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\begin{document}

\title{The General Relevance of the Modified Cosmological Model}

  \author{Jonathan W. Tooker}
\date{\today}


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\begin{quotation}
    The stone the builders rejected has become the cornerstone.

    -- Psalm 118:22
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\section{{\textsf{\LARGE Introduction }}}
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In lieu of an abstract each chapter in this book will have a description of its contents.  There is no chapter dedicated to the general relevance of the modified cosmological model to particles \cite{QS} because high-energy physics is logically after the material that has heretofore been considered in this research program.  This book is focused on recapping, consolidating, streamlining, and annotating previous work related to gravitation and non-relativistic quantum theory while adding a few new insights when they are modest.  Throughout this book the reader's familiarity with the modified cosmological model is assumed.

The focus of the first section in this chapter is a review of geometry.  Section two gives a preliminary overview of an algorithm that will violate conservation of information.  In section three we propose to modify Feynman's application of the action principle by replacing the least action complex field trajectories with maximum action hypercomplex field trajectories that still satisfy the action principle.

\subsection{An Abstract Psychological Dimension}


It is shocking that after this many years of work on the theory of infinite complexity that the associated material calculated and referred to here is not already well known with the entire field of all possible linear nuance being mapped out to the n$^{\text{th}}$ degree.  It is surprising that there is no Wikipedia article regarding the modified cosmological model (MCM) or the theory of infinite complexity (TOIC) that spells out all of the trivially derived properties.  To that end, consider a cube spanned by $\hat x$, $\hat y$, and $\hat z$.  The slices of constant $z$ are the subspaces spanned by $\hat x$ and $\hat y$ at each value of $z$.  Every curve that can be constructed using $\hat x$ and $\hat y$ will be confined to some slice of $z$.  Any curve leaving the slice would have a component in the $\hat z$ direction.  Likewise any curve constructed from just $\hat x$ and $\hat y$ will have its tangent vectors confined to that single slice of constant $z$.  The curve's cotangent space is the first place we could possibly come across vectors with a non-vanishing $\hat z$ component.  We state these obvious truths because the MCM describes de Sitter (dS) and Anti-de Sitter (AdS) spacetimes as slices of a 5D cube and we want to show the exceptional behavior of our flat universe when it sews together two 5D spaces but is not itself a slice of any 5D space.


\begin{figure}[t]
    \makebox[\textwidth][c]{
        \includegraphics[scale=.5]{unitcell2.png}}
    \captionsetup{format=hang}\caption{The region between two adjacent moments of psychological time: $\mathcal{H}_1$ and $\mathcal{H}_2$.  The arrangement immediately suggests a gravitational pilot wave formulation as the path to evolve through the discontinuity of the as-yet-undescribed region inside the null interval between $\Omega_1$ and $\aleph_2$.  We will introduce new $\chi^A_\varnothing$ coordinates to accommodate this representation wherein $x^\mu\in\mathcal{H}$ are moved away from the center of the MCM unit cell we have depicted them in previous work.  $\chi^5$ is the horizontal direction across this figure.  This figure uses the values $\Phi^2$, $\Phi$, and 1 to demonstrate $\Sigma^\varnothing$ but there are many such arrangements.}
    \label{fig:unitcell2}
\end{figure}

Now consider flat empty 5-spaces $\Sigma^\pm$ where general relativity  in the absence of 5D matter-energy leads to the desired dynamics in the 4D slices through the Kaluza-Klein (KK) metric

\begin{equation}\label{eq:KKmetric762}
\Sigma_{AB}=\left(\begin{matrix}
g_{\alpha\beta}+\phi^2B_\alpha B_\beta  & ~& \phi^2B_\alpha \\
~& ~&~\\
\phi^2B_\beta & ~& \phi^2 \\
\end{matrix}\right)~~.
\end{equation}\newline

\noindent In this book we will use the Greek letter $\chi$ for the 5D coordinates where we have used $\xi$ previously.  Where Latin indices $A$ have previously run from $0$ to $4$, here they will run from $1$ to $5$ so $\xi^4\to\chi^5$.  We will add a layer of complexity when we take $\mu,\nu\in\{0,1,2,3\}$ in the usual way but then add a subtle convention for $\alpha,\beta\in\{1,2,3,4\}$.  In 5D we have $A,B\in\{0,1,2,3\}$ and $a,b\in\{0,1,2,3,4\}$.  Taking the coordinates of $\Sigma^\pm$ as $\chi^A_\pm$ we will call the bulk metrics $\Sigma_{AB}^\pm$ and they will have the form of equation (\ref{eq:KKmetric762}).  Curves in the flat slices of constant $\chi^5_\pm$ can never have tangent vectors that point to the left or right in the cosmological unit cell.  Figure 1 shows that cell.  The slices $\aleph$ and $\Omega$ are flat slices of $\chi^5$ but they appear curved in this figure to demonstrate the curvature associated with the embedded metric of the $x^\mu_\pm\neq\chi^\alpha_\pm$ de Sitter coordinates.

The oft-lamented ``cylinder condition'' that the MCM both embodies and motivates from first principles \cite{KK,MOD,GC} says that physics in the 4D worldsheets spanned by $\chi^\mu_\pm$ can never depend on the fifth coordinate.  This can be accomplished via a generalized disallowance of the appearance of $\chi^5_\pm$ in any equations of motion but we can accomplish the same thing by taking our 4D spacetimes as surfaces of constant $\chi^5_\pm$ in the 5D bulk \cite{GC}.  The ordinary limitation of the cylinder condition on physics is that position and momentum measured in $x^\mu$ can never depend on $x^5$.  However, that doesn't say anything about the abstract coordinates $\{\chi_+^A,\chi_\varnothing^A,\chi_-^A \}$ or vice versa.

Here we begin to develop the complex behavior that can be derived by modeling our universe of $x^\mu$ at the intersection of two 5D spaces.  This is a key point to notice: observables will always be defined on $x^\mu$ which can, in principle depend on all of the $\chi^A$ coordinates.  This contrasts the normal application of KK theory which says $x^\mu$ cannot depend on $x^4$.  Therefore, even at this early stage, it is apparent that the MCM is very different from the standard cosmological model and other KK models.  One well known issue with standard KK theory is that the field equations indicate that the electromagnetic field strength tensor must always vanish with respect to 4D general relativity, but by adding the 15 \textbf{chirological coordinates} $\{\chi_+^A,\chi_\varnothing^A,\chi_-^A \}$ we have a lot of room to develop novel workarounds.  For instance, if the Kaluza--Klein requirement for vanishing electromagnetic strength tensors applies to the chirological coordinates then that puts only a loose constraint on what we do with the $x^\mu$, $x^\mu_\varnothing$, and $x^\mu_\pm$ coordinates.

Let $\chi^5$ be a non-relativistic psychological dimension with identical topological flatness.  The identical topological flatness of $\chi^5_\pm$ does not hold for $\chi^5_\varnothing$ which can have an arbitrary non-linear curvature with tangent vectors pointing anywhere because it has no width in the path from $\mathcal{H}_1$ to $\mathcal{H}_2$, as in figure \ref{fig:unitcell2}.  $\Sigma^\varnothing$ exists only to sew $\Sigma^\pm$ together with a single point so we are not concerned with the overall curvature there.  There is no constrained object anywhere in the vector bundle of $\Sigma^\varnothing$ so everything about that bundle is introduced as a new MCM degree of freedom.  The only constraint on $\Sigma^\varnothing$ is that it has at least one point where we can construct a Lorentz frame and then use that point to ensure smooth transport of a Lorentz frame from $\mathcal{H}_1$ to $\mathcal{H}_2$.  The 4D slices of flat 5-space are flat but $\aleph$ and $\Omega$, themselves slices, are curved and what's more: the only flat space we do have, $\mathcal{H}$, isn't even a slice of a 5-space \cite{GC}.  How can we get a curved slice out of a flat space?  These new degrees of freedom will be helpful.  Quantum field theory is known to explode when a certain solution shows the divergent energy density of the vacuum.  To avoid that problem and others, we want to define our observables in a way that does not ever depend on unintegrable factors (such as $|\vec r-\vec r'|^{-2}$ at the tip of $\vec r'$ in $\int d\vec r$) and never explodes on account of our newly introduced hypercomplex-valued field variables.  Integrals explode when they integrate over singular points and even general relativity breaks down at the point-like ``singularity in center of a black hole'' but, as we will show, all of these ideas have totally different interpretations when the fields are hypercomplex.

The addition of only one new degree of MCM freedom to go through larger infinity in the \textbf{hyperreal number system} $^*\mathbb{R}$ (via $\hat{\Phi}^n\rightarrow\hat{\Phi}^{n+1}$) leads to two new degrees of freedom: the two dimensions of $\mathbb{C}$ become hyperreal and \textbf{hyperimaginary}.  We will name this system the \textbf{hypercomplex number system}\footnote{There is already another number system named the hypercomplex numbers, but it is not $^\star\mathbb{C}$.  We have previously used the name $\mathbb{C}^3$ in this regard which was also already taken.} $^\star\mathbb{C}$.  The analytical factors of QFT that would make integrals explode if integrated over at the wrong time will be hypercomplex and we will make special rules for integrating over them.\footnote{This is something we will do in the future $\Omega$, not in this book.}  We point to hyperimaginarity as the reason for the fourth ontological vector $\hat2$ (or rather we might call choose to call the fourth one $\hat i$ because it more precisely corresponds to hyperimaginarity.)  Our initial desire to add a single degree of freedom in a longitudinal mode along $\hat{\Phi}$ showed that $\{\hat{\pi},\hat{\Phi},\hat i\}$ was incomplete.  Luckily we found $\hat2$ already there in the plane wave solutions whose periodic topology goes like $\psi=e^{2\pi i\theta}$.  We assemble all the ontological numbers when the first place we want to look in the phase space is where $\theta=|\hat{\Phi}|=\Phi$ and we find $\psi=e^{2\pi i\Phi}$.

The $\hat{\Phi}^{n+1}$-site shall be located at the tip of the $\hat{\Phi}$ vector as pointing from the $\hat{\Phi}^n$-site and this point must belong to what is ordinarily called future timelike infinity in general relativity.  Due to the special MCM boundary condition that $\chi^5$ is always flat we can retain the non-relativistic notion of a vector connecting two points in a single manifold.  $\hat \Phi$ points in the direction of $\chi^5$ and all the $\hat \Phi^n$ lie end to end in an infinite 1D manifold.\footnote{This manifold is infinite in the sense of an infinite stack of unit cells, each of finite width in $\chi^5$.}  All the other structure is foliated from there.  While there are many different discrete $\hat \Phi$ vectors in any significant volume of the cosmological lattice, the string of observations that can be made by one observer is always a straight line in the cosmological lattice.  This is what we mean when we say that $\chi^5$ is identically flat.  The idea has an analogue in special relativity when the orientation of the observer's proper time axis is always vertical and the slope of the null interval is relative to that fixed proper time axis.


\begin{figure}[t]
    \makebox[\textwidth][c]{
        \includegraphics[scale=1.37]{hyperreal.png}}
    \captionsetup{format=hang}\caption{From Wikipedia: an example of how hyperreal numbers $^*\mathbb{R}$ work.  Note that, in general (not pictured), there is more than one level of infinitude and more than one level of infinitesimality.  These levels of infinitesimality and infinitude are what we will call tiers of infinitude when referring to $^*\mathbb{R}$ and levels of $\aleph$ when referring to the exact same principle in the context of $\hat{\Phi}^n$, though we will interchange the terms informally.  This figure shows three levels of hyperreal infinitude but the complete hypercomplex number system $^\star\mathbb{C}$ goes all the way up and all the way down.  The rightward direction in figure 1 is strictly associated with the upward direction in this figure.  }
    \label{fig:hyperreal}
\end{figure}

Feynman considered it a flaw in his own approach \cite{FEYN} that he was forced to choose an arbitrarily finite amount of chronological time $t$ within an infinite natural duration of time $t\in[-\infty,\infty]$ so as to avoid divergent integrals and in this chapter we will say a lot about Feynman's framework.  Note that when an observer's lifetime is comprised of a finite number of observations we naturally have a way to impose Feynman's mathematical constraint with a more realistic philosophical predicate.  Since the theory of infinite complexity regards the observer's ability to test his own theory, and an observer can only make finitely many observations in a lifetime, it is likely that we can impose a constraint based on finite chirological time even when the proper chronological time of the universe has no inherent constraint to finiteness.  Feynman's theory works even for artificially finite time so we are able to begin to build the MCM by dividing the real line $\mathbb{R}$ into three non-specific regions \cite{DE}

\begin{equation}\label{eq:timetime}
\mathrm{Past}\in[t_{\text{min}},t_0)~~,\quad\qquad\mathrm{Present}\in[t_0]~~,\quad\qquad\mathrm{and}\quad\qquad\mathrm{Future}\in(t_0,t_{\text{max}}]~~.
\end{equation}\newline

\noindent Notice that the actual values of $t_{\text{min}}$ and $t_{\text{max}}$ do not enter into consideration.  In the MCM we will presume that $t$ is infinite in extent when $t\equiv x^0$ but finite when $t\equiv\chi^5$.  $\chi^5$ is built from some stack of $\hat \Phi$ vectors that point from one observation to the next.  Since an observer can only make finitely many observations in a lifetime, it will be never become necessary to consider the implications of an infinitely long $\chi^5$ dimension.  This shows what we mean when we say $\chi^5$ is an abstract psychological dimension.

We go into a lot of detail regarding the mathematical analysis of concepts of infinite complexity in reference \cite{ZETA} and we will also do so in chapter four.  We introduce $^\star\mathbb{C}$ in a rigorous way by labeling each tier of infinitude with some unique $\hat{\Phi}^n$ and refer to them as \textbf{levels of aleph} \cite{OP}.  We may efficiently use integers for logical ordering of tiers of hypercomplex infinitude so we can use integers to refer to them as levels of $\aleph$.  Figure \ref{fig:hyperreal} shows three levels of $\aleph$.  \textbf{Hyperspacetime} will refer to an object constructed from 3D position space by the addition of, first, relativistic chronological time $x^0$ to make spacetime, and then flat chirological time $\chi^5$ to make 5D hyperspacetime.  The \textbf{hypercosmos} will consist of many hyperspacetimes on many levels of $\aleph$.  We will use the \textbf{ontological basis} $\{\hat 2,\hat \pi,\hat \Phi,\hat i\}$ as a non-coordinate basis for tensor analysis but we do not yet require that the 5D set $\{\hat i,\hat 1,\hat \Phi,\hat 2,\hat \pi\}$ invented in reference \cite{IC} will be the analogue geometric basis in 5D.  One way to think about this is to assume that any 5D basis of a geometric manifold is not the ontological basis which is 4D.  Rather, the 5D construction $\{\hat i,\hat 1,\hat \Phi,\hat 2,\hat \pi\}$ will refer to an \textbf{ontological group}
$\aleph^\Omega(5,4,3)$ where the arguments five, four, and three represent a standard group theoretical labeling similar to how O(3) distinguishes the group of rotations from the O(1,3) group of Lorentz rotations\footnote{We remind the reader that O(1,3) is the natural topology of $\{\hat 2,\hat \pi,\hat \Phi,\hat i\}$.}.  However, this book is only about the general relevance of the MCM and we won't develop many new concepts here.

In previous publications detailing the MCM, $\mathcal{H}$ refers both to the 4D manifold that is flat Minkowski space and also to a Hilbert space $\mathcal{H}'$.  Here we will prime the letter when it refers to a vector space and leave it unprimed when it describes a manifold: $\aleph'$ and $\Omega'$ are members of a rigged Hilbert space with $\mathcal{H}'$; $\aleph$ and $\Omega$ are anti-de Sitter and de Sitter spaces respectively.  We have chosen an arbitrary convention putting AdS in the past since the only thing definite is that $\aleph'$ is a subspace $\mathcal{H}'$ and that $\Omega'$ is the dual of $\aleph'$.  We don't yet know for sure which of $\aleph$ and $\Omega$ \emph{must} be dS or AdS but by turning the crank on the theoretical constructions presented in this book, it should be possible to determine that one configuration or the other describes time moving in the forward direction.

\subsection{The Dual Tangent Space}

We have previously described a need to expand the phase space from $2N$ dimensions to $3N$ \cite{GC} because we have revamped the theory as

\begin{equation}
\mathbb{C}\quad\equiv\quad\{\hat i,\hat1\}\quad\qquad\longrightarrow\quad\qquad\mathbb{C}^3\quad\equiv\quad\{\hat i,\hat \pi,\hat \Phi\}~~.
\end{equation}\newline

\noindent We find that it was wise not to immediately cease exploration in an attempt to calculate $\mathbb{C}^3$ because by adding $\hat 2$ we have likely moved to $4N$ dimensional phase space with

\begin{equation}
^\star\mathbb{C}\quad\equiv\quad\{\hat i,\hat\pi,\hat \Phi, \hat 2\}~~.
\end{equation}\newline

In ordinary quantum mechanics, in either the position or momentum space representations, the other of the observable operators $\hat x$ or $\hat p$ is represented as the partial derivative operator.  Through the derivative operator the quantum mechanics is connected to the continuum in the sense that a manifold is connected to its tangent space through the gradient.  Classical phase space is $2N=2\times3=6$ dimensional because it is comprised of three dimensions of space and three dimensions of momentum space.  The conjugate nature of position and momentum is not quite the same as the duality between the unprimed geometric manifolds $\{\aleph,\mathcal{H},\Omega\}$ and the abstract vector spaces $\{\aleph',\mathcal{H}',\Omega'\}$ whose states have position and momentum space representations written in the coordinates of different manifolds which are described as lattice sites.  Multiple simultaneous avenues of complexity in duality and conjugation are the expected source of new complexity in the theory of infinite complexity.  Each lattice site has a localized bubble of physically realizable phase space when, for example, the $x^\mu_{\{j\}}$ coordinates are beyond infinity in phase space with respect to the $x^\mu_{\{n<j\}}$ coordinates (and infinitesimal with respect to $x^\mu_{\{n>j\}}$.)

The process of evolution in the lattice is expected to differ from pure Schr\"odinger evolution as follows: the first derivative operator that appears in quantum mechanics only allows us to go back and forth between $\hat p\equiv-i\partial_x$ and $\hat x\equiv i\partial_x$ but in $4N$ dimensional phase space we can likely find complex representational loops that achieve novel effects such as arbitrage of information.  Perhaps we can use the addition of two new layers of complexity called hyperreality and hyperimaginarity to transfer information from position space into momentum space, then into the dual tangent space, then into the momentum space of a different position space such that $\hat M^3$ takes initial conditions in the position space of $\mathcal{H}_1$ and returns the expectation value in the momentum space representation of the $\mathcal{H}_2$ coordinates.  Recall the underlying process

\begin{equation}
\hat M^3\quad:\quad\mathcal{H}_1~~\mapsto~~\Omega_1~~\mapsto~~\aleph_2~~\mapsto~~\mathcal{H}_2~~,
\end{equation}\newline

\noindent and keep in mind that we want to use the dual tangent space to bridge the gap across $\Sigma^\varnothing$ that appears in figure \ref{fig:unitcell2}.  The coordinates $x^\mu_{\{n\}}$ of each level of $\aleph$ $\hat \Phi^n$ are separated by infinity but the frequencies associated with the eigenstate basis vectors are separated by finite width in phase space through $\omega_{i+1}=\Phi\omega_i$ \cite{OP}.  Therefore in momentum space we can expect tunneling between regions that are separated by infinity in position space. The equality $\omega_{i+1}=\Phi\omega_i$ follows from the idea that $\mathcal{H}'$ is always the same when it is attached to any $\mathcal{H}_n$.  One of the remaining nebulosities of the theory is how to actually move beyond infinity when translating across the unit cell.  Therefore consider how it will be useful make the Fourier transform between coordinates that can't make it to infinity and wave numbers which don't know anything about ``the surface at infinity'' and don't have to ``reach it'' before they can describe what happens behind it.

To begin to lay the groundwork for infinitely complex quantum states that have new degrees of freedom hidden in the dual tangent spaces of their position and momentum space representations consider that a generic set of non-ontological 4D basis vectors $\hat e^\alpha\equiv\hat e^\mu$ can be inherited from the generic 5D basis $\hat e^A$ exactly via suppression of the component $\hat e^5$ that would point in the direction of increasing or decreasing $\chi^5_\pm$.  The four basis vectors of the observer's lab frame $\hat x^\mu$ do not point into the bulk hyperspacetime between each instance of $\mathcal{H}$ because $x^4\not\in x^\mu$.  When evolving some qubit encoded on a Lorentz frame across the unit cell we will rely heavily upon the fact that the flat metric of the Lorentz frame is always the same and does not depend on the global topology specified by the $\{+,\varnothing,-\}$ scripting.  It is very important to note that it will be possible to take the four unit vectors $\hat\chi^\mu_\pm$ of $\Sigma^\pm$ which span slices of $\chi^5_\pm$ as the basis of a local Lorentz frame defined by an observer living in a manifold with \textbf{arbitrary global topology} as long as the curvature is mild enough for the Lorentz approximation.  We will therefore restrict ourselves to the weak field limit

\begin{equation}
g_{\mu\nu}=\eta_{\mu\nu}+h_{\mu\nu}~~,
\end{equation}\newline

\noindent where $\eta_{\mu\nu}$ is the Minkowski metric and $h_{\mu\nu}$ is a small perturbation.  Specifically we will use $\hat x^\mu_+$ for the basis of 4D de Sitter space and $\hat x^\mu_-$ as the basis of Anti-de Sitter space.  When we consider the unperturbed cosmological solution $h_{\mu\nu}=0$, and let the intergalactic magnetic field go to zero with $B_\mu=0$ in equation (\ref{eq:KKmetric762}), we have a cosmological KK metric

\begin{equation}
\Sigma_{AB}^\pm=\left(\begin{matrix}
\eta_{\mu\nu} & 0 \\
~& ~\\
  0 & \phi^2(\chi^5_\pm) \\
\end{matrix}\right)~~,
\end{equation}\newline

\noindent that will serve as the basis for modification in the sense of the first M in MCM.  This metric describes the singularity-free cosmos postulated in reference \cite{DE}.  The no-singularities condition was initially assumed on the basis of loop quantum cosmology but the condition was independently motivated later.  Neither of LQC or LQG have any direct relevance to the MCM machinery.

