text:
\section{Introduction}

Crotons are pregeometric objects that emerge both as labels encoded
on the boundary of a ``volume'' and as complementary aspects of
geometric fluctuations within that volume. To express their multifacetedness,
the name \emph{croton} was chosen, after Crotos, son of Pan and Eupheme,
who, once a mortal $3D$ being, was put in sky by Muses as the celestial
fixture Sagittarius. The term volume is normally linked to the categories
space or space-time. In a pre-geometric setting, more basic categories
are needed $-$ Mersenne fluctuations and qphyla. Both require various
stages of \emph{pregeometric refinement} which, expressed through
the complementary aspects croton amplitude and phase, combine to eventually
form -- or dissolve -- real geometric objects. Advancing from mark
$n$ to $n+1$ thus, in what follows, means a time-like refinement
$2^{-n}c\;\mapsto\;2^{-n-1}c$ (roughly the analog of an exponential
increase of temporal resolution in tenacious time), and an increase
from $\alpha$ to mark $\alpha+1$ a space-like refinement ${\displaystyle {\displaystyle {\textstyle \frac{a}{b_{\alpha}}}}}\;\mapsto\;{\displaystyle {\displaystyle {\textstyle \frac{a}{b_{\alpha}+\frac{1}{b_{\alpha+1}}}}}}$
$(b_{\alpha},b_{\alpha+1}>0)$ (analoguous to increase of spatial
resolution in sustained space). On the boundary, these increases find
expression in additionally encoded labels.

In a previous work \cite{Merkel}, basic croton components have been
identfied, though at the time the name croton was not yet used. The
starting point was the equivalence between a Mersennian identity,
destilled from the special case that two parafermi algebras \cite{Green}
are of neighboring orders $p=2^{i}-1$, $p'=p^{i+1}\!-1$ (order marked
by parenthesized superscript):\vspace{-0.25cm}

\begin{equation}
\frac{1}{2}\textrm{\large\{}\boldsymbol{b}^{(p')},1\!\!\!\boldsymbol{1}^{\otimes n}\otimes\boldsymbol{b}^{(1)}\textrm{\large\}}=\boldsymbol{b}^{(p)}\otimes1\!\!\!\boldsymbol{1},\label{eq:inter-b}
\end{equation}
\vspace{-0.5cm}
and the identities\negthinspace{}
\footnote{\ \ where $\boldsymbol{f}^{(1)}\equiv\boldsymbol{b}^{(1)}=\left(\begin{array}{cc}
0 & 0\\
1 & 0
\end{array}\right)$,$\; c_{3}=\left(\begin{array}{cc}
0 & 1\\
-1 & 0
\end{array}\right)$, $c_{2}=\left(\begin{array}{cc}
0 & 1\\
1 & 0
\end{array}\right),1\!\!\!\boldsymbol{1}=\left(\begin{array}{cc}
1 & 0\\
0 & 1
\end{array}\right)$: \\
$\boldsymbol{\qquad f}^{(p)}\equiv1\!\!\!\boldsymbol{1}^{\otimes i-1}\otimes\boldsymbol{b}^{(1)}+(G_{\mu\nu}^{(p)})\otimes c_{3}$,$\quad\boldsymbol{h}^{(p)}\equiv1\mathbf{\!\!\!1}^{\otimes i-1}\otimes\boldsymbol{b}^{(1)}+(J_{\mu\nu}^{(p)})\otimes c_{2}$,
$\quad(i=2,3,\ldots)
}

\begin{equation}
(\boldsymbol{f}^{(p')})^{2}=\boldsymbol{f}^{(p)}\otimes1\mathbf{\!\!\!1},\label{eq:inter-f}
\end{equation}
\vspace{-0.9cm}

\begin{equation}
(\boldsymbol{h}^{(p')})^{2}=\boldsymbol{h}^{(p)}\otimes1\mathbf{\!\!\!1}.\label{eq:inter-f-1}
\end{equation}

\noindent Leaving the details to \ref{sec:Crotons-as-boundary},
the way croton base numbers are derived and how they are subdivided
into bases pop out naturally when the matrix elements of $\boldsymbol{f}^{(p)}$
and $\boldsymbol{h}^{(p)}$ are constructed. Crotons, conceived of
as linear combinations, use the following croton base numbers as scalars
(underlining explained later): for $i=2$, $G^{(3)}=1$, $J^{(3)}=1$;
for $i=3$, $G^{(7)}=\underline{1}$, $(J_{\varrho}^{(7)})=(-\underline{1},3)$;
for $i=4$, $(G_{\rho}^{(15)})=(3,\underline{5},11,17,41,113)$, $(J_{\varrho}^{(15)})=(-\underline{5},15,-43,149),$
to name only the first few (singletons and bases). They are instructive
enough to show how label encoding works on the boundary.

\section{Crotons on the boundary}

We first concentrate on order $p=15$, dropping the parenthesized
superscript and just asking the reader to bear in mind that the crotons
examined belong to $i=\log_{2}(p+1)=4$. Our boundary is then defined
by the $3^{T}-1$ outer nodes of a $T$-cube complex, $T$ being the
number of croton base numbers to handle: $T=6$ for $(G_{\rho})=(3,\underline{5},11,17,41,113)$,
and $T=4$ for $(J_{\varrho})=(-\underline{5},15,-43,149)$. Let the
$x$-th node out of the $728=3^{6}-1$ of the first boundary bear
the label $\Gamma_{x}=E_{x}^{\rho}G_{\rho}$, and, correspondingly,
the $y$-th node out of the $80=3^{4}-1$ of the second boundary the
label $\chi_{y}=E_{y}^{\varrho}J_{\varrho}$(summation convention,
and $E$ denoting all non-null $T$-tuples out of $3^{T}$ possible
from $-1,0,1$). It's easy to see that the total of labels form a
croton field in either case: $\Gamma$ and $\chi$. The fact aside
that nodes can be grouped into pairs bearing values of opposite sign,
field values may occur multiply, for instance $6=(0,-1,1,0,0,0)\cdot G^{t}=(0,0,-1,1,0,0)\cdot G^{t}$.
With each field defined on its own boundary, it's far from obvious
they should have anything in common. Yet, as we assume either one
deals with a distinct crotonic aspect, we have to find a way of considering
them side by side.

\subsection{\label{sub:Croton-field-duality}Croton field duality and complementarity}

We may, for instance, ask how many distinct labels there can be expressed
\emph{potentially}, neglecting mere sign reversals. Counting from
1 on and taking as the highest conceivable value the sum of croton
base numbers in absolute terms, we arrive at the number $190$ of
potential labels from $G$. Out of these, 170 are realized as node
labels $\Gamma_{x}$. Those not realizable are 20 in number: $7,34,48,\ldots,189$.
The converse holds true for the $J$ case. Of 212 potentially attainable
labels, 40 are realized by $\chi_{y}$ (sign-reversals included, that's
the stock of nodes), leaving 172 labels in potential status: $1,2,3,4,6,\ldots,211$.

\noindent A comparable situation arises when we bunch together croton
base numbers that are rooted in neighboring Mersenne orders, a process
we have previously termed \emph{interordinal} to express this kind
of hybridization. We now have $T=7$ for $(G_{\rho}^{(7,15)})$ $=(\underline{1},3,\underline{5},11,17,41,113)$,
and $T=6$ for $(J_{\rho}^{(7,15)})=(-\underline{1},3,-\underline{5},15,-43,$
$149)$. Neglecting sign reversals and counting again from 1 on, we
get 191 potential labels from the enlarged $G$ and 216 from the enlarged
$J$. All of the 191s' bunch are realized as $\Gamma_{x}$ on the
expanded boundary's nodes; but a singularity also springs up, $0=(1,0,1,1,-1,0,0)\cdot G^{t}$.
By contrast, 202 out of the 216s' bunch are realized as $\chi_{y}$,
on another expanded boundary's nodes and with no singularity popping
up, leaving 14 in potential status: $68,69,\ldots,81$. The conclusion
is that the fields are dual to each other with respect to realizability
of labels on the boundary, the reason being that they encode complementary
aspects of crotonicity $-$ croton amplitude and phase in the volume.
The duality is controlled by two quantities, Catalan number $C_{q\pm1}$
and the number $5\cdot2^{i-r}$$\:(q\in\{1,3\},r\in\{2,3\})$: \medskip{}

\noindent \emph{Intraordinal case}:\vspace{-0.25cm}
\begin{equation}
\begin{array}{ccc}
 & _{C_{2}}\\
\#\:\Gamma_{x}=170 & \longleftrightarrow & \#\:_{{\textstyle \urcorner}}\chi_{y}=172\\
 & _{5\cdot2^{2}}\\
\#\:\chi_{y}=40 & \longleftrightarrow & \#\:_{{\textstyle \urcorner}}\Gamma_{x}=20.
\end{array}\label{eq:control1}
\end{equation}
\newpage{}\noindent \emph{Interordinal case}:

\begin{equation}
\begin{array}{ccc}
 & _{5\cdot2}\\
\#\:\Gamma_{x}=192^{^{*}} & \longleftrightarrow & \#\:\chi_{y}=202\\
 & _{C_{4}}\\
\#\:_{{\textstyle \urcorner}}\chi_{y}=14 & \longleftrightarrow & \#\:_{{\textstyle \urcorner}}\Gamma_{x}=0.
\end{array}\label{eq:control2}
\end{equation}
($^{{\scriptscriptstyle *}}$The singularity assignment included.)
The key role in that duality is taken by the quantity $C_{q}$ $(q=(p-3)/4)$
around which the croton base numbers belonging to a basis of order
$p$ are built (hence the underlining):

\emph{\vspace{1cm}
}\\
\emph{Intraordinal case}:\emph{ }
\begin{equation}
\begin{array}{ccc}
 & _{C_{3}\textrm{\,\ sign\,\ reversal}}\\
G_{\rho}^{(15)} & \longleftrightarrow & J_{\varrho}^{(15)}
\end{array}
\end{equation}
\emph{Interordinal case}:

\begin{equation}
\begin{array}{ccc}
 & {\scriptstyle C_{1},C_{3}\textrm{\,\ sign\,\ reversals}}\\
G_{\delta}^{(7,15)} & \longleftrightarrow & J_{\vartheta}^{(7,15)}.
\end{array}
\end{equation}

\section{Crotons in the volume}

For ``volume'' as the term is used here, a multitude of Mersenne
fluctuations are constituive. They assume a descriptive $\bigwedge$
shape when amplitude is plotted versus ``time''. Nodes on legs of
a `$\bigwedge$' each bear a croton amplitude $\varphi_{\alpha_{n\mp r}}^{(n\mp r)}$
$\in\mathbb{N}$ that emerges with a specific time-like and space-like
refinement $-$ on the left leg $n-r,\alpha_{n-r}$, on the right
$n+r,\alpha_{n+r}$ $-$ and the peak amplitude is reached at $n,\alpha_{n}$.
The left-leg structure is given by
\begin{equation}
\varphi_{\alpha_{n-r+1}}^{(n-r+1)}=2\varphi_{\alpha_{n-r}}^{(n-r)}+\delta+\epsilon,\label{eq:left-leg}
\end{equation}
the right-leg structure by
\begin{equation}
\varphi_{\alpha_{n+r+1}}^{(n+r+1)}=\left\lfloor \varphi_{\alpha_{n+r}}^{(n+r)}/2\right\rfloor -\delta\label{eq:right-leg}
\end{equation}
$(\delta\in\{0,1\},\epsilon\in\{-1,0,1\})$, under the constraint
of a maximal croton-amplitude shift between $n-r$ and $n+r$ of 1:
$\left|\varphi_{\alpha_{n-r}}^{(n-r)}-\varphi_{\alpha_{n+r}}^{(n+r)}\right|\leq1$
$(r<n,r\in\mathbb{N}_{0})$. A typical Mersenne fluctuation is shown
in Fig. \ref{fig:A-prototype-geometric}:
\begin{figure}[H]
\caption{A geometric fluctuation of Mersenne type\label{fig:A-prototype-geometric}}
\bigskip{}
\begin{tabular*}{12cm}{@{\extracolsep{\fill}}cccccccccccccc>{\centering}p{1cm}>{\centering}p{1cm}}
 &  &  &  &  &  &  &  &  &  &  &  &  &  &  & \tabularnewline
 &  &  &  &  &  &  &  &  &  &  &  &  &  &  & \tabularnewline
 &  &  &  &  &  &  &  & \begin{sideways}
${\scriptstyle \varphi_{352}^{(212)}}{\scriptstyle =918}$
\end{sideways} &  &  &  &  &  &  & \tabularnewline
 &  &  &  &  &  &  & \begin{sideways}
${\scriptstyle \varphi_{366}^{(211)}=458}$
\end{sideways} &  & \begin{sideways}
${\scriptstyle \varphi_{362}^{(213)}=459}$
\end{sideways} &  &  &  &  &  & \tabularnewline
 &  &  &  &  &  & \begin{sideways}
${\scriptstyle \varphi_{338}^{(210)}=228}$
\end{sideways} &  &  &  & \begin{sideways}
${\scriptstyle \varphi_{350}^{(214)}=229}$
\end{sideways} &  &  &  &  & \tabularnewline
 &  &  &  &  & \begin{sideways}
${\scriptstyle \varphi_{344}^{(209)}=114}$
\end{sideways} &  &  &  &  &  & \begin{sideways}
${\scriptstyle \varphi_{328}^{(215)}=114}$
\end{sideways} &  &  &  & \tabularnewline
 &  &  &  & \begin{sideways}
${\scriptstyle \varphi_{338}^{(208)}=56}$
\end{sideways} &  &  &  &  &  &  &  & \begin{sideways}
${\scriptstyle \varphi_{336}^{(216)}=56}$
\end{sideways} &  &  & \tabularnewline
 &  &  & \begin{sideways}
${\scriptstyle \varphi_{326}^{(207)}=27}$
\end{sideways} &  &  &  &  &  &  &  &  &  & \begin{sideways}
${\scriptstyle \varphi_{328}^{(217)}=28}$
\end{sideways} &  & \tabularnewline
\begin{sideways}
\end{sideways} &  & \begin{sideways}
${\scriptstyle \varphi_{336}^{(206)}=13}$
\end{sideways} &  &  &  &  &  &  &  &  &  &  &  & \begin{sideways}
${\scriptstyle \varphi_{324}^{(218)}=14}$
\end{sideways} & \tabularnewline
 & $\vdots$ &  &  &  &  &  &  &  &  &  &  &  &  &  & $\vdots\qquad\qquad$\tabularnewline
 &  &  &  &  &  &  &  &  &  &  &  &  &  &  & \tabularnewline
\end{tabular*}
\end{figure}
\noindent We can stay in the (``time'',amplitude) coordinate system
and observe how fluctuations which share amplitudes that differ maximally
by $\delta$ at each node but peak at different heights, organize
into what we have previously termed \emph{qphylum.
\footnote{\emph{\label{fn:One-such-qphylum}}~One such qphylum would for instance
house (peaks in boldface) the Mersenne fluctuations\\
($\ldots,$17,35,72,145,291,584,$\mathbf{1170}$,585,292,145,72,35,17,$\ldots$),\\
($\ldots,$18,36,72,145,291,584,1169,$\mathbf{2340}$,1170,585,292,145,72,36,18,$\ldots$),\\
($\ldots,$17,35,72,146,292,584,1169,2340,$\mathbf{4681}$,2340,1169,584,292,145,72,36,17,$\ldots$)
etc. However, the Mersenne fluctuation ($\ldots,$584,1168,2337,4676,$\mathbf{9351}$,4675,2336,1168,583,$\ldots$)
definitely belongs to a different qphylum $-$ one worth mentioning
also because it is one of the rare instances where overt inversion
of the $\epsilon$ term occurs: increase from 4676 to 9351 implies
$\delta=0,\epsilon=-1$ (see Eq. (\ref{eq:left-leg}))
}~
\begin{figure}[H]
\caption{A prototype qphylum\label{fig:A-prototype-qphylum}}
\vspace{2cm}
\hspace{2.5cm}\includegraphics[scale=0.8]{left-complete-binary-tree}
\end{figure}
\bigskip{}
\noindent Seen top-down, a qphylum is a left-complete binary tree,
that is: a rooted tree whose root node and left child nodes have left
and right child nodes, while right child nodes have only right child
nodes, as shown in Fig. \ref{fig:A-prototype-qphylum}. Typically,
qphyleticly related amplitudes are rooted in different time-like and
space-like refinements; nodes of a qphylum thus are associated with
a set of frozen-in pregeometric ``time'' and ``space'' signatures.
``Volume'' then becomes the assembly of all distinct qphyla. But
let us go back one step and ask what it means when an amplitude in
a given fluctuation reaches a certain level. If that level coincides
with $L_{m}+1$ or $L_{m}$, where $L_{m}$ denotes the kissing number
of $m$-dimensional Euclidean space, it could mean that a chunk of
space containing an $(m-1)$-sphere packing \emph{with} or \emph{without}
centerpiece was created in that fluctuation $-$ or dissolved if the
amplitude did not peak: crotons which wax and wane. As an example,
Fig. \ref{fig:A-prototype-geometric} shows a fluctuation that peaks
at 918, a quantity considered to be the proper kissing number of 13-dimensional
Euclidean space. A chunk of 13$D$-space containing 918 12-spheres
certainly is hard to visualize, so a 2$D$ version may suffice to
give a first impression (see Fig. \ref{fig:1-sphere-packing-with}):\newpage{}
\begin{figure}[th]
\caption{\label{fig:1-sphere-packing-with}1-sphere packing with our without
centerpiece}

