.00559}

So, rather startlingly, the extended Rishon Model not only predicts the
existence of two new quarks (ultra-up and ultra-down), but also predicts
the composition of the W and Z Bosons to effectively be ultra-heavy flavours
of Pions, predicts the Higgs Boson to be effectively an ultra-heavy proton
\textit{and} also predicts the existence of a \textit{second} Higgs Boson, named the Higgs-0,
as being an ultra-heavy neutron.

How could any of these have been missed, after decades of research?  Firstly:
the Rishon Model has two types of matter, whereas all research into particle
physics has taken place from the perspective of only one of those types of
matter.  Secondly, the Extended Rishon Model only began to fall into place once 
the existence of the Higgs was confirmed.  Without a third level of hierarchy
to confirm (from a reverse-engineering or knowledge-inference perspective) the
existence of the ultra-quarks and the tau would have remained an anomaly, and
against that lack of information, the identity and make-up of the W and Z
could not really have been confirmed.  The clue is, ultimately, in the mass
ratios of all the ultra-heavy particles (including the tau).

\section{Rishon colour}

The term "colour" is believed to refer to phases.  The hypothesis currently
under investigation is that a rishon triplet is actually the exact same
wave present in three close physically-proximate locations, each presence
separated by exactly 120 degree phases \textit{(note: "present" being the most convenient
word to describe the possibility of a multi-dimensional entity having more
fold-back points into a 3 dimensional perspective than would normally be
considered).}  Two potential candidates present themselves as to how to account
for T and V matter types.  The first is simply that V matter has a complex
component.  The second is O(6) phase space using Clifford Algebra as outlined
by Piotr Zencyzkowski \textit{(note: it is worth bearing in mind that the complex
numbers may be expressed in Clifford Algebraic terms).}

Either way, out of the two types of matter plus their anti-particles, the
addition of three phases would give twelve states.  How would these particles
interact?  Recall earlier that it was believed that a Rishon triplet is
the same wave-form presented in three close positions, each 120 degrees
apart.  This would present a triplet that could rotate about the middle
of the three points.

In the case of a six-dimensional universe (folded back to three), even
particles comprised of both T and V matter would be able to remain
coherent and in close proximity, as the T charge would on one cycle
of the wave-form move into the other three dimensions whilst the V charged
matter made its presence felt in the opposing three, and vice-versa.
\textit{(Note: the exact details are unclear and the author lacks the
mathematical background to provide models).}

Leaving aside details of mathematical models and dimensions, the most pressing
requirement is to resolve the role of colour in the phase-transitions, to
ensure that the rules stack up.  Recall the rule that opposing phase transforms
must take place in pairs?  The "colour" extension of this rule is that, because
energy must be conserved, the phases (i.e. colours) must \textit{also} match.
It is no good for example exchanging a red T particle with a green anti-T
particle: that would imply that a wave in the "T" field resonating at a
specific frequency was attempting to exchange places with another wave that
was 120 degrees out of phase with it.  The energy to do that has to come from
somewhere (read: nowhere), so it cannot happen.

What is believed to occur is best illustrated with an example.  Beginning
again with the simplest phase transition: neutron plus neutrino to proton
and electron:

\noindent%
\begin{minipage}{\linewidth}
\makebox[\linewidth]{%
  \includegraphics[width=0.8\textwidth]{rishon_neutron_transition_colour.png}}
\captionof{figure}{neutron transition with colour (3 phases)}
\end{minipage}
\linebreak

Examining the transformation of the neutron and down quark, if the two
particles are XORed together, what remains is a green V particle and a
green anti-T particle.  These two particles remain in place as part of the
V$\overline{T}$0 phase shift.  On the other side of the transformation, we assume that
the reconstruction of the particle permits the introduction of two pairs
of T and anti-T particles, each pair with red and green phase.  As they both
spin about the central green particle, the order is immaterial (except as an
indicator of the \textit{direction} of spin).  Red could have been on the left,
blue on the right, or vice-versa.

The remaining phase-shift is deceptive in that it is there merely to switch
the order of the two quarks.  In doing so however the phase order of each
of the three is retained.  Despite the swap, after the pair of transforms
the colours (phases) match up correctly.  In order to ensure that the whole
I-Frame retains attraction (cohesion), it is \textit{critical} that not only the
particle-anti-particle rule be respected (middle of end quarks attracts to
ends of middle quark) but also that the phase of those particles involved in
attraction also match.  i.e. the middle Rishon of the left quark must match the
phase of the left Rishon of the middle quark, and the middle Rishon of the
right quark must match the phase of the right Rishon of the middle quark.