The tangent vectors to the geodesics associated with the perturbation $h_{\mu\nu}$, such as the local deformation of spacetime due to an ordinary cathode ray tube, while very small, are not strictly confined to the slices of $\chi^5_\pm$.  We can say the same thing about the tangent vectors to some set of mega-scale cosmological geodesics in de Sitter space: they point outside of the flat slices of $\chi^5_+$ and the same can be said about anti-de Sitter space and $\chi^5_-$.  In both cases: vectors near the origin associated with some perturbation $h_{\mu\nu}$ and vectors very far from the origin tangent to some cosmological geodesics that show the global topology, the tangent vectors point outside of the 4D worldsheet to some other place in the bulk.  A vector in curved space is an object defined at a point and nothing more but $\chi^5$ is not curved so its tangent vectors always point to other points along $\chi^5$.  All the Lorentz frame vectors remain in the slice.  If the slice is Minkowski space then the Lorentz frame defines it identically.  Likewise, the tangent vectors to the cosmological geodesics in Minkowski space remain in the slice but, as we have shown, there are still other important vectors in the MCM that do point outside of the slices.

\begin{figure}[t]
    \makebox[\textwidth][c]{
        \includegraphics[scale=.6]{PHImagic.png}}
    \captionsetup{format=hang}\caption{This figure describes the MCM topological configuration.  The geodesic that traverses the MCM unit cell and whose tangent vectors include the three vectors $\{V^\mu_+,V^\mu,V^\mu_-\}$ shown here could be akin to a periodic orbit of the Hopf fibration.  Note the connective vectors are never derived from the 5D coordinates $\chi^A$.}
    \label{fig:PHImagic}
\end{figure}


How do we know the vectors that point outside of the flat slices are pointing to other slices of $\mathcal{H}\cup\{\Sigma^+,\Sigma^\varnothing,\Sigma^-\}$ and not outside of hyperspacetime altogether?  This is where we take advantage of the flat topology that we have assigned to our psychological dimension $\chi^5$.  It only has one collinear tangent space.  Vectors tangent to $\chi^5$ everywhere point in the direction of $\hat\chi^5$.  It is a straight line.  When we need some vector $V^\mu$ to be aligned correctly in the cosmological lattice, as in figure \ref{fig:PHImagic}, we can begin $\hat M^3$ by defining $\chi^5_+$ as pointing in the direction of $V^\mu$ which is defined in terms of $x^\mu\in\mathcal{H}$ only.  When we need to continue the process on $\chi^5_-$ we can choose a vector $V^\mu_+$ from $\Omega$ and use it to the define the direction of $\hat\chi^5_-$.  If $V^\mu_+$ is the parallel transport of $V^\mu$ onto $\Omega$, and $V^\mu_-$ is the parallel transport of $V^\mu_+$ onto $\aleph$, then the parallel transport of $V^\mu_-$ onto $\mathcal{H}_2$, call it $W^\mu$, will be very nearly the same as $V^\mu$ due to the weak field condition.  If the Lorentz approximation was perfect then $V^\mu$ would be equal to $W^\mu$ but in reality there will be small differences between the flat, spherical, and hyperbolic spaces that lead to small ``errors'' in each step of parallel transport from one Lorentz frame to the next in $\mathcal{H}\mapsto\Omega\mapsto\aleph\mapsto\mathcal{H}$.  We expect that the sum of these errors $Q=|V^\mu-W^\mu|$ will look like quantum decoherence.

Here is an important point.  Consider an infinite number of 4D de Sitter and anti-de Sitter spaces that have an infinite number of curvature parameters, all unique, that form a continuum of monotonically increasing curvature: curvature parameters in $[-a,0)$ for AdS and $(0,b]$ for dS.  In the worldsheet representation we can order all the like de Sitter branes by increasing curvature parameter and then abut them to construct the smooth 5-spaces $\Sigma^\pm$.  Since each 4D slice is curved individually, the 5D spaces will also be curved.  Physics in a curved manifold always has tangent vectors pointing outside of the manifold.  Compare this to the case mentioned above when we start with flat 5-space and chop it into slices so that it is only the cotangent vectors that point outside of the manifold.  The slices are all flat in the mathematical sense of the most natural geometry but in the physical sense of distance we are imposing an embedded metric on the 4D slices that makes them seem curved to an observer within.  We can do this because the 4D coordinates $x^\mu$ are an independent object from the 5D ones $\chi^A$ even if there are certain cases where they are the same.  Discerning between either tangent vectors or cotangent vectors leaving the slices of constant coordinate will be a critical distinction used to construct a bridge across $\Sigma^\varnothing$.

\begin{figure}
    \makebox[\textwidth][c]{
        \includegraphics[scale=.6]{PHImagic2.png}}
    \captionsetup{format=hang}\caption{This figure shows our proposal to avoid computing the geodesics in figure 3 through the use of an appropriately defined $\hat \Phi$ vector or set of $\hat \Phi$ vectors.  This figure shows the single instance one 5D $\hat{\Phi}$ vector.  In terms of the 4D vector it might be better illustrated as $\hat{\Phi}^3$ and we begin to see an origin for the $(\Phi\pi)^3$ term in $\alpha_{M\!C\!M}$ when each $\hat{\Phi}$ has an accompanying $\hat{\pi}$.  We have the option to say that $\hat{\Phi}$ points from one 5D manifold to $p$ in the next or it can act in 4D (more likely) where it points to successive $p$'s in $\{\aleph,\mathcal{H},\Omega\}$ and where the third 4D $p$ would be the same point as the first 5D $p$.  This figure shows the 5D $\hat \Phi$, the two poles of the sphere in figure \ref{fig:polarray} are connected by the 4D $\hat \Phi$ that we will work with in general.  (Figure \ref{fig:Riemann} shows three 4D $p$'s.)}
    \label{fig:PHImagic2}
\end{figure}

If we are relying on the topological disconnection of $\mathcal{H}$ from $\Sigma^\pm$ to motivate quantum weirdness \cite{GC,QS} then we need to show how they are still connected in the non-topological sector.  If they are not sufficiently connected then the action that steers geodesics through figure \ref{fig:PHImagic} will not exist.  If they are connected, then the action is guaranteed to exist and the trick (figure \ref{fig:PHImagic2}) will work.  It is very easy to demonstrate the non-topological connection.  Beginning with an initial state in $\mathcal{H}_1$, parameterize some curve in the Minkowski metric whose tangent vectors are collinear with $\chi^5_+$ and its tangent space (which will be accomplished by defining $\chi^5_+$ accordingly)\footnote{The method of obtaining this first relativistic 4-vector from some quantum state is suggested loosely in section II.6 but that will be beyond the scope of this section.}. The statement $\chi^5_+\hat{\Phi}=V^\mu\hat{\pi}$ is a non-topological, purely algebraic statement.  Take note that $\chi^5_+$ having the same direction as $V^\mu$ is the only constraint on $\Sigma^+$ so there is no other constraint that could make it impossible to construct a unit cell around $\mathcal{H}$ when it preexists in a void.  The $\chi^\alpha_+$ are attached to $\chi^5_+$ strictly to ensure our ability to transport qubits defined in a local Lorentz frame along $\chi^5$.  Above we introduced the $\hat x^\mu$ as a subset of the $\hat{\chi}^A$ but a more physical approach is to begin with observable $\hat x^\mu$ and then define $\hat{\chi}^A$ as a superset.  Since 5-space has infinitely many 4D subspaces, there are infinitely many $\chi^A$ that can be constructed to contain $x^\mu$.
Keep in mind that there is no requirement for anything to ``slide'' along $\chi^5_\pm$.  The qubit in $\mathcal{H}_1$ can just as easily be described as tunneling onto $\Omega$, then $\aleph$, and then onto $\mathcal{H}_2$.  In fact, Kaluza--Klein theory relies on the vanishing 5D Ricci tensor $R_{AB}=0$ so the qubit \emph{must} tunnel directly from one brane onto the advanced brane in $\mathcal{H}\mapsto\Omega\mapsto\aleph\mapsto\mathcal{H}$.  A current of massive particles in the bulk would be the opposite of a vanishing Ricci tensor.  Once $\chi^5_+$ establishes a place to put $\chi^\alpha_+$, we may define the coordinates $x^\mu_+$ in the dS metric of the $\Omega$ manifold.  In $\aleph$ and $\Omega$ it is likely that we can always take $x^\mu_\pm=\chi^\alpha_\pm$ when considering an unperturbed ground state.

\begin{figure}[t]
    \makebox[\textwidth][c]{
        \includegraphics[scale=1]{potential.png}}
    \captionsetup{format=hang}\caption{This figure seeks to demonstrate how dark energy is a solution to the Poisson equation inside the cosmological unit cell.}
    \label{fig:potential}
\end{figure}


Describing figure 3, we parallel transport $V^\mu$ onto $\Omega$, call it $V^\mu_+$, and then define $\chi^5_-$ so it is pointing in the direction of $V^\mu_+$ along the 1D manifold of stacked $\hat \Phi$ vectors, possibly something like $\chi^5_-\hat{i}:=V^\mu_+\hat{\Phi}$.   To avoid conflict with the collinearity of $V^\mu$ and $\chi^5_+$, meaning that the topological arrangement should prevent any linear superposition of $V^\mu$ and $V^\mu_-$ once $V^\mu$, we say $\chi^5_-$ is out of phase with both $V^\mu$ and $V^\mu_+$ \footnote{We follow the convention that $\hat x$ is out of phase with $\hat y$ by $\pi/2$ radians in the Cartesian plane.}.  We can accomplish this in the obvious way with the orthogonality of the ontological basis vectors but there is another avenue available when the level of $\aleph$ increases $n\rightarrow n+1$ between $\Omega$ and $\aleph$.  The MCM forbids the interference of vectors on different levels of $\aleph$ \cite{OP} and this is already the established behavior for hyperreal quantities on different tiers of infinitude in $^*\mathbb{R}$.  With two number lines \cite{IC}, chronos $x^0$ and chiros $\chi^5$, we can introduce the concept of \textbf{double orthogonality}.  Consider Hamiltonian systems which only consider the canonical conjugate coordinates wherein any physical system is determined once any two variables are known.  There single orthogonality (regular orthonormalism) can only halfway decouple two elements of a closed system.  Double orthogonality will allow complete decoupling of elements.

It is known that every gravitational manifold has at least one point $p$ in the manifold at which it is possible to construct a coordinate system wherein the metric is locally the Minkowski metric and its first derivatives all vanish at $p$.  Considering figure \ref{fig:PHImagic} let each vector $\{V^\mu,V^\mu_+,V^\mu_-\}$ point to the special point $p$ in the future-adjacent manifold as in $\mathcal{H}\mapsto\Omega\mapsto\aleph\mapsto\mathcal{H}$.  This means that $V^\mu_+$ passing through $p\in\Sigma^\varnothing$ is the only constraint on $\Sigma^\varnothing$.  Since $V^\mu_+$ was already separated from $\mathcal{H}$ by only appearing in its tangent space, and $\Sigma^-$ is separated by not having the inverse vector to $V^\mu_+$ anywhere in its vector bundle\footnote{The inverse vector would go as $1/\infty$ since $\Sigma^-$ is on a higher level of $\aleph$ than $\Sigma^+$.}, $V^\mu_-\in\Sigma^-$ is doubly separated from $V^\mu\in\mathcal{H}$, or doubly orthogonal.  To begin the three-fold process of observation, calculation, and observation again \cite{TER} we start with a vector $V^\mu$ defined at a point in Minkowski space $\mathcal{H}$ and say it points to $p$ not in $\mathcal{H}$, and then we construct $p$'s manifolds $\Sigma^+$ and $\Omega$  around $p$.  Then we do the same thing with $V^\mu_-$: construct $\chi^5_-$ from $V^\mu_+$ and then create $V^\mu_-$ so that it points to $\mathcal{H}_2$.\footnote{To put this on a computer it may be necessary to do this one more time with $p$ whose manifolds are $\varnothing$ and $\Sigma^\varnothing$ being an intermediate location between $V^\mu_+$ and $V^\mu_-$.  This could require another object $V^\mu_\varnothing$.}

We point out that if $\{V^\mu_-,V^\mu,V^\mu_+\}$ are all identically the 4D $\hat \Phi$ vector when written in the various coordinates systems on $\{\aleph,\mathcal{H},\Omega\}$, but $Q=|V^\mu-W^\mu|\neq0$ then we have an algorithm that violates \textbf{conservation of information}.

Every observable state we could consider has some associated energy density that is unambiguously a perturbation $h_{\mu\nu}$ on the background Minkowski metric $\eta_{\mu\nu}$.  $\mathcal{H}$ is globally flat but with the perturbation it is not precisely Minkowski space locally so it has local geodesics with tangent vectors that point outside of the slice, into the bulk.  All this allows us to construct the Riemann sphere with poles at successive $p$ and perform the inversion operation that has been described for transporting Hilbert space along with the geometry from one moment to the next as in figure \ref{fig:polarray} \cite{TER}.\footnote{The reader might ask, ``If the vector in Hilbert space exists independently of the position space representation, then why transport it all?  Surely we can use the same Hilbert space everywhere.''  If 4D physics is independent of the fifth dimension because the branes are slices of constant $\chi^5_\pm$, and $\chi^5_\pm$ increases with increasing $\mathcal{H}_n$, then we can mitigate the increase of $\chi^5$ with an appropriate linear reduction factor attached to the position space representations of $\mathcal{H}'$ vectors in the successive position spaces $\mathcal{H}_n$.  This reduction operation can be inconsequentially included in then inversion operation on the Riemann sphere that occurs between observations \cite{TER,IC}.}  It is likely that the ordinary inversion map between the two coordinate charts on $\mathbb{S}^2$ will be sufficient for the purposes of carrying out what we have called ``the inversion operation on the Riemann sphere.''  When the sphere is situated between two branes $\mathcal{H}_1$ and $\mathcal{H}_2$ (or $\mathcal{H}_1$ and $\Omega$) then it is natural to associate the coordinate chart that covers one of the sphere's poles with the brane that touches that pole, and likewise for other pole and the other coordinate chart on $\mathbb{S}^2$.  Between $\mathcal{H}_1$ and $\mathcal{H}_2$ these would be the $x^\mu_{\{1\}}$ and $x^\mu_{\{2\}}$ coordinates shown in figure \ref{fig:unitcell2} or between $\mathcal{H}$ and $\Omega$ we could have the $x^\mu$ and $\{x^\mu_+,x^\alpha_+\}$ coordinates.

\begin{figure}[t]
	\makebox[\textwidth][c]{
		\includegraphics[scale=.47]{Stereographic1.png}}
	\captionsetup{format=hang}\caption{This figure shows the bijection between a sphere and two planes: two flat slices of some bulk.  $\mathcal{O}$ marks the origin of coordinates on each chart.  This figure differs from figure \ref{fig:PHImagic2} because here the sphere connects $\mathcal{H}$ to $\Omega$ but in figure \ref{fig:PHImagic2} $\hat \Phi$ connects $\mathcal{H}_1$ to $\mathcal{H}_2$.  Where figure \ref{fig:PHImagic2} shows what we called the 5D $p$, the point $\mathcal{O}$ on the right of this figure would be the first 4D $p$.  Since the Gelfand triple is constructed by first taking a subspace of $\mathcal{H}$ this figure suggests that we should examine a possible reversal of the association of $\aleph'$ and $\Omega'$ with $\aleph$ and $\Omega$.}
	\label{fig:polarray}
\end{figure}

Consider $\mathcal{H}$ and its vector space $\mathcal{H}'$.  If we are going to send information across the unit cell from $\mathcal{H}_1$ to  $\mathcal{H}_2$ without a non-vanishing Ricci tensor in the bulk then we need to use an alternate channel.  Hilbert space $\mathcal{H}'$ is defined on $\mathbb{C}$ which is topologically equivalent to either of the two charts on $\mathbb{S}^2$.  The Riemann sphere is the portion of $\mathbb{S}^2$ that can be covered by a single chart, and the map between $\mathbb{S}^2$'s two chart coverings, call them $\xi$ and $\zeta$, is the inversion map

\begin{equation}
\zeta\quad\equiv\quad\frac{1}{\xi}~~.
\end{equation}\newline


\noindent There are normally two maps between the plane and the Riemann sphere: one where the plane intersects the equator of the sphere and another where the sphere sits on top of the plane and we will use the latter.  The two representations completely mirror the choice to put either $\mathcal{H}$ or $\Sigma^\varnothing$ in the center of the MCM unit cell.  When the sphere rests on the plane, the bijection is performed by polar ray tracing as in figure \ref{fig:polarray}.  After we use the polar ray to trace the plane onto the sphere, we can use the inversion map to invert the information encoded on the sphere and then use the polar ray (which now points always upward instead of always downward) to trace the information back onto a new plane.  This new plane can be some other flat slice of 5D bulk with its own unrelated embedded metric that is de Sitter space in some coordinates $x^\mu_+$ so that the slice is what we call $\Omega$.  The inversion map is exactly what is required to go to a higher level of $\aleph$.  Points on the sphere very near the pole at the base of the polar ray have planar coordinates that approach infinity so the inversion map between $\zeta$ and $\xi$ will give something like $\infty\to1/\infty$ which is axiomatically zero.  However, there is some wiggle room where we can transform the coordinates as $x^\mu\to dx^\mu$ and $\chi^A\to d\chi^A$ which are axiomatically non-zero.  In this way we can build the one- and two-forms of general relativity as well as whatever $N$-forms are required for the differential geometry of the hypercosmos.  The reader is referred to reference \cite{ZETA} for further specifications on the Riemann sphere as it relates to the MCM/TOIC.

In reference \cite{KN} we showed an alternative formulation where $\hat{\Phi}$ points to an intermediate point somewhere inside the bulk of $\Sigma^+$ but here the reader should not go in with too much bias regarding $\hat{\Phi}$.  We will develop a 4D $\hat{\Phi}$ now and we will discuss this very important alternative formulation from reference \cite{KN} in section V.1 but we will only use the 5D version (figure \ref{fig:PHImagic2}) to demonstrate how $\hat \Phi$ has at least the potential to provide a shortcut.  The shortcut might be that the symmetry of the $\hat \Phi$ vectors connecting the poles of these spheres that are the successive $p$ where it is possible in every manifold to construct a local Lorentz frame.  The 4D version is preferred at this time so that the framework is guaranteed to accept input in the classical Lorentz approximation.  Since we know the bulk curves in figure 3 are guaranteed to exist, we may be able to simply define the vector $\hat{\Phi}$ such that it points from $\mathcal{H}_1$ to $\mathcal{H}_2$.  All of the determinism can be written in coordinate independent tensors so it is also possible to choose the coordinates where $\hat\Phi$ has the correct behavior and then write the tensor equations in those coordinates.  To some extent we are cheating by taking $\chi^5_\pm$ to be in the direction of an arbitrary tangent vector but to a further extent that is exactly what one would expect using $\hat2\pi=\hat\pi+\hat\pi=\hat{\pi}+\varphi\pi\hat{\Phi}$ \cite{IC}.  Obviously the \textit{copy} assigned to the $\hat\Phi$ component will have the same direction as the original component: it is a co-$\hat \pi$.  They can point in the same direction from the same point and not interfere with each other if we require that $\hat{\pi}$ and $\hat{\Phi}$ are orthogonal and we have already done so.  If there are two quantities associated with the single direction indicated by $V^\mu$ then that is essentially the system of two number lines described in reference \cite{IC}.  Double orthogonality means vectors can have the same direction and still be orthonormal due to the appended ontological basis vectors, \textit{e.g.}: $|\psi\rangle\,\hat \pi_n$.

We can define the \textbf{ontological gauge} to be the one where $\hat{\Phi}^n$ is a vector in $\mathcal{H}_n$ that points to the termination of the worldline on $\Omega$ which is then carried forward by the MCM boundary condition.  This holds for any $n$ but not necessarily more than one $n$ at a time.  By ``termination of the worldline on $\mathcal{H}_{n+1}$'' we mean the observer has computed the mathematical evolution of the initial state in $\mathcal{H}_1$ which correctly describes what is observed when the theory is tested at $\mathcal{H}_2$.  In many realistic applications this will be repeated millions of times iteratively because fine-grained simulation might require millions of time steps, each calculated with $\hat M^3$.  The ontological gauge is very complicated but the \textit{theory} of the TOIC is that $\hat{\Phi}$ agrees with experiment and there will be no need to actually compute the curves in figure \ref{fig:PHImagic}, which would bring us to the same solution arrived at through $\hat \Phi$ (or three $\hat \Phi$'s).  This is illustrated in the comparison between figures \ref{fig:PHImagic} and \ref{fig:PHImagic2}. In any case, if we did try to compute the curves in figure 3 we would probably bump into ourselves where we have required that chiros is non-relativistic because the full hypergeometry of the cosmological cell including $\Sigma^\varnothing$ would likely require $\chi^5$ to be non-flat.  Such a system has proven too complex even to represent schematically and rightly so if the computation represented by the diagram is technically impossible.  By that we mean to imply that $\hat \Phi$ is not an alternative means by which to compute MCM processes, it will be the only way and the reader my be disappointed to learn that no such computation appears in this book.  Furthermore, if we are somehow accomplishing an impossible computation then that gives some loose understanding of the principle of violation of conservation of information.