\vspace{2cm}

\qquad{}\qquad{}\qquad{}\qquad{}\qquad{}\qquad{}\includegraphics[scale=0.5]{kissing+centerp}
\end{figure}

\subsection{Connecting boundary and volume croton data}

We may ask if and how the peak amplitude 918 is related to a label
$\Gamma_{x}$ on the boundary. Certainly it is a realizable label,
and one realizable \emph{intra}ordinally: All kissing numbers lying
$-$ on the basis of $(G_{\rho}^{(15)})$ $-$ in the range of potentially
attainable labels can be seen to be realizable intraordinally, and
this holds true too for the current basis, the 18-tupl
\footnote{~in full length, the tuple reads $(G_{\rho}^{(31)})=(19,43,115,155,\underline{429},1275,1595,1633,4819,4905,$
$15067,15297,18627,58781,189371,227089,737953,2430289)$; its origin
and the origin of the tuple $(J_{\rho}^{(31)})=(13,-41,117,143,-\underline{429},1319,1343,1547,-4823,-4903,15547,17989,18269,$
$-58791,\,194993,\,223573,-747765,$ $2886235)$ are elucidated in
\ref{sec:Crotons-as-boundary}; \ref{sec:Crotons-in-the} tabularizes
various kissing-number related croton amplitudes, among them also
amplitude 91
} $(G_{\rho}^{(31)})=(19,43,115,155,\underline{429},\ldots,1275,\ldots,$
$4819$, $4905,\ldots)$ where 918 belongs: $918=(0,1,1,0,-1,0,{\scriptstyle \ldots},1,0,{\scriptstyle \ldots},1,-1,0,{\scriptstyle \ldots})\cdot(G^{(31)})^{t}$.
The crucial question is, Do we require \emph{all} croton amplitudes
from a given Mersenne fluctuation with one of them ``geometrizing''
to have counterparts in intraordinally realizable labels on the boundary,
in a narrow interpretation of the holographic principle? Amplitudes
``on the way/from there'' may at least in principle be amenable
to an answer. And, what does this mean for Mersenne fluctuations ``making
detours'' which presumably are by far in the majority? If one of
the croton amplitudes, call it \emph{pivotal}, comes only close and
does not ``geometrize'', it is because some residual Mersenne fluctuations
co-evolve in \emph{different} qphyla. Yet, with sufficiently tight
space-like and time-like refinement constraints, fluctuations that
are inter-qphyleticly linked to the pivotal fluctuation can be identified
and examined. See the example below where one of the residual partial
amplitudes is $102$ as $n-2=1556$, and the pivotal amplitude $5219$,
together with a second residual partial amplitude $24$, is closing
in on $L_{17}(=5346)$ as $n=1558$:
\begin{figure}[H]
\caption{\label{fig:Pivotal-amplitude-closing}Pivotal amplitude closing in
on $L_{17}(=5346)$ plus two residual partial amplitudes}
\vspace{2cm}

\qquad{}\qquad{}\qquad{}\qquad{
\begin{tabular}{cccccc}
 &  &  &  &  & \tabularnewline
\cline{3-3}
 & \multicolumn{1}{c|}{} & \multicolumn{1}{c|}{$\begin{array}{c}
{\scriptstyle \varphi_{401}^{(1558)}}\\
{\scriptstyle =5219}
\end{array}$} &  &  & \tabularnewline
\cline{3-3}
 & $\begin{array}{c}
{\scriptstyle \varphi_{442}^{(1557)}}\\
{\scriptstyle =2609}
\end{array}$ &  & $\begin{array}{c}
{\scriptstyle \varphi_{407}^{(1559)}}\\
{\scriptstyle =2609}
\end{array}$ &  & \tabularnewline
$\begin{array}{c}
{\scriptstyle \varphi_{448}^{(1556)}}\\
{\scriptstyle =1304}
\end{array}$ &  &  &  & $\ddots$ & \tabularnewline
 &  & $\begin{array}{c}
{\scriptstyle \varphi_{390}^{(1558)}}\\
{\scriptstyle =411}
\end{array}$ &  &  & \tabularnewline
 & $\begin{array}{c}
{\scriptstyle \varphi_{380}^{(1557)}}\\
{\scriptstyle =205}
\end{array}$ &  & $\begin{array}{c}
{\scriptstyle \varphi_{398}^{(1559)}}\\
{\scriptstyle =205}
\end{array}$ &  & \tabularnewline
\cline{1-1}
\multicolumn{1}{|c|}{$\begin{array}{c}
{\scriptstyle \varphi_{404}^{(1556)}}\\
{\scriptstyle =102}
\end{array}$} &  &  &  & $\ddots$ & \tabularnewline
\cline{1-1}
 &  &  &  &  & \tabularnewline
$\begin{array}{c}
{\scriptstyle \varphi_{406}^{(1556)}}\\
{\scriptstyle =100}
\end{array}$ &  &  &  &  & \tabularnewline
 & $\begin{array}{c}
{\scriptstyle \varphi_{382}^{(1557)}}\\
{\scriptstyle =49}
\end{array}$ &  &  &  & \tabularnewline
\cline{3-3}
 & \multicolumn{1}{c|}{} & \multicolumn{1}{c|}{$\begin{array}{c}
{\scriptstyle \varphi_{392}^{(1558)}}\\
{\scriptstyle =24}
\end{array}$} &  &  & \tabularnewline
\cline{3-3}
 &  &  & $\begin{array}{c}
{\scriptstyle \varphi_{400}^{(1559)}}\\
{\scriptstyle =12}
\end{array}$ &  & \tabularnewline
 &  &  &  & $\ddots$ & \tabularnewline
\end{tabular}
\end{figure}
\noindent Another example is shown in Fig. \ref{fig:Pivotal-amplitude-closing-1}
where two residual partial amplitudes attain the levels $58$ and
$193$ respectively as $n-2=1003$, allowing a pivotal amplitude $207679$
to close in on $L_{29}(=207930)$ as $n=1005$:
\begin{figure}[H]
\caption{\label{fig:Pivotal-amplitude-closing-1}Pivotal amplitude closing
in on $L_{29}(=207930)$ plus two residual partial amplitudes}
\vspace{2cm}

\qquad{}\qquad{}\qquad{}\qquad{
\begin{tabular}{cccccc}
 &  &  &  &  & \tabularnewline
\cline{3-3}
 & \multicolumn{1}{c|}{} & \multicolumn{1}{c|}{$\begin{array}{c}
{\scriptstyle \varphi_{438}^{(1005)}}\\
{\scriptstyle =207679}
\end{array}$} &  &  & \tabularnewline
\cline{3-3}
 & $\begin{array}{c}
{\scriptstyle \varphi_{442}^{(1004)}}\\
{\scriptstyle =103839}
\end{array}$ &  & $\begin{array}{c}
{\scriptstyle \varphi_{442}^{(1006)}}\\
{\scriptstyle =103839}
\end{array}$ &  & \tabularnewline
$\begin{array}{c}
{\scriptstyle \varphi_{448}^{(1003)}}\\
{\scriptstyle =51919}
\end{array}$ &  &  &  & $\ddots$ & \tabularnewline
 &  &  &  &  & \tabularnewline
 &  &  &  &  & \tabularnewline
\cline{1-1}
\multicolumn{1}{|c|}{$\begin{array}{c}
{\scriptstyle \varphi_{449}^{(1003)}}\\
{\scriptstyle =193}
\end{array}$} &  &  &  &  & \tabularnewline
\cline{1-1}
 & $\begin{array}{c}
{\scriptstyle \varphi_{443}^{(1004)}}\\
{\scriptstyle =96}
\end{array}$ &  &  &  & \tabularnewline
 &  & $\begin{array}{c}
{\scriptstyle \varphi_{439}^{(1005)}}\\
{\scriptstyle =48}
\end{array}$ &  &  & \tabularnewline
 & $\begin{array}{c}
{\scriptstyle \varphi_{477}^{(1004)}}\\
{\scriptstyle =117}
\end{array}$ &  & $\begin{array}{c}
{\scriptstyle \varphi_{427}^{(1006)}}\\
{\scriptstyle =23}
\end{array}$ &  & \tabularnewline
\cline{1-1}
\multicolumn{1}{|c|}{$\begin{array}{c}
{\scriptstyle \varphi_{479}^{(1003)}}\\
{\scriptstyle =58}
\end{array}$} &  & $\begin{array}{c}
{\scriptstyle \varphi_{465}^{(1005)}}\\
{\scriptstyle =58}
\end{array}$ &  & $\ddots$ & \tabularnewline
\cline{1-1}
 &  &  & $\begin{array}{c}
{\scriptstyle \varphi_{449}^{(1006)}}\\
{\scriptstyle =29}
\end{array}$ &  & \tabularnewline
 &  &  &  & $\ddots$ & \tabularnewline
 &  &  &  &  & \tabularnewline
\end{tabular}
\end{figure}
\noindent The pivotal amplitudes in Figs. \ref{fig:Pivotal-amplitude-closing}
and \ref{fig:Pivotal-amplitude-closing-1} each coincide with the
peak of their paternal fluctuation, but peak amplitude is not a necessary
condition. Fig. \ref{fig:Pivotal-edge-amplitudes} describes a situation
where an $(n\mp r)$-pair of pivotal amplitudes on a fluctuation's\emph{
}legs are about to close in on $L_{29}$; here, since only one time-like
refinement lies between each candidate and the peak, one further time-like
refinement also suffices to determine the residual partial amplitudes,
where they originate and which of the two `leggy' candidates $207646$
and $207647$ would have succeeded in filling the bill had it peaked:
\begin{figure}[H]
\caption{\label{fig:Pivotal-edge-amplitudes}Leggy pivot closing in on $L_{29}(=207930)$
plus two residual partial amplitudes}
\vspace{0.5cm}
\qquad{}\qquad{}\qquad{}\qquad{
\begin{tabular}{cccccc}
 &  &  &  &  & \tabularnewline
 &  & $\begin{array}{c}
{\scriptstyle \varphi_{421}^{(992)}}\\
{\scriptstyle =415294}
\end{array}$ &  &  & \tabularnewline
\cline{4-4}
 & $\begin{array}{c}
{\scriptstyle \varphi_{417}^{(991)}}\\
{\scriptstyle =207646}
\end{array}$ & \multicolumn{1}{c|}{} & \multicolumn{1}{c|}{$\begin{array}{c}
{\scriptstyle \varphi_{423}^{(993)}}\\
{\scriptstyle =207647}
\end{array}$} &  & \tabularnewline
\cline{4-4}
$\begin{array}{c}
{\scriptstyle \varphi_{433}^{(990)}}\\
{\scriptstyle =103823}
\end{array}$ &  &  &  & $\ddots$ & \tabularnewline
 &  &  &  &  & \tabularnewline
 &  &  &  &  & \tabularnewline
$\begin{array}{c}
{\scriptstyle \varphi_{421}^{(990)}}\\
{\scriptstyle =1178}
\end{array}$ &  &  &  &  & \tabularnewline
 & $\begin{array}{c}
{\scriptstyle \varphi_{409}^{(991)}}\\
{\scriptstyle =589}
\end{array}$ &  &  &  & \tabularnewline
 &  & $\begin{array}{c}
{\scriptstyle \varphi_{413}^{(992)}}\\
{\scriptstyle =294}
\end{array}$ &  &  & \tabularnewline
\cline{1-1} \cline{4-4}
\multicolumn{1}{|c|}{$\begin{array}{c}
{\scriptstyle \varphi_{485}^{(990)}}\\
{\scriptstyle =136}
\end{array}$} &  & \multicolumn{1}{c|}{} & \multicolumn{1}{c|}{$\begin{array}{c}
{\scriptstyle \varphi_{415}^{(993)}}\\
{\scriptstyle =147}
\end{array}$} &  & \tabularnewline
\cline{1-1} \cline{4-4}
 & $\begin{array}{c}
{\scriptstyle \varphi_{465}^{(991)}}\\
{\scriptstyle =68}
\end{array}$ &  &  & $\ddots$ & \tabularnewline
 &  & $\begin{array}{c}
{\scriptstyle \varphi_{473}^{(992)}}\\
{\scriptstyle =33}
\end{array}$ &  &  & \tabularnewline
 &  &  & $\begin{array}{c}
{\scriptstyle \varphi_{471}^{(993)}}\\
{\scriptstyle =16}
\end{array}$ &  & \tabularnewline
 &  &  &  & $\ddots$ & \tabularnewline
\end{tabular}
\end{figure}