So although some principles of enquiry have been established there is not
enough information here to deduce the full extent of colour rules, although
some suspicions are beginning to form.  Looking again at the Transformation
Chart (Figure ~\ref{fig:transformchart}) but adding in colour, we notice something interesting:

\noindent%
\begin{minipage}{\linewidth}
\makebox[\linewidth]{%
  \includegraphics[width=0.8\textwidth]{rishon_phase_colour_map.png}}
\captionof{figure}{phase transition marking unchanged Rishons}
\end{minipage}
\linebreak

In each phase transition, two Rishons remain unchanged.  The other four (two
from each Rishon) cancel out and can thus be replaced with something completely
different (as long as they are also polar opposites and of the same phase).
What is fascinating is that in those phase-shifts involving the neutrino
(VVV), positron (TTT) and their anti-particles it is \textit{always} the central
Rishon that remains in place.  Only in the case of the up and down quark
(and anti-up and anti-down) phase transforms is it possible to pick either
the left-hand Rishon of each triplet as the "unchanging" one \textit{or} the right
hand Rishon of each triplet as being unchangeable.

A suspicion is therefore forming that could end up, if it is true, as being a
significant discovery.  The suspicion is that in \textit{all} matter since the birth
of the Universe, the phase of the central Rishon of all neutrinos, all
electrons, all anti-neutrinos and all positrons has \textit{always} been the same.

For the sake of an arbitrary but sensible choice (red-shift -120 phase,
blue-shift +120 phase) we pick "green" as being that central hypothetical
constant phase, and continue the analysis.  The implications of this would be
fascinating, because it would mean that any phase transformations involving
the four identical-Rishon particles would \textit{always} be green.  Whether
there are any implications in the sub-structure of I-Frames that guarantee
that the green-phased Rishon, even when in an up or down quark, always comes
out into the central place when phase-shifting into a solitary VVV or TTT
particle, remains to be seen.

The next example is different from the neutron phase transform in that it
involves a "gluon".  Consider again the pion+ phase transform into a positron
and a neutrino:

\noindent%
\begin{minipage}{\linewidth}
\makebox[\linewidth]{%
  \includegraphics[width=0.8\textwidth]{rishon_pion_simple_colour.png}}
\captionof{figure}{pion+ simple phase transition involving colour}
\end{minipage}
\linebreak

In this example, we immediately see a problem.  Consider the two phase
transforms together.  The fact that the gluon's temporary existence has to
have two quarks with conflicting colours indicates that something is wrong
with the original assumptions.  sadly!  It would be a fascinating universe.

Instead we are forced to review the options.  Would it be reasonable for
the phase (colour) of the V in each matched VT0 phase-transform pair to be
the same, and likewise for T?

TODO: COMPLETE

\section{Proton-Neutron proximity: stability within nuclei}

Much earlier in this paper the issue was raised of how 9 Rishon I-Frame
particles could remain in close proximity to each other, when the concept of
a doubled-up I-Frame using muons in each of three positions was considered.
Due to the close proximity of positrons and electrons within the I-Frame it
was considered extremely unlikely that such a closely-bonded particle would
work.  However, with careful consideration it would appear that there should
be a balance point where the forces of attraction and repulsion would level
out.  Taking the example of a neutron and proton, how this would work is
shown below, in principle \textit{(caveat: the drawing below is 2 dimensional,
and should be taken as a close approximation to demonstrate the principle,
which could well need to be considered in at least 6 dimensions for a
proper analysis or simulation).}

\noindent%
\begin{minipage}{\linewidth}
\makebox[\linewidth]{%
  \includegraphics[width=0.8\textwidth]{rishon_neutron_proton_rotation.png}}
\captionof{figure}{Showing a neutron and proton stably rotate in close proximity}
\end{minipage}
\linebreak

Walking through this diagram, the quarks are shown with three colours (phases)
identified with Red, Green and Blue on a proton (left) and neutron (right).
Attraction between proximate quarks is shown in purple (1).  Repulsion shown
in black (2).  Direction of rotation about the central quark in each I-Frame
shown in yellow (3).