We have introduced $\{\varnothing,\Sigma^\varnothing\}$ as a new degree of topological freedom.  In general in physics, each degree of freedom is a channel for the flow of information so when we say $\hat M^3$ is incomputable, perhaps we can compute $\tilde M^4$

\begin{equation}
\tilde M^4\quad:\quad\mathcal{H}_1~~\mapsto~~\Omega_1~~\mapsto\varnothing~~\mapsto~~\aleph_2~~\mapsto~~\mathcal{H}_2~~,
\end{equation}\newline

\noindent with the additional point $p\in\varnothing\subset\Sigma^\varnothing$.  $\varnothing$ is a 4D Poincar\'e section of $\Sigma^\varnothing$ where we apply the \textit{ad hoc} operation of discrete translation to the higher level of $\aleph$.  $p$ is a point in $\Sigma^\varnothing$ and $\varnothing$ is a submanifold that contains $p$.  We want to sidestep the divergent blowups of integrals that do not return finite values by making a copy of the initial cubit $|\psi\rangle\hat \Phi^0$ and placing it at infinity \textit{a priori} as $|\psi\rangle\hat \Phi^1$.  The definition of this process is to be such that

\begin{equation}
x^0\in\{0,\infty\}\quad\qquad\implies\quad\qquad \chi^5_+\in\big\{0,|\psi\rangle\hat\Phi\big\}
\end{equation}\newline

\noindent where the $\sqrt{i}$ channel will be sufficient to create an adequate dimensional transposing parameter for $|\psi\rangle\hat \Phi\to\hat1$.

We may use the two central steps of $\tilde M^4$ to convert to the 1-form basis (description of the manifold in terms of it dual tangent vectors) on the manifold with $\Omega\mapsto\varnothing$ and then convert back to tangent vectors in $\Sigma^-$ with $\varnothing\mapsto\aleph$.  When the information is in AdS space we create a well defined qubit in the hyperbolic geometry that prevents any information flowing in that could disrupt the integrity of the qubit.  In the spherical geometry of $\Sigma^+$ it is impossible to define initial value problems that preclude the possibility of information flowing in from infinity.  While we seem to have the correct convention for dS and AdS spaces on $\Omega$ and $\aleph$, we should, at some point, consider the reverse ordering of the elements of the Gelfand triple $\{\aleph',\mathcal{H}',\Omega\}$ with respect to $\{\aleph,\mathcal{H},\Omega\}$.  Since the first step is $\mathcal{H}\mapsto\Omega$ it is likely that the first algebraic step is to select the subspace of Hilbert space which we have been calling $\aleph$', whereas we have previously assigned the abstract superspace $\Omega'$ to $\Omega$.  If we do change this convention, we will simply make a revision in the definitions of $\{\aleph',\Omega'\}$ to retain the intuitive object associations.

The interpretation of $\tilde M^4$ is not meant to be physical.  It is a specification of a way to use numerical algorithms across the dual tangent space to derive a good approximation for $\hat M^3$ which we have theorized as the real physical process of Nature.





\subsection{Feynman, Functions, and Functionals}\addtocontents{toc}{\protect\vspace{22pt}}

Now we have explained, in principle, how a geodesic can pass from one disconnected space $\mathcal{H}_1$ to another one $\mathcal{H}_2$ as if they were connected.  We just have to find the correct algorithm for the computation that makes use of the vectors in the dual tangent bundles to these cosmological manifolds.  As a preliminary for what will follow, insofar as the modified cosmological model is the boundary condition in which we propose to unify general relativity with quantum electrodynamics via the theory of infinite complexity, and by extension, eventually, with the standard model, we will now consider an excerpt\footnote{We change a few variable names here to enforce consistency between Feynman's notation and the present conventions.} from Feynman's seminal paper which he wrote in part as a response to a paper of Dirac's  \cite{FEYN,DIRAC}.\footnote{At the end of Dirac's paper he points out that his theory still has some problems with it because it returns a complex-valued probability whose only physical utility is to make a hand-waving (but valid!) association between a very small complex number and a very low probability.  Large complex numbers were apparently unintelligible.  Feynman even dedicates an entire section of his much longer paper \cite{FEYN} to describing the inadequacies of his own formulation.  Together, these two war era papers serve to sharply contrast contemporary editorial standards in comparable modern journals.  In reference \cite{TER} we make a statement similar to Dirac's when we ignore the problems associated with the sum of a vector and a tensor: ``Assume an evolution operator that is the sum of a   vector part and a tensor part so that $\hat{\Upsilon}\equiv\hat U+\hat M^3$.  We ignore the difficulties associated with adding a vector to a tensor and for now it will suffice to say that $\hat \Upsilon$ is a strange mathematical object.  The operator $\partial$ is a unit vector and $\hat M^3$ takes on unitary properties in chronos.  Using the convention to denote tensor states $|\psi\rangle\hat{\pi}$, we outline a new quantum theory.''  Then in reference \cite{ICT} we discovered that $\hat{\Upsilon}$ is the higher rank, complexified analogue of a strange representation of the quaternions $\textbf{q}=v_0+\vec v$.  \newline\indent Why are papers about partially formed ideas that only kind of work no longer allowed?  When such papers appear in the literature, other papers that fill in the gaps often appear in short order.  It is only the papers that don't work at all which should be disallowed.  Given the current \textit{total standstill} in the pace of discovery in theoretical physics (or recent rather if we \textit{reject the false narrative} about how no one is studying the TOIC/MCM), one would assume that editors would encourage the kinds of papers that might inspire others to pursue new research directions but, alas, 'tisn't so.  The vast bounties of low hanging fruit pointed to by the present research (as yet undescribed even now in 2017) will continue to be ignored (officially) by those who would seemingly rather build Rube Goldbergs on the backs of giraffes in the hopes of randomly grabbing some fruit from the clutches of a bird that might fly by even though no one has seen a bird with any fruit like that for decades.}

\begin{quotation}
    ``We shall see that it is the possibility [\emph{of expressing the action}] $\mathcal{S}$ as a sum, and hence $\Phi$ as a product\footnote{$\Phi[x(t)]$ is the contribution to the complex phase from the action associated with a given path $x(t)$.}, of contributions from successive sections of the path, which leads to the possibility of defining a quantity having the properties of a wavefunction.


    \begin{figure}[t]
    	\makebox[\textwidth][c]{
    		\includegraphics[scale=.60]{FeynR.png}}
    	\captionsetup{format=hang}\caption{This figure shows the division of the time interval described by Feynman.  The red square here is necessarily the same as the one in figure \ref{fig:PHImagic2} because Feynman's relativistic region $R$ is spanned by $x$ and $t$.  $R$ is Minkowski space. }
    	\label{fig:FeynR}
    \end{figure}

    ``To make this clear, let us imagine that we choose a particular time $t_0$ and divide the region $R$ [(\textit{figure \ref{fig:FeynR}})] into pieces, future and past relative to $t_0$.  We imagine that $R$ can be split into: (a) a region $R'$, restricted in any way in space, but lying entirely earlier in time than some $t'$, such that $t'<t_0$; (b) a region $R''$ arbitrarily restricted in space but lying entirely later in time than $t''$, such that $t''>t_0$; (c) the region between $t'$ and $t''$ in which all the values of the $x$ coordinates are unrestricted, i.e., all of space-time between $t'$ and $t''$.  The region (c) is not absolutely necessary.  It can be taken as narrow in time as desired.  However, it is convenient in letting us consider varying $t$ a little without having to redefine $R'$ and $R''$.  Then $|\varphi(R',\,R'')|^2$ is the probability that the path occupies $R'$ and $R''$.\footnote{Feynman writes that the region of variation can be taken as thin as desired but that also means that it can be taken as thick as wanted as well.  Feynman said that it can even be taken as $0$ and when we use the inversion map on $\mathbb{S}^2$ we show that it can even be taken as $\infty$.  The inversion map is what we will use to switch between the description of the MCM unit cell with $\mathcal{H}$ or $\Sigma^\varnothing$ in the center, and it will also mark the changing level of $\aleph$ where some yet-to-be-defined transfinite renormalization induces $\infty\to\epsilon$.}  Because $R'$ is entirely previous to $R''$, considering the time $t$ as the present, we can express this as the probability that the path had been in region $R'$ and will be in region $R''$.  If we divide by a factor, the probability that the path is in $R'$, to renormalize the probability we find: $|\varphi(R',\,R'')|^2$ is the (relative) probability that if the system were in region $R'$ it will be found later in $R''$.

    ``This is, of course, the important quantity in predicting the results of many experiments.  We prepare the system in a certain way (e.g., it was in region $R'$) and then measure some other property (e.g., will it be found in region $R''$?)\footnote{Note how Feynman describes the three-fold process of observation, calculation, and then observation again.}  What does [\textit{equation (\ref{eq:FEYNM1})}] say about computing this quantity, or rather the quantity $\varphi(R',\,R'')$ [\emph{which is the square of equation (\ref{eq:FEYNM1})}]?

    \begin{equation}\label{eq:FEYNM1}
    \varphi(R)=\lim\limits_{\epsilon\rightarrow0} \int_{R} \exp \left[ \frac{i}{\hbar}\sum_{i}\mathcal{S}(x_{i+1},x_i)\right]...\frac{dx_{i+1}}{A}\frac{dx_i}{A}...
    \end{equation}

    ``Let us suppose in [\emph{equation (\ref{eq:FEYNM1})}] that the time $t$ corresponds to one particular point $k$ of the subdivision of time into steps $\epsilon$, i.e., assume $t=t_k$, the index $k$, of course, depending on the subdivision $\epsilon$.  Then, the exponential of a sum may be split into a product of two factors

    \begin{equation}\label{eq:FEYNM2}
    \exp\left[\frac{i}{\hbar}\sum_{i=k}^{\infty}\mathcal{S}(x_{i+1},x_i)\right] \cdot \exp\left[\frac{i}{\hbar}\sum_{i=-\infty}^{k-1}\mathcal{S}(x_{i+1},x_i)\right] ~~.
    \end{equation}

    ``The first factor contains only coordinates with index $k$ or higher, while the second contains only coordinates with index $k$ or lower.  This split is possible because [\emph{the representation of the action as a sum of actions}]

    \begin{equation}\label{eq:FEYNM3}
    \mathcal{S}=\sum_i\mathcal{S}(x_{i+1},x_i)~~,
    \end{equation}

    \noindent results essentially from the fact that the Lagrangian is a function only of positions and velocities.  First, the integration on all variables $x_i$ for $i>k$ can be performed on the first factor resulting in a function of $x_k$ (times the second factor).  Next, the integration on all variables $x_i$, for $i<k$ can be performed on the second factor also, giving a function of $x_k$.  Finally, the integration on $x_k$ can be performed.  That is, $\varphi(R',R'')$ can be written as the integral over $x_k$ of the product of the two factors.  We will call these $\vartheta^*(x_k,t)$ and $\psi(x_k,t)$:

    \begin{equation}\label{eq:FEYNM4}
    \varphi(R',R'')=\int\vartheta^*(x_k,t)\psi(x_k,t) dx~~,
    \end{equation}

    \noindent where

    \begin{equation}\label{eq:FEYNM5}
    \psi(x_k,t)=\lim\limits_{\epsilon\rightarrow0}\int_{R'}\exp\left[\frac{i}{\hbar}\sum_{i=-\infty}^{k-1}\mathcal{S}(x_{i+1},x_i)\right]\frac{dx_{k-1}}{A}\frac{dx_{k-2}}{A}...~~,
    \end{equation}

    \noindent and

    \begin{equation}\label{eq:FEYNM6}
    \vartheta^*(x_k,t)=\lim\limits_{\epsilon\rightarrow0}\int_{R''}\exp\left[\frac{i}{\hbar}\sum_{i=k}^{\infty}\mathcal{S}(x_{i+1},x_i)\right]\frac{dx_{k+1}}{A}\frac{dx_{k+2}}{A}...~~.
    \end{equation}

    ``The symbol $R'$ is placed on the integral for $\psi$ to indicate that the coordinates are integrated over the region $R'$, and, for $t_i$ between $t'$ and $t$, over all space.  In like manner, the integral for $\vartheta^*$ is over $R''$ and over all space for those coordinates  corresponding to times between $t$ and $t''$.  The asterisk on $\vartheta^*$ denotes complex conjugate, as it will be found more convenient to define [\textit{equation (\ref{eq:FEYNM6})}] as the complex conjugate of some quantity $\vartheta$.

    ``The quantity $\psi$ depends only upon the region $R'$ previous to $t$, and is completely denned if that region is known.  It does not depend, in any way, upon what will be done to the system after time $t$.  This latter information is contained in $\vartheta$.  Thus, with $\psi$ and $\vartheta$ we have separated the past history from the future experiences of the system.  This permits us to speak of the relation of past and future in the conventional manner\footnote{For example, see definitions (\ref{eq:timetime})}.  Thus, if a particle has been in a region of space-time $R'$ it may at time $t$ said to be in a certain condition, or state, determined only by its past and described by the so-called wavefunction $\psi(x,t)$.  This function contains all that is needed to predict future probabilities. [\emph{sic}]

    ``Thus, we can say: the probability of a system in state $\psi$ will be found by an experiment whose characteristic state is $\vartheta$ (or, more loosely, the chance that a system in state $\psi$ will appear to be in $\vartheta$) is

    \begin{equation}\label{eq:FEYNM7}
    \left|\int \vartheta^*(x,t)\psi(x,t) dx \right|^2~~.
    \end{equation}


    ``These results agree, of course, with the principles of ordinary quantum mechanics.  They are a consequence of the fact that the Lagrangian is a function of position, velocity, and time only.''\newline
\end{quotation}


When we include the advanced potential, Feynman's nice machinery of the classical formalism will fail because the MCM Lagrangian depends on position, the first derivative of position: velocity $\dot x$, and also at least the third derivative of position $\dddot x$ that is almost unique to the theory of advanced (and retarded) potentials.  We expect to be able to make the extension in the ontological formalism but emphasize that it will not be a direct extension of the classical action formalism.  To achieve direct extension of the classical action formalism we will consider the case when the advanced and retarded potentials $A^\mu_\pm$ vanish and thereby remove the dependence on $\dddot x$.  $A^\mu_\pm$ represent an esoteric electromagnetic effect whose technical details will mostly be beyond the scope of this paper which aims (mostly) to present the modestly technical details of the general relevance of what has been previously reported regarding the MCM.  However, we will briefly look at the advanced and retarded potentials $A^\mu_\pm$ in section III.11.

Feynman didn't invent QED with the above quoted paper.  QED's greatest success, arguably, was the positive result obtained by Schwinger who wrote the following about \textit{eleven six two} in reference \cite{SCHW}.

\begin{quotation}
    ``The simplest example of a radiative correction is that for the energy of an electron in an external magnetic field.  The detailed application of the theory shows that the radiative correction to the magnetic interaction energy corresponds to an additional magnetic moment associated with the electron  spin, of magnitude  $\delta\mu/\mu=(1/2\pi)e^2/hc=0.001162$.  It is indeed gratifying
    that recently acquired experimental data confirm this prediction.  Measurements on the hyperfine splitting of the ground states of atomic hydrogen  and deuterium 2 have yielded values that are definitely larger than those to be expected from the directly measured nuclear moments and an electron moment of one Bohr magneton.  These discrepancies can be accounted for by a small additional electron spin magnetic moment.''\newline
\end{quotation}

We eventually need to show how the radiative corrections mentioned by Schwinger are the natural ones expected in the MCM but we will go that way at a later date since it is about twenty years on down the line from the historical period with which the MCM has been preoccupied.  We will go into lot of detail examining Feynman's method for dividing the time interval but first note the process we described in reference \cite{TER}.

\begin{quotation}
    ``The observer is fixed in the present (at the origin) with the inclusion of $\delta(t)$ and since this function returns an undefined value at $t=0$ it is impossible to integrate directly from early times to late times.  To use an integrand of the form $f(t)\delta(t)$ we must employ the method from complex analysis $f(t)\delta(t)\mapsto g(r,\theta)$.  The integral over all times will trace a path through $\aleph$, $\mathcal{H}$, and $\Omega$.'' \newline
\end{quotation}

The method is the Cauchy integral formula

\begin{equation}\label{eq:CauchyINT}
f(z_0)=\frac{1}{2\pi i}\oint_C \frac{f(z)}{z-z_0}dz~~.
\end{equation}\newline

\noindent Normally it is possible in physics to ignore boundary terms at infinity because the action principle chooses the minimum of the action that goes from the past to the future along $\aleph\mapsto\mathcal{H}\mapsto\Omega$.  That is the process Feynman considered by building a continuum of all possible paths from the limit of infinitely many screens.  Here we consider another process when the MCM boundary condition blocks the path of least action with a topological obstruction at the origin so that the two sums in equation (\ref{eq:FEYNM2}) are joined on $i=\infty$ instead of $i=k$.  We expect the \textbf{maximum action path} (which is a perfectly good solution to the equations of motion) is the one that goes around infinity like $\mathcal{H}\mapsto\Omega\mapsto\aleph\mapsto\mathcal{H}$ as in figure \ref{fig:CauchyC} where boundary at infinity can not be ignored.

\begin{figure}[t]
	\makebox[\textwidth][c]{
		\includegraphics[scale=0.7]{delta3.png}}
	\captionsetup{format=hang}\caption{We can understand the path symbolically labeled $\hat M^2$ as the path through $\Sigma^\varnothing$.  The two path segments on the real line labeled $x$ very clearly correspond to $\chi^5_+$ and $\chi^5_-$.}
	\label{fig:CauchyC}
\end{figure}

We can see one kind of topological obstruction at the origin when we set $z_0=0$ in equation (\ref{eq:CauchyINT}) to obtain

\begin{equation}\label{eq:Cauch2}
f(0)=\frac{1}{2\pi i\Phi}\oint \frac{f(z)}{z}dz\,\hat \Phi~~.
\end{equation}\newline

\noindent where we have included $\hat \Phi$ to show how $f(0)$ can describe the advanced time solution at the origin of coordinates of $\mathcal{H}_{n+1}$.  Also note how the normalization of equation (\ref{eq:Cauch2}) shows all four ontological numbers.  Clearly with $f(0)$ there is no continuous path of integration through the point where $z=0$.  However, by using the path around the outside of the complex plane we can compute the paths like $\lim_{\epsilon\to0}\,\int^\infty_\epsilon dz$ knowing that the result is the same as if the central point $z=0$ had contributed.

Instead of the most direct path, we integrate along the path in figure \ref{fig:CauchyC} that is exactly of the form

\begin{equation}
\chi^5\quad\equiv\quad\chi^5_+\otimes\chi^5_\varnothing\otimes\chi^5_-~~.
\end{equation}\newline

\noindent The point $p$ in $\Sigma^\varnothing$ can be the point $z_0$ that appears in the Cauchy formula, or any other special point that we might need it to be, such as perhaps the location of the observer at the origin.  The important thing is that the Cauchy formula defines three piecewise path lengths around the origin and it is obvious that we have already represented those paths with $\chi^5_+$, $\chi^5_-$, and $\chi^5_\varnothing$.  The reader should note the excellent qualitative agreement between the specific concept of $\Sigma^\varnothing$ and the path at infinity, and in general between $\hat M^3$ and the three pieces of the integral over $C$.

We proposed in reference \cite{TER} to use equation (\ref{eq:CauchyINT}) to write

\begin{equation}\label{eq:COMPLEX11}
\int_{-\infty}^{\infty}f(t)\delta(t)dt=\int_{0}^{\infty}g\left(r,0\right)dr+\int_{0}^{\tfrac{1}{\alpha_{M\!C\!M}}}g\left(\infty,\theta\right)d\theta+\int_{-\infty}^{0}g\left(r,\tfrac{1}{\alpha_{M\!C\!M}}\right)dr~~.
\end{equation}\newline

\noindent In more recent iterations the mechanism that we expected to derive from $1/\alpha_{M\!C\!M}$ hyperradians in equation (\ref{eq:COMPLEX11}) has been offloaded onto aspects of hypercomplexity elsewhere and we refer the reader to reference \cite{IC} for fuller details on that offloading.

Here is a good place to notice an asymmetry between chronological and chirological time.  In chronos, the future $t=\infty$ is perfectly balanced with a symmetric value at $t=-\infty$.  However, the hyperreal infinity-valued chirological qubit at $t=\infty$ is balanced with a hyperreal infinitesimal-valued qubit at the non-negative value $t=1/\infty\to0$.  Certainly any number of principles can be attached to this asymmetry.  The matter/anti-matter imbalance comes to mind because big and little hyperreal infinity are both positive numbers and they contrast the notion of infinity composed from only positive and negative ``big infinity.''  This symmetric concept of infinity might dominate the analyses that predict global baryon neutrality.  Also note that in plane polar coordinates there is no such thing as minus infinity because $r\in[0,\infty]$ for $\theta\in(0,2\pi]$.

Originally we put the rotation in equation (\ref{eq:COMPLEX11}) to be through $1/\alpha_{M\!C\!M}$ ``hyperradians'' to show that the method allows a free parameter when we complexify the geometry of the complex plane.  Instead we have complexified the underlying real analysis. Now that we have a different origin for the free parameter $1/\alpha_{M\!C\!M}=2\pi+(\Phi\pi)^3$ \cite{IC} the $\pi$ ordinary radians required for the normal piecewise expansion of $C$ will suffice.  Then equation (\ref{eq:COMPLEX11}) becomes


\begin{equation}\label{eq:COMPLEX12}
\int_{-\infty}^{\infty}\psi(x,t)\delta(t)dt=\int_{0}^{\infty}\psi\left(r,0\right)dr+\int_{0}^{\pi}\psi\left(\infty,\theta\right)d\theta+\int_{-\infty}^{0}\psi\left(r,\pi\right)dr~~.
\end{equation}\newline

We will eventually need to show that the proposed representation in equation (\ref{eq:COMPLEX12}) can support a non-exploding integration scheme that solves the problem Feynman encountered when he tried to integrate over all of spacetime: his integrals would invariably explode.  It was for the reason of exploding integrals that Feynman imposed his finite time constraint.  In developing the associated argument Feynman used one idea of ``all of spacetime'' that perhaps relies on an unassumed differentiation between the theory of functions and the theory of functionals.  When the probability is a function of $x$ and $t$

\begin{equation}\label{eq:PROBAA1}
P\quad\equiv\quad P(x,t)~~,
\end{equation}\newline

\noindent because the wavefunction is a function of $x$ and $t$, then Feynman's thought experiment does produce in its limit the complete space of all possible paths.  However, in functional analysis the probability

\begin{equation}\label{eq:PROBAA}
P\quad\equiv\quad P[\psi]~~,
\end{equation}\newline

\noindent clearly implies that the continuum of all possible paths includes the Cauchy $C$ curve.  Unlike equation (\ref{eq:PROBAA1}), the $P$ in equation (\ref{eq:PROBAA}) depends on a complex variable because $\psi$ is complex-valued.  Therefore the ordinary rules of complex analysis must apply and those rules say $C$ needs to be included in the definition of all of spacetime.