\subsection{Croton phase and its inter-qphyletic role}

``Phase'' in the pre-geometric setting assumed here just means `having
an ordinate value fluctuate between positions above and below an imaginary
baseline,' with consecutive marks on that line corresponding to stepwise
increases of space-like refinement. That ordinate value, let us call
it $\psi_{\alpha_{\psi}}^{(n)}$ for a given ``time'' $n$, is linked
to the croton amplitude $\varphi_{\alpha_{\varphi}}^{(n)}$ by the
condition:
\begin{equation}
\textrm{if}\;\left|\psi_{\alpha_{\psi}}^{(n)}\right|=\varphi_{\alpha_{\varphi}}^{(n)}+\delta\;\textrm{then}\enskip\psi_{\alpha_{\psi}}^{(n)}=\left\{ \begin{array}{cl}
\!\!\!-\varphi_{\alpha_{\varphi}}^{(n)}-1 & \!\!\!(\alpha_{\psi}\:\textrm{even})\\
\!\!\!\varphi_{\alpha_{\varphi}}^{(n)}+\delta & \!\!\!(\alpha_{\psi}\:\textrm{odd})
\end{array}\right.(\delta\in\{0,1\},\alpha_{\psi}\geq\alpha_{\varphi}).
\end{equation}
Whilst introducing croton phase in the volume now, the discussion
will be limited to the fluctuations already considered in order to
keep things as coherent as possible. Let us first follow two Mersenne
fluctuations' amplitudes (Fig. \ref{fig:Pivotal-edge-amplitudes})
and their associated $\psi^{(n)}$, one steering a pivotal, one a
selected residual's course:

\begin{table}[H]
\caption{\label{tab:Pivotal-and-residual-fig5}Pivotal and inter-qphyleticly
accompanying residual fluctuation for $987\leq n\leq997$}
\medskip{}
\begin{tabular}{|c|cc|cc|c|cc|cc|c|}
\hline
\multicolumn{1}{|c}{\vspace{-0.2cm}
} & \multicolumn{4}{c}{} &  & \multicolumn{5}{c|}{}\tabularnewline
\multicolumn{6}{|c|}{Pivot} & \multicolumn{5}{c|}{Residue}\tabularnewline
$n$ & $\varphi_{P}$ & $\alpha_{\varphi}$ & $\psi_{P}$ & $\alpha_{\psi}$ & $\triangle\alpha$ & $\varphi_{R}$ & $\alpha_{\varphi}$ & $\psi_{R}$ & $\alpha_{\psi}$ & $\triangle\alpha$\tabularnewline
\vspace{-0.2cm}
 &  &  &  &  &  &  &  &  &  & \tabularnewline
\hline
987 & 12977 & 407 & $1$2977 & 437 & 30 & 9434 & 397 & -9435 & 426 & 29\tabularnewline
\hline
988 & 25955 & 411 & -25956 & 418 & 7 & 4716 & 399 & -4717 & 410 & 11\tabularnewline
\hline
989 & 51911 & 397 & 51911 & 449 & 52 & 2357 & 385 & -2358 & 438 & 53\tabularnewline
\hline
990 & 103823 & 433 & 103824 & 441 & 8 & 1178 & 421 & -1179 & 430 & 9\tabularnewline
\hline
\noalign{\vskip0.5mm}
991 & 207646 & 417 & 207647 & 457 &
\framebox{\begin{minipage}[t][0.6\totalheight]{0.03\columnwidth
4
\end{minipage}} & 589 & 409 & 589 & 449 &
\framebox{\begin{minipage}[t][0.6\totalheight]{0.03\columnwidth
4
\end{minipage}}\tabularnewline[0.5mm]
\hline
\noalign{\vskip0.5mm}
992 & 415294 & 421 & 415294 & 469 &
\framebox{\begin{minipage}[t][0.6\totalheight]{0.03\columnwidth
4
\end{minipage}} & 294 & 413 & 294 & 461 &
\framebox{\begin{minipage}[t][0.6\totalheight]{0.03\columnwidth
4
\end{minipage}}\tabularnewline[0.5mm]
\hline
993 & 207647 & 423 & 207647 & 473 & 50 & 147 & 405 & 147 & 461 & 56\tabularnewline
\hline
994 & 103823 & 443 & 103823 & 457 & 14 & 73 & 435 & -74 & 446 & 11\tabularnewline
\hline
995 & 51911 & 425 & 51912 & 453 & 28 & 36 & 413 & 36 & 445 & 32\tabularnewline
\hline
996 & 25955 & 431 & 25955 & 477 & 46 & 17 & 417 & 17 & 461 & 44\tabularnewline
\hline
997 & 12977 & 439 & 12978 & 469 & 30 & 8 & 427 & 8 & 457 & 30\tabularnewline
\hline
\end{tabular}
\end{table}
\noindent From the table one can glean that, as the $991${\small -th}
time-like refinement is reached, the offsets $\triangle\alpha(\equiv\alpha_{\psi}-\alpha_{\varphi})$
under consideration get correlated for pivot and residual $-$ first
in $\triangle\alpha=40$, then in $\triangle\alpha=48$ $-$ signalling
the residual amplitude's share 147 to the target bill and concomitant
decorrelation of $\triangle\alpha$ as $n=993$. The same holds true
for the pivot's and the largest residual's amplitudes from Fig. \ref{fig:Pivotal-amplitude-closing}
and their associated $\psi^{(n)}$: (1) correlation in $\triangle\alpha=20$
as $n=1003$; (2) correlation in $\triangle\alpha=22$ as $n=1004$;
(3) residual amplitude's (belated) contribution 193 and decorrelation
of $\triangle\alpha$ as $n=1005$ (see Table \ref{tab:Pivotal-and-residual-fig4}).
So a first conclusion is that, from a volume point of view, target-seeking
implies phase correlation irrespective of a pivotal amplitude's coincidence
with a peak or not.

\begin{table}[H]
\caption{\label{tab:Pivotal-and-residual-fig4}Pivotal and inter-qphyleticly
accompanying residual fluctuation for $1003\leq n\leq1005$}

\medskip{}
\
\begin{tabular}{|c|cc|cc|c|cc|cc|c|}
\hline
\multicolumn{1}{|c}{\vspace{-0.2cm}
} & \multicolumn{4}{c}{} &  & \multicolumn{5}{c|}{}\tabularnewline
\multicolumn{6}{|c|}{Pivot} & \multicolumn{5}{c|}{Residue}\tabularnewline
$n$ & $\varphi_{P}$ & $\alpha_{\varphi}$ & $\psi_{P}$ & $\alpha_{\psi}$ & $\triangle\alpha$ & $\varphi_{R}$ & $\alpha_{\varphi}$ & $\psi_{R}$ & $\alpha_{\psi}$ & $\triangle\alpha$\tabularnewline
\vspace{-0.2cm}
 &  &  &  &  &  &  &  &  &  & \tabularnewline
\hline
\noalign{\vskip0.5mm}
1003 & 51919 & 448 & -51920 & 468 &
\framebox{\begin{minipage}[t][0.6\totalheight]{0.03\columnwidth
2
\end{minipage}} & 193 & 449 & 194 & 469 &
\framebox{\begin{minipage}[t][0.6\totalheight]{0.03\columnwidth
2
\end{minipage}}\tabularnewline[0.5mm]
\hline
\noalign{\vskip0.5mm}
1004 & 103839 & 442 & 103839 & 464 &
\framebox{\begin{minipage}[t][0.6\totalheight]{0.03\columnwidth
2
\end{minipage}} & 96 & 443 & 96 & 465 &
\framebox{\begin{minipage}[t][0.6\totalheight]{0.03\columnwidth
2
\end{minipage}}\tabularnewline[0.5mm]
\hline
\noalign{\vskip0.5mm}
1005 & 207679 & 438 & 207679 & 443 & 5 & 48 & 439 & 48 & 445 & 6\tabularnewline[0.5mm]
\hline
\end{tabular}
\end{table}

\noindent  There is more to croton phase than just that. Let us once
more go back one step and consider the target-matching case first.
If a croton amplitude reaches a level $L_{m}+1$ or $L_{m}$, it was
assumed a chunk of $m$-dimensional Euclidean space containing an
$(m-1)$-sphere packing \emph{with} or \emph{without} centerpiece
was created in that fluctuation (or dissolved if the amplitude did
not peak). Whether that creation succeeded depends on the quantity
$\delta\in\{0,1\}$: only if the amplitude $\varphi$ peaks on $L_{m}$
and the phase $\psi$, in absolute terms, on $L_{m}+\delta\:(\delta\in\{0,1\})$
can we be sure of successful creation; if $\varphi<L_{m}$, or if
$\left|\psi\right|>L_{m}+1$, we'd be uncertain whether to settle
on success or state failure. There are situations where that criterion
applies to more than one Mersenne fluctuation. See Table \ref{tab:Pivotal-and-residual-fig4-1}
which illuminates the stance of three detouring fluctuations:

\vspace{-0.2cm}
\begin{table}[H]
\caption{\label{tab:Pivotal-and-residual-fig4-1}Co-occurrent fluctuations
targeted at $L_{29}(=207930)$, $L_{10}(=336)$ and $L_{8}(=240)$}

\medskip{}
\quad{
\begin{tabular}{|c|cc|cc|c|cc|cc|c|}
\hline
\vspace{-0.2cm}
 & \multicolumn{4}{c}{} &  & \multicolumn{5}{c|}{}\tabularnewline
 & \multicolumn{5}{c|}{Pivot} & \multicolumn{5}{c|}{Residue 1}\tabularnewline
$n$ & $\varphi_{P}$ & $\alpha_{\varphi}$ & $\psi_{P}$ & $\alpha_{\psi}$ & $\triangle\alpha$ & $\varphi_{R_{1}}$ & $\alpha_{\varphi}$ & $\psi_{R_{1}}$ & $\alpha_{\psi}$ & $\triangle\alpha$\tabularnewline
\vspace{-0.2cm}
 &  &  &  &  &  &  &  &  &  & \tabularnewline
\hline
\noalign{\vskip0.5mm}
609 & 208430 & 72 & -208431 & 72 & 0 & 10 & 58 & -11 & 52 &
\framebox{\begin{minipage}[t][0.6\totalheight]{0.03\columnwidth

\end{minipage}}\tabularnewline[0.5mm]
\hline
\multicolumn{6}{|c|}{\vspace{-0.2cm}
} & \multicolumn{5}{c|}{}\tabularnewline
\multicolumn{1}{|c}{} & \multicolumn{5}{c|}{} & \multicolumn{5}{c|}{Residue 2}\tabularnewline
\multicolumn{1}{|c}{} &  & \multicolumn{1}{c}{} &  & \multicolumn{1}{c}{} &  & $\varphi_{R_{2}}$ & $\alpha_{\varphi}$ & $\psi_{R_{2}}$ & $\alpha_{\psi}$ & $\triangle\alpha$\tabularnewline
\multicolumn{1}{|c}{\vspace{-0.2cm}
} &  & \multicolumn{1}{c}{} &  & \multicolumn{1}{c}{} &  &  &  &  &  & \tabularnewline
\hline
\noalign{\vskip0.5mm}
\multicolumn{1}{|c}{609} & \multicolumn{5}{c|}{} & 66 & 78 & 66 & 83 &
\framebox{\begin{minipage}[t][0.6\totalheight]{0.03\columnwidth

\end{minipage}}\tabularnewline[0.5mm]
\hline
\end{tabular}
\end{table}
\noindent Contrary to the former examples of detouring, in the above
there is only one time-like refinement that counts because the largest
amplitude (still called pivot) overshoots as $n=609$. The three fluctuations
could, in a ``covert conspiracy'', strive concurrently after three
targets, $\varphi_{P}+\varphi_{R_{1}}+\varphi_{R_{2}}=L_{29}+L_{10}+L_{8}$,
where $L_{29}=207930,$ $L_{10}=336$ and $L_{8}=240$. Residuality
not only assumes a different meaning here, the space-like refinements
get also symmetrized, one residual's being lower than the pivot's,
the other one's higher, and the offsets in question get correlated
as $n=609$. Offset equality obviously is uncertain by a factor $\delta=\left|\triangle\alpha_{R_{2}}-\triangle\alpha_{R_{1}}\right|\;(\delta\in\{0,1\})$,
and $\delta=1$ above since the phase inversions that enter at $n=609$
are followed only by two of the three contributors. A very similar
example is shown in Table \ref{tab:Pivotal-and-residual-fig4-1-1}:\vspace{-0.2cm}

\begin{table}[H]
\caption{\label{tab:Pivotal-and-residual-fig4-1-1}Co-occurrent fluctuations
targeted at $L_{12}(=756)$, $L_{10}(=336)$ and $L_{2}(=6)$}

\medskip{}
\quad{}\quad{
\begin{tabular}{|c|cc|cc|c|cc|cc|c|}
\hline
\vspace{-0.2cm}
 & \multicolumn{4}{c}{} &  & \multicolumn{5}{c|}{}\tabularnewline
 & \multicolumn{5}{c|}{Pivot} & \multicolumn{5}{c|}{Residue 1}\tabularnewline
$n$ & $\varphi_{P}$ & $\alpha_{\varphi}$ & $\psi_{P}$ & $\alpha_{\psi}$ & $\triangle\alpha$ & $\varphi_{R_{1}}$ & $\alpha_{\varphi}$ & $\psi_{R_{1}}$ & $\alpha_{\psi}$ & $\triangle\alpha$\tabularnewline
\vspace{-0.2cm}
 &  &  &  &  &  &  &  &  &  & \tabularnewline
\hline
\noalign{\vskip0.5mm}
1000 & 758 & 239 & 758 & 269 & 30 & 335 & 356 & -336 & 388 &
\framebox{\begin{minipage}[t][0.6\totalheight]{0.03\columnwidth
3
\end{minipage}}\tabularnewline[0.5mm]
\hline
\multicolumn{6}{|c|}{\vspace{-0.2cm}
} & \multicolumn{5}{c|}{}\tabularnewline
\multicolumn{1}{|c}{} & \multicolumn{5}{c|}{} & \multicolumn{5}{c|}{Residue 2}\tabularnewline
\multicolumn{1}{|c}{} &  & \multicolumn{1}{c}{} &  & \multicolumn{1}{c}{} &  & $\varphi_{R_{2}}$ & $\alpha_{\varphi}$ & $\psi_{R_{2}}$ & $\alpha_{\psi}$ & $\triangle\alpha$\tabularnewline
\multicolumn{1}{|c}{\vspace{-0.2cm}
} &  & \multicolumn{1}{c}{} &  & \multicolumn{1}{c}{} &  &  &  &  &  & \tabularnewline
\hline
\noalign{\vskip0.5mm}
\multicolumn{1}{|c}{1000} & \multicolumn{5}{c|}{} & 5 & 135 & -6 & 168 &
\framebox{\begin{minipage}[t][0.6\totalheight]{0.03\columnwidth
3
\end{minipage}}\tabularnewline[0.5mm]
\hline
\end{tabular}
\end{table}

\noindent A natural question to ask is if the aforementioned amplitudes
and phases and the conclusions drawn from them stand a boundary check.