So as the neutron and proton align at 90 degrees to each other, one outer
down quark of the neutron comes into close proximity to the central down
quark of the proton.  Assuming a square law of attraction and repulsion,
the down quarks repel each other (2) whilst simultaneously the down quark is
\textit{attracted} to the two up quarks at the ends of the very same proton (1).
This seems at first consideration to be paradoxical and an arrangement
ill-equipped to produce stability (the nuclear force), but considered purely
from a mechanical perspective, amazingly it actually works.

As the two particles rotate, the repulsion and attraction remain in play, but
now, additionally, one outer up quark of the proton begins to come into
proximity with the inner up quark of the neutron as well as the other outer
down quark of the same.  So as this occurs there is a less strong repulsive
force (5) betweeen two up quarks, as well as a less strong attractive force (4)
between two up and down quarks, both of which become stronger as the \textit{other}
down quark rotates away from the proton, resulting in its paradoxical dual
repulsive and attractive forces being lessened.  Obviously, by the time the
two particles reach a 45 degree rotation from their original starting point,
all those forces become equal.  Another 45 degrees later the positions are
reversed, and that quadrant of the cycle will have been covered.  In this way
the two particles remain in an extremly closely-linked interlocking and stable
relationship.

The exact details of the relevance of the colour (phases) is as of yet unclear,
but is suspected to result in a coherent phase-locked loop that keeps the "spin"
of the outer triplets in absolute synchronised lock-step with the rotation of
the neutron and proton's rotational cycle.

One important question arises: how can the two opposing forces of attraction
and repulsion result in the "nuclear force"?  In other words, why do the
neutron and proton remain in such an amazingly-stable orbit at a constant
distance away from each other?  Consider the mechanics of the arrangement
shown by the forces (1) and (2).  If the force is a simple square law, then
if the two down quarks become too close together, the repulsion (2) would
simply push them apart.  With the two attractive forces (1) being further
away they are less effective.  However, should the distance between the
particles become larger, the two attractive forces (1) would exceed that
of the repulsive force (2), resulting in an overall attraction.

Thinking of this purely in mechanical terms that would allow an actual
physical and practical demonstration:
\begin{enumerate}
\item Take 12 small permanent magnets of equal strength.
\item Tape or glue 6 of them back-to-back with North-to-North
\item Mark the North-North magnets with the letter "u"
\item Tape or glue 6 of them back-to-back with South-to-South
\item Mark the South-South magnets with the letter "d"
\item Take 2 rigid sticks of equal length.
\item On stick 1, tape or glue a "u" double-magnet at each end
\item On stick 1, tape or glue a "d" double-magnet at its centre
\item On stick 2, repeat this but with "d" at ends and "u" in centre
\item Take some string, wrap it round the centre of each stick
\item Suspend the two sticks at right-angles and see what happens.
\end{enumerate}
The diagram below illustrates this with some arbitrary numbers and distances:

\noindent%
\begin{minipage}{\linewidth}
\makebox[\linewidth]{%
  \includegraphics[width=0.8\textwidth]{strong_force_illustrated.png}}
\captionof{figure}{Illustrating the repulsion and attraction resulting in "Strong Force"}
\end{minipage}
\linebreak

In this illustration there is a proton with an up quark at A and D, and
a down quark at C.  There is also a neutron with a down quark at B.  The
top diagram has the neutron and proton close together; the bottom diagram
has them further apart.

Here it is clear that in the top diagram, the sum of the inverse-square-law
forces (A-B, C-D and D-B) when the neutron's down quark is close to the
proton's down quark, results in a net total \textit(repulsion), and
yet at distances slightly further away, as shown in the bottom diagram,
the sum of those exact same forces is a net total \textit(attraction).

So it is quite interesting to note that this would actually result in some
harmonic oscillation - "ringing" - where individual neutrons and protons would
continuously bounce back and forth around a happy medium.  That central point
could approximately be calculated - ignoring other effects such as centripetal
force, and only taking one group of triplets into account at a time - by
sin(angle) * 2 = cos(angle).  Or, more specifically, if as suspected a square
law is involved:

    \textit{1/sqrt(sin(angle) * 2) = 1/sqrt(cos(angle)).}

A quick 
\hyperref{http://lkcl.net/reports/rishon_model/angle_search.py}{}{}{python script}
and one iterative search later shows this angle
to be approximately 26.6 degrees, no other quarks taken into consideration,
no centripetal forces or other factors.  In other words: below 26.6 degrees
between the line of the outer up quark of the proton and the outer down quark
of the neutron and there is a repulsive force keeping the proton and neutron
apart, whilst above 26.6 degrees the balance turns the other way and there is
a net attractive force.