We need to decide if we will use the Cauchy integral formula to expand the probability, as examined above, or the probability amplitude which usually depends on purely real variables but there are some subtleties.  In equation (\ref{eq:COMPLEX11}) $g(r,\theta)$ is a function of a point in the complex plane and that is to say that $g$ is a function of a complex number.  On the other hand, $\psi(x,t)$ appears to be a complex-valued function of two real variables $x$ and $t$.  This would imply that the Cauchy integral formula would have no natural application to the integrals Feynman used to build the quantum mechanical wavefunction but the reader must recognize that we can force $\psi$ to be a function of a complex variable.  How can we force this other possibility?  The analytical origin of the minus sign in the line element of Lorentzian spacetime

\begin{equation}
ds^2=-c^2(dt)^2+(dx)^2+(dy)^2+(dz)^2~~,
\end{equation}\newline

\noindent can come from a definition $x^0=ict$ which means that the region $R$ considered by Feynman is actually the complex plane $\mathbb{C}$ because it is spanned by one real axis and one imaginary axis.  The dimensional transposing parameter $c$ is a trivial coefficient that can be ignored so

\begin{equation}
z \quad\equiv\quad x+it\in\mathbb{C}~~.
\end{equation}\newline

\noindent If $\{x,t\}$ is a point in $\mathbb{C}$ then $\psi(x,t)\equiv\psi(z)$ must be a function of a complex variable and the Cauchy formula applies.

Feynman builds up his space of all possible paths by adding increasing numbers of screens to a hypothetical double slit experiment.  The probability is a function of the wavefunction, which is a complex-valued function so the path in the Cauchy integral formula is completely contextually correct.  In quantum physics the thing that the observer actually tests at the endpoints of $\hat M^3$ is the real valued probability $P$ which, in turn, is a function of the complex-valued wavefunction.  We have $f(z_0)\equiv P(\psi_0)$ but what is $\psi_0$?  A general question in quantum mechanics is to ask, when given an initial state $\psi_i$, what is the probability of observing another state later.  Call that state $\psi_0$ so the probability of observing it is $P(\psi_0)$.

Feynman has not introduced hypercomplex field variables so he considered the neighborhood around one level of $\aleph$ and the minimum of the action is favored.  When all the future levels of $\aleph$ have gravitating objects whose amplitudes (masses) are defined with hyperreal infinities of successively increasing infinitude, the trajectory of a test mass is intuitively pulled out of the Euclidean minimum of action into the maximum action path that heads off toward future infinity.  Notice the consistency here.  An intuitive way to test gravitational equations is to consider the motions of test particles.  If we have equations of motion for gravity near the surface of the Earth, they should show that an ordinary object will fall.  Now we are proposing to radically modify the entire paradigm of gravitation by adding this new path $C$ and different hypercomplex infinites, and we expect the reader to believe that it works when we have no significant accompanying calculation similar to Einstein's prediction for the deflection of light from Mercury as it passes deep through the sun's gravitational well.\footnote{This work resulted from Einstein's collaborations with Grossmann and Besso so it contrasts greatly with the MCM which did not result from any collaborations.}  However, we do have many other results including one very much like Schwinger's derivation of the first order correction to the electron's magnetic moment.  He used perturbation theory that depends on an expansion in the magical fine structure constant which is the number we have derived from first principles with help from God.  The author expresses his gratitude to God.


What possible reason could there be for a trajectory in which stationary states seem stationary but constantly zoom off to positive timelike infinity in $\mathcal{H}\mapsto\Omega\mapsto\aleph\mapsto\mathcal{H}$?  The infinitely large hyperreal qubit at the end of $\mathcal{H}_n$'s $\hat \Phi$ vector is the most massive thing in the local hypercosmos so when the trajectory always goes straight to timelike infinity, it is making a beeline for the most massive object in its local neighborhood.  This is exactly what is expected of a theory of gravity.  There is a lot of consistency.  The main push of this book will be to demonstrate familiar aspects of unification between quantum theory and general relativity but we very much urge a third party undertaking of a rigorous survey of MCM/TOIC objects/methods to check if they are useful in any cutting edge experimental applications or if there is any interesting hypercomplexity hidden somewhere within the infinite complexity.  The existence of this complexity has been convincingly demonstrated throughout this research program.

It has been very many decades since Feynman wrote hits paper titled ``Space-Time Approach to Non-Relativistic Quantum Mechanics,'' and some of the problems with that approach had not been resolved until the introduction of the MCM wherein we have proposed to explore the exotic but allowed maximum action equations of motion.  Feynman's Ph.D. thesis was about the principle of least action in quantum theory so he discarded the maximum action paths \textit{a priori}, but we will not do so.  Since the action principle only allows two solutions, and Feynman has already described one of them, the MCM must rely on the other one.\newline\newline

\begin{figure}[p!]
	\makebox[\textwidth][c]{
		\includegraphics[scale=.5]{Riemann2.png}}
	\captionsetup{format=hang}\caption{This figure illustrates the region of figure \ref{fig:FeynR} between $t_0$ and $t''$.  If one considers the third $\hat{\pi}$ component in the middle as $x^\mu_\varnothing\hat \pi$ then it is very easy to see where the $(\Phi\pi)^3$ term comes from in $\alpha_{M\!C\!M}$, and in a sense we can say $\hat \pi_{1/2}$ is out of phase with the other two $\hat \pi$'s (the normal sense of $\pi/2$) and it is $\hat \pi_0$ and $\hat \pi_1$ that give the $2\pi$ for $\alpha_{M\!C\!M}=2\pi+(\Phi\pi)^3$.  }
	\label{fig:Riemann}
	\makebox[\textwidth][c]{
		\includegraphics[scale=.7]{DE.png}}
	\captionsetup{format=hang}\caption{This is the the mechanism from reference \cite{DE} through which we claimed to have unified gravity and electromagnetism, and here we point out that reference would have been better titled \emph{Dark Energy in the Modified Cosmological Model with Ancillary Takeaways}.  We use the ``in'' $\odot$ and ``out'' $\otimes$ diagrammatic notation of EM to show the places where the piecewise bulk geodesics are joined across different tiers of infinitude (levels of $\aleph$) in $^*\mathbb{C}$.  The original caption for the diagram was, ``The Feynman diagrams of gauge theory generate surfaces which represent interacting strings \cite{BOER}.  On the left: electromagnetic pair creation near the horizon.  On the right: polarized gravitational pair creation.''  The right side of this figure is taken from reference \cite{CARR} and it went unfortunately uncited when we used it in reference \cite{DE}.}
	\label{fig:DE}
\end{figure}



\clearpage







\begin{figure}[t]
    \makebox[\textwidth][c]{
        \includegraphics[scale=.57]{peach.png}}
    \captionsetup{format=hang}\caption{There are a lot of ways that $\aleph$ and $\Omega$ could be connected.}
    \label{fig:peach}
\end{figure}



\renewcommand\thesection{\Roman{section}}
\section{{\textsf{\LARGE  General Relevance with Emphasis on Gravitation}}}
\renewcommand\thesection{\arabic{section}}

The first section of this chapter is an elementary review.  Sections two is dedicated to general relativity.  In section three we make a nice comment about the Bekenstein--Hawking formula for the entropy of black holes.  Section four describes how coordinates can be defined so that the corresponding metric becomes either Lorentzian or Euclidean while maintaining a valid general relativity.  This method can force the condition that causes the wavefunction to be a function of a complex variable.  Section five describes a method by which we may insert qubits onto a geometric manifold.  This section emphasizes the integration of quantum mechanics into MCM quantum gravity/quantum cosmology and we provide some modest but original insights into the structure of perturbation theory with a new definition for $\hat\varphi$.  Sections six and seven deal with twistors, spinors, dyads, and quaternions.  The final section in this chapter is about the MCM mechanism of unification.

Whereas the bulk of the research conducted in this program has focused on the quantum sector, here we also investigate the gravitational sector.  This chapter applies general relativity to derive properties of the MCM and the reader should keep in mind that GR is not called ``general'' relativity because it is a high-ranking theory but rather because it is generic.  Special relativity to the contrary, is not generic.  This book is more comprehensive than previous work and is therefore more rigorous but is also very reiterative.  The mathematical process defined by the MCM boundary condition is periodic in the collapse of the wavefunction and the tools of both hyperreal and complex analyses provide synergy for new tools of \textbf{hypercomplex analysis} in the theory of infinite complexity.

The entire history of physics shows that the rigorous mathematical connection of modes is sufficient to demonstrate inter-modal energy transfers in representative experiments and research regarding the MCM has uncovered a novel new mathematical connection.  There is no guarantee that a ``correct'' mathematical demonstration of a principle will show the principle in the lab but if it does not then that will be the first time.  The emphasis of course is on correctness but due to the simplicity of the TOIC principles it is totally obvious that they are correct.  Any defect cited in this research program is of at most genus errata.\footnote{During the preparation of this book we discovered and corrected an important erratum in reference \cite{IC}.}  As Jesus said in John 18:23, ``If I said something wrong, testify as to what is wrong.''  Jesus immediately goes on to ask, ``But if I spoke the truth, why did you strike me?,'' which can also be taken, very much so, in the context of this banned research.


\subsection{Relevant Aspects of Classical Physics}

General relativity has a Newtonian limit that satisfies the Poisson equation in the same way as the classical electric force.  Those solutions are commonly written as

\begin{equation}\label{eq:NEWTON}
\vec F_g=-\dfrac{GMm}{r^2}\hat r~~,\quad\qquad\text{and}\quad\qquad \vec F_e=-\dfrac{e\,Qq}{r^2}\hat r~~,
\end{equation}\newline

\noindent where $\vec F_g$ and $\vec F_e$ are Newton's law for gravity and the Lorentz force law with no moving charges, a.k.a.$~$Coulomb's law.  If quantum electrodynamics (QED) is one kind of multiplectic expansion of the Poisson equation then all we have to do to unify gravity with electromagnetism (EM) is to show that GR is another multiplectic expansion of the Poisson equation using the same set of basis vectors for each in a common lattice (see figure \ref{fig:DE}).

A common representation of the classical force law is

\begin{equation}\label{eq:Newtonold}
\vec F_{jk}=\frac{\beta}{r^2_{jk}}\left(\frac{\vec r_{jk}}{r_{jk}}\right)~~,
\end{equation}\newline

\noindent where $\beta$ is an electric or gravitational coupling constant.  The multiplectic expansion from Newton's vector gravity to Einstein's tensor theory brings in tensor indices to track curvature, the metric to define the distance along $\vec r_{jk}$, and it throws out the idea of a vector such as $\vec r_{jk}$ connecting two points in a curved manifold.  To accommodate dynamical spacetime geometry it is required to use metrical and other tensors, and non-tensorial connections to fully describe gravity but even those tensor equations have to reduce in the Newtonian limit to representations of the form of equations (\ref{eq:NEWTON}) and (\ref{eq:Newtonold}).  These formulae are derivable from the Poisson equation

\begin{equation}\label{eq:Poisson762}
\nabla^2\phi=4\pi G\rho~~,
\end{equation}\newline

\noindent when the gravitational potential $\phi$ at a distance $r$ away from mass $M$ is

\begin{equation}\label{eq:POT1}
\phi(r)=-\frac{GM}{r}~~.
\end{equation}\newline

Classical gravitation uses the potential $\phi$ to determine the equations of motion but general relativity has no gravitational potential and instead uses the connection $\Gamma^\mu_{\rho\sigma}$ to compute geodesics with

\begin{equation}\label{eq:GEODESIC}
\frac{d^2x^\mu}{d\tau^2}+\Gamma^\mu_{\rho\sigma}\frac{dx^\rho}{d\tau}\frac{dx^\sigma}{d\tau}=0~~.
\end{equation}\newline

\noindent The connection, also called the Christoffel symbol, the Christoffel connection, or the affine connection is determined from the metric via

\begin{equation}\label{eq:connection}
\Gamma^\lambda_{\mu\nu}=\frac{1}{2}g^{\lambda\sigma}\big(\partial_\mu g_{\nu\sigma}+\partial_\nu g_{\sigma\mu}-\partial_\sigma g_{\mu\nu}\big)~~.
\end{equation}\newline

\noindent The inverse metric $g^{\lambda\sigma}$ appears in equation (\ref{eq:connection}) so when we convert to 5D we will make use of the inverse Kaluza--Klein metric $\Sigma^{AB}$ that has, thankfully, already been calculated and posted on Wikipedia.  We can simply plug it into the formula for the 5D connection coefficients where $g^{\lambda\sigma}$ appears in equation (\ref{eq:connection}).  Other important objects include the (4D) Riemann tensor which is

\begin{equation}\label{eq:RRRRRR1}
R^\rho_{\sigma\mu\nu}=\partial_\mu\Gamma^\rho_{\nu\sigma}-\partial_\nu\Gamma^\rho_{\mu\sigma}+\Gamma^\rho_{\mu\lambda}\Gamma^\lambda_{\nu\sigma}-\Gamma^\rho_{\nu\lambda}\Gamma^\lambda_{\mu\sigma}~~,
\end{equation}\newline

\noindent and one of its contractions: the Ricci tensor

\begin{equation}\label{eq:RRRRRR2}
 R_{\mu\nu} \quad\equiv\quad  R^\lambda_{\mu\lambda\nu} ~~.
\end{equation}\newline

\noindent Equations (\ref{eq:RRRRRR1}) and (\ref{eq:RRRRRR2}) generalize to 5D by changing the Greek indices to Latin.  We will consider the 5D variants in chapter three.

General relativity can become a very complex theory when the Ricci tensor and the metric are not linearly dependent on each other and we will discuss related nuance throughout chapter three.  To date we have only worked with Einstein's equation in the form

\begin{equation}
8\pi T_{\mu\nu}=G_{\mu\nu}+g_{\mu\nu}\Lambda~~,
\end{equation}\newline

\noindent but the avenue toward greatest complexity is shown when the Einstein tensor

\begin{equation}
G_{\mu\nu}\quad\equiv\quad R_{\mu\nu}-\tfrac{1}{2}Rg_{\mu\nu}~~,
\end{equation}\newline

\noindent is decomposed to give Einstein's equation as

\begin{equation}\label{eq:longEE}
8\pi T_{\mu\nu}=R_{\mu\nu}-\frac{1}{2}Rg_{\mu\nu}+g_{\mu\nu}\Lambda~~,
\end{equation}\newline

\noindent with the Ricci scalar $R$ being

\begin{equation}
 R  \quad\equiv\quad g^{\mu\nu}R_{\mu\nu} ~~.
\end{equation}\newline

The main result of the MCM has been to develop a new connection between gravitation and quanta.  When the ontological basis was considered to have three elements\footnote{The fourth element $\hat 2$ first appeared in references \cite{QG} and \cite{ICT}.} $\{\hat \pi,\hat \Phi, \hat i\}$ we showed a one-to-one unification


\begin{align}
f^3\big|\psi;\hat\pi\big\rangle&~~\mapsto~~ T_{\mu\nu}\\
&~\nonumber\\
\big|\psi;\hat\Phi\big\rangle&~~\mapsto~~ G_{\mu\nu}\\
&~\nonumber\\
i\big|\psi;\hat i\big\rangle&~~\mapsto~~ g_{\mu\nu}\Lambda~~,\label{eq:imappp}
\end{align}\newline

\noindent but now the ontological basis is $\{\hat i,\hat \Phi, \hat 2,\hat \pi\}$.  This is better suited to Einstein's equation as it appears in equation (\ref{eq:longEE}) but there are several choices for what $\hat 2$ should map to.  This freedom to choose compounds the original arbitrariness imposed when we chose mappings for $\{\hat \pi,\hat \Phi, \hat i\}$.  When there were only three objects in Einstein's equation, we chose one particular set of maps to demonstrate that the maps exist.  Now that there is a fourth object we will not make a guess about how the four tensors in Einstein's equation are connected to the objects in the ontological basis.  Instead we simply point out that since there are four of each, there exist some other one-to-one mappings even if we don't pick one right now.  We will explore mapping in the four-fold representation in the section II.3.  The important thing isn't the form of the specific maps.  The most important feature is that we have obtained the correct dimensionless coefficient of proportionality: $8\pi$.  We will discuss the origin of $8\pi$ in the MCM in chapter three.

The Newtonian gravitational potential, by construction, does not include the gravitational self-force of a particle when it deforms the spacetime of its own inertial frame.  Likewise with the scalar electromagnetic potential: when we probe the field of a strong magnet with a weak test charge $e$ the equations of motion, though completely valid, are only an excellent approximation for the equations of motion that include the small deformation of magnetic field lines around $e$ as it moves through the magnetic field.  The geodesics of general relativity also do not account for the gravitational self force (which is very complicated for the case of an infinitely dense point mass) but they do allow other generalized energy densities to change the course of particles in a way that the classical gravitational potential cannot.  It is for this reason among others that we say Einstein's formulation of gravitation is superior to Newton's.

The electromagnetic self-force is called the Abraham--Lorentz force and it involves the third time derivative of position as

\begin{equation}
F_{\text{self}}:=q^2\dddot x~~.
\end{equation}\newline

\noindent Normally the electromagnetic force is the just the Lorentz force

\begin{equation}
\vec F_L=m\vec{\ddot x}=q\big(\vec E+\vec{\dot x}\times\vec B\big)~~,
\end{equation}\newline

\noindent but the force equation becomes third order in $\partial_t$ as the Lorentz force is taken in superposition with the Abraham--Lorentz force.  In general the approximation to the real electric force with the second order Lorentz force is valid but there are extreme regimes such as plasma physics where the third order equations

\begin{equation}
\ddot x\quad\equiv\quad\ddot x(\dot x)\quad\qquad\longrightarrow\qquad\quad \ddot x \quad\equiv\quad \ddot x(\dot x, \dddot x)=q\big(\vec E+\vec{\dot x}\times\vec B\big)+q^2\vec{\dddot x}~~,
\end{equation}\newline

\noindent must be considered if a realistic answer is to be obtained.  As discussed in the previous chapter, equations of this form are not directly compatible with Feynman's formalism.  The Lagrangian that gives the Abraham--Lorentz force is not solely a function of position and velocity.

Consider Carroll's words from reference \cite{CARR}.

\begin{quotation}
    ``The primary usefulness of geodesics in general relativity is that they are the paths followed by unaccelerated test particles.  A \textbf{test particle} is a body that does not itself influence the geometry through which it moves\footnote{This is the gravitational analogue of the Abraham--Lorentz force.  It is called the gravitational self-force or sometimes the gravitational backreaction.} -- never perfectly true, but often an excellent approximation.  This concept allows us to explore, for example, the properties of the gravitational field around the Sun, without worrying about the field of the planet whose motion we are considering.  The geodesic equation can be thought of as the generalization of Newton's law $\mathbf{f}=m\mathbf{a}$, for the case of $\mathbf{f}=0$, to curved spacetime.  It is also possible to introduce forces by adding terms to the right-hand side [\textit{of equation (\ref{eq:GEODESIC})}],

\begin{equation}\label{eq:EXeqnnn}
    \frac{d^2x^\mu}{d\tau^2}+\Gamma^\mu_{\rho\sigma}\frac{dx^\rho}{d\tau}\frac{dx^\sigma}{d\tau}=\frac{q}{m}F^\mu_{~\nu}\frac{dx^\nu}{d\tau}~~.\mathrm{"}
    \end{equation}\newline
\end{quotation}

Equation (\ref{eq:EXeqnnn}) is a second order force law because, as stated, it does not account for the self-acceleration of test particles.  The full relativistic force law is more complicated and Dirac is credited with working out those details in 1938.  The familiar object $F^\mu_{~\nu}=g_{\rho\nu}F^{\mu\rho}$ is the electromagnetic field strength tensor. It is the tensor that most directly influences observables.  It is defined via

\begin{equation}
\partial_\mu F^{\mu\nu}=J^\nu~~,\quad\qquad\mathrm{or}\quad\qquad F^{\mu\nu}=\partial^\mu A^\nu-\partial^\nu A^\mu~~,
\end{equation}\newline

\noindent where the 4-vector potential $A^\mu$ defines the current through

\begin{equation}\label{eq:dnbfuwtevfib}
\eta^{\mu\nu}\partial_\mu\partial_\nu A^\mu=4\pi J^\mu~~.
\end{equation}\newline

Note how equation (\ref{eq:dnbfuwtevfib}) (the tensor Poisson equation) has a dimensionless coefficient $4\pi$ is also the coefficient of the leading term of the theory of infinite complexity's ontological resolution of the identity

\begin{equation}\label{eq:ontological}
\hat 1\quad\equiv\quad\frac{1}{4\pi}\,\hat \pi-\frac{\varphi}{4}\,\hat\Phi +\frac{1}{8}\,\hat2 -\frac{i}{4}\,\hat i~~.
\end{equation}\newline

\noindent It is really striking that the leading coefficient of a logical ordering of the the ontological resolution of the identity is $1/4\pi$.  The dimensionless constants that effortlessly fall out of the MCM include: exactly the electromagnetic coupling coefficient $4\pi$, exactly the gravitational coefficient $8\pi$ in Einstein's equation, and the fine structure constant to within $0.4\%$.  On the last it is nearly certain that we can reformulate quantum theory so that the inverse fine structure constant is exactly $2\pi+(\Phi\pi)^3$ and then the $0.4\%$ disagreement will be shuffled into some other quantum fuzziness elsewhere.  Quantum electrodynamic perturbation theory works because $\alpha_{Q\!E\!D}$ is a very small number and another number that differs by $0.4\%$ is also very small.