\subsection{Boundary check}

Clearly, the boundary must be checked at this stage because it has
yet to be decided if crotons from Mersenne fluctuations making a detour
around a kissing number get encoded \emph{intra}ordinally or \emph{inter}ordinally.
We may put together the relevant facts here by starting with a recollection
and extrapolating from there:

(1) Out of all $\Gamma^{(15)}$ labels realizable on the basis of
$(G_{\rho}^{(15)})$ (the intraordinal case), one subset of labels
can be extracted that encode croton amplitudes coincident with $\pm L_{m},\pm(L_{m}+1)$
$(m=1,2,\ldots7)$. Not $\pm(L_{2}+1)=\pm7$, however. The complete
realization (interordinal case) demands an enlarged basis $(G_{\varrho}^{(7,15)})$
that brings singular labels in its wake. On the basis of $(J_{\rho}^{(15)})$,
no subset of $\chi^{(15)}$ labels is able to encode $L_{m},L_{m}+1$
$(m=1,2,\ldots7)$ or sign reversed versions thereof; that encoding
only catches up when the basis is enlarged to $(J_{\varrho}^{(7,15)})$,
facing us with two adamant cases yet: $\pm(L_{7}+1)=\pm127$ (realizable
before) and $\pm L_{6}$, $\pm(L_{6}+1)$ (unrealizable after); no
singular labels are entailed. Altogether a complex picture.

(2) In contrast, singular labels spring up directly on the basis of
either $(G_{\rho}^{(31)})$ or $(J_{\rho}^{(31)})$ (see \ref{sec:Crotons-as-boundary});
a simplification that, in turn, pays off with \emph{twofold}-realizable
labels all the way up for croton amplitudes coincident with $\pm L_{m}$,
$\pm(L_{m}+1)$ $(m=8,9,\ldots,31)$, also making up leeway to the
former special cases $m=2,6,7$.

\noindent So what can, on top of that, be checked is the realizability
of our example \emph{pivots} from Figs.\ref{fig:Pivotal-amplitude-closing}
to \ref{fig:Pivotal-edge-amplitudes}, including their associated
$\varphi_{P}$. Because of what they are targeted at, it is decidable
unequivocally where their images are to be sought: $\Gamma^{(31)}$
and $\chi^{(31)}$, based on the 18-tuples $(G_{\rho}^{(31)})$ and
$(J_{\rho}^{(31)})$. The targets under discussion behave as one would
expect: $\pm L_{17}(=\pm5346)$, $\pm(L_{17}+1)$ as well as $\pm L_{29}(=\pm207930)$,
$\pm(L_{29}+1)$ are realizable altogether by $\Gamma^{(31)}$ and
$\chi^{(31)}$. The same statement holds true for our example pivots
$-$ 5219, 5220. And, peaking or not, amplitudes 207646, 207647 and
207679 are perfectly twofold-realizable either. Surprisingly, twofold-realizability
holds out for the whole pivotal and residual stopovers and co-occurrent
targets mentioned in the discussion of detouring fluctuations. The
holographic principle, according to which all volume quantities $\varphi_{P}$,
$\psi_{P}$ and $\varphi_{R}$, $\psi_{R}$ from Tables \ref{tab:Pivotal-and-residual-fig5}
to \ref{tab:Pivotal-and-residual-fig4-1-1} should have boundary counterparts,
$\varphi$ in $\Gamma^{(31)}$, $\psi$ in $\chi^{(31)}$, is exceedingly
satisfied $-$ the aforesaid quantities are invariably twofold-realizable
(see \ref{sec:Crotons-as-boundary}). Since the same encompassing
holographicity also obtains for the stopovers in the target matching
Mersenne fluctuation of Fig. \ref{fig:A-prototype-geometric}, one
can in summary say that amplitude and phase data from target-seeking
Mersenne fluctuations in the volume have a perfect image on the boundary.\newpage{}

\section{\label{sec:Application-to-subatomic}Application to subatomic physics}

The only ingredient that sounded physical thus far is the complementarity
of amplitude and phase. Complementarity provides an excellent starting
point for studying the conceptual link between Mersenne fluctuations,
or qphyla $-$ our pregeometric categories linked to increases in
time-like and space-like refinement $-$, and the particle creation/lifetime
patterns that arise with increases in temporal and spatial resolution
for sustained geometry. As a prelude invoking atomic physics, let
us examine the ratio of the electrical to the gravitational forces
between a proton and an electron, where a first kind of conplementarity
comes into play. Consider the collections formed by $m$ Magnus terms
$M_{k}\equiv(2k+1)^{2}(-x)^{k(k+1)/2}\quad(0<x<1)$ and the estimated
number of protons in the universe, $N=10^{80}$,
\begin{equation}
\sum^{(m)}M_{k}(N-m+1),\label{eq:Magnus}
\end{equation}
from which $x$ is to be determined. The electrical force $F_{e}$
is considered independent of $N$; thus $m=1$, \emph{i.e}., only
one Magnus term is there to account for $x_{e}$. Assuming that the
boundaries $\Gamma^{(15)}$, $\chi^{(15)}$ are sufficient for the
proton-electron system, we make a choice of the triple $\left(k,2k+1,x_{e}^{-1}\right)$
such that expression (\ref{eq:Magnus}) forms a least upper bound
to the observed ratio $F_{e}/F_{g}$ under the constraint that only
successive Mersenne numbers are being used. This is fulfilled for
$M_{7}(x_{e})=225(-x_{e})^{28}$ with $k=7$, $2k+1=15$, $x_{e}^{-1}=31$
where (\ref{eq:Magnus}) assumes just the reasonable value $M_{7}(1/31)\times10^{80}\approx3.92\times10^{40}$.
Thus,
\[
x_{e}=\frac{1}{31}.
\]
In contrast, the gravitational force according to Mach's principle
is dependent on all other gravitating bodies in the universe so that,
in this case, $m=N$ and expression (\ref{eq:Magnus}) reduces to
$10^{80}$ Magnus sum terms, starting with $k=0$, that are going
to account for $x_{g}$. To good approximation,
\[
x_{g}\approx\Lambda=0.10765\ldots,
\]
the so-called 'one-ninth' constant which is the unique exact solution
of the full Magnus equation $\sum_{k=0}^{\infty}(2k+1)^{2}(-x)^{k(k+1)/2}=0\quad(0<x<1)$.
Next, we come to the aforementioned croton complementarity, which
plays the part of fine-tuning: 170 croton field values are representable
on $\Gamma^{(15)}$, 40 of them representable also on $\chi^{(15)}$;
but only field values $\Gamma_{x}^{(15)}$ that are \emph{not} representable
as $\chi_{y}^{(15)}$ seem to be allowed, leaving $2^{130}$ combinations
for the power set of crotons compatible with charged particles. The
only gravity-related croton, on the other hand, is $\varphi=13$,
each occurrence of which corresponds to a chunk, endowed with centerpiece,
of the very $3D$ space whose curvature according to general relativity
is equivalent to gravity. The power set allowing just two combinations
in this case, one finds
\begin{equation}
\frac{2^{130}}{x_{e}}{\displaystyle \:\div\:\frac{2}{x_{g}}\approx2.27123\times10^{39},}\label{eq:ratio}
\end{equation}
which coincides with the measured ratio $F_{e}/F_{g}$ to five decimal
places.

\noindent From the above we behold: $\varphi=13$ is key to providing
a $3D$ scenery. In order for ever-expanding Mersenne fluctuations
to allow for particulates
\footnote{\,a generalization of amplitudes to trajectories is proposed in \emph{Sect}.
\ref{sub:Mersenne-fluctuations-and
} a cloud of crotons of amplitude $\leq13$ has to keep company with
them which administer a background of 1-$3D$ space chunks. The denser
that cloud, the more convincing the impression of a persistent $3D$
continuum.

\noindent Now we are prepared for subatomic physics. We have seen
that complementarity of bound\-ary-field values affects the power
set of croton combinations admissible in a situation, the theoretical
upper bound for combinations of order 31 being $2^{3^{18}-1}$. It
seems reasonable to associate $\Gamma_{x}^{(31)}\notin\chi^{(31)}$
with nuclear phenomena $-$ whose fundamental laws and constants are
unknown and pecularities such as the EMC effect and SRC plateaux \cite{Higin}
have remained puzzling to this day $-$ and reserve non-complementary
croton combinations $-$ among them all those identified in the previous
section (and summarized in \ref{sec:Crotons-as-boundary}) as doubly
representable on $\Gamma^{(31)}$ and $\chi^{(31)}$ $-$ to quarks
and preons. Magnus-type considerations cannot be expected to apply
without qualification. A safe starting point is to presume that preons
carry electric charge (see \ref{sec:Crotons-on-the-b}), an assumption
that allows to associate crotonic activity to (para)fermionic forms
of particulates, from superordinate levels such as protons and neutrons
to quarks and quark constituents.

\noindent Oscar Wallace Greenberg envisaged a parafermionic nature
of quarks. But with the advent of QCD, and the experimental findings,
valid to this day, that quarks are pointlike down to $10^{-20}\,$m,
preons, parafermionic or otherwise, have not found much acclaim among
physicists. The consequence of the experimental standoff is that preons,
if they exist, must inhabit extradimensions, do aggregate there and
betray their origin only in short-lived resonances known as quark
flavors. It is known that the up quark carries more momentum than
the down quark, which makes it likely that even the two of them are
not of the same dimensional origin. The following is not meant to
be a worked out general model of hadronic matter $-$ it just contemplates
on the possible mathematical structure of the subatomic onion in the
light of crotonic activity. In what follows we use the notation $f_{n+1}(=2^{n+1}-1)$
to denote (para-)fermionic order and the symbols $\mathrm{p}_{\textrm{up}}^{(f_{n+1})}$
and $\mathrm{p}_{\textrm{down}}^{(f_{n+1})}$ for up-type and down-type
preons of that order, respectively.
\begin{conjecture}
\label{con:Preons-of-order}Preons of order $f_{n+1}$ are either
up-type or down-type, $\mathrm{p}_{\textrm{\emph{up}}}^{(f_{n+1})}$
or $\mathrm{p}_{\textrm{\emph{down}}}^{(f_{n+1})}$. The electric
charge (in \emph{e}) of up-type items is given by the expressions
$\mathrm{c}_{\mathrm{up}}^{(f_{n+1})}=(f_{n+1}-\sum_{s=0}^{n}f_{s})/\prod_{r=1}^{n+1}f_{r}=(n+1)/\prod_{r=1}^{n+1}f_{r}$,
while down-type items\emph{ }have the charge $\mathrm{c}_{\mathrm{down}}^{(f_{n+1})}=-\sum_{s=0}^{n}f_{s}/\prod_{r=1}^{n+1}f_{r}$
(see Table \ref{tab:Interordinal-preon-model-1-1-1} below). The charge
of up-type items transforms as\emph{ $\mathrm{c}_{\mathrm{up}}^{(f_{n})}=(f_{n+1}-1)\mathrm{c}_{\mathrm{up}}^{(f_{n+1})}+\mathrm{c}_{\mathrm{down}}^{(f_{n+1})}$}
and the charge\emph{ }of down-type items as $\mathrm{c}_{\mathrm{down}}^{(f_{n})}=(f_{n}+1)\mathrm{c}_{\mathrm{down}}^{(f_{n+1})}+f_{n}\mathrm{c}_{\mathrm{up}}^{(f_{n+1})}$\textup{.}

\begin{table}[H]
\caption{\label{tab:Interordinal-preon-model-1-1-1}Mersennian preon charge
model}
\vspace{0.2cm}
\quad{}\quad{}\quad{}\quad{}\quad{}\quad{}\quad{}\quad{
\begin{tabular}{|c|c|c|}
\hline
\multirow{2}{*}{$f_{n+1}$} & \multirow{2}{*}{up-type charge} & \multirow{2}{*}{down-type charge}\tabularnewline
 &  & \tabularnewline
\hline
\hline
\multirow{2}{*}{1} & \multirow{2}{*}{1} & \multirow{2}{*}{0}\tabularnewline
 &  & \tabularnewline
\hline
\multirow{2}{*}{3} & \multirow{2}{*}{$\frac{2}{3}$} & \multirow{2}{*}{$-\frac{1}{3}$}\tabularnewline
 &  & \tabularnewline
\hline
\multirow{2}{*}{7} & \multirow{2}{*}{$\frac{3}{21}$} & \multirow{2}{*}{$-\frac{4}{21}$}\tabularnewline
 &  & \tabularnewline
\hline
\multirow{2}{*}{15} & \multirow{2}{*}{$\frac{4}{315}$} & \multirow{2}{*}{$-\frac{11}{315}$}\tabularnewline
 &  & \tabularnewline
\hline
\multirow{2}{*}{31} & \multirow{2}{*}{$\frac{5}{9765}$} & \multirow{2}{*}{$-\frac{26}{9765}$}\tabularnewline
 &  & \tabularnewline
\hline
\multirow{4}{*}{$\begin{array}{c}
63\\
\vdots
\end{array}$} & \multirow{4}{*}{$\begin{array}{c}
\frac{6}{615195}\\
\vdots
\end{array}$} & \multirow{4}{*}{$\begin{array}{c}
-\frac{57}{615195}\\
\vdots
\end{array}$}\tabularnewline
 &  & \tabularnewline
 &  & \tabularnewline
 &  & \tabularnewline
\hline
\end{tabular}
\end{table}
\textup{}\\
\textup{\medskip{}
}
\end{conjecture}
\noindent  For preon configurations enabling these charge transformations
the shorthand