It is anticipated that examining a more comprehensive version of this equation
with respect to the distance between the proton and neutron would result in
a derivation of the nuclear force.  Factors to take into consideration in
deriving such would include:
\begin{itemize}
\item the strain placed on the quarks in each of the
proton and neutron, causing them to no longer be in linear alignment
\item the counterbalancing centripetal force of rotation of outer quarks around
their respective centres
\item the strength of the attractive force between the individual Rishons
\item the distance between all Rishons
\item the speed of rotation of the neutron and proton
\end{itemize}

Interestingly, the frequency of Rishon waveform oscillation is unlikely to be
involved in the above.  Relativistic effects however \textit{might} have to be taken
into consideration.

The above is quite involved.  It first requires a model of the relationships
of the forces between Rishons: without that it would be challenging at
best to get the model to work.  However, if the underlying model of Rishon
particle force relationship were correctly ascertained (Clifford Algebra being
at the top of the list), then deriving the Nuclear Force using the above
considerations as the basis would be a clear validation of the whole Extended
Rishon Model.

\subsection{Possible stable anti-muon / muon rotation}

With the possible exception of the interaction between the V and T matter
in quarks producing some stabilising spin characteristic, there is no
compelling reason why instead of neutrons and protons, that anti-muons and
muons may not be held in the same stable patterns, i.e. be subject to the
Nuclear Force to create nuclei.  Without proper simulations or analysis it
would be hard to say if this would result in a stabilising or destabilising
effect on the muons.  The proximity of a large number of electrons and
protons, solely comprised of T matter, does however give some cause for
concern!

The only caveat is that it would clearly be improbably or simply impossible
for an anti-muon to take the place of a proton (or a muon-neutrino to take
the place of a neutron), for a number of reasons:

Firstly, the difference in the orbit, size, energy and rotational speed
when compared to either a neutron or a proton would badly mess up the
careful balance of the attractive and repulsive forces: contrast this
to the situation in which in the muon and anti-muon combination clearly
have identical characteristics.  In fact, looking at the above equation
showing the sine-cosine relationship there may not be a stable point \textit{at all}
at which the muon (or anti-muon) would reach equilibrium with a proton (or
neutron) before getting so close as to cause huge disruption to the structure
of both particles, activating strong force (VT0 phase-changes) in the process.

Secondly: the +2/3 and -1/3 charges of the neutron and proton quarks, at such
close range, would be in direct conflict with the +1 and -1 charges of the
muon and anti-muon.

Thirdly: muons would already have had some considerable difficulty overcoming
the barrier which typically places them in orbit around the nucleus.

Overall, then, whilst it seems reasonable to consider that muon-anti-muon
combinations may exhibit nucleic structure and be subject to what is termed
"Nuclear Force", it is flat-out impossible for combinations of muons or
muon neutrinos to combine together to create nuclei via the "Nuclear Force".
Yet to be determined is the stability (or otherwise) of larger muonic nuclei.
\textit{(Note: the degenerate case of a muonic atom - muonic hydrogen - has already
been observed).}

Also it is worthwhile pointing out that due to the right-angular nature of
the interaction between T and V matter, it is considered impossible for muon
and muon neutrino to combine to form nuclei.  An overall summation of the
force interactions between muon and muon neutrino simply do not result in a
stability point (aka "Nuclear Force") occurring.

\section{Summary}

The list of enquiries that this paper generates is almost overwhelming.  At
least two new quarks are identified; the composition of the Bosons and the
gluon are identified, and one new Boson (Higgs-0) is predicted.

Summarising the points made so far:

\begin{enumerate}

\item The original Rishon Model is altered to invert the neutrino's charge to
   VVV (positive) in order to match the remaining triplet particles and to
   respect the rules regarding multi-triplet particles.