The special behavior of the ontological basis hails from the new notation that specifies a manifold and a vector space for state vectors

\begin{align}
\big|\psi;\hat i\big\rangle\quad\quad\equiv\quad&\quad\psi(x^\mu_-)\label{eq:MAP00}\\
&~\nonumber\\
\big|\psi;\hat \pi\big\rangle\quad\quad\equiv\quad&\quad\psi(x^\mu)\label{eq:MAP11}\\
&~\nonumber\\
\big|\psi;\hat \Phi\big\rangle\quad\quad\equiv\quad&\quad\psi(x^\mu_+)\label{eq:MAP22}~~,
\end{align}\newline

\noindent that would otherwise all live in $\mathcal{H}'$ and have position space representations in the $x^\mu$ coordinates of Minkowski space $\mathcal{H}$ due to the implicit assumptions of quantum mechanics.  Equation (\ref{eq:MAP00}) says $\psi$ lives in $\aleph'$ and equation (\ref{eq:MAP22}) says $\psi$ lives in $\Omega'$, and the coordinate definitions on the RHS specify the coordinates of the relevant manifold.  $x^\mu_-$ is in $\aleph$, $x^\mu_+$ is in $\Omega$, \textit{etc}.  Equation (\ref{eq:MAP11}) shows $|\psi;\hat \pi\rangle\in\mathcal{H}$.  $\mathcal{H}$ is where the theory is required to match observations and $\hat \pi$ is the element of the ontological basis whose coefficient is the electromagnetic coupling constant in equation (\ref{eq:Poisson762}).  We expect to extend equations (\ref{eq:MAP00}-\ref{eq:MAP22}) to include the complete ontological basis with

\begin{equation}\label{eq:MA44}
\big|\psi;\hat 2\big\rangle \quad\equiv\quad \psi(x^\mu_\varnothing)~~.
\end{equation}\newline

\noindent This equation says that $|\psi;\hat 2\rangle$ lives in the same $\mathcal{H}'$ that $|\psi;\hat \pi\rangle$ lives in but the position space representation is in the $x^\mu_\varnothing$ coordinates.  The original three maps were associated with the three vector spaces of a Gelfand triple and this fourth map should be associated with the dual space to Hilbert space $\mathcal{H}'$ which is also $\mathcal{H}'$.  Since $\mathcal{H}'$ is self-dual the fourth map for $\hat2$ is easily accommodated in the Gelfand triple and there is likely an interpretation where $|\psi;\hat\pi\rangle$ and $|\psi;\hat2\rangle$ experience Schr\"odinger evolution on a pair of co-$\hat \pi$'s.  $2\pi$ radians constitute the circle that generates the U(1) symmetry of the classical electromagnetic theory that has already been successfully adapted to quantum theory.

A fourth map for $|\psi;\hat2\rangle$ implies a new topological component associated with $\{\aleph,\mathcal{H},\Omega\}$, which we will call simply $\varnothing$ and say it is a 4D section of the 5D manifold $\Sigma^\varnothing$.  $\Sigma^\varnothing$ lies around the outside of the old unit cell (figure \ref{fig:unitcell}) and is centered in the most recent depiction: figure \ref{fig:unitcell2}\footnote{We use a loose definition of manifold because it is possible to leave the MCM ``manifolds'' along the $\chi^5$ direction.}.  The description of the MCM unit cell has always been as a \textit{partial} unit cell, because we had never even begun to examine all the diagrammatic subtleties that would fully include $\Sigma^\varnothing$.  In figure \ref{fig:unitcell2} we put $\Sigma^\varnothing$ in the center to emphasize that it is the main unknown at this point but the figure still only shows a partial cosmological unit cell because it does not show how $\Sigma^\pm$ are actually connected, which is vital.  We have been able to get away with not stating the relationship between $\hat 2$, $\varnothing$, and $\Sigma^\varnothing$ because, as indicated by the ``$\varnothing$'' notation, it does not directly contribute to observables.  However, before the set of MCM observables that would be of interest to experimentalists can be derived, the indirect dependence on $\Sigma^\varnothing$ must be clarified and we have mentioned $\tilde M^4$ as a next step in that direction.

Regarding $\varnothing$, note that the path over $d\theta$ on the right hand side of

\begin{figure}[t]
	\makebox[\textwidth][c]{
		\includegraphics[scale=.77]{unitcell.png}}
	\captionsetup{format=hang}\caption{$\mathcal{H}$ is not at the point $p\in\Sigma^\varnothing$.  $\Sigma^\varnothing$ lies all around the outside of this representation.}
	\label{fig:unitcell}
\end{figure}

\begin{equation}\label{eq:CAUCHY762}
\int_{-\infty}^{\infty}\psi(x,t)\delta(t)dt=\int_{0}^{\infty}\psi\left(r,0\right)dr+\int_{0}^{\pi}\psi\left(\infty,\theta\right)d\theta+\int_{-\infty}^{0}\psi\left(r,\pi\right)dr~~,
\end{equation}\newline

\noindent is zero because $\psi(\infty)=0$ and similarly the path over $\chi^5_\varnothing$, which lies entirely at infinity, has zero length in our ``affine'' parameter $\chi^5$ (in the representation in figure \ref{fig:unitcell2}).  In equation (\ref{eq:CAUCHY762}) the motivation for the $d\theta$ path with a null contribution

\begin{equation}
\int_{0}^{\pi}\psi\left(\infty,\theta\right)d\theta=0~~,
\end{equation}\newline

\noindent is the implication of quantum mechanics' square integrability condition that $\psi(r\!=\!\infty)=$~0.  In the MCM we say the $x^\mu_\varnothing$ coordinates contribute nothing because the corresponding measure $dx^\mu_\varnothing$ is such that $\int dx^\mu_\varnothing=0$.  The entire manifold $\Sigma^\varnothing$ exists only to guarantee the existence of an exceptional point where the level of $\aleph$ can increase during $\Omega\mapsto\aleph$, as in figure \ref{fig:oldMMM}.  For now we simply say that there is some manifold $\Sigma^\varnothing$ which has at least one point that can sew together $\Sigma^+$ and $\Sigma^-$, and then we introduce a non-trivial parameter


\begin{equation}\label{eq:chi5notafff}
\chi^5 \quad\equiv\quad \chi^5_+\otimes\chi^5_\varnothing\otimes\chi^5_-~~.
\end{equation}\newline

\noindent We can use this definition to draw diagrams that show a smooth affine parameter $\chi^5$ even when the manifolds it passes through have been disconnected by $\mathcal{H}$- and $\varnothing$-branes.


\begin{figure}[t]
	\makebox[\textwidth][c]{
		\includegraphics[scale=.6]{mmm333.png}}
	\captionsetup{format=hang}\caption{This figure makes reference to figure \ref{fig:DE} and describes where the fourth step in $\tilde M^4$ could be added to the existing process $\hat M^3$.  While we will not develop the method here, the general idea is to use $\hat 2$ as the first step to create a copy of $\mathcal{H}$ and then put it in between $\aleph$ and $\Omega$ where the level of $\aleph$ increases.  If in some sense we can ``leave $\hat \Phi$ behind'' when evolving through $\varnothing$, possibly by folding it into the coefficient of a Fourier transform, then we have a good clue about how to restore unitarity after the non-unitary MCM operations: use modified Fourier coefficients.}
	\label{fig:oldMMM}
\end{figure}


Why is being able to represent the non-smooth topology on a smooth affine parameter important?  It is important because affine parameters are the best way to convert a diagram's lines into equations.  No matter how complex a given curve is, be it geodesic or field line, once the curve is known it is possible to parameterize the curve in a way that makes it trivial.  There are other, more complex parameterizations but the simplest one uses an affine parameter.  A good definition for an affine parameter $\lambda$ is that it is a linear rescaling of the proper time $\tau\rightarrow\lambda=a\tau+b$ that would be experienced by an observer traveling along the path.  Once we have constructed $\chi^5$ as in equation (\ref{eq:chi5notafff}) we can put an affine parameter on that construction and retain the label $\chi^5$.  $\chi^5$ is the parameter that allows us to represent the MCM unit cell in a smoothly connected diagram.

Why are diagrams important?  As we pointed out in reference \cite {DE}, ``Diagrams are used in physics to transmit information with a clarity that is not present in excessively quantified arguments.''  Diagrams are important because they allow us to visualize the degrees of freedom of a system in a way that lets us easily write down the kinetic and potential energy functions, which then let us formulate the Lagrangian or Hamiltonian.  Thereafter, known methods are applied to derive a solution to either the Euler-Lagrange equation or the action principle and equations of motion result.  For instance, while we have spent a lot of time developing the diagrammatic representation which led to the principle of most action, other researchers may have neglected the diagrammatic representation in favor of some algebraic descriptions of their models and then become bogged down trying to squeeze the theory onto a software module that only computes the path of least action.

It is quite popular in the contemporary physics scene to assume a Hamiltonian without first sketching out a system's diagram.  We completely shy away from this on philosophical grounds.  This is why we have never made any statements like, ``Let there be a diagonal Hamiltonian matrix operator,'' and it is why we have not considered arbitrarily specific objects like the Einstein--Cartan--Kibble--Schiama action or similar.  The TOIC represents only one single point in parameter space where the normal analysis is to consider gauge theories as entire equivalence classes, or broad hypersurfaces in parameter space.  By guessing random Hamiltonians, the idea is that the guesses should intersect these planar equivalence classes in phase space.  However, the TOIC with its infinitely precise irrational coefficients is pointlike and singular in parameter space so it is highly unlikely be intersected in this way.  It must be developed from the diagram.  Furthermore, many classical methods for probing theoretical parameter space only use one kind of time and therefore could never find the parameters of a two time model.

In the course of doing physics, only when we have developed the diagrams enough to extract one intuitive, unambiguous Hamiltonian should we begin to compute the equations of motion.  The diagrammatic aspect is the infamous ``underlying conceptual component'' \cite{DE} whose illusory nature is currently the main bottleneck preventing progress in theoretical physics.  Diagrams have been among the most important contributions of the TOIC and we have been lucky enough to uncover several irrefutable non-diagrammatic quantitative results as well.  However, we still have not written the MCM Hamiltonian so that will be an important step to return to in the future.  The purpose of this research has been to develop a method by which one could calculate a new Hamiltonian that has remained elusive to very many other researchers that are looking for it.  While Mathematica or Matlab are happy to give the equations of motion for a given Hamiltonian, we philosophers of Nature, we physicists, should only be happy with equations of motion that come from a diagram so guessing random Hamiltonians can never lead to true happiness.

Reference \cite{MANN} lists five general categories for theories of gravitation.  Mann writes, ``By setting various conditions on [\textit{the manifold}] M and choosing an appropriate Lagrangian $\mathcal{L}$ we can construct a variety of theories of gravity.''  They are: general relativity, torsion theories, Kaluza--Klein theories, supergravity theories, and algebraically extended theories.  Mann defines the last as follows \cite{MANN}.

\begin{quotation}
    \noindent ``[\textit{Algebraically extended theories are}] theories in which all geometric objects take their values in an algebra $\mathcal{A}$, instead of the real numbers $\mathbb{R}$.  [\textit{sic}]  The spacetime manifold M is still real; by requiring the reality of physically measurable quantities on M it has been shown that the only allowable algebras which may be introduced are the real numbers (general relativity), the complex and hypercomplex numbers and the quaternions and hyperquaternions.''\footnote{Mann goes on to make an aside about how the theory based on hypercomplex numbers is testable in the solar system, but those hypercomplex numbers, as mentioned in an earlier footnote, are likely not $^\star\mathbb{C}$.}  \newline
\end{quotation}



The MCM combines aspects of general relativity, torsion theories, Kaluza--Klein theories, and algebraically extended theories.  Infinite complexity is too much complexity to consider and luckily there is a simplifying constraint built into Kaluza-Klein theory.  The 5D Ricci tensor $R_{AB}$ has to vanish at all times.  This severely restricts the complexity of the bulk hyperspacetime in $\Sigma^\pm$, but it still allows rather complex solutions to the 5D metric because gravitational radiation is a non-trivial solution to $R_{AB}=0$.  We will therefore not be constrained to say that the bulk hyperspacetime is a void; we are only constrained to say that the 5D Ricci tensor $R_{AB}$ always vanishes and the hyperspacetime bulk is void of any points where the wavefunction collapses.  (By definition collapse inducing measurements happen in $\mathcal{H}$ and never in the bulk.)  The infinite number of $\mathcal{H}$-branes in the hypercosmos can imply a physical multiverse, or there is a more philosophically untouchable interpretation when the hypercosmos is only a mathematical potential.  It can be the observer's non-physical tool for making abstract calculations to derive observable predictions in spacetime.


 \begin{figure}[t]
 	\makebox[\textwidth][c]{
 		\includegraphics[scale=.9]{Downhill.png}}
 	\captionsetup{format=hang}\caption{The hypercomplex gravitational potential $\bar B$ that replicates the motion along hypercomplex geodesics, if it exists when $m=n+1$, will be a function of potentially very many variables.}
 	\label{fig:hyperdown}
 \end{figure}

Figure \ref{fig:hyperdown} shows the expected gravitational potential energy curve between adjacent levels of $\aleph$ $\hat{\Phi}^n$ and $\hat{\Phi}^{m}$.  As levels of $\aleph$ increase in the future, the arrow of time points strictly in the direction of increasing magnitude in $^\star\mathbb{C}$, which is the normal downhill energy condition seen in figure \ref{fig:hyperdown}.  Downhill leads toward the infinitely greater mass-energy that exists in the future under the assumptions of the MCM.  We can say that the gravitational potential energy landscape in the time direction is completely determined by the object in the future because the gravitational amplitude of the object in the past, being on a lower level of $\aleph$, is on the order of an infinitesimal with respect to the future object.  However, since the future is out there at future infinity, it will be difficult to make the direct extension of the Newtonian potential from spatial dependence to temporal as


\begin{equation}\label{eq:POT2222}
\phi(r)=-\dfrac{GM}{r}\quad\qquad\longrightarrow\quad\qquad\phi'(\chi^5)=-\dfrac{GM}{\chi^5}~~.
\end{equation}\newline


\noindent If an object is infinitely far away, or at least approximately infinitely far away then $\phi'(\chi^5)$ gives $\lim_{\chi^5\to\infty} \Phi(r)=0$ but if we use hypercomplex numbers to get $M>\infty$ there exists a likely workaround involving infinite mass divided by infinite distance.  We know the Newtonian gravitational potential describes a limit of general relativity but there is no guarantee that a new scalar potential function can represent hypercomplex gravitation.  On the other hand we would be remiss not to at least look for the case where it is possible to encode the entire thing on a Newtonian potential which could then later be shown to be a limit of the the MCM but here we will work in the post-Newtonian tensor language.  In chapter four we will build a quantum analogue of the energy curve in figure \ref{fig:hyperdown}.  The notion will be to assemble a periodic array of $\delta$-valued energy wells $\mathcal{H}_n$ whose infinite depths monotonically increase as tiers of infinitude in $^\star\mathbb{C}$ and then take the limit of infinitely many discrete wells to construct the hypercomplex version of the continuous $1/r$ gravitational potential well.

Now we will examine the metric in the bulk of the MCM unit cell.  In reference \cite{BAILLOVE} Bailin and Love report the Kaluza-Klein metric as follows and once again we change certain variable names for global consistency.

\begin{quotation}
    ``We adopt coordinates $\tilde \chi^A$, with $A=1,...,5$ with

    \begin{equation}\label{eq:KKBL1}
    \tilde\chi^\mu=\chi^\mu\qquad\quad \mu=0,1,2,3
    \end{equation}

    \noindent being coordinates for ordinary four-dimensional spacetime, and

    \begin{equation}
    \tilde\chi^5=\theta
    \end{equation}

    \noindent being an angle to parameterize the compact dimension with the geometry of a circle.\footnote{This is what we have suggested by mapping Minkowski space onto a cylinder \cite{MOD}.  While we will refer to our 5D metric as the Kaluza--Klein metric, the KK model is the one with the compactified circular topology of the fifth dimension.  In the MCM, the fifth dimension has flat Euclidean topology so it is not the ordinary KK model but since the KK metric is consistent with the MCM/TOIC we will still speak of the KK metric.  Rather than a fifth dimension compactified on a tiny circle or spiral, the MCM compactifies the 4D gravitational manifold $\mathcal{H}$ as a hypercomplex infinitesimal in an unending, monotonic march of infinitude in $^\star\mathbb{C}$ where some hypercomplex normalization restores finite unitarity at the end of each unit cell.}  The ground-state metric after compactification is

    \begin{equation}
    \tilde \Sigma_{AB}=\mathrm{diag}\{\eta_{\mu\nu},-\tilde \Sigma_{55}\}
    \end{equation}

    \noindent where

    \begin{equation}
    \eta_{\mu\nu}=(1,-1,-1,-1)
    \end{equation}

    \noindent is the metric of Minkowski space, $M_4$, and

    \begin{equation}\label{eq:KKBL2}
    \tilde \Sigma_{55}=\tilde R^2
    \end{equation}

    \noindent is the metric of the compact manifold $\mathbb{S}^1$, where $\tilde R$ is the radius of the circle.\footnote{Dolce has extensively documented how the dynamical topological radius can be used to scale gauge theories.  We we refer the reader to reference \cite{DOLCE}.}

    ``The identification of the gauge field arises from an expansion of the metric about the ground state.  Quite generally, we may parameterize the metric in the form

    \begin{equation}
    \tilde \Sigma(\chi,\theta)=
    \left(
    \begin{matrix}
    g_{\mu\nu}(\chi,\theta)-B_\mu(\chi,\theta) B_\nu(\chi,\theta)\phi(\chi,\theta)\quad&\quad B_\mu(\chi,\theta)\phi(\chi,\theta)\\
    &~\\
    B_\nu(\chi,\theta) \phi(\chi,\theta)\quad&\quad -\phi(\chi,\theta)
    \end{matrix}
    \right)~~.
    \end{equation}

    \noindent To extract the graviton and the Abelian gauge field theories it proves sufficient to replace $\phi(\chi,\theta)$ by its ground-state value $\tilde \Sigma_{55}$, and to use the ansatz without $\theta$ dependence:

    \begin{equation}
    \tilde \Sigma(\chi)=
    \left(
    \begin{matrix}
    g_{\mu\nu}(\chi)-B_\mu(\chi) B_\nu(\chi)\tilde\Sigma_{55}\quad&\quad B_\mu(\chi)\tilde\Sigma_{55}\\
    &~\\
    B_\nu(\chi) \tilde\Sigma_{55}\quad&\quad -\tilde\Sigma_{55}
    \end{matrix}
    \right)~~.
    \end{equation}

    \noindent We write

    \begin{equation}
    B_\mu(x)=\xi A_\mu(\chi)
    \end{equation}

    \noindent where $\xi$ is a scale factor we shall choose later so that $A_\mu(\chi)$ is a conventionally normalized gauge field.''\newline
\end{quotation}

We see unification between classical gravity and classical electromagnetism beginning to occur when the electromagnetic potential appears in the KK metric.  However, we know KK theory doesn't work by itself (without MCM modifications \cite{GC}.)  Furthermore, the Newtonian gravitational potential is a scalar field whereas the the electromagnetic potential transforms as a 4-vector so there is some fundamental imbalance of the formalism that remains to be corrected through unification.

Where GR calculates geodesics, EM calculates field lines and we need to unify these approaches.  To that end, consider what Zangwill\footnote{Zangwill is credited, and in fact lauded, as the person who first explained the Abraham--Lorentz force to this writer along with its anomalous reliance on the  $\partial_t^{\,3}$ operator.} says about chaotic magnetic field lines in reference \cite{ZANG}.

\begin{quotation}
    ``The complexity of a large class of magnetic field line configurations can be appreciated using a field constructed from a constant $B_0$ and an arbitrary scalar function $f(\mathbf{r})$:

    \begin{equation}
    \mathbf{B}(\mathbf{r})=B_0\hat z+\hat z\times\nabla f(\mathbf{r})~~.
    \end{equation}

    \noindent The field satisfies $\nabla\cdot\mathbf{B}=0$ by construction. [\textit{sic}]  The equation for the field lines is

    \begin{equation}
    \frac{B_x}{dx}=\frac{B_y}{dy}=\frac{B_z}{dz}=\lambda
    \end{equation}

    \noindent or

    \begin{equation}\label{eq:preH1}
    \frac{dx}{dz}=-\frac{1}{B_0}\frac{\partial f}{\partial y}\quad\qquad\mathrm{and}\quad\qquad\frac{dy}{dz}=+\frac{1}{B_0}\frac{\partial f}{\partial x}~~.
    \end{equation}

    \noindent Now, change the variables in [\textit{equation (\ref{eq:preH1})}] so $x=q$, $y=p$, and $z=t$.  If, in addition, we let $f=-B_0H$, the two equations above are exactly Hamilton's equations of classical mechanics,

    \begin{equation}
    \dot q=\frac{\partial H}{\partial p}\quad\qquad\mathrm{and}\quad\qquad \dot p=-\frac{\partial H}{\partial q}
    \end{equation}

    \noindent Therefore the magnetic field lines are the `time'-dependent trajectories in $(q,p)$ phase space of a `particle' with Hamiltonian $H=-f/B_0$.  Since most Hamiltonians are non-integrable and produce chaotic trajectories,\footnote{We expect the simplest MCM Hamiltonian to be of the non-integrable variety but after the hypercomplex formalism is fully developed it may be simpler than expected.} the magnetic field line configuration will be very complex indeed.''\newline
\end{quotation}

In GR we can derive a a set of differential equations in Hamiltonian form whose solutions are geodesics.  Zangwill has shown how field lines can be derived in the same way and yet at some level the two approached to physics, geodesics and field lines, are not compatible.  EM field lines can be derived as a limit of a quantum theory but geodesics cannot.  The main push of the MCM is to identify and resolve these outstanding discrepancies.  Luckily this research program has no aim to calculate any field lines or non-trivial geodesics, or to calculate an expectation value or anything like that.  Where we have seen other researchers stuck with conceptual difficulties, the main contribution of this research program to the total body of human knowledge has been to suggest to those other researchers, ``Try doing it this other way.''  The difficult problem was not to make the other calculation, the difficult part was to \textit{suggest} the other calculation which we have now identified as computing the path of maximum action.  Before we can do that we must first develop the field of hypercomplex analysis in $^\star\mathbb{C}$ and chapter four deals with related topics.