\[
\mathrm{\mathrm{Q}}_{\mathrm{up}}^{(f_{n})}=(f_{n+1}-1)\mathrm{\mathrm{p}}_{\mathrm{up}}^{(f_{n+1})}+\mathrm{\mathrm{p}}_{\mathrm{down}}^{(f_{n+1})},\quad\mathrm{\mathrm{\bar{Q}}}_{\mathrm{up}}^{(f_{n})}=(f_{n+1}-1)\mathrm{\mathrm{\bar{p}}}_{\mathrm{up}}^{(f_{n+1})}+\mathrm{\mathrm{\bar{p}}}_{\mathrm{down}}^{(f_{n+1})},
\]
\[
\mathrm{Q}_{\mathrm{down}}^{(f_{n})}=(f_{n}+1)\mathrm{p}_{\mathrm{down}}^{(f_{n+1})}+f_{n}\mathrm{p}_{\mathrm{up}}^{(f_{n+1})},\quad\mathrm{\bar{Q}}_{\mathrm{down}}^{(f_{n})}=(f_{n}+1)\mathrm{\bar{p}}_{\mathrm{down}}^{(f_{n+1})}+f_{n}\mathrm{\bar{p}}_{\mathrm{up}}^{(f_{n+1})},
\]
is used where the bar in $\mathrm{\bar{p}}$ indicates an antipreon
of opposite electric charge. The Magnus formalism suggests a connection
between $f_{n+1}$ (or $f_{n}$) and $x_{e}^{-1}=31$. This does not
only suffice for the proton and the neutron $-$ if the greatest assignment
eligible is $f_{n+1}:=x_{e}^{-1}$, it suffices for three generations
of quarks; and if $f_{n}:=x_{e}^{-1}$ is eligible, for a fourth generation
as well. Proton and neutron are assigned the least root order: $f_{n+1}=3$.
They are named (1) configurations here:
\[
\mathrm{\mathrm{Q}}_{\mathrm{up}}^{(1)}\quad\textrm{proton ({\tt p}) with charge 1,}
\]
\[
\mathrm{\mathrm{Q}}_{\mathrm{down}}^{(1)}\quad\textrm{neutron ({\tt n}) with charge 0.}
\]
 The valence quarks or quarks of generation 1 are considered (3) configurations:
\[
\mathrm{\mathrm{Q}}_{\mathrm{up}}^{(3)}\quad\textrm{up quark ({\tt u}) with charge }\frac{2}{3},
\]
\[
\mathrm{\mathrm{Q}}_{\mathrm{down}}^{(3)}\quad\textrm{down quark ({\tt d}) with charge }-\frac{1}{3}.
\]
Charm quark and strange quark (generation 2) have ($7^{2}3)$ configurations:
\[
\mathrm{\mathrm{Q}}_{\mathrm{up}}^{(7)}+\mathrm{\mathrm{\bar{Q}}}_{\mathrm{down}}^{(7)}+\mathrm{\mathrm{\bar{Q}}}_{\mathrm{down}}^{(3)}\quad\textrm{charm quark ({\tt c}) with charge }\frac{2}{3},
\]
\[
\mathrm{\mathrm{\mathrm{Q}}_{\mathrm{up}}^{(7)}+\mathrm{\bar{Q}}}_{\mathrm{down}}^{(7)}+\mathrm{\mathrm{\bar{Q}}}_{\mathrm{up}}^{(3)}\quad\textrm{strange quark ({\tt s}) with charge }-\frac{1}{3},
\]
and top quark and bottom quark (generation 3), with root order $f_{n+1}\!=\! x_{e}^{-1}\!=\!31$,
are represented by ($15^{2}7^{2}3^{2}$) configurations
\footnote{$\,$ The top quark's charge, for instance, ensues from the ($15^{2}7^{2}3^{2}$)
transformations

\hspace{-0.3cm}$\frac{2}{3}\!=\!\left(\frac{30\cdot(-5)+1\cdot26}{9765}+\frac{16\cdot(-26)+15\cdot5}{9765}+\frac{14\cdot(-4)+1\cdot11}{315}\right)$$+\!\left(\frac{14\cdot(-4)+1\cdot11}{315}+\frac{6\cdot3+1\cdot(-4)}{21}+\frac{4\cdot4+3\cdot(-3)}{21}\right)$;
corresponding transformations yield the charges of the remaining quarks
}
\[
\mathrm{\mathrm{(\bar{Q}}}_{\mathrm{up}}^{(15)}+\mathrm{\mathrm{Q}}_{\mathrm{down}}^{(15)}+\mathrm{\mathrm{\bar{Q}}}_{\mathrm{up}}^{(7)})+\mathrm{\mathrm{(\bar{Q}}}_{\mathrm{up}}^{(7)}+\mathrm{\mathrm{Q}}_{\mathrm{up}}^{(3)}+\mathrm{\mathrm{\bar{Q}}}_{\mathrm{down}}^{(3)})\quad\textrm{top quark ({\tt t}) with charge }\frac{2}{3},
\]
\[
\mathrm{\mathrm{(\bar{Q}}}_{\mathrm{up}}^{(15)}+\mathrm{\mathrm{Q}}_{\mathrm{down}}^{(15)}+\mathrm{\mathrm{\bar{Q}}}_{\mathrm{up}}^{(7)})+\mathrm{\mathrm{(\bar{Q}}}_{\mathrm{down}}^{(7)}+\mathrm{\mathrm{\bar{Q}}}_{\mathrm{up}}^{(3)}+\mathrm{\mathrm{\bar{Q}}}_{\mathrm{down}}^{(3)})\quad\textrm{bottom qu.({\tt b}) with charge }-\frac{1}{3}.
\]
\noindent How these configurations relate to targets of crotonic
activity is propounded in our second conjecture:
\begin{conjecture}
\label{con:To-serve-as}To qualify as constituents of a superordinate
preon, hyperspheres must be used in numbers that divide the kissing
number of the space they live in.
\end{conjecture}
\noindent This principle is best understood as a simile to the Magnus
ansatz, where the intra-generational quark-mass ratios $m_{\textrm{u}}/m_{\textrm{d}}$,
$m_{\textrm{c}}/m_{\textrm{s}}$ and $m_{\textrm{t}}/m_{\textrm{b}}$
replace the dimensionless force ratio $F_{e}/F_{g}$. Quark mass is
assumed to result from crotonic activity, and the configurations {\tt c},{\tt s}
and {\tt t},{\tt d} make it clear that this activity has to cover
extended spans of orders. Here, only leading-order crotonic activity
is considered in deriving bounds for intra-generational mass ratios.
This implies identifying where leading-order crotonic activity singles
out space chunks that suit the up-type quark of a generation and other
space chunks suiting the down-type quark. The kissing number
\footnote{$\,$ large kissing numbers are an active field of research \cite{Cohn}; those used here are taken from {\small\tt http://www.math.rwth-aachen.de/Gabriele.Nebe/LATTICES/kiss.html
} of the target spaces, $L_{\textrm{up}}$ and $L_{\textrm{down}}$,
must in turn show the divisibility properties demanded in Conjecture
\ref{con:To-serve-as}. But that's only a necessary condition. In
the Magnus ansatz, assignment of successive Mersenne numbers to the
triple $(k,2k+1,x_{e}^{-1}$) is essential to getting a handle on
bounds. Here, the clue to successful bounds comes from a divisibility
postulate for a generation's $L_{\textrm{up}}$: in addition to being
divisible by $f_{n+1}-1$, $L_{\textrm{up}}$ must contain a genuine
prime factor $P_{\mu}>13$ (larger than the croton amplitude of $3D$
space) such that
\begin{equation}
\left|P_{\mu}-L_{\mu_{0}+\mu}\right|=1\quad(\mu=1,2,3).\label{eq:postulate}
\end{equation}
The task is for the $\mu$th generation completed when all its kissing
numbers $L_{\textrm{down}}$ $-$ there are several $-$ which are
divisible by $f_{n}+1$ and have but prime factors

\begin{table}[H]
\caption{\label{tab:Prime-factors-of}Prime factors of $(x_{e})^{-1}$ kissing
numbers; characterstic divisors determine up-type and down-type kissing
numbers that bound measured intra-generational quark mass ratios from
below}
\vspace{0.2cm}
\begin{tabular}{|l|l|c|c|c|c|}
\hline
\multirow{2}{*}{$m$} & \multirow{2}{*}{$L_{m}$} & \multirow{2}{*}{prime factorization} & \multirow{2}{*}{$\begin{array}{c}
\textrm{divisors}\\
\textrm{{\tt t}}\:[\textrm{{\tt b}}]
\end{array}$} & \multirow{2}{*}{$\begin{array}{c}
\textrm{divisors}\\
\textrm{{\tt c}}\:[\textrm{{\tt s}}]
\end{array}$} & \multirow{2}{*}{$\begin{array}{c}
\textrm{divisors}\\
\textrm{{\tt u}}\:[\textrm{{\tt d}}]
\end{array}$}\tabularnewline
 &  &  &  &  & \tabularnewline
\hline
1 & 2 & 2 &  &  & \tabularnewline
\hline
2 & 6 & $2\times3$ &  &  & \tabularnewline
\hline
3 & 12 & $2^{2}\times3$ &  &  & \tabularnewline
\hline
4 & 24 & $2^{3}\times3$ &  &  & $[2^{2}]$\tabularnewline
\hline
5 & 40 & $2^{3}\times5$ &  &  & $[2^{2}]$\tabularnewline
\hline
6 & 72 & $2^{3}\times3^{2}$ &  &  & $[2^{2}]$\tabularnewline
\hline
7 & 126 & $2\times3^{2}\times7$ &  &  & \tabularnewline
\hline
8 & 240 & $2^{4}\times3\times5$ & $[2^{4}]$ & $[2^{3}]$ & $[2^{2}]$\tabularnewline
\hline
9 & 272 & $2^{4}\times17$ & $[2^{4}]$ & $[2^{3}]$ & $[2^{2}]$\tabularnewline
\hline
10 & 336 & $2^{4}\times3\times7$ & $[2^{4}]$ & $[2^{3}]$ & $[2^{2}]$\tabularnewline
\hline
11 & 438 & $2\times3\times
\framebox{\begin{minipage}[t][0.6\totalheight]{0.03\columnwidth
7
\end{minipage}} &  &  & $2\times3$\tabularnewline
\hline
12 & 756 & $2^{2}\times3^{3}\times7$ &  &  & \tabularnewline
\hline
13 & 918 & $2\times3^{3}\times17$ &  &  & \tabularnewline
\hline
14 & 1422 & $2\times3^{3}\times79$ &  &  & \tabularnewline
\hline
15 & 2340 & $2^{2}\times3^{2}\times5\times13$ &  &  & \tabularnewline
\hline
16 & 4320 & $2^{5}\times3^{3}\times5$ & $[2^{4}]$ &  & \tabularnewline
\hline
17 & 5346 & $2\times3^{5}\times11$ &  &  & \tabularnewline
\hline
18 & 7398 & $2\times3^{3}\times137$ &  &  & \tabularnewline
\hline
19 & 10668 & $2^{2}\times3\times7\times
\framebox{\begin{minipage}[t][0.6\totalheight]{0.04\columnwidth
12
\end{minipage}} &  & $2\times7$ & \tabularnewline
\hline
20 & 17400 & $2^{3}\times3\times5^{2}\times29$ &  &  & \tabularnewline
\hline
21 & 27720 & $2^{3}\times3^{2}\times5\times7\times11$ &  &  & \tabularnewline
\hline
22 & 49896 & $2^{3}\times3^{4}\times7\times11$ &  &  & \tabularnewline
\hline
23 & 93150 & $2\times3^{4}\times5^{2}\times23$ &  &  & \tabularnewline
\hline
24 & 196560 & $2^{4}\times3^{3}\times5\times7\times13$ &  &  & \tabularnewline
\hline
25 & 197040 & $2^{4}\times3\times5\times821$ &  &  & \tabularnewline
\hline
26 & 198480 & $2^{4}\times3\times5\times827$ &  &  & \tabularnewline
\hline
27 & 199912 & $2^{3}\times24989$ &  &  & \tabularnewline
\hline
28 & 204188 & $2^{2}\times51047$ &  &  & \tabularnewline
\hline
29 & 207930 & $2\times3\times5\times29\times
\framebox{\begin{minipage}[t][0.6\totalheight]{0.04\columnwidth
23
\end{minipage}} & $2\times3\times5$ &  & \tabularnewline
\hline
30 & 219008 & $2^{7}\times29\times59$ &  &  & \tabularnewline
\hline
31 & 230872 & $2^{3}\times28859$ &  &  & \tabularnewline
\hline
\multirow{2}{*}{} & \multirow{2}{*}{$\frac{L_{\textrm{up}}}{\Sigma L_{\textrm{down}}}$} & \multirow{2}{*}{} & \multirow{2}{*}{$\thickapprox40.23$} & \multirow{2}{*}{$\thickapprox12.58$} & \multirow{2}{*}{$\thickapprox0.45$}\tabularnewline
 &  &  &  &  & \tabularnewline
\hline
\multirow{2}{*}{} & \multirow{2}{*}{$\frac{m_{\textrm{up}}}{m_{\textrm{down}}}$} & \multirow{2}{*}{} & \multirow{2}{*}{$\thickapprox41.86$} & \multirow{2}{*}{$\thickapprox13.58$} & \multirow{2}{*}{$\thickapprox0.48$}\tabularnewline
 &  &  &  &  & \tabularnewline
\hline
\end{tabular}
\end{table}
 \noindent less $P_{\mu}$ are identified. Then for each $L_{\textrm{down}}$
found the single ratio $\frac{L_{\textrm{up}}}{L_{\textrm{down}}}$
could be considered a bound to the ratio $\frac{m_{\textrm{up}}}{m_{\textrm{down}}}$
in question. However, true to the Magnus ansatz, getting a fine-tuned
result requires taking all contributors into account. The above table
shows how, for each generation, a ratio $\frac{L_{\textrm{up}}}{\Sigma L_{\textrm{down}}}$
can be deduced that bounds the respective measured ratio $\frac{m_{\textrm{up}}}{m_{\textrm{down}}}$
from below.

\noindent Let us clear up the inner workings of the table, beginning
with the third-generation quarks, $\textrm{{\tt t}}$ and $\textrm{{\tt b}}$.
With the top quark, $f_{5}-1=30$ hyperspheres must fit in a chunk
of space such that $f_{5}-1$ and a prime factor $P_{3}$ satisfying
postulate (\ref{eq:postulate}) for some $\mu_{0}$ divide its kissing
number without rest. Both is the case for the spaces $23D$ and $29D$,
with the candidates $(93150,23)$ and $(207930,239)$ for $(L_{\textrm{up}},P_{3})$,
respectively. Only by observing that the same selection rules must
apply to the other generations are we able to decide that $(207930,239)$
(framed in the table) is the appropriate pair $-$ a $P_{3}=L_{4}-1$
would not leave place for a $P_{2}$ exceeding 13: $L_{3}\pm1=P_{2}\ngtr13$.
The bottom quark is collectively realized by all subspace chunks in
which there are $f_{4}+1=16$ hyperspheres such that $f_{4}+1$ divides
their kissing numbers without rest (marked by $[\,]$) and the prime
factors involved are less $P_{3}=239$. For the next lower generation,
the pair suiting the up-type quark is from $19D$, $(10668,127)$,
and for the first generation, that pair is from $11D$, $(438,73)$.
In accordance with postulate (\ref{eq:postulate}), the $P_{\mu}$
($\mu=1,2,3$), specify a triple of successive kissing numbers, $(72,126,240)$.
Unsurprisingly
\begin{equation}
\mu_{0}=\log_{2}\left(x_{e}^{-1}+1\right).
\end{equation}
 The kissing numbers of the subspace chunks corresponding to down-type
quarks {\tt s} and {\tt d} too satisfy the required divisibilities
(again marked by $[\,]$).