\item \textbf{All} particles including the gluon, W, Z and Higgs Bosons, including
   the tau, are identified in terms of Rishons by way of some simple recursive
   rules.

\item A series of rules for particle transformation are given - none of them
   incompatible with Standard Models, nor violating conservation of energy,
   that redefine particle "decay" as being energy-conserving matched phase
   shifts, including in some (most) cases the ultra-short-lived introduction
   of temporary particles.  The temporary particles are involved in matched
   phase-transforms that \textit{simultaneously} create \textit{and} destroy them,
   as a bridge to more stable wave functions better suited to the conditions
   than the particles' previous states.
   
\item Two new heavy quarks - the ultra-up and ultra-down - are identified as
   being extensions of the neutron and proton with pairs of anti-up and
   anti-down quarks either side of each, in a short-lived arrangement that
   has the startling effect of cancelling out spin \textit{within} the ultra-up
   and ultra-down quarks themselves (to spin 0).

\item The W and Z Bosons are identified as being simply heavy flavours of Pions
   using ultra-up and ultra-down quarks.  Not that it matters (because the
   quarks would counterbalance, resulting in zero spin anyway) but the zero
   spin of the ultra quarks results in the W and Z Bosons having zero spin.

\item In a clarification of interpretation: there is effectively no such thing
   as the "Weak Force"; there is only the \textit{carrier} of what is
   \textit{believed} to be the "Weak Force": the W and Z Bosons. These are merely
   heavy flavours of pions; they are particles in their own right.

\item The Higgs Boson is identified as simply being an ultra-heavy flavour of an
   anti-proton, using ultra-up and ultra-down quarks.  Again the zero spin
   of the ultra-quarks results in the Higgs Boson having zero spin despite
   a +1 charge.

\item A \textit{second} Higgs Boson - the Higgs-0 - is predicted to exist in \textit{two}
   flavours: anti-Higgs-0 (an ultra-heavy anti-neutron) and Higgs-0 (an ultra
   heavy neutron).  It is anticipated that the Higgs-0 is the heavier of the
   two observed masses (126.0 GeV) whilst the Higgs (renamed to Higgs+) is
   the lower, at 125.3 GeV.  This prediction is based on the observation of
   the neutron being slightly heavier than the proton.

\item The Higgs-0 is anticipated to undergo oscillation, in exactly the same
   way as the Pion-0 (and others such as the B-Meson-0).

\item The neutron is predicted to have orbital shells comprising neutrinos.
   A line of enquiry of several straightforward theoretical and practical
   experiments are given to help establish if this hypothesis is correct.

\item Rishon V matter is tentatively predicted to be involved in magnetism.
   Although it is extremely hard to prove \textbf{or disprove} that hypothesis, a
   number of potential lines of enquiry have been outlined.

\item Rishon V matter is tentatively believed to be responsible for "absorption"
   ("deflection" would be better) of electrical charge when used as a means to
   influence the velocity of particles.  The rule under investigation is that
   the more V matter is present in a particle, the heavier the particle \textit{appears}
   to be.  However it could well simply be the case that the particles
   genuinely are that heavy, just with the majority of that energy having
   gone into unavoidably increasing the rotational speed of V particles within
   the sub-structure instead of increasing the velocity of the whole, giving
   the \textit{impression} that the particles are significantly heavier than they
   would be at "rest".  There is much here to still be investigated.

    TODO: shorten this.  move fuller explanation elsewhere.

\item A large range of additional particles are predicted to exist: combinations
    of ultra-quarks with top, bottom, strange, charm, up and down in one of
    the two (or three) spots, all above 17 GeV but well below 80 GeV for the
    two-quark variants, and all above 17 GeV but well below 125.3 GeV for
    the three-quark variants.  Due to the extreme size of the ultra-quarks
    it is anticipated that asymmetric three-quark variants with ultra-quarks
    as part of the makeup will not feature strongly (if at all), nor
    three-quark variants where significantly lighter quarks (at least up and
    down) are in the centre position: these are not anticipated to exist at all.

    TODO: shorten this.  move fuller explanation elsewhere.

\item Rishon "colour" is a reflection - indication - of an underlying order
    involving three 120 degree separate phases.  The order in which triplet
    phases apply critically affects the way in which particles attract (or
    repel).