If someone wants to see a calculation that does not appear in this research, they should make it themselves or ask someone to do it for them.  Detractors can continue to harrumph that they would have used a computer to solve some equations and then copied their software's output into LaTeX, and then sent their paper to a publisher other than viXra, but waiting for this writer to abandon his research in fundamental physics in favor of some problem in applied physics will not prove fruitful.  We do not mean to disparage applied physics in any way.  Rather our aim is to, however unlikely, get one or more detractors to finally clear the conceptual hurdle where they begin to understand that this writer's research program is his own and not his detractors', that it is already experimentally verified, and that said detractors have received the new theory stupidly.  How can a body of research be both ``experimentally verified,'' as this research is, and also ``pending peer-review'' as this research is not?  The widespread desire to ignore or obfuscate this issue pending a change in this writer's research direction will be a dire mark of shame on the history of the professional conduct of science by supposedly reasonable men.

To move forward with the present considerations in classical physics, consider Carroll's further words from reference \cite{CARR}.

\begin{quotation}
    ``As we shall see, the metric tensor contains all the information we need to describe the curvature of the manifold (at least in what is called Riemannian geometry [\emph{sic}]).  In Minkowski space we can choose coordinates in which the components of the metric are constant; but it should be clear that the existence of curvature is more subtle than having the metric depend on the coordinates [\emph{sic}].  Later, we shall see that the constancy of the metric components is sufficient for a space to be flat, and in fact there always exists a coordinate system on any flat space in which the metric is constant.  But we might not know how to find such a coordinate system, and there are many ways for a space to deviate from flatness; we will therefore want a more precise characterization of the curvature [\emph{sic}].

    ``A useful characterization of the metric is obtained by putting $g_{\mu\nu}$ into its \textbf{canonical form}.  In this form the metric components become

    \begin{equation}\label{eq:diag762}
    g_{\mu\nu}=\mathrm{diag}(-1,-1,...,-1,+1,+1,...,+1,0,0,...,0)~~,
    \end{equation}

    \noindent where [\emph{equation (\ref{eq:diag762})}] means a diagonal matrix with the given elements.  The \textbf{signature} of the metric is the number of both positive and negative eigenvalues; we speak of ``a metric with signature minus-plus-plus-plus'' for Minkowski space, for example.  If any of the eigenvalues are zero, the metric is ``degenerate,'' and the inverse metric will not exist; if the metric is continuous and nondegenerate, its signature will be the same at every point.  We will always deal with continuous nondegenerate metrics\footnote{We, however, cannot always do this because the MCM has piecewise discontinuous and degenerate metrics.}.  If all of the signs are positive, the metric is called \textbf{Euclidean} or \textbf{Riemannian} (or just positive definite), while if there is a single minus sign it is called \textbf{Lorentzian} or \textbf{pseudo-Riemannian}, and any metric with some $+1$'s and some $-1$'s is called indefinite.  [\emph{sic}]  The spacetimes of interest in general relativity typically have Lorentzian metrics.

    ``We have not yet demonstrated how it is always possible to put the metric in canonical form.  In fact it is always possible to do so at some point $p\in M$, but in general it will only be possible at that single point, not in any neighborhood of $p$.  Actually we can do slightly better than this; it turns out that at any point $p$ there exists a coordinate system $x^{\hat{\mu}}$ in which $g_{\hat{\mu}\hat{\nu}}$ takes its canonical form and the first derivatives $\partial_{\hat{\sigma}}g_{\hat{\mu}\hat{\nu}}$ all vanish (while the second derivatives $\partial_{\hat{\rho}}\partial_{\hat{\sigma}}g_{\hat{\mu}\hat{\nu}}$ cannot be made to all vanish):

    \begin{equation}
    g_{\hat{\mu}\hat{\nu}}(p)=\eta_{\hat{\mu}\hat{\nu}}~~,\qquad \partial_{\hat{\sigma}}g_{\hat{\mu}\hat{\nu}}(p)=0~~.
    \end{equation}

    ``\noindent Such coordinates are known as \textbf{locally inertial coordinates}, and the associated basis vectors constitute a \textbf{local Lorentz frame}; we often put hats on the indices when we are in these special coordinates.  Notice that in locally inertial coordinates the metric at $p$ looks like that of flat space to first order.  This is the rigorous notion of the idea that ``small enough regions of spacetime look like flat (Minkowski) space.''  Also, there is no difficulty in simultaneously constructing sets of \emph{basis vectors} at every point in $M$ such that the metric takes its canonical form, the problem is that in general there will not be a \emph{coordinate system} from which this basis can be derived. [\textit{sic}]

    ``The idea is to consider the transformation law for the metric

    \begin{equation}\label{eq:CARReq}
    g_{\hat{\mu}\hat{\nu}}=\dfrac{\partial x^\mu}{\partial x^{\hat{\mu}}}\dfrac{\partial x^\nu}{\partial x^{\hat{\nu}}}g_{\mu\nu}~~,
    \end{equation}

    \noindent and expand both sides in Taylor series in the sought after coordinates $x^{\hat{\mu}}$.  The expansion of the old coordinates $x^\mu$ looks like

    \begin{align}\label{eq:CARReq22}
    x^\mu=\left(\dfrac{\partial x^\mu}{\partial x^{\hat{\nu}}}\right)_p\! x^\mu&+\dfrac{1}{2}\left( \dfrac{\partial^2 x^\mu}{\partial x^{\hat{\mu}_1}\,\partial x^{\hat{\mu}_2}}\right)_p\!x^{\mu_1}x^{\mu_2}
    \\&~~~~~~+\dfrac{1}{6}\left( \dfrac{\partial^3 x^\mu}{\partial x^{\hat{\mu}_1}\,\partial x^{\hat{\mu}_2}\,\partial x^{\hat{\mu}_3}}\right)_p\! x^{\mu_1}x^{\mu_2}x^{\mu_3}+...~~,\nonumber
    \end{align}

    \noindent with the other expansions proceeding along the same lines.  For simplicity we have set $x^\mu(p)=x^{\hat{\mu}}(p)=0$.  Then, using some extremely schematic notation the expansion of [(\ref{eq:CARReq})] to second order is

    \begin{align}\label{eq:CARReq2}
     (\hat g)_p&+(\hat{\partial}\hat g)_p\hat x+(\hat{\partial}\hat{\partial}\hat g)_p\hat x\hat x\nonumber\\
     &~\nonumber\\
     &=\left( \frac{\partial x}{\partial \hat x}\frac{\partial x}{\partial \hat x}g\right)_p+\left( \frac{\partial x}{\partial \hat x}\frac{\partial^2 x}{\partial \hat x\,\partial \hat x}g+\frac{\partial x}{\partial \hat x}\frac{\partial x}{\partial \hat x}\hat\partial g\right)_p\!\hat x\\
     &~\nonumber\\
     &~~~~~~
     +\left( \frac{\partial x}{\partial \hat x}
        \frac{\partial^3 x}{\partial \hat x\,\partial \hat x\,\partial \hat x}g
             +\frac{\partial^2 x}{\partial \hat x\,\partial \hat x}
             \frac{\partial^2 x}{\partial \hat x\,\partial \hat x} g
             +\frac{\partial x}{\partial \hat x}
             \frac{\partial^2 x}{\partial \hat x\,\partial \hat x}\hat\partial g  +\frac{\partial x}{\partial \hat x}
             \frac{\partial x}{\partial \hat x}\hat\partial\hat\partial g \right)_p\! \hat x\hat x\nonumber
    \end{align}

    ``\noindent We can set terms of equal order in $\hat x$ on each side equal to each other.  Therefore, the components $g_{\hat{\mu}\hat{\nu}}(p)$, 10 numbers in all (to describe a two-index tensor\footnote{This is true for a symmetric two-index tensor.}), are determined by the matrix $(\partial x^\mu/\partial x^{\hat{\mu}})_p$.  This is a $4\times4$ matrix with no constraints; thus, we are free to choose 16 numbers.  Clearly this is enough freedom to put the 10 numbers of $g_{\hat{\mu}\hat{\nu}}(p)$ into canonical form, at least as far as having enough degrees of freedom is concerned.  (In fact there are some limitations--if you go through the procedure carefully, you find for example that you cannot change the signature.)  The six remaining degrees of freedom can be interpreted as exactly the six parameters of the Lorentz group; we know that these leave the canonical form unchanged.  At first order we have the derivatives $\partial_{\hat{\sigma}}g_{\hat{\mu}\hat{\nu}}(p)$, four derivatives of ten components for a total of 40 numbers.  But looking at the right hand side of [\textit{equation (\ref{eq:CARReq2})}]\footnote{We will discuss equations (\ref{eq:CARReq22}) and (\ref{eq:CARReq2}) in chapter 4.} we see that we now have additional freedom to choose $(\partial^2 x^\mu/\partial x^{\hat{\mu}_1}\partial x^{\hat{\mu}_2})_p$.  In this set of numbers there are ten independent choices of the indices $\hat{\mu}_1$ and $\hat{\mu}_2$ (it's symmetric since partial derivatives commute) and four choices of $\mu$ for a total of 40 degrees of freedom.  This is precisely the number of choices we need to determine all of the first derivatives of the metric, which we can therefore set to zero.  At second order, however, we are concerned with $\partial_{\hat{\rho}}\partial_{\hat{\sigma}}g_{\hat{\mu}\hat{\nu}}(p)$; this is symmetric in $\hat{\rho}$ and $\hat{\sigma}$ as well as $\hat{\mu}$ and $\hat{\nu}$, for a total of $10\times10$ numbers. Our ability to make additional choices is contained in $(\partial^3 x^\mu/\partial x^{\hat{\mu}_1}\partial x^{\hat{\mu}_2}\partial x^{\hat{\mu}_3})_p$ [\textit{\textbf{!!!}}].  This is symmetric in the three lower indices, which gives 20 possibilities, times four for the upper index gives us 80 degrees of freedom--20 fewer than we require to set the second derivatives of the metric to zero.  So in fact we cannot make the second derivative vanish; the deviation from flatness must therefore be measured by the 20 degrees of freedom representing the second derivatives of the metric tensor field.  We will see later how this comes about, when we characterize the curvature using the Riemann tensor which will turn out to have 20 independent components in four dimensions.

    ``Locally inertial coordinates are unbelievably useful.  Best of all, their usefulness does not generally require that we do the work of constructing such coordinates [\emph{sic}], but simply that we know they do exist\footnote{It is this line of reasoning that leads us to believe it will be possible to make a vastly simplified calculation of trajectories across the MCM unit cell whose various coordinate systems might be very difficult to construct.  Figure \ref{PHImagic2} gives an idea of what the simplified calculation might look like.}.''\newline
\end{quotation}

Wow!  Rather than using tensor language for all of that we can use commas to separate all the items needed to make a calculation, put them in curly brackets, and call it a multiplex.  Is the best way to keep track of the 20 curvature components of the Riemann tensor really in an object that transforms like a tensor?  Why not use a $4\times5$ matrix, or a $20\times1$ matrix that more directly represents the idea of a multiplex which we will define in section III.5?  Quantum mechanics does fine with matrices instead of tensors so we know that in some sense there exists an analytical channel for physics other than tensor analysis on manifolds.  There are already hundreds of multiplectic positions needed just to cover tensor analysis in Lorentzian 4-space so two copies of the 14D MCM system (the requirement for two copies of the system appears in reference \cite{IC}) will need many thousands of positions to describe the general relativity of one MCM qubit.  Due to the sphere theorem and well-known methods of Ricci flow, the whole thing boils down to large systems of partial differential equations but report format is not the correct venue in which to describe possibly hundreds of thousands or millions of constraint equations that would rigorously demonstrate how the curvature is related to the quantized electric current.  Whoever has the facility to make those calculations should do so and upload his result file to a public mirror such as Wikipedia.  Instead of specific constraint equations we have a powerful argument that goes something like, ``You see how these two things connect?  There is only one way that we can pull an answer out of these objects.''  As an example consider the TOIC operator $\partial^3/\partial \chi^5_+\,\partial \chi^5_\varnothing\, \partial\chi^5_-$ \cite{IC} and how, among all the objects in GR, it can only connect to the $\partial^3/\partial x^{\hat{\mu}_1}\,\partial x^{\hat{\mu}_2}\,\partial x^{\hat{\mu}_3}$ operator that appears in equation (\ref{eq:CARReq22}).  This is the object Carroll identifies as containing ``our ability to make additional choices.''

To avoid confusion with very many other hats we will not use hats to specify locally inertial coordinates.  It will be the convention that $\{x^\mu_-,x^\mu,x^\mu_+\}$ are always locally inertial coordinates on $\{\aleph,\mathcal{H},\Omega\}$ and we have a lot of freedom to use $x^\mu_{\varnothing}$ as required.  The reader must take extreme care to note that Carroll's setting $\partial_\sigma g_{\mu\nu}=0$ at the origin to reduce the complexity of the equations is what we have referred to as \textbf{the MCM condition}.  The MCM condition states that the observer is always at the origin and any coordinate system that places him elsewhere will make the TOIC contraption falter.  When evolution moves the observer from $t_i=0$ to $t_{i+1}\neq0$, that is simply $t_{i+1}'=0$ in the coordinates of the time advanced manifold.



\subsection{General Relativity}

We have firmly established the requirement for four ontological vectors $\{\hat2,\hat{\pi},\hat{\Phi},\hat{i}\}$ \cite{QG,ICT,IC} as opposed to the three-fold subset $\{\hat{\pi},\hat{\Phi},\hat{i}\}$ proposed in reference \cite{TER}.  Therefore, we shall consider Einstein's equation resolved into four components

\begin{equation}\label{eq:preTzOIC}
8\pi T_{\mu\nu}~=~R_{\mu\nu}-\frac{1}{2}Rg_{\mu\nu}+\Lambda g_{\mu\nu}~~,
\end{equation}\newline

\noindent and how they connect to $\{\mathcal{H},\aleph,\varnothing,\Omega\}$ via something along the lines of

\begin{equation}\label{eq:TOICeq}
8\pi
f^3\big|\psi;\hat{\pi}\big\rangle~=~\frac{\Phi}{4}|\psi;\hat2\big\rangle-\frac{1}{2}|\psi;\hat{\Phi}\big\rangle+i\big|\psi;\hat{i}\big\rangle~~.
\end{equation}\newline

\noindent The reader should note that we have chosen this arbitrary position for $\hat 2$ only to show matching  between equations (\ref{eq:preTzOIC}) and (\ref{eq:TOICeq}) on the $1/2$ coefficient and the  $8\pi$ coefficient.  This begins to show that we might assign the geometry $R_{\mu\sigma\nu}^\lambda$ to the $\hat i$ component and then use the fundamental incompatibility between irrational $\pi$ and maximally irrational $\Phi$ to establish the computational topology of the topological incompatibility between dS and AdS.  When apply the symmetry operator that replaces $\pi/2\approx1.57$ with $\Phi\approx1.62$ all of the periodicity in $2\pi$ vanishes when the trigonometric functions are represented as distinct infinite series that are assumed to all have an even or odd number of infinity terms.  It is the fundamental property of sine that it is an even function and the fundamental property of cosine that it is an odd function so we have good reason to consider the case when the topology must be such that it can accomodate a remainder term after operations pairing two even infinite numbers of terms.  This is often neglected in practice because this idea is programmed into modern software that does not account for the analytical component of the remainder.  In this way we may represent two infinities in $\Sigma^\pm$ and then use the remainder to define something in $\mathcal{H}$ or $\varnothing$.  We will return to this issue in chapter four.  Evidently this newish $\hat 2$ component corresponds to $\Sigma^\varnothing$.  The original depiction of the hypercosmological neighborhood around one instance of $\mathcal{H}$ \cite{GC} had $\Sigma^\varnothing$ all around the outside and figure \ref{fig:unitcell2} shows it in the center to emphasize that there is still a missing component.  Whatever the connection between the four tensors and the four manifolds, it is constrained to connect to the four vector spaces $\{\mathcal{H}',\{\aleph',\varnothing',\Omega'\}\}$ where we have put $\varnothing'$ in the Gelfand triple with the understanding that Hilbert space is its own dual space so $\varnothing'\equiv\mathcal{H}'$.  In general we will refer to $\mathcal{H}'$ but not $\varnothing'$ because it is totally redundant other than to show that there are four vector spaces.

Previous work developing the TOIC has drifted in the direction of quantum theory from its cosmological roots as the MCM and here we swing back in the other direction toward the physics of the continuum described (with random assignments) by

\begin{align}
f^3\big|\psi;\hat\pi\big\rangle&~~\mapsto~~ T_{\mu\nu}\label{eq:GR1}\\
&~\nonumber\\
\big|\psi;\hat\Phi\big\rangle&~~\mapsto~~ G_{\mu\nu}\label{eq:GR2}\\
&~\nonumber\\
i\big|\psi;\hat i\big\rangle&~~\mapsto~~ g_{\mu\nu}\Lambda~~.\label{eq:GR3}
\end{align}\newline

\noindent Various iterations of these maps are derived in reference \cite{FSC,TER,KN,QG,IC}.  See appendix A for a synopsis of previous work.  In this section we will examine what needs to be done to split the Einstein tensor into the components that appear in equation (\ref{eq:preTzOIC}).  Throughout this book we will ignore the signs of terms when they seem irrelevant and as justification we refer the reader to the as yet undiscussed channel for $\sqrt{i}$ (or $\sqrt[n]{i}$) which can be added later to generate an arbitrary sign convention.  We continue to choose arbitrary matching for objects rather than represent them in the most general fashion.  For instance, the $\hat 2$ term in equation (\ref{eq:TOICeq}) could be the one that connects to $T_{\mu\nu}$ or any of the tensors in Einstein's equation but we choose a particular (and reasonable) ansatz purely for convenience.  Even the selection of the spherical and hyperbolic spaces $\Omega$ and $\aleph$ as belonging to the future or past was arbitrary.  Likewise the assignment of the members of our Gelfand triple $\aleph'$ and $\Omega'$ to either dS or AdS manifolds was arbitrary.  Such ansatzes are fine and if something about equations (\ref{eq:GR1}-\ref{eq:GR3}) is found to be non-optimal in the final analysis then later we can switch it or perhaps we will complement the work of Benjamin Franklin and end up with an electric current vector that does not point in the opposite direction to the motion of electrons.  It may even be that the other form of Einstein's equation $R_{\mu\nu}=T_{\mu\nu}-\frac{1}{2}Tg_{\mu\nu}$ is the one that shows the most natural mapping of the objects in maps (\ref{eq:GR1}-\ref{eq:GR3}).

Throughout this research program we have made many arbitrary choices when they make it easier to analyze an idea than would be possible with a fully rigorous but overly general case.  Reasonable choices have consistently pointed the way to better choices such as the replacement of the original basis vector $\hat{\varphi}$ \cite{DE} with $\hat{\Phi}$.  In this book we use a different $\hat \varphi$ and the reader should understand that this is a completely new object unrelated to the original notation for what we now call $\hat \Phi$.  By choosing three ontological vectors we discovered that three are not enough to do all of the heavy lifting and now there are four objects in the ontological basis.  Since $\hat2$ now appears along side $\{\hat{\pi},\hat{\Phi},\hat i\}$ we should examine what role it might play in relating the quantum and gravitational sectors as in equations (\ref{eq:GR1}-\ref{eq:GR3}).  To that end, consider the Einstein tensor

\begin{equation}
G_{\mu\nu}\quad\equiv\quad R_{\mu\nu}-\frac{1}{2}Rg_{\mu\nu}~~,
\end{equation}\newline

\noindent where we modify equation (\ref{eq:GR2}) with $1=1/2+1/2$ and $\hat\Phi=\Phi/2~\hat2$ as

\begin{equation}\label{eq:eq762}
\dfrac{1}{2}\big|\psi;\hat\Phi\big\rangle+\frac{\Phi}{4}\big|\psi;\hat2\big\rangle~~\mapsto~~ R_{\mu\nu}-\frac{1}{2}Rg_{\mu\nu}~~.
\end{equation}\newline

\noindent There are multiple ways to assign the components of $G_{\mu\nu}$ because the expanded map is from two objects to two objects.  The first obvious one-to-one possibility of non-mixed linear dependence is

\begin{equation}\label{eq:Rconst1}
\frac{1}{2}\big|\psi;\hat\Phi\big\rangle~~\mapsto~~ R_{\mu\nu}~~,
\qquad\quad\mathrm{and}\qquad\quad
\frac{\Phi}{4}\big|\psi;\hat2\big\rangle~~\mapsto~~ \frac{1}{2}Rg_{\mu\nu}~~,
\end{equation}\newline

\noindent which gives $R=\Phi/2$ and the second is

\begin{equation}\label{eq:Rconst2}
\frac{1}{2}\big|\psi;\hat\Phi\big\rangle~~\mapsto~~ \frac{1}{2}Rg_{\mu\nu}~~,
\qquad\quad\mathrm{and}\qquad\quad
\frac{\Phi}{4}\big|\psi;\hat2\big\rangle~~\mapsto~~ R_{\mu\nu}~~,
\end{equation}\newline

\noindent which gives $R=1$.  Either way we obtain a nice constraint on the Ricci tensor $R_{\mu\nu}$.