\noindent We now want to hint at the possible existence of a fourth
quark generation.

\begin{table}[H]
\caption{\label{tab:Quantities-:=00003D-;}Quantities $S_{\mu_{0}+\mu-1}$:=
$\sum_{m=1}^{\mu_{0}+\mu-1}(2^{m}-1)$; $L_{\mu_{0}+\mu}$; $P_{\mu}$;
$\kappa(\mu)$ := $6\,\textrm{Prime}(\mu)+(-1)^{\mu}$}
\bigskip{}
\qquad{}\qquad{}\qquad{}\qquad{}\qquad{
\begin{tabular}{|c|c|c|c|c|}
\hline
$\mu$ & $S_{\mu_{0}+\mu-1}$ & $L_{\mu_{0}+\mu}$ & $P_{\mu}$ & $\kappa(\mu)$\tabularnewline
\hline
\hline
1 & 57 & 72 & 73 & 11\tabularnewline
\hline
2 & 120 & 126 & 127 & 19\tabularnewline
\hline
3 & 247 & 240 & 239 & 29\tabularnewline
\hline
4 & 502 & 272 & 271 & 43\tabularnewline
\hline
5 & 1013 & 336 & 337 & 65\tabularnewline
$\vdots$ &  &  &  & \tabularnewline
\hline
\end{tabular}
\end{table}

\noindent The entries of Table \ref{tab:Quantities-:=00003D-;} may
be used to define quark family characteristics: Where $\chi_{\textrm{prime}}(\cdot)$
is the characteristic function of prime numbers, $L_{\textrm{up}}$
can be said to belong to the family, and be identified with $L_{\kappa(\mu)}$,
if $\chi_{\textrm{prime}}(\kappa(\mu))=1$. This is obvious for $\mu=1,2,3$.
One further notes that the signum function values $\textrm{sgn}(P_{\mu}-L_{\mu_{0}+\mu})$
and $\textrm{sgn}(S_{\mu_{0}+\mu-1}-P_{\mu})$ cancel each other out
for $\mu=1,2,3$. But said observations hold out for $\mu=4$: The
equations

\begin{equation}
\begin{array}{c}
\textrm{sgn}(P_{\mu}-L_{\mu_{0}+\mu})-\textrm{sgn}(P_{\mu}-S_{\mu_{0}+\mu-1})=0,\\
\chi_{\textrm{prime}}(\kappa(\mu))-1=0
\end{array}\quad(\mu_{0}=\log_{2}\left(x_{e}^{-1}+1\right))\label{eq:conservation law}
\end{equation}
are not violated until at $\mu=5$. What might thus constitute the
quark family's conservation law would predict that $L_{43}$ have
2, 31 and 271 among its prime factors and serve as the $L_{\textrm{up}}$
of a fourth-generation quark {\tt t}$'$.

\noindent It should be noted that the bounds given in Table \ref{tab:Prime-factors-of}
for the first three generations (next-to-last row) are valid for intra-generational
mass ratios only (last row). Ratios of mass for quarks that belong
to different generations are distinctly different. One form of heuristics,
whose effectiveness would be understood by degrees (cf. \ref{sec:Crotons-on-the-b}),
draws a link between quark masses and duality controls. These we use
as a way of expressing inter-generational quark mass ratios, with
the modification that Catalan numbers $C_{x}$$\ngtr13$ are mapped
to $\hat{C_{x}}\equiv1$:
\begin{equation}
\begin{array}{llll}
\textrm{measured} & m_{\textrm{t}}/m_{\textrm{c}}\thickapprox135.64 & \textrm{control} & C_{6}=132\\
\:\quad\textrm{ "} & m_{\textrm{c}}/m_{\textrm{u}}\thickapprox560.87 & \textrm{\quad\textrm{ "}} & 5\cdot2^{3}\cdot C_{4}=560\\
\:\quad\textrm{ "} & m_{\textrm{b}}/m_{\textrm{s}}\thickapprox44 & \quad\textrm{ "} & C_{5}=42\\
\:\quad\textrm{ "} & m_{\textrm{s}}/m_{\textrm{d}}\thickapprox19.79 & \quad\textrm{ "} & 5\cdot2^{2}\cdot\hat{C_{3}}=20
\end{array}\label{eq:intermass}
\end{equation}
\noindent That, in the light of the above discussion, Eqs.(\ref{eq:intermass})
dictate the hypothetical {\tt t}$'$ weigh either $C_{7}(=429)$
or $5\cdot2^{4}\cdot C_{5}$ $(=3360)$ top masses is an educated
guess at best. Apart from {\tt t}$'$ mass, how the two types of
mass ratios, inter-generational and intra-generational, intertwine
could be ascertained by bound states. It is known that the top quark
has too short a lifetime ($5\times10^{-25}\,\mathrm{s}$) to hadronize.
Instead of forming a bound state$\begin{array}{c}
\bar{\textrm{{\tt t}}}\\
\textrm{{\tt b}}
\end{array}$, say, the top antiquark decays into the bottom quark. Table \ref{tab:Pivotal-and-residual-fig4-1}
may be interpreted as a crotonic picture of that failure. First, there
is a minority problem: there are only two co-occurring residues to
conspire with the pivot, with two down-type targets at their disposal.
At $n=69$, two more residues, $\varphi_{R_{3}}=233$ and $\varphi_{R_{4}}=39$
at $\alpha=438$ and $\alpha=453$, are available. Their inclusion
would solve the minority problem, but yield only one further $L_{\textrm{down}}$(=
272); the main down-type target $L_{\textrm{down}}=4320$ stays out
of reach. If this interpretation is correct, we can address the co-occurrence
example of Table \ref{tab:Pivotal-and-residual-fig4-1-1}. According
to Table \ref{tab:Prime-factors-of}, the up-type kissing number of
the charm quark is $L_{\textrm{up}}=10668$, the main down-type kissing
number of the strange quark $L_{\textrm{down}}=336$. Yet the pivot's
target in Table \ref{tab:Pivotal-and-residual-fig4-1-1} is $L_{12}=756$
$-$ which is about $10668/C_{4}$. On the other hand we have this
inter-generational mass ratio charm-quark-to-up-quark $\approx5\cdot2^{3}\cdot C_{4}$
which helps translate ``charmed'' mass to ``up'' mass, thusly
allowing to consider $756$ the $L_{\textrm{up}}$ of a meson $\thickapprox5\cdot2^{3}$
times as heavy as the average of an up quark and a down antiquark
$-$ a description that suits the pion $\pi^{+}$.

\noindent One further observation may round out this section. In
Table \ref{tab:Pivotal-and-residual-fig5}, $\varphi_{P}^{(991)}=207646$
connects to $L_{\textrm{up}}=207930$ and $\varphi_{R}^{(988)}=4716$
to the main $L_{\textrm{down}}=4320$, with a flipped phase $\psi_{R}^{(988)}=-4717$.
Can this be interpreted as an attempted bound state $\begin{array}{c}
\textrm{{\tt t}}\\
\textrm{{\tt \ensuremath{\bar{\textrm{{\tt b}}}}}}
\end{array}$? The minority problem can be lifted. We only have to include $\varphi_{R^{+}}^{(988)}$
$(=\psi_{R^{+}}^{(988)}$= 50 at $\alpha_{\varphi_{R^{+}}}$= 437,
$\alpha_{\psi_{R^{+}}}$= 449), $\varphi_{R^{++}}^{(991)}$ $(=\psi_{R^{++}}^{(991)}$=
589 at $\alpha_{\varphi_{R^{++}}}$= 409, $\alpha_{\psi_{R^{++}}}$=
449) and $\varphi_{R^{+++}}^{(991)}$(\,= 97, $\psi_{R^{+++}}^{(991)}$=
98 at $\alpha_{\varphi_{R^{+++}}}$= 377, $\alpha_{\psi_{R^{+++}}}$=
409) which together with the amplitudes $\varphi_{R}^{(988)}$ and
$\varphi_{P}^{(991)}$ sum up to $L_{29}+(L_{16}+L_{10}+L_{9}+L_{8})$.
Yet there remains a difference. Table \ref{tab:Pivotal-and-residual-fig4-1}
has only co-occurrent fluctuations, Table \ref{tab:Pivotal-and-residual-fig5}
enhanced with the new residues has two fluctuations that contribute
at time-like refinement level 988 and three that contribute at 991.
For all remarkable correlations arising in the levels of space-like
refinement, this might be the footprint of crotonic activity ending
in virtual particles $-$ peak amplitude $\varphi_{P}^{(992)}=415294$
makes a $\textrm{{\tt t} \negthinspace\ensuremath{\bar{\textrm{{\tt t}}}} }$sea
quark pair plausible. By contrast, less-than-perfect correlations
but within one level of time-like refinement as in Tables \ref{tab:Pivotal-and-residual-fig4-1}-\ref{tab:Pivotal-and-residual-fig4-1-1}
seem to be the hallmark of real particles.

\section{Sources of Mersenne fluctuations}

Thus far, examples of Mersenne fluctuations have been alleged without
specifying their sources. What we expect from actual sources is that
they reveal the conditions under which Mersenne fluctuations (1) develop
and (2) grow into qphyla that in turn define ``volume'' in a pregeometric
context. The apparatus employed here is continued fractions
\begin{equation}
b_{0}+\frac{\left.a_{1}\right|}{\left|b_{1}\right.}+\frac{\left.a_{2}\right|}{\left|b_{2}\right.}+\frac{\left.a_{3}\right|}{\left|b_{3}\right.}+\cdots
\end{equation}
where the shorthand $\left[b_{0};b_{1},b_{2},\ldots\right]$ is used
for the regular case $(a_{\alpha}=1)$; a shorthand for the case $a_{0}=a_{2\mu-1}=1,$
$a_{2\mu}=-1$ will be given soon.

\begin{figure}[h]
\caption{\label{fig:Sqare-spiral-of}Square spiral representation of $\mathbb{N}_{0}$}

\hspace{-1.6cm}\includegraphics[scale=0.33]{square-spiral-timelike-new}
\end{figure}

\subsection{The role of continued fractions in refinement}

To illustrate the role of contined fractions in refinement, let us
start with a time-honored example, the square spiral formed by the
numbers $\mathbb{N}_{0}$ accompanied by a generalization of $p=2^{i}-1$
$(i=1,2,\ldots5)$ to Mersenne numbers $p_{n}\equiv2^{n}-1$ $(n\in\mathbb{N})$.
As indicated in Fig. \ref{fig:Sqare-spiral-of}, with $\mathrm{B(,)}$
the Beta function, the terms $(C_{p_{n}}\mathrm{B}(p_{n},p_{n}+1))^{-1}=p_{n}(p_{n}+1)$
figure as marks on a subset of corners along the number pattern's
diagonal: For $p_{1}$, this is one corner away from the origin, for
$p_{2}$ two corners, and for $p_{n}$, $p_{n-1}+1$ corners generally.
Taking the number of corners as a measure, we can say the square spiral
is endowed with an \emph{expansion} parameter: $(p_{n-1}+1)/\sqrt{2}$,
the radius of an inscribed circle of a square with side length $(p_{n}+1)/\sqrt{2}$.
That in turn is equivalent to saying a fixed irrational quantity $\sqrt{2}$
gets refined in steps of powers of two, $\left(\frac{2^{n}}{\sqrt{2}}\right)^{-1}$.
The denominators from a convergent's regular continued fraction representation
$\left(\frac{2^{n}}{\sqrt{2}}\right)^{-1}\rightarrow\left[b_{0}^{(n)};b_{\alpha}^{(n)}\right]$
then unveil the time-like and space-like aspects of refinement: One
just proceeds from $n$ to $n+1$ in the superscript of the denominators
to follow the convergent's \emph{time}-like refinement, and follows
its respective \emph{space}-like refinements by proceeding from $\alpha=1$
to $\alpha=2$ to further increments of $\alpha$ in the denominator
subscripts ad infinitum.

\subsection{\label{sub:Mersenne-fluctuations-and}Mersenne fluctuations and randomness }

The continued fraction representations $\left(\frac{2^{n}}{\sqrt{2}}\right)^{-1}\rightarrow\left[b_{0}^{(n)};b_{\alpha}^{(n)}\right]$
yield orderly Mersenne fluctuations. Moreover, allowing for (CFR)
$\left(\frac{2^{n}}{\sqrt{2}}\right)^{-1}\pm\mathbb{N}\rightarrow\left[\tilde{b}_{0}^{(n)};\tilde{b}_{\alpha}^{(n)}\right]$,
amplitudes can conveniently be generalized to trajectories across
lattice points via the identities $b_{\alpha}^{(n)}=\tilde{b}{}_{\alpha}^{(n)}\;(\alpha>2)$.
In both representations, however, denominators are confined to a period
after which they repeat and lead to relatively modest target matching:
Results are, at least for online CFR calculators with the typical
limits $n\leq3324,\:\alpha\leq499$ restricted to the kissing numbers
$L_{m}$, $m=1,\ldots,13$ (see Table \ref{tab:Specific-fractions-in}).