\item Aside from either a -120 or +120 phase delay, absolutely every T and V
    particle's wave-form beats in absolutely synchronous harmonic oscillation
    with every other particle in the universe, regardless of location.
    without exception.

    TODO: add to TODO below - explanation of above!

\item Matter is considered to be merely waves that propagate outwards at the
    speed of light, causing constructive and destructive interference that
    at very short distances gives the mathematical appearance of "Strong Force",
    short distances the mathematical appearance of "Weak Force", and even
    greater distances when sufficient particles combine to average out their
    wave-forms: gravity.  

    TODO: add to TODO below - explanation of above!

\item An explanation is offered as to how the Nuclear Force paradoxically
    switches from attraction to repulsion, in terms of simple mechanics.
    Some direction for a more comprehensive derivation of the Nuclear
    Force is outlined.

\item A prediction is made that negative muons - on their own - \textit{might} be 
    able to form more complex nuclei (for short durations) but that they would
    \textbf{never} be part of nuclei that also included either protons or neutrons:
    this is flat-out impossible.  \textit{(The simplest example of this has already 
    been observed: a muonic atom, aka muonic hydrogen).}  Clarification:
    muons may exclusively be able to form nucleons; muon-neutrinos likewise,
    but muon and muon-neutrino combinations are also as impossible.
\end{enumerate}

\section{TODO:}

\begin{itemize}
\item colour. start from 3 particles.

\item \textbf{Rishon I-Frame "spin"}

  to describe how particles attract one another.  related to proton-neutron
  stability.  mention colour.

\item Explanation of gravity (phase-locking)

\item mass of muon in ratios (9 T particles)

\item possible explanation for muon not decaying to e-:e+:e- is the particles
  are simply too closely bound together (in the I-Frame configuration).
  Lack of presence in the V domain means no stability there, thus the
  particle decays, but the probability of the constituent particles
  in the e-:e+:e- configuration being able to "escape" unscathed is
  negligeable.  This probably sounds a lot better when described in
  terms of "phase transitions", of which there are a limited number.
  i.e. because phase transitions occur at the single-rishon level,
  the probability of three such phase transitions occurring simultaneously
  in order to allow the constituent particles to "jump out" (i.e
  phase-transition with other matter, the end result being a decay
  pattern with 2 electrons and one positron) is vanishingly small.

\item The inverse of the difference between the muon's experimentally observed
  and theoretically predicted magnetic dipole moment is 857.69.  This is
  suspiciously close to the half-life (in seconds) of a neutron.

\item alternative perspective on how "gluons" aka virtual Pions come into
  existence, even for the ultra-brief durations of time needed for them
  to be part of a back-to-back (self-annihilating) matched pair of
  VT0 phase transforms: wave-based constructive interference diffraction
  patterns.  simple as that.  at a location - which should actually be
  quite easy to mathematically calculate based on relative position and
  3 (6?) dimensional axis of rotation of two particles - the constructive
  interference of two wave fronts *should* combine to an amount of energy
  equal to that of the relevant particles of a Pion.  in the correct
  phases, and everything.  

\item paragraph that needs some explanation and a place to put it... if anywhere.

Why the neutrino-anti-neutrino pair come into existence itself requires some
explanation.  Examining the overall V-charge of the neutron the first hint
as to why this could happen becomes clear.  The V-charge of the neutron
is +1.  The sum total V-charge and T-charge of all particles in the universe
respectively is zero.  All surrounding particles are therefore balanced
at zero charge.  A stray neutron with a +1 V-charge is therefore hugely
disruptive (effectively a V-ionic particle).  Logically therefore a
bare neutron would be more likely to influence the universe's continuous
creation of particle-anti-particle pairs for a sufficient duration such
that the paired phase-transitions can occur.  This is just one possible
explanation, however with further examples (the most elegant of which
is the Pion phase-transition) we see more support for the above hypothesis.

\end{itemize}

* for cut/paste purposes:

    VT0
    $\overline{V}$T0 
    V$\overline{T}$0 

    up
    TVT

    anti-up

    $\overline{T}$$\overline{V}$$\overline{T}$

    down
    $\overline{V}$$\overline{T}$$\overline{V}$

    anti-down
    VTV

    electron
    $\overline{T}$$\overline{T}$$\overline{T}$ 

    anti-neutrino
    $\overline{V}$$\overline{V}$$\overline{V}$

\end{document}