In reference \cite{GC} we connected the curvature of the cosmos to the golden ratio by defining embedded hyperboloids $\{\aleph,\Omega\}$ in 5D hyperspacetime with curvature parameters equal to

\begin{equation}
\Phi=\frac{1+\sqrt{5}}{2}\approx1.62~~,\quad\qquad\mathrm{and}\quad\qquad\varphi=\frac{1-\sqrt{5}}{2}\approx-0.62~~.
\end{equation}\newline

\noindent The reader should be aware of this odd notation that $\varphi$ is a negative number, or what Gauss would have called an inverse number.  We initially defined hyperboloids directly in the 5D bulk \cite{GC} but now we take flat slices of bulk in the $\chi^A$ coordinates and then impose the hyperboloidal geometry for the $x^\mu_\pm$ coordinates with a 4D embedded metric $g_{\mu\nu}^\pm$ on the flat slice.  The embedded metric is assigned by the MCM boundary condition so it is not inherited from the 5D metric.  The slice stays flat in the 4D metric $\Sigma^\pm_{\alpha\beta}$ that is inherited from the 5D metric $\Sigma^\pm_{AB}$.

We already have two places where the Riemann tensor $R^\sigma_{\,\mu\lambda\nu}$ and the golden ratio $\Phi$ connect: the first is in the embedded hyperboloid constraints

\begin{equation}\label{eq:DoubleNEG}
\Phi^2=-\big(x^0_+\big)^2+\sum_{i=1}^{3}\big(x^i_+\big)^2~~,\quad\qquad\mathrm{and}\quad\qquad-\varphi^2=-\big(x^0_-\big)^2+\sum_{i=1}^{3}\big(x^i_-\big)^2~~,
\end{equation}\newline

\noindent for the $x^\mu_\pm$ coordinates.  The new second connection is derived from equations (\ref{eq:Rconst1}) and (\ref{eq:Rconst2}) as a constraint on the Ricci scalar

\begin{equation}\label{eq:Rconstrain}
R\in~\left\{1,~\tfrac{\Phi}{2}\right\}~~.
\end{equation}\newline

\noindent The first connection through the hyperboloid constraints is known because the Riemann tensors of dS and AdS are known.  The second new connection is known because the Ricci scalar $R$ might be equal to $\Phi/2$ and it is the double contraction of the Riemann tensor.  When $R=\Phi/2$ as per equation (\ref{eq:Rconstrain}) it becomes the coefficient in the tensor transformation law

\begin{equation}\label{eq:firstTENS}
x^{\mu}_\varnothing=\frac{\Phi}{2}~x^\mu_+~~,
\end{equation}\newline


 ============================

 RECALCULATE AND DEMONSTRATE MORE CLEARLY

 ============================

\noindent associated with a change of coordinates from the $\hat\Phi$-site to the $\hat2$-site (via $\hat\Phi=\Phi/2~\hat2$).  The form $R=\Phi/2$ is very natural looking and it seems reasonable but we have another nice possibility as well.  We may set the Ricci scalar $R$ to unity and let that be a scale factor that enforces unitarity in lieu of the \emph{ad hoc} normalization that is usually applied.  Both choices can be interpreted as an obvious (non-contrived) scale factor so we have good evidence of the \emph{reasonableness} of sticking $\hat2$ into the maps that were already doing fine without it.

Moving on, if we're going to analyze the MCM and TOIC with the methods of general relativity then its objects better satisfy the tensor transformation law.  Mirroring equation (\ref{eq:firstTENS}) the tensor transformation associated with a change of coordinates from $x^\mu$ to $x^\mu_+$ for some vector $A^\mu$ is

\begin{equation}
A^{\mu}=\frac{\partial x^{\mu}}{\partial x^{\mu}_+}A^{\mu}_+~~.
\end{equation}\newline

\noindent  To consider a change of coordinates from $x^\mu$ associated with $\hat{\pi}$ to $x^\mu_+$ associated with $\hat{\Phi}$ let $A^\mu\equiv A\hat{\pi}$ so that

\begin{equation}\label{eq:TENmap}
A^\mu=\left(\begin{matrix}
A\\
0\\
0\\
0\\
\end{matrix}\right)~~,\quad\qquad\text{and}\quad\qquad\hat{\Phi}\,:\,A^\mu\quad\mapsto\quad A^\mu_+~~.
\end{equation}\newline

\noindent By construction we have $\hat{\pi}=-\varphi\pi\hat{\Phi}$ so


\begin{equation}\label{eq:approx2}
x^{\mu}\,\hat \pi=-\varphi\pi x^\mu \,\hat \Phi\quad\qquad\implies\quad\qquad x^\mu_+=-\varphi\pi x^\mu ~~.
\end{equation}\newline

\noindent Therefore

\begin{equation}
\frac{\partial x^{\mu}_+}{\partial x^{\mu}}=-\varphi\pi ~~,
\end{equation}\newline

\noindent and evidently the tensor transformation law is

\begin{equation}
A^{\mu}_+=-\varphi\pi A^\mu~~.
\end{equation}\newline

\noindent   It is good that the $x^\mu$ coordinates of $\hat{\pi}$ have a different scale than the $x^\mu_+$ coordinates in the future  specified by $\hat{\Phi}$.  The change of linear scale should have a direct association with non-unitarity.  Furthermore we should explore what it means for the quantity $-\varphi\pi$ to be inverse (prefaced with a negative sign) but also positive with $-\varphi\pi>0$.

To solve for the new hypercomplex coordinates we have to solve equation (\ref{eq:approx2}).   In the usual way we can write


\begin{equation}\label{eq:implies1}
\partial x^\mu_+=-\varphi\pi\partial x^\mu~~,\quad\qquad\text{and}\quad\qquad
\int\frac{\partial x^\mu_+}{\partial x^\mu}  ~dx^\mu=-\varphi\pi\int~dx^\mu
\end{equation}\newline

\noindent which gives

\begin{equation}\label{eq:implies2}
x^\mu_+=-\varphi\pi x^\mu~~.
\end{equation}\newline



\noindent To solve for $\chi^A$ we set

\begin{equation}\label{eq:CHIdef1}
\chi^\alpha_+=x^\mu_+\quad\qquad\text{so}\quad\qquad\chi^\alpha_+=-\varphi\pi x^\mu~~.
\end{equation}\newline

\noindent Then we simply add flat $\chi^5_+$ and we have an abstract universe from scratch.




The boundary condition given in reference \cite{GC} for determining $\mathcal{H}$'s metric $\eta_{\mu\nu}$ (or equally $g_{\mu\nu}=\eta_{\mu\nu}+h_{\mu\nu}$) was

\begin{equation}\label{eq:limitBC}
\lim_{\chi^5_+\to 0^+} \Sigma^+_{AB}+\lim_{\chi^5_-\to 0^-} \Sigma^-_{AB}=\left(\begin{matrix}
\eta_{\mu\nu}& 0\\
0 & 0
\end{matrix}\right)~~.
\end{equation}\newline

\noindent The RHS of equation (\ref{eq:limitBC}) shows the 5D analogue of the 4D metric in $\mathcal{H}$.  It has a vanishing determinant and only one sign different than the other four so it is degenerate definite and the metrics $\Sigma^\pm_{AB}$ are nondegenerate because $\Sigma^\pm_{55}\in\Sigma^\pm\neq0$.  Among $\Sigma^\pm_{AB}$ one of them is a Lorentzian metric and one is an indefinite metric because if $\chi^5$ is timelike in one of $\Sigma^\pm$ it is necessarily spacelike in the other.  Whichever of $\Sigma^\pm$ has two timelike dimensions is indefinite and the other is Lorentzian.

We chose the definition in equation (\ref{eq:limitBC}) to show that the present needs to be considered a superposition of the past and future which would not have been specified from a simpler boundary condition such as

\begin{equation}\label{eq:276}
\lim_{\chi^5_+\to 0^+} \Sigma^+_{AB}=\lim_{\chi^5_-\to 0^-} \Sigma^-_{AB}.
\end{equation}\newline

\noindent Equation (\ref{eq:276}) does not show that the metric in $\mathcal{H}$ is \emph{always} the flat Minkowski metric $\eta_{\mu\nu}$ but equation (\ref{eq:limitBC}) does show that requirement.  However, if we suppose to define the 5D metric as

\begin{equation}\label{eq:hypereta}
\eta_{AB} \quad\equiv\quad \left(\begin{matrix}
\eta_{\mu\nu}&0\\
0&0\\
\end{matrix}\right)~,
\end{equation} \newline

\noindent then no inverse metric will exist ($\eta_{AB}$ is degenerate) and it will be impossible to compute the connection coefficients according to the ordinary Christoffel prescription.  That formula relies on the inverse metric as in

\begin{equation}\label{eq:connection2}
\Gamma^\lambda_{\mu\nu}=\frac{1}{2}g^{\lambda\sigma}\big(\partial_\mu g_{\nu\sigma}+\partial_\nu g_{\sigma\mu}-\partial_\sigma g_{\mu\nu}\big)~~.
\end{equation}\newline

\noindent  To get the object $\eta_{AB}$ in equation (\ref{eq:limitBC}) into invertible form we can consider


\begin{equation}\label{eq:nonunitmetric}
\lim_{\chi^5_+\to 0^+} g^+_{\mu\nu}+\lim_{\chi^5_-\to 0^-} g^-_{\mu\nu}=
\eta_{\mu\nu}~~,
\end{equation}\newline

\noindent where $g^\pm_{\mu\nu}$ is the non-flat embedded metric on each 4D slice of constant $\chi^5$.  Equation (\ref{eq:nonunitmetric}) shows that the ``size'' or linear scale, of $g^\pm_{\mu\nu}$ is not equal to $\eta_{\mu\nu}$ and this is exactly what we would expect from the tensor transformations shown in the previous section.  This non-uniformity of scale must be associated with the non-unitary component of the MCM.  We found that the chronological coordinates associated with each cosmological lattice site are linearly scaled with respect to each other.  If the scale of different sectors was the same then we would have


\begin{equation}\label{eq:nonunitmetric2}
\dfrac{1}{2}\lim_{\chi^5_+\to 0^+} g^+_{\mu\nu}+\dfrac{1}{2}\lim_{\chi^5_-\to 0^-} g^-_{\mu\nu}=
\eta_{\mu\nu}~~,
\end{equation}\newline

\noindent but this is not the stated relationship.

The inherited metric on each slice of constant $\chi^5$ is $\Sigma_{\alpha\beta}^\pm$ and the embedded metric of the chronological coordinates is $g^\pm_{\mu\nu}$.  The MCM is such that $\Sigma^\pm_{55}\equiv\pm\chi^5_\pm$ so it is easy to see that the embedded metric is defined by setting the parameter of curvature in dS and AdS to the values of $\chi^5_+$ and $\chi^5_-$ on each respective slice of $\chi^5$.  Therefore the embedded metric on each slice of $\chi^5$ has subtle curvature near $\chi^5=0$ and greater curvature where the absolute value of $\chi^5$ is larger.  When we take the limit $\chi^5\to0$, it means curvature approaches zero on each side of $\mathcal{H}$.  The difference of the two metrics will be very small because AdS with infinitesimal negative curvature is approximately indistinguishable from dS with infinitesimal positive curvature.  Note the critical distinction that dS with positive curvature and AdS with negative curvature are topologically irreconcilable but all the flat slices of $\chi^5$ share the same flat topology.

Equation (\ref{eq:nonunitmetric}) shows how the slices of $\Sigma^\pm$ approach a matching condition at $\mathcal{H}$, but we can just as easily put the matching condition at $\Sigma^\varnothing$, as in figure \ref{fig:unitcell2a}.  Then equation (\ref{eq:nonunitmetric}) becomes

\begin{equation}\label{eq:nonunitmetric3}
g^\varnothing_{\alpha\beta}=g^\Omega_{\mu\nu}\bigg|_{\chi^5_+=\Phi}+\,g^\aleph_{\mu\nu}\bigg|_{\chi^5_-=\varphi}~~.
\end{equation}\newline

\noindent  This equation shows that $\Sigma^\pm$ do contain their boundary at $\Sigma^\varnothing$ to contrast the definition that $\Sigma^\pm$ are infinite half spaces that do not contain their boundary at $\chi^5_\pm=0$.  If they did not contain the other boundary, which is $\Omega$ or $\aleph$ in $\Sigma^+$ or $\Sigma^-$ respectively, we would write

\begin{equation}
g^\varnothing_{\alpha\beta}=\lim_{\chi^5_+\to \Phi^-} g^+_{\mu\nu}+\lim_{\chi^5_-\to \varphi^+} g^-_{\mu\nu}~~.
\end{equation}\newline



\begin{figure}[t]
	\makebox[\textwidth][c]{
		\includegraphics[scale=0.5]{unitcell2a.png}}
	\captionsetup{format=hang}\caption{This iteration of the MCM unit cell emphasizes that $\mathcal{H}$ does not connect to $\Sigma^+$ and $\Sigma^-$ simultaneously.}
	\label{fig:unitcell2a}
\end{figure}

\noindent When we include the boundary we can define $g^\aleph_{\mu\nu}$ and $g^\Omega_{\mu\nu}$

In the arrangement described by figure \ref{fig:unitcell2a} we see that $\mathcal{H}$ touches only one of $\Sigma^\pm$ at a time so simultaneous matching on both sides of $\mathcal{H}$ is not implied in any way.  We are joining $\Sigma^\pm$ with $\varnothing$ instead of $\mathcal{H}$ so the new matching condition in equation (\ref{eq:nonunitmetric3}) is very much implied.  The original condition about the smoothness of the bulk hyperspacetime has now been dropped.  It had to be smooth across $\mathcal{H}$ in the picture where $\mathcal{H}$ was in the center of the unit cell but it does not have to be smooth across $\mathcal{H}$ in figure \ref{fig:unitcell2a} because there is no ``across $\mathcal{H}$'' in the that representation of the MCM unit cell.  Normally the notion of something going across $\mathcal{H}$ is required for physics but the MCM considers the regions beyond timelike and spacelike infinity to construct this alternate representation where it is not required.  Hyperspacetime does not need to be smooth across $\mathcal{H}$ because the non-trivial parameter $\chi^5$ is only defined between two adjacent instances of $\mathcal{H}$ that are doubly piecewise disconnected through $\Sigma^\pm$ and $\Sigma^\varnothing$.  We might fill in the piecewise disconnection between $\Omega$ and $\aleph$ with a smooth interface but it might not be necessary to do that to compute $\psi\in\mathcal{H}_{j+1}$ given $\psi\in\mathcal{H}_{j}$.  Therefore we can define the MCM bulk-boundary correspondence \cite{GC} to be the one in equation (\ref{eq:nonunitmetric2}) rather than equation (\ref{eq:nonunitmetric}).  The big bar notation implies that boundary is included and the limit notation implies that the boundary is not included.  The $\eta_{\mu\nu}$ in equation (\ref{eq:nonunitmetric}) has been replaced with $g^\varnothing_{\mu\nu}$ because $\Sigma^\varnothing$ is not defined \textit{a priori} to be flat and also everything we say about $\eta_{\mu\nu}$ will have to generalize to $g_{\mu\nu}$ in the dynamical limit of physical systems.  However, here we will make use of the fact that every manifold has at least one point where we can make the metric be the Minkowski metric with vanishing first derivatives to ensure a smooth connection across $\varnothing$.  The reader will recall from the earlier excerpt that Carroll\footnote{Carroll's general relativity textbook \cite{CARR} has an appendix dedicated to analysis of embedded metrics.} has associated the second derivatives with the curvature and we expect there to be some discontinuity in the manifolds as the topology changes between O(1,4) and O(2,3) in $\Sigma^\pm$.

We still have not exactly specified the how the size of $g^+_{\mu\nu}$ relates to $g^-_{\mu\nu}$ but it is likely related to the numbers in the ontological basis.  Regarding the other two $g$ metrics: $\hat2g_{\mu\nu}\equiv g_{\mu\nu}^\varnothing$ a natural assumption.  The convention in classical EM is to say that the potential in the present is $1/2$ times the advanced potential in superposition with $1/2$ times the retarded potential with $1/2+1/2=1$ but the MCM more likely relies on the non-trivial case $\Phi+\varphi=1$.  When we have these two different metrics $g^\pm_{\mu\nu}$ it might provide a channel through which to apply complexity directly to the weak field limit of GR

\begin{equation}\label{eq:WWeaKK}
g_{\mu\nu}=\eta_{\mu\nu}+h_{\mu\nu}~~,
\end{equation}\newline

\noindent so that equation (\ref{eq:limitBC}) becomes

\begin{equation}\label{eq:limitBCg}
\lim_{\chi^5_+\to 0^+} \Sigma^+_{AB}+\lim_{\chi^5_-\to 0^-} \Sigma^-_{AB}=\left(\begin{matrix}
\eta_{\mu\nu}+h_{\mu\nu}& 0\\
0 & 0
\end{matrix}\right)~~.
\end{equation}\newline

This gives us a good place to make a distinction between $\hat M^3$ and $\tilde M^4$.  The former might be impossible to compute because it is impossible for the different topologies of O(1,4) and O(2,3) to smoothly merge in the way that would be required for them to both perfectly and simultaneously approach an observable state that is exactly the Minkowski metric $\eta_{\mu\nu}$.  Perhaps the computation of the sum of the two limits is guaranteed to never perfectly sum to the degenerate 5D extended Minkowski metric $\eta_{AB}$.  Perhaps due to the numerical approximations that are required (those now assigned to the extra step of $\tilde M^4$) the two limits in equations (\ref{eq:limitBC}) and (\ref{eq:limitBCg}) can never sum to $\eta_{AB}$ but instead sum to

\begin{equation}
g_{AB}\quad\equiv\quad\left(\begin{matrix}
\eta_{\mu\nu}+h_{\mu\nu}& 0\\
0 & 0
\end{matrix}\right)~~.
\end{equation}\newline

\noindent We may take the small perturbation that arises naturally in the attempt to approximate spherical and hyperbolic manifolds as equal objects (by taking the Lorentz approximation in $\aleph$ and $\Omega$) to be the difference of two large perturbations as

\begin{equation}\label{eq:smallpert777}
h_{\mu\nu} \quad\equiv\quad H^+_{\alpha\beta}-H^-_{\alpha\beta}~~,
\end{equation}\newline

\noindent where $H^\pm_{\mu\nu}$ are perturbations in $\Sigma^\pm$.  Similarly to equation (\ref{eq:smallpert777}) there exists another interpretation wherein we may write the real metric including perturbations as

\begin{equation}
g_{\mu\nu}\quad\equiv\quad H^+_{\mu\nu}-H^-_{\mu\nu}       =\eta_{\mu\nu}+h_{\mu\nu}~~.
\end{equation}\newline


This idea probably can be applied to the hierarchy problem but we are not there yet.\footnote{Some particle physics applications of the MCM unit cell, along with its most specific experimental prediction are found in reference \cite{QS}.}  Wikipedia says, ``In theoretical physics, the hierarchy problem is the large discrepancy between aspects of the weak force and gravity,''  and in general it has to do with the large relative scale of many of the empirical parameters of the standard model.   In the MCM we get large relative scale by taking nested resolutions of the identity to generate arbitrarily small numbers such as those that appear in the standard model and also as the amplitudes of arbitrarily unlikely quantum processes.  Perhaps the irreconcilable topological element, represent it with some arbitrarily large perturbation $H_{\alpha\beta}^\varnothing$, can be decomposed into two perturbative modes that live in $\Sigma^\pm$.

\begin{equation}
H_{\alpha\beta}^\varnothing\quad\equiv\quad H^+_{\mu\nu}\hat\Phi^1-H^-_{\mu\nu}\hat\Phi^2~~.
\end{equation}\newline

\noindent For this we must note that $\Sigma^+$ and $\Sigma^-$ exist on two adjacent levels of $\aleph$ when the unit cell is centered on $\varnothing$ but they exist on the same level of $\aleph$ in the representation centered on $\mathcal{H}$.  We will not discuss it here but there may be a connection between $\varphi<0$ and the $\Sigma^-$ perturbation being on the higher level of $\aleph$.  Mirroring what we have done in equation (\ref{eq:hypereta}) we can embed the 4D perturbation in a larger perturbation $H_{AB}^\varnothing$, and then use the nine extra matrix positions to be the numerical representation of the remainder of the topological incompatibility between $\Sigma^\pm$.  In a practical sense, it may be possible to reverse engineer what these numerical coefficients must be and thereby complete the ``impossible calculation'' without knowing them before hand.  This will another stark departure from the almost two dozen experimentally determined parameters that must be manually inserted to prop up the standard model of particle physics.  In chapter four we call some specific attention to the idea that

\begin{equation}
\tilde M^4\quad\equiv\quad\hat \Upsilon =\hat U+\hat M^3~~,
\end{equation}\newline

\noindent and also that $\hat M^3$ was never meant to be the complete operation \cite{FSC}.  We have added $\hat 2$ as an accommodation for a fourth component and even the first MCM representation of fine structure in reference \cite{FSC} used two when the the ratio as $\Phi D=2L$.

We have been modestly specific about what all of the MCM coordinates are and now we will list all of the metrics.  The real metric $g_{\mu\nu}$ in $\mathcal{H}$ given by equation (\ref{eq:WWeaKK}) is the idealized Minkowski metric $\eta_{\mu\nu}$ plus a small perturbation $h_{\mu\nu}$.  $\Sigma^\pm_{AB}$ are the metrics of the 5D manifolds $\Sigma^\pm$.  The slices of $\chi^5_\pm$ in $\Sigma^\pm$ are flat in the $\chi^\alpha$ coordinates associated to the inherited metric $\Sigma^\pm_{AB}$.  The embedded metric of the $x^\mu_\pm$ coordinates on any slice is $g_{\mu\nu}^\pm(\chi^5)$.  $g_{\mu\nu}^\pm(\chi^5)$ is the dS or AdS metric when $\chi^5$ is positive or negative, and the metrics on $\aleph$ and $\Omega$ are $g_{\mu\nu}^-(\varphi)\equiv g^\aleph_{\mu\nu}$ and $g_{\mu\nu}^+(\Phi)\equiv g^\Omega_{\mu\nu}$ respectively.  When it comes to the metric for $x^\mu_\varnothing$ we will probably rely on the mechanism described by Carroll when he wrote, ``Best of all, their usefulness does not generally require that we do the work of constructing such coordinates [\emph{sic}], but simply that we know they do exist.''