\noindent It was mentioned in the introduction that the bases $G^{(p_{n})}$,
$J^{(p_{n})}$ are rooted in the matrix representations of the operators
$\boldsymbol{f}^{(p_{n})}$, $\boldsymbol{h}^{(p_{n})}$. In the same
previous work that introduced them it was further noted that $\left(\boldsymbol{f}^{(p_{n})}\right)^{p_{n}+1}\!\!=0$,$\left(\boldsymbol{h}^{(p_{n})}\right)^{p_{n}+1}\!\!=0$,
and that the length of an arc on a cardioid parametrized by $p_{n}$
shows similar behavior: After $n$ steps taken in reverse, the cardiod's
arclength too becomes zero:
\begin{equation}
(A_{n},A_{n})\equiv2\,\frac{A_{n}\cdot\textrm{co-}A_{n}}{A_{n}+\bar{A}_{n}}=A_{n-1}\qquad\textrm{etc.}\label{eq:f-squared}
\end{equation}
where

\begin{equation}
\begin{array}{c}
\begin{array}{cr}
A_{n}=4c\sin\frac{\pi}{p_{n}+1},\\
\textrm{co-}A_{n}=4c\cos\frac{\pi}{p_{n}+1}, & \:(c\;\textrm{a}\:\textrm{parameter})\\
\bar{A}_{n}=4c-A_{n}.
\end{array}\end{array}\:\label{eq:arcs-per-se}
\end{equation}
Normalizing to $c=\frac{1}{4}$ in Eqs. (\ref{eq:arcs-per-se}) and
associating a regular (CFR) $A_{n}\rightarrow\left[\tau_{0}^{(n)};\tau_{1}^{(n)},\tau_{2}^{(n)},\ldots\right]$
to $A_{n}$, one notes that the denominator $\tau_{1}$ satisfies
\begin{equation}
\tau_{1}^{(n+1)}=2\tau_{1}^{(n)}+\delta_{\tau}^{(n)}\qquad(\delta_{\tau}^{(n)}\in\{0,1\},n>2).\label{eq:generalized-neigh}
\end{equation}
 In particular, $\tau_{1}^{(n)}$ is the integer part of ${\scriptstyle {\displaystyle \frac{2^{n}}{\pi}}}$:

\begin{equation}
\begin{array}{cccccc}
 & \tau_{1}^{(4)}=5 & \tau_{1}^{(5)}=10\quad & \tau_{1}^{(6)}=20\quad & \tau_{1}^{(7)}=40\quad & \tau_{1}^{(8)}=81\quad\\
\\
 & {\scriptstyle {\scriptstyle {\displaystyle \frac{2^{4}}{\pi}}}}=5.09\quad & {\scriptstyle {\displaystyle \frac{2^{5}}{\pi}}}=10.18\quad & {\scriptstyle {\displaystyle \frac{2^{6}}{\pi}}}=20.37\quad & {\scriptstyle {\displaystyle \frac{2^{7}}{\pi}}}=40.74\quad & {\scriptstyle {\displaystyle \frac{2^{8}}{\pi}}}=81.48\quad
\end{array}\cdots\label{eq:rows}
\end{equation}
with ${\scriptstyle {\displaystyle \frac{2^{n}}{\pi}}}$ as a new
expansion parameter. Again, we have an irrational quantity that allows
for refinement in steps of powers of two, and the denominators from
a convergent's regular (CFR) $\left(\frac{2^{n}}{\pi}\right)^{-1}\rightarrow\left[b_{0}^{(n)};b_{\alpha}^{(n)}\right]$
show all the the time-like and space-like refinement characteristics
constitutive for Mersenne flucuations
\footnote{With the option of treating denominators as croton amplitudes and
generalizing them to trajectories via (CFR) $\left(\frac{2^{n}}{\pi}\right)^{-1}\pm\mathbb{N}_{0}\rightarrow\left[\tilde{b}_{0}^{(n)};\tilde{b}_{\alpha}^{(n)}\right]$
}$^{,}
\footnote{the clamp which connects the sources $\left(\frac{2^{n}}{\pi}\right)^{-1}$
and $\left(\frac{2^{n}}{\sqrt{2}}\right)^{-1}$ is the quantity $\delta\in\{0,1\}$
which links a specific denominator at $n$ to the first denominator
at $n+1$; in Eq. (\ref{eq:generalized-neigh}), $\delta_{\tau}^{(n)}=1-\left\lceil {\displaystyle \frac{\tau_{2}^{(n)}-1}{\tau_{2}^{(n)}}}\right\rceil $,
while for (CFR) $\left(\frac{2^{n}}{\sqrt{2}}\right)^{-1}\rightarrow\left[b_{0}^{(n)};b_{\alpha}^{(n)}\right]$
the analog to Eq. (\ref{eq:generalized-neigh}) is $b_{1}^{(n+1)}=2b_{\textrm{last}}^{(n)}+\delta_{b}^{(n)}$
and $\delta_{b}^{(n)}=1-\left\lceil {\displaystyle \frac{b_{\textrm{ntl}}^{(n)}-1}{b_{\textrm{ntl}}^{(n)}}}\right\rceil $,
with $b_{\textrm{ntl}}^{(n)}$ and $b_{\textrm{last}}^{(n)}$ respectively
denoting the next-to-last and last denominator terminating a period
($\sqrt{2}$ is an algebraic number).
} But contrary to the case of the square spiral, the cardiod has a
\emph{second} expansion parameter, $c$. It was previously shown \cite{Merkel}
that the croton base number $C_{q_{n}}$ $(q_{n}\equiv(p_{n}-3)/4)$
comes with the identity
\begin{equation}
{\scriptscriptstyle -}\left\lceil {\scriptscriptstyle {\textstyle n/2}}\right\rceil +\Sigma_{i=1}^{n-2}p_{i}=\left\lfloor \log_{2}C_{q_{n}}\right\rfloor \;(n>3),\label{eq:identity}
\end{equation}
which makes for an ideal candidate regarding second parametrizing
via $c$. We can therefore conceive of the irrationals
\begin{equation}
\begin{array}{lc}
\textrm{Type}\:\textrm{I}: & \left\lfloor \log_{2}C_{q_{s}}\right\rfloor \end{array}\left(\frac{2^{n}}{\pi}\right)^{-1},\label{eq:full-mers-1}
\end{equation}

\begin{equation}
\begin{array}{lc}
\textrm{Type}\:\textrm{II}: & \log_{2}(C_{q_{s}})\end{array}\left(\frac{2^{n}}{\pi}\right)^{-1},\label{eq:full-mers-2}
\end{equation}
and

\begin{equation}
\begin{array}{lc}
\textrm{Type}\:\textrm{III}: & \log_{2}(C_{q_{s}})\left(\left\lfloor {\displaystyle \frac{2^{n}}{\pi}}\right\rfloor \right)^{-1}(=\log_{2}(C_{q_{s}})/\tau_{1}^{(n)}).\end{array}\label{eq:trunc-mers}
\end{equation}
The symmetry is not a perfect one: Mersenne fluctuations of type I
or II (a shorthand saying they have their habitat in the CFR of irrationals
of type I and II) are fully traceable $-$ as are those based on $\frac{2^{n}}{\sqrt{2}}$.
In contrast, rounding $\frac{2^{n}}{\pi}$ to $\left\lfloor \frac{2^{n}}{\pi}\right\rfloor (=\tau_{1}^{(n)})$
in the CFR of irrationals of type III leads to truncated Mersenne
fluctuations, referred to as Mersenne fluctuations of type III. Truncation
occurs whenever a sequence of like $\delta$'s that determine the
upper row of Eq.(\ref{eq:rows}) above breaks: $\delta_{\tau}^{(n)}=\delta_{\tau}^{(n+1)}=\cdots=\delta_{\tau}^{(n+i-1)}\neq\delta_{\tau}^{(n+i)}$.
Conversely, a Mersenne fluctuation of type III is given birth when
a like delta sequence is initiated $-$ to stay in the picture, when
$\delta_{\tau}^{(n-1)}\neq\delta_{\tau}^{(n)}$. Truncated fluctuations
can be hard to assign to a qphylum. The longer the fragment that coincides
with a connected path in a qphylum the lesser the risk of misassignment;
if only few predecessor and successor nodes are available to escort
the insertion, assignment is fraught with uncertainty. Mersenne fluctuations
of type III thus occupy a middle position between randomness and qphyleticly
founded ``volume'' definition.

\subsection{CFR aspect of the examples shown in Sect. 3}

In agreement with the symbol choices of Sect. 3, a regular CFR associated
with irrationals of type I, II or III will be denoted $\left[\varphi_{0};\varphi_{1},\varphi_{2},\ldots\right]$,
while the alternating case corresponding to $b_{0}+\frac{\left.1\right|}{\left|b_{1}\right.}+\frac{\left.-1\right|}{\left|b_{2}\right.}+\frac{\left.1\right|}{\left|b_{3}\right.}+\cdots$
is denoted $\left[\psi_{0};\psi_{1},\psi_{2},\ldots\right]$. With
$\log_{2}C_{q_{s}}\:(q_{s}\equiv(2^{s}-1))$ modulating the outcome
of refinements, the start value of $s$ must be 2 to guarantee a nonvanishing
$\log_{2}C_{q_{s}}$ term; otherwise, the natural numbers $s$ and
$n$ can be chosen freely due to a remarkable property: As was shown
in Sect. 3, the amplitudes and phases produced by Mersenne fluctuations
are by the holographic principle (hgp) linked to the croton bases
crucial to boundary definition. The croton bases $G^{(15)}$, $J^{(15)}$,
$G^{(31)}$, $J^{(31)},\ldots$ are in turn rooted in the square-matrix
reprentations of $\boldsymbol{f}^{(15)}$, $\boldsymbol{h}^{(15)}$,
$\boldsymbol{f}^{(31)}$, $\boldsymbol{h}^{(31)},\ldots$ If peak
amplitudes fit with boundary $\Gamma^{(p)}$ at $\log_{2}(p+1)=n_{\textrm{hgp}}$
(the same reasoning applying to $\chi^{(p)}$), they can nevertheless
originate in Mersenne fluctuations for which the margins $n_{\textrm{min}}$,$n_{\textrm{max}}\gg n_{\textrm{hgp}}$
because of an in-built recursivity. Using the shorthands LL and UL
for lower left and upper left square-matrix quadrants, the quadrant
$\textrm{LL}(\boldsymbol{f}^{(p)})$ can be shown to coincide with
the subquadrant $\textrm{UL}(\textrm{LL}(\boldsymbol{f}^{(p')}))$,
a recursivit
\footnote{there are subsubquadrantal (SSQ) relationships that owe their existence
to recursivity too and can be classified in accordance with the equations'
(\ref{eq:arcs-per-se}) leading/next-to-leading CF denominator behavior:
\emph{interordinal} SSQ identities forming structural analogs of $p'=2p+1$
can be classified as sine-like, and \emph{intraordinal} SSQ identities
forming structural analogs of $p''=2p'+1=2(2p+1)+1$ as cosine-like
(see \cite{Merkel} for details
} that allows for arbitrarily large assignments $n_{\textrm{min}}$,$n_{\textrm{max}}$
without impeding the amplitude's membership to $\Gamma^{(p)}$. The
assignment of $s$ is similarly open-ended; for technical reasons
\footnote{~online CF calculators command a scope of 499 denominators; with
a value of $\pi$ accurate to 1000 decimals, that makes for a limit
$n_{\textrm{max}}\approx1024$ at full coverage, and $n_{\textrm{max}}\approx3030$
at lesser and lesser denominator productio
} however, only fluctuations with $n_{\textrm{max}}\leq3330$ and modulations
$\log_{2}C_{q_{s}}$ with a domain $2\leq s\leq9$ have been considered.

\noindent In what follows, Mersenne fluctuations are listed in the
order they occurred in the text. The fluctuation shown in Fig. \ref{fig:A-prototype-geometric}
is taken from the CFR of $\left\lfloor \log_{2}C_{3}\right\rfloor \cdot$
$\left(\frac{2^{n}}{\pi}\right)^{-1}(=\frac{\pi}{2^{n-1}})$. The
values $\varphi_{\alpha}^{(n)}$ at $n<206$ and $n>218$ have been
omitted. A principal limitation of the CFR approach becomes apparent
at this point: The more frequent and closer to unity $\varphi$ gets,
the less can we tell its affinity
\footnote{\,In \emph{Sect}.\ref{sec:Application-to-subatomic}, we have postulated
a mere 1-$3D$-space relatedness of $\varphi_{\alpha}\leq13$ $-$
quasi as the \emph{conditio sine qua non} of continuum illusion
} The fact that $\alpha\,\textrm{mod}\,2$ must be invariant and offsets
in $\alpha$ not become too decorrelated from one stopover $n$ to
the next $n+1$ (see Fig. \ref{fig:A-prototype-geometric}) is a help
in telling right from wrong candidates, but that criterion fails if
candidates satisfying $\alpha\,\textrm{mod}\,2$ equivalence come
close to one another. Worst are instances of $\varphi_{\alpha}=1$
with like $\alpha\,\textrm{mod}\,2$ $-$ they are truly legion.

\noindent The rest of the examples mentioned are, in the right order,
based on the CFR of:\\
\begin{tabular}{ll}
\hspace{-0.2cm}Footnote \ref{fn:One-such-qphylum}: & $\left\lfloor \log_{2}C_{31}\right\rfloor $ $\left(\frac{2^{n}}{\pi}\right)^{-1}$$(1488\leq n\leq1500)$;\tabularnewline
 & $\log_{2}C_{31}\left(\frac{2^{n}}{\pi}\right)^{-1}$$(1068\leq n\leq1082)$;\tabularnewline
 & $\log_{2}C_{31}\left(\frac{2^{n}}{\pi}\right)^{-1}$$(2012\leq n\leq2028)$; \tabularnewline
 & $\left\lfloor \log_{2}C_{31}\right\rfloor $ $\left(\frac{2^{n}}{\pi}\right)^{-1}$$(1951\leq n\leq1959)$; \tabularnewline
\hspace{-0.2cm}Fig. \ref{fig:Pivotal-amplitude-closing}:  & $\left\lfloor \log_{2}C_{127}\right\rfloor $ $\left(\frac{2^{n}}{\pi}\right)^{-1}$$(1556\leq n\leq1559)$; \tabularnewline
\hspace{-0.2cm}Fig. \ref{fig:Pivotal-amplitude-closing-1}, Table
\ref{tab:Pivotal-and-residual-fig4}:  & $\log_{2}C_{3}\left(\frac{2^{n}}{\pi}\right)^{-1}$$(1003\leq n\leq1006)$; \tabularnewline
\hspace{-0.2cm}Fig. \ref{fig:Pivotal-edge-amplitudes}, Table \ref{tab:Pivotal-and-residual-fig5}: & $\log_{2}C_{3}\left(\frac{2^{n}}{\pi}\right)^{-1}$$(987\leq n\leq997)$;\tabularnewline
\hspace{-0.25cm}Table \ref{tab:Pivotal-and-residual-fig4-1}:  & $\log_{2}C_{511}\left\lfloor \left(\frac{2^{n}}{\pi}\right)\right\rfloor ^{-1}$$(n=609)$; \tabularnewline
\hspace{-0.25cm}Table \ref{tab:Pivotal-and-residual-fig4-1-1}: & $\log_{2}C_{127}\left\lfloor \left(\frac{2^{n}}{\pi}\right)\right\rfloor ^{-1}$$(n=1000)$. \tabularnewline
\end{tabular}

\section{Conclusions}

The notable thing about CFR-based Mersenne fluctuations is that whether
the underlying irrational quantity to be refined is an algebraic or
a transcendental number does not matter, as long as there exists an
\emph{interordinal} connection $b_{1}^{(n+1)}=2b_{x}^{(n)}+\delta^{(n)}\:(\delta^{(n)}\in\{0,1\}$
\footnote{For (CFR) $\left(\frac{2^{n}}{\sqrt{2}}\right)^{-1}\rightarrow\left[b_{0}^{(n)};b_{\alpha}^{(n)}\right]$,
$b{}_{x}$ is the last denominator in a finite period, while the CFR
of the cardiod-arclength-function argument $\left(\frac{2^{n}}{\pi}\right)^{-1}$
has an infinite period, hence $\tau{}_{x}=\tau_{1}$ in Eq. (\ref{eq:generalized-neigh})
} The tables shown in \ref{sec:Crotons-in-the} are the outcome of
an in-depth study of denominators emerging with CFR-based Mersenne
fluctuations. Table \ref{tab:Specific-fractions-in-1} summarizes
the results of scanning the CFRs $\left(\frac{2^{n}}{\sqrt{2}}\right)^{-1}\rightarrow\left[b_{0}^{(n)};b_{\alpha}^{(n)}\right]$
for kissing-number matches and hit frequencies, while Table \ref{tab:Specific-fractions-in}
summarizes the corresponding figures for Eqs. (\ref{eq:full-mers-1})-(\ref{eq:trunc-mers}).
Results for closest pivots and largest peaks are set in parentheses.
A note on largest peaks: While those given in Table \ref{tab:Specific-fractions-in}
for type I and III, 12\,986\,152 and 9\,996\,953, are definitely
beyond $\Gamma^{(31)}$, $\chi^{(31)}$ realizabilities
\footnote{~the largest number realizable in $\Gamma^{(31)}$ being 3\,707\,462,
the largest in $\chi^{(31)}$, 4\,177\,840
} the largest peak in Table \ref{tab:Specific-fractions-in-1}, 2\,445\,930,
and the largest type-II peak in Table \ref{tab:Specific-fractions-in},
3\,614\,855, are twofold-realizable in $\Gamma^{(31)}$, $\chi^{(31)}$.
That does not mean that a kissing number $L_{m}$ they may have as
target must have an image in both boundaries $\Gamma^{(31)}$ and
$\chi^{(31)}$. For example, many numbers realizable in $\Gamma^{(15)}$,
$\chi^{(15)}$ may in theory become pivotal with respect to $L_{6}(=72)$;
but, although the numbers $L_{m}$ $(m=1,\ldots,7)$ are realizable
in $\Gamma^{(15)}$, $L_{6}$ is not in $\chi^{(15)}$ $-$ it becomes
(twofold-)realizable in $\Gamma^{(31)}$, $\chi^{(31)}$ at last.
By the same token, it may be that peaks such as 2\,445\,930 and
3\,614\,855 become true pivots not until a target $L_{m}$ for them
realizable in $\Gamma^{(63)}$, $\chi^{(63)}$ is found. Unfortunately,
kissing number candidates $L_{m}\:(m>31)$ are, with the exception
$L_{48}(=52\,416\,000)$, notoriously uncertain. For the time being,
more matches with $L_{m}\:(m\leq31)$ and interesting pivots may only
be obtained by enlarging the scope of $s$.

\noindent At the very beginning, however, there is a master Mersenne
fluctuation that pauses at $n=1$: If we take the geometrizing hypothesis
via Mersenne fluctuations literally, the continued fraction $[0;1,\bar{2}]$
$-$ a special case of (CFR) $\left(\frac{2^{n}}{\sqrt{2}}\right)^{-1}\!\!\rightarrow\!\left[b_{0}^{(n)};b_{\alpha}^{(n)}\right]$
for $n=1$ $-$ means ``recursive geometrization into a centerpiece-free
pair of a 0-spheres'' or, a self-similar laminar pattern ``dash-space-dash''
for all space-like refinements.


text:
\section*{Abstract}{\small
Hierarchical models predict that massive early-type galaxies (mETGs) derive from the most massive and violent merging sequences occurred in the Universe. However, the role of wet, mixed, and dry major mergers in the assembly of mETGs is questioned by some recent observations. We have developed a semi-analytical model to test the feasibility of the major-merger origin hypothesis for mETGs, just accounting for the effects on galaxy evolution of the major mergers strictly reported by observations. The model proves that it is feasible to reproduce the observed number density evolution of mETGs since $z\sim 1$, just accounting for the coordinated effects of wet/mixed/dry major mergers. It can also reconcile the different assembly redshifts derived by hierarchical models and by mass downsizing data for mETGs, just considering that a mETG observed at a certain redshift is not necessarily in place since then. The model predicts that wet major mergers have controlled the mETGs buildup since z$\sim$1, although dry and mixed mergers have also played an essential role in it. The bulk of this assembly took place at 0.7$<z<$1, being nearly frozen at z$\lesssim$0.7 due to the negligible number of major mergers occurred per existing mETG since then. The model suggests that major mergers have been the main driver for the observational migration of mass from the massive end of the blue galaxy cloud to that of the red sequence in the last $\sim 8$\,Gyr.
\normalsize}
\end{minipage}
\section{Introduction \label{sec:intro}}

Recent observations show that massive galaxies ($\log(\mathcal{M}_*/\mathcal{M}_\odot)\geq$11) have been in place since z$\sim$0.7 (\cite{2008ApJ...687...50P}). This epoch also coincides with the moment at which early-type galaxies (E-S0a's, ETGs) start to dominate the massive end of the galaxy luminosity function (LF), suggesting that there must exist "a dominant mechanism that links the shutdown of star formation and the acquisition of a spheroidal morphology in massive systems" (\cite{2010ApJ...709..644I}). According to current hierarchical models of galaxy formation, this mechanism has been major merging. The hierarchical scenario predicts that present-day mETGs have resulted from the most massive and violent merging sequences occurred in the Universe. Nevertheless, this also implies that they have been the latest systems to be in place into the cosmic scenario (at $z\sim 0.3$, see \cite{2006MNRAS.366..499D}), a prediction that conflicts directly with observations evidencing galaxy mass downsizing. Therefore, mETGs have turned into privileged laboratories to test hierarchical theories. However, studies on their physical properties have not provided conclusive results yet (see references in \cite{2010arXiv1003.0686E}), questioning or supporting the major-merger origin of mETGs (\cite{2006ApJ...644...54M,2010ApJ...718.1158L}).

If mETGs derived mainly from major mergers, these galaxies should reside preferably in environments where mergers could have been relevant. Nevertheless, the evolutionary processes controlling the galaxy evolution in each environment are still poorly understood. In clusters, the S0 fraction is observed to increase with time at the expense of the spirals, but the fraction of ellipticals has remained constant since z$\sim$0.8 (\cite{2007ApJ...660.1151D,2009ApJ...697L.137P,2010arXiv1010.4442V}). This has been interpreted as a sign of the poor role played by major mergers in the evolution of cluster ellipticals, because it suggests the conversion of cluster spirals into S0's (but not into ellipticals) through mechanisms gentler than major mergers, contrary again to hierarchical expectations. However, the majority of present-day mETGs do not reside in clusters, but in groups (\cite{2006ApJS..167....1B}), an environment where mergers and tidal interactions determine galaxy evolution (\cite{2008ApJ...688L...5K}). In fact, the evolutive trend observed in groups is quite different to the one reported in clusters: old evolved groups evidence an evolution of spirals into S0's, and of these later into ellipticals with time  (\cite{2009AJ....138..295F,2010ApJ...713..637B}), suggesting a possible pre-processing of galaxies in groups before falling into clusters (\cite{2009AN....330.1059W}). So, the role played by the environment in galaxy evolution (and in particular, in the assembly of mETGs) is still unsettled.

Mixed scenarios have been proposed for the formation of mETGs, in which blue galaxies quench their star formation through gas-rich mergers, migrate to the red sequence, and merge further through mixed and dry mergers to give place to the most massive ETGs (\cite{2007ApJ...665..265F}). Depending on the gas content of the encounter, the outcome of the major merger would be a S0 or an elliptical (\cite{2008ApJS..175..390H}). However, a direct verification of the feasibility of this scenario accounting strictly for the wet/mixed/dry major mergers reported by observations at each redshift has not been carried out yet.

\section{The models \label{sec:models}}

We have approached this question directly through semi-analytical models, studying how the present-day mETGs would have evolved backwards-in-time assuming that they derive exclusively from the major mergers strictly reported by observations (\cite{2010A&A...519A..55E,2010arXiv1003.0686E}). We have accounted for the different phases, timescales, and progenitor galaxies of a major merger depending on its type (wet/mixed/dry), and for the number of major mergers of each type observed at each redshift. In particular, transitory phases in which gas-rich major mergers become dust-reddened star-forming galaxies (DSFs) have been considered. The model predictions apply to the global population of mETGs (averaged over all environments), as the local LFs and observational major merger fractions used in the model basically trace group and field environments. The effects of minor mergers have been neglected in the model (see comments concerning this topic in \cite{2010A&A...519A..55E}).

\begin{figure}[t!]
\center
\includegraphics[scale=.98]{elichecF1.pdf}
\includegraphics[scale=.98]{elichecF2.pdf}
\caption{Evolution of the $B$-band LFs with redshift of red and blue galaxies up to $z\sim 1.2$. \emph{Symbols}: Observational data (see references in \cite{2010A&A...519A..55E}). \emph{Lines}: Model predictions. \emph{Black lines}: Model predictions considering that only ETGs are classified as red galaxies. \emph{Green lines}: Considering that also DSFs are identified as red galaxies (beside ETGs). \emph{Red thin lines}: Predictions of a pure luminosity evolution (PLE) model for all galaxies. \emph{Dotted shaded regions}: Magnitude range of model validity (more information in \cite{2010A&A...519A..55E}). Adapted from \cite{2010A&A...519A..55E}.
\label{fig:fig1}}
\end{figure}

\section{Reproducing the evolution of the galaxy LFs since z=1 \label{sec:observations}}

The model can reproduce the observed evolution of the galaxy LFs at z$\lesssim$1, simultaneously for different rest-frame bands ($B$, $I$, $K$) and selection criteria (based on color or morphology, see \cite{2010A&A...519A..55E}). Figure\,\ref{fig:fig1} shows the expected evolution of the LFs of red and blue galaxies in the $B$-band. The model predicts an increase of the number density of mETGs by a factor of $\sim$2-2.5 since z$\sim$1, in agreement with some studies (compare the black line and the pentagons in panel d of red galaxies). Moreover, if the DSFs predicted by the model are also considered as red galaxies, the model can also reproduce the apparent negligible number evolution of red galaxies reported by other authors (compare the PLE model, the model predictions including DSFs in the red sample, and the data -triangles and crosses- in the same panel). So, the model provides a framework in which conflicting observational results on the amount of number evolution experienced by massive red galaxies can be reconciled, just accounting for the typical contamination by DSFs that many red galaxy samples exhibit at $z>0.7$. The gas-rich progenitors of these "recently" assembled mETGs reproduce naturally the observational excess of $\sim$40\% of blue galaxies at 0.8$<z<$1, as compared to PLE models (see panel d of blue galaxies in the same figure). The model predicts a proportional rise in the number density of mETGs since z$\sim$1 similar for all mass bins (this also applies to mETGs with $\log(\mathcal{M}_*/\mathcal{M}_\odot)\gtrsim$11.7 at z$=$0). So, it is remarkable that, even predicting a noticeable buildup of the most massive ETGs (apparently in contradiction with mass downsizing), the LF of these systems predicted by the model at 0.8$<z<$1 lies on top of the PLE model (accordingly to observational data supporting mass downsizing). Consult \S\ref{sec:reconciling}.

\begin{figure}[t!]
\center
\includegraphics[scale=1]{elichecF3.pdf}
\caption{Model predictions on the fraction of present-day mETGs observed, really in place, and still being assembled at each redshift \emph{z}. \emph{Arrows}: Assembly redshift of mETGs according to observations (\emph{open blue}, \cite{2008ApJ...687...50P}), to the model (\emph{thick solid black}), and average $z$ between the assembly redshifts of 50\% and 80\% of their stellar content according to the models by \cite{2006MNRAS.366..499D} (\emph{red thin solid}). Our model reproduces the global observed buildup of mETGs since z$\sim$1 (triangles), predicting an assembly redshift for mETGs in good agreement with hierarchical models (\cite{2006MNRAS.366..499D}) at the same time. Adapted from \cite{2010arXiv1003.0686E}.
\label{fig:fig2}}
\end{figure}

\section{Reconciling the hierarchical scenario with mass downsizing\label{sec:reconciling}}

An ETG is in place since a given redshift $z$ basically if it does not undergo any major merger since then (not accounting for minor mergers, see \cite{2010arXiv1003.0686E}). This implies that a mETG observed at $z$ is not necessarily in place since then, as it depends on if it has undergone a major merger since then down to the present (contrary to what is usually assumed). Considering this, the model proves that mass downsizing and hierarchical scenarios can be reconciled easily. In fact, it predicts that more than $\sim$90\% of the mETGs existing at z$\sim$1 are going to be involved in a dry or mixed event since then (see Fig.\,\ref{fig:fig2}). This means that these mETGs are not the passively-evolved high-z counterparts of present-day ones (as usually interpreted in many studies), but their gas-poor progenitors instead. This implies that very few present-day mETGs have been really in place since z$\sim$1 ($\lesssim$5\%), contrary to the $\sim$40-50\% reported by traditional interpretations of observations. Accounting for this, the model is capable of deriving a lower assembly redshift for mETGs (z$\sim$0.5), in excellent agreement with hierarchical models, reproducing global mass downsizing trends at the same time (see Fig.\,\ref{fig:fig2}).

\begin{figure}[t!]
\center
\includegraphics[scale=.95]{elichecF4.pdf}
\includegraphics[scale=.95]{elichecF5.pdf}
\caption{Predicted average number of wet, mixed, dry, and all major merger events occurred at each redshift bin $[$z-0.1,z$]$ and accumulated since each redshift $z$ down to the present, per present-day $L>L^*$ galaxy (\emph{top and bottom left panels, respectively}) and per local mETG (\emph{top and bottom right panels, respectively}). \emph{Open diamonds}: For all major mergers. \emph{Open triangles}: Wet mergers. \emph{Open squares}: Mixed mergers. \emph{Open circles}: Dry mergers. \emph{Remaining symbols}: Observational estimates derived by different studies (consult the legend in each panel). Adapted from \cite{2010arXiv1003.0686E}.
\label{fig:fig3}}
\end{figure}

\section{The relative role of wet, mixed, and dry major mergers}

Accordingly to observations, the model predicts that wet major mergers have controlled the mETGs buildup since z$\sim$1, although dry and mixed mergers have also contributed significantly to it (see Fig.\,\ref{fig:fig3}). The bulk of this assembly took place during a $\sim$1.4\,Gyr time period elapsed at 0.7$<z<$1, being nearly frozen at z$\lesssim$0.7. This "frostbite" in the global buildup of mETGs is due to the negligible number of major mergers occurred since z$\sim$0.7, as compared to the mETGs population in place at those redshifts (see the right panels of Fig.\,\ref{fig:fig3}). Therefore, the contribution of major mergers (and, in particular, of dry events) to the global assembly of mETGs is predicted to have been irrelevant during the last $\sim$6.3\,Gyr. Notice that this is not necessarily extendable to clusters (as our models are biassed to field and group environments). The model predictions concerning to the numbers of wet/mixed/dry major mergers at each redshift are in excellent agreement with independent estimates (see Fig.\,\ref{fig:fig3}) and with observational studies also pointing to the fact that $z\sim 0.8$ is a transition epoch in the assembly of mETGs (\cite{2010ApJ...709..644I}). According to the model, major mergers are responsible for nearly doubling the stellar mass at the massive end of the red sequence since z$\sim$1, being dry events responsible of the accretion of $\sim$18\% of the stellar mass of present-day mETGs during the last $\sim$8\,Gyr.
\small
\section*{Acknowledgments}
Supported by the Spanish Ministry of Science and Innovation (MICINN) under project AYA2009-10368 and by the Madrid Regional Government through the AstroMadrid Project (CAM S2009/ESP-1496). Funded by the Spanish MICINN under the Consolider-Ingenio 2010 Program grant CSD2006-00070: "First Science with the GTC".

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