We will go into more detail about $\aleph$ and $\Omega$ in the next chapter but for now note that since equation (\ref{eq:nonunitmetric}) suggests a scale factor of $2$ when we take $g_{\mu\nu}^\pm$ to be equal in size, we are compelled to show that

\begin{equation}
\varphi\pi\simeq1.94~~,\quad\qquad\mathrm{and}\quad\qquad   2=2~~, \quad\qquad\implies\quad\qquad \Delta\aleph\simeq3\%~~,
\end{equation}\newline

\noindent with the intention of reminding the reader that

\begin{equation}
\frac{\pi}{2}\simeq1.57~~,\quad\qquad\mathrm{and}\quad\qquad   \Phi\simeq1.62~~, \quad\qquad\implies\quad\qquad \Delta\aleph\simeq3\%~~,
\end{equation}\newline

\noindent and even pointing out that

\begin{equation}
\Phi^2\simeq2.62~~,\quad\qquad\mathrm{and}\quad\qquad   e\simeq2.72~~, \quad\qquad\implies\quad\qquad \Delta\aleph\simeq3\%~~.
\end{equation}\newline

\noindent If the scale of $\{g_{\mu\nu}, g^+_{\mu\nu}, g^-_{\mu\nu}\}$ were the same then we would have to multiply the LHS of equation (\ref{eq:nonunitmetric}) by $1/2$ and this is how we come to say that the simplest scale factor is two but we also need to consider scales relative to $1=\Phi+\varphi$ and similar.

Here we clearly have a mechanism that can lead to fuzzy geometry when the topology of the boundary conditions on the plane waves in the cosmological lattice changes from O(1,4) to O(2,3) across $\Sigma^\varnothing$ in the MCM unit cell.  Perhaps what is $e$ on one side of $\mathcal{H}$ becomes $\Phi^2$ on the other and we can use the extra freedom of $H^\varnothing_{AB}$ to make those transformations.  $\chi^5_+$ has the opposite sign to $\chi^5_-$ so therefore $\Sigma^\pm$ have different numbers of spacelike and timelike dimensions.  We aim to pass this all off with a topological twist on $\chi^5=0$ and at $p\in\Sigma^\varnothing$ as in figure \ref{fig:twist}.  When we use the $\hat 2$ operator to generate two topological spaces from one, the operation that separates the co-$
\hat\pi$'s can scale the ratio of their lengths from 1 to $\Phi$ such that they can no longer be combined to form equal halves of a circle.  They can still be combined into a circle but one co-$\hat\pi$ will have more than $\pi$ radians of arc length.  This is nothing but a conformal transformation but it is an additional beyond twisting that breaks the symmetry retained throughout figure \ref{fig:twist}.

 \begin{figure}[t]
 	\makebox[\textwidth][c]{
 		\includegraphics[scale=1]{twist.png}}
 	\captionsetup{format=hang}\caption{The twisting mechanism for modularizing co-$\hat \pi$'s and creating paths that connect various sites in the cosmological lattice.}
 	\label{fig:twist}
 \end{figure}

Lorentzian metrics only have one timelike dimension.  This means that when we write the metric as a diagonal matrix, all the eigenvalues are positive except one, or the whole thing is shifted and they are all negative except one: $\{\pm\mp\mp\mp\}$.  A new problem that will eventually need to be tackled is to compute 5D geodesics of the form

\begin{equation}
\frac{d^2\chi^C}{(d\chi^5)^2}+\Gamma^C_{AB}\frac{d\chi^A}{d\chi^5}\frac{d\chi^B}{d\chi^5}=0~~,
\end{equation}\newline

\noindent when the background topology of piecewise $\chi^5$ is O(1,4) on one side of $\chi^5_\varnothing$ and O(2,3) on the other side.  We will begin to treat that problem in the next section but it is likely the these geodesics are in the impossible regime of $\hat M^3$ and will be replaced by some new technique in $\tilde M^4$.



\subsection{An Entropic Application}


Here we state a new facet of the model.  Clearly $|\varphi\pi|=1.94$ so we can use equation (\ref{eq:CHIdef1}) to write

\begin{equation}\label{eq:ppojohgogi}
x^\mu_+=|\varphi\pi x^\mu|\approx2x^\mu~~,\quad\qquad\mathrm{and}\quad\qquad  x^\mu_\varnothing=2x^\mu~~.
\end{equation}\newline

\noindent This is a new definition for $x^\mu_\varnothing$: it is twice as big as $x^\mu$ so it is just bigger than $x^\mu_+$.  We have already defined the tensor transformation law between $x^\mu$ and $x^\mu_+$ that gives $|\varphi\pi|\approx1.94\approx2$ and now we define the $x^\mu_\varnothing$ coordinates as exactly twice as big as the observable $x^\mu$ coordinates.  This is similar to what we have done making chirological orthogonality on $\Phi\approx1.62$ just larger than the ordinary orthogonality of $\pi/2\approx1.57$.

When we want to write a local Lorentz frame on $\varnothing$ so that we can sew $\Sigma^\pm$ together, we find

\begin{equation}
x^0_\varnothing=2x^0~~,\quad\qquad\mathrm{and}\quad\qquad x^i_\varnothing=2x^i~~.
\end{equation}\newline

\noindent This leads to a Minkowski diagram in $\varnothing$ that has four times the area of the same diagram written in $\mathcal{H}$.  Since $\hat M^3$ is expected to preserve analytical qubits in the ground state along geodesics, the point density of the fabric of the diagram is rarefied by a factor of four as in figure \ref{fig:rare4}.  We could say that entropy increases in physics because there is some boundary condition misalignment between how the present connects to the past and future respectively, possibly on the order of

\begin{equation}
2-1.94\approx0.06<<1~~,
\end{equation}\newline

\noindent This is about ten times larger than

\begin{equation}
\alpha_{Q\!E\!D}\approx\alpha_{M\!C\!M}\approx1/137\approx0.00729927007299270072992700729927007299270072992700729927007299270072992700729927007299270072992700729927007299270072992700729927007299270072992700729927007299270072992700729927007299270072992700729927...~~,
\end{equation}\newline

\noindent so have good reason for this mode to dominate as decoherence does over determinism in the quantum sector. Perhaps this mismatch being constantly inflated from one moment to the next results in the second law of thermodynamics: $dS>0$ where $S$ is the entropy.  Note that the area $A$ and the number $4$ are two of the main components for the Bekenstein--Hawking formula for the entropy of a black hole.  In units where $c=G=\hbar=1$ that formula is

\begin{equation}\label{eq:BEKHawk}
S_{B\!H}=\frac{A}{4}~~.
\end{equation} \newline

\noindent This could be irrelevant but since we have criticized Hawking's own analysis of dynamics near black holes as missing a single point \cite{ICT}, and $\Sigma^\varnothing$ has been created to supply a single point, there could be some profound connection.



\begin{figure}[t]
	\makebox[\textwidth][c]{
		\includegraphics[scale=.6]{rare4.png}}
	\captionsetup{format=hang}\caption{The non-unitary properties of the $\hat M^3$ operator require that the Minkowski diagram is stretched.  In this map the area increases exactly by a factor of 4 but in the map $\hat \Phi:\mathcal{H}\mapsto\Omega$ the area of $\Omega$ only increases by about $(\varphi\pi)^2\simeq1.94^2\simeq3.7$.  }
	\label{fig:rare4}
\end{figure}

\subsection{Complex Coordinates}

It is quite common to write the metric as a diagonal matrix and here we will present a few of the ordinary definitions used for tensor analysis in gravitational manifolds.  For the Minkowski metric where we don't set $c=1$, we must choose from the two allowed Lorentzian signatures $\{\pm\,\mp\,\mp\,\mp\}$ that are out of phase by $e^{i\pi}$.  The signature that implies distance in the $x^0$ direction is imaginary but gives real-valued spacelike distance is

\begin{equation}
\eta_{\mu\nu}=\left(\begin{matrix}
-c^2& 0& 0&0 \\
0 & 1& 0 & 0 \\
0 & 0 & 1&0  \\
0 &0  &0  & 1\\
\end{matrix}\right)~~.
\end{equation}\newline

\noindent The eigenvalues of the matrix define the line element in the manifold

\begin{equation}
ds^2=-c^2(dx^0)^2+(dx^1)^2+(dx^2)^2+(dx^3)^2~~,
\end{equation}\newline

\noindent and all of this can be condensed by writing the line element as

\begin{equation}
ds^2=\eta_{\mu\nu}dx^\mu dx^\nu~~,
\end{equation}\newline

\noindent and of course the proper time is

\begin{equation}
d\tau^2=-\eta_{\mu\nu}dx^\mu dx^\nu=-ds^2~~.
\end{equation}\newline


\noindent Since 3-velocity is a measure of time and proper time always passes at the same rate, we see the reason for the surprisingly ordinary normalization of the 4-velocity $U^\mu$ of a an object in motion in spacetime

\begin{equation}\label{eq:4vel}
g_{\mu\nu}U^\mu U^\nu=-c^2~~.
\end{equation}\newline

\noindent The 4-velocity is always normalized as in equation (\ref{eq:4vel}) regardless of the metric.  The passage of time recorded by non-inertial clocks is universally relative to this normalization convention.  The 3-velocity $\vec v$ always has to be measured with respect to some external clock since the thing that is moving with $\vec v$ is stationary in its own inertial frame.  Therefore the object of relativity is the external clock.  It has to slow down if observers are going to agree about distances.

Within the normalization of the 4-velocity, the freedom to move around phase space among a range of momenta $p^i$ and to move throughout space $x^i$ at relativistic speeds is counterbalanced by time dilation in $x^0$.  Since the 4-velocity includes the real velocity $\vec v$ and the relativistic kinetic energy $E$ we are able to show that

\begin{equation}\label{eq:onshell}
p^\mu=mU^\mu\quad\qquad\implies\quad\qquad p^\mu p_\mu=-E^2+\big|\vec p\big|^2=m^2~~,
\end{equation} \newline

\noindent where $m$ is a particle's proper mass.  Hypersurfaces in phase space lead to the concepts of ``on shell'' real particles, and ``off shell'' virtual particles.  When QFT shows a particle with energy and momentum not related as in equation (\ref{eq:onshell}) then we say the particle is ``off shell'' in some kind of bulk of virtual states.  This is why the shell in the on- or off-shell condition is called the mass shell.  The implication of equation (\ref{eq:onshell}) is exactly a hyperboloid condition in the manner of equations (\ref{eq:DoubleNEG}).


Given the coordinates $x^\mu$ and the Minkowski metric $\eta_{\mu\nu}$, we have assembled all the pieces of the flat 4D line element

\begin{equation}\label{eq:lineELEM}
(ds)^2=\eta_{\mu\nu}dx^\mu dx^\nu~~,
\end{equation}\newline

\noindent which is a good starting point for making calculations in general relativity.  We have used $x^\mu:\{ct,x,y,z\}$ with $\eta_{\mu\nu}=\mathrm{diag}(-\,+\,+\,+)$ but it is equally valid to build Lorentzian spacetime with $x^0\equiv ict$ which requires a different metric to preserve equation (\ref{eq:lineELEM}).  In that case we have the 4D Euclidean metric

\begin{equation}
\eta_{\mu\nu}'=\left(\begin{matrix}
c^2&0 &0 & 0\\
0& 1&0 &0 \\
0& 0&1 &0 \\
0& 0& 0& 1~
\end{matrix}\right)~~,
\end{equation}\newline

\noindent which leads to a significantly different formulation of the theory.  The spacetime is still Lorentzian -- that is an immutable aspect of Nature -- but now the metric is Euclidean because we have made the change $ct\to ict$.  With complex coordinates that use a Euclidean metric there will be no distinction between raised and lowered tensor indices and any eventual calculations will be vastly simplified.

The most general form of the 4D line element for a given diagonal metric $\bar Z_{\mu\nu}$ is

\begin{equation}
ds^2=\bar Z_{00}\big(dZ^0\big)^2+\bar Z_{11}\big(dZ^1\big)^2+\bar Z_{22}\big(dZ^2\big)^2+\bar Z_{33}\big(dz^3\big)^2~~,
\end{equation}\newline

\noindent where $Z^\mu$ is some set of coordinates.  If we set

\begin{equation}
Z^\mu \quad\equiv\quad x^\mu:\begin{cases}
x^0\quad\equiv\quad ct\\
x^1\quad\equiv\quad x\\
x^2\quad\equiv\quad y\\
x^3\quad\equiv\quad z\\
\end{cases}~~,
\end{equation}\newline

\noindent and we want to make $\bar Z_{\mu\nu}$ give the Minkowski line element

\begin{equation}\label{eq:regline}
ds^2=-c^2(dt)^2+(dx)^2+(dy)^2+(dz)^2~~,
\end{equation}\newline

\noindent then evidently the Lorentzian metric $\text{diag}(-1,1,1,1)$ is required.

However!  We can alter the $Z^\mu$ coordinates as

\begin{equation}
Z^\mu \quad\equiv\quad y^\mu:\begin{cases}
y^0\quad\equiv\quad ict\\
y^1\quad\equiv\quad x\\
y^2\quad\equiv\quad y\\
y^3\quad\equiv\quad z\\
\end{cases}~~,
\end{equation}\newline

\noindent and now

\begin{equation}
dZ^0\quad\equiv\quad ic\,dt\qquad\quad\implies\qquad\quad \big(dZ^0\big)^2=-c^2\big(dt\big)^2~~,
\end{equation}\newline

\noindent so if we want to write the line element in the form of equation (\ref{eq:regline}) we have to use the Euclidean metric.  Here we have the convenient option to finally make a definition for $\hat i$ with

\begin{equation}\label{eq:I1}
\hat Z^0\quad:=\quad ct\hat i~~.
\end{equation}\newline

\noindent This looks very much like $\hat i$ should be the arrow of time but there is another option for using $\hat i$ as the arrow of time, one that nicely demonstrates the principle of complexity.  We can use equation (\ref{eq:I1}) with the Lorentzian metric when we take

\begin{equation}\label{eq:wher9th3foh40hf}
\hat Z^\mu \quad\equiv\quad  z^\mu\,\hat i~~,\qquad\quad\mathrm{with}\qquad\quad z^\mu:~~\begin{cases}
z^0\quad\equiv\quad ct\\
z^1\quad\equiv\quad ix\\
z^2\quad\equiv\quad iy\\
z^3\quad\equiv\quad iz
\end{cases}~~,
\end{equation}\newline

\noindent Note that definition (\ref{eq:wher9th3foh40hf}) is also the one that naturally eliminates the need for Wick rotation of the time component $z^0\mapsto-iz^0$ during the computation of QFT action integrals \cite{WICK}.  Wick rotation will be unnecessary if we replace the differential volume element of four real dimensions $dx^\mu$ the volume element of one real dimension and three imaginary.  When we integrate over $dz^\mu$ as in equation (\ref{eq:wher9th3foh40hf}) we will recover three factors of $i$ giving $i^3=-i$ which is exactly the term inserted during analytic continuation via Wick rotation.  Another configuration, one that seeks to demonstrate the concept of $\hat \pi$ as an arrow of space is

\begin{equation}\label{eq:timearrowspace}
\hat Z^\mu \quad\equiv\quad \hat z^\mu~~,\qquad\quad\mathrm{with}\qquad\quad \hat z^\mu:~~\begin{cases}
z^0\quad\equiv\quad ct\,\hat i\\
z^1\quad\equiv\quad ix\,\hat \pi\\
z^2\quad\equiv\quad iy\,\hat \pi\\
z^3\quad\equiv\quad iz\,\hat \pi
\end{cases}~~.
\end{equation}\newline

\noindent Due to infinite complexity, there are very many schemes that can be explored.

To close this section we will point out another general relativistic artifact related to quantum theory.  By making the revision $ct\rightarrow ict$ we have shifted the complex phase of $x^0$ by $\pi/2$ radians but the corresponding revision to the metric $\eta_{00}=1\rightarrow\eta_{00}=-1$ is a rotation through a full $\pi$ of arc in the complex plane.  This is the same behavior seen in rotations of the half-integer spin vectors in relation to the locally inertial lab frame: rotating one makes the other one rotate twice (or half) as much.


\subsection{What is $\hat \varphi$?}


In section I.2 we introduced the vector $V^\mu$ that will be transported through $\mathcal{H}\mapsto\Omega\mapsto\aleph\mapsto\mathcal{H}$ and figure \ref{fig:Phi762} gives the general idea.  The object $\hat \varphi$ will be the initial $V^\mu$ written in ontological notation.  The  $\hat \varphi$ component shall complement the perturbation tensor $h_{\mu\nu}$ as the computational source for what happens across a given MCM unit cell.  We described how $\hat M^3$ pushes a vector $V^\mu$ through the MCM unit cell and now we begin to clarify those preliminary definitions as

\begin{equation}
V^\mu\quad\equiv\quad \psi\big(x;t_{\text{initial}}\big)\,\hat \varphi~~,\quad\qquad\text{and}\quad\qquad W^\mu\quad\equiv\quad \psi\big(x;t_{\text{final}}\big)\,\hat \varphi~~.
\end{equation}\newline


We can imagine a method in which $\hat M^3$ takes $\psi(x;t_{\text{initial}})\hat\varphi$ and returns a qubit in the unitary sector like $\psi(x;t_{\text{final}})\hat1$ which then becomes the seed term for evolution across the next level of $\aleph$ when we make the change $\psi(x;t_{\text{final}})\,\hat 1\to\psi(x;t_{\text{initial}})\,\hat \varphi$.  We will implement a transfinite normalization $\hat\Phi^2\to\hat \Phi^1$ as part of the ontological rescaling at the boundary of each unit cell so we can see how we might also come across $\hat \Phi^0\!\equiv\!\hat1\to\hat\Phi^{-1}\!\equiv\!\hat\varphi$: as the level of $\aleph$ increases the magnitude of infinitude must be decreased to maintain finite normalization.  Since $\hat\varphi$ is new we will not be too specific about it other than to mention that it should exist in some form.  If $\{\hat i,\hat \Phi,\hat2,\hat \pi\}$ are going to be a basis for 4D spacetime then we need some other informatic channel for the quantum component of our quantum gravity.  We propose to send that channel on $\hat \varphi$.  It will be the object that holds the quantum initial condition which will be operated upon with $\hat M^3:\mathcal{H}_1\mapsto\mathcal{H}_2$.  Since the TOIC is a theory of quantum gravity and not just gravity we need to evolve the dynamical geometry of $\mathcal{H}$ across the unit cell and also whatever quantum information from $\mathcal{H}'$ was encoded on $\mathcal{H}$ through its representations in the coordinates $x^\mu$ or momenta $p^\mu$.


The utility of $\hat \varphi$, without being specific about one exact mathematical definition or another, will be as follows.  The true history is the string of earlier $\hat \Phi$ vectors that precede an observer on a given level of $\aleph$ but $\hat \varphi$ encodes the record of that history in the present $\mathcal{H}$.  For this reason we will attach the new object to $\mathcal{H}$.  The qubit on $\hat \varphi$ is $\psi\in\mathcal{H}'$.  It defines the entire quantum mechanical sector because states $\psi$ are complete boundary conditions in quantum theory under the Schr\"odinger equation.  We can also take $\hat \varphi$ as the source of torsion as in reference \cite{ICT}.  The torsion can be encoded with a potential so if we use the wavefunction as the potential then we see how the quantum gravity acts by sending qubits into the geometry through the torsion.  We would like to show that the probability density on $\mathcal{H}_2$ exhibits decoherence with respect that on $\mathcal{H}_1$ because of the Ricci flow or some other geometric flow acting on the quantum information during its geometric representation in hyperspacetime.   When we discussed torsion in reference \cite{ICT} we set $\mathcal{H}$ as a torsion free boundary surface in the MCM unit cell.  We have also defined $\mathcal{H}$ as the place where measurements of quantum states happen and that corresponds to what is called the collapse of the wavefunction.  It is an intuitive arrangement when the wavefunction is encoded in the torsion and then it collapses on the torsion-free surfaces $\mathcal{H}$.  One might liken the situation to a sinusoid.


\begin{figure}[t]
	\makebox[\textwidth][c]{
		\includegraphics[scale=.5]{Phimagic762.png}}
	\captionsetup{format=hang}\caption{The new object $\hat \varphi$ encodes the quantum sector in the MCM unit cell along side the ontological basis $\{\hat i,\hat \Phi,\hat 2,\hat \pi \}$ which are used, among other things, as a non-coordinate basis for general relativity.  The repeated measurement non-decoherence mode is such that all the $V$ are linearly dependent on $\hat\Phi$.  Then $Q=|V^\mu-W^\mu|=|\hat\Phi-\hat\Phi|=0$.}
	\label{fig:Phi762}
\end{figure}

One of quantum mechanics' many odd empirical results is that a state will not exhibit decoherence if the state is measured sufficiently often.  If the geometric flow is always present between successive measurements then we will always see decoherence which is not the desired behavior.  Instead we will need to introduce some parameter\footnote{This parameter might describe some experimental configuration and not Nature itself fundamentally.} that sets a transition between laminar flow across the unit cell and then some supercritical non-laminar phase.  In the case of laminar flow, decoherence will not be observed because the laminar flow corresponds to each of $\{V^\mu,V^\mu_+,V^\mu_-\}$ being $\hat \Phi$ giving $Q=|V^\mu-W^\mu|=0$.  When the parameter, which might be the length of chronological time $x^0$ between successive measurements, becomes too large then there must be a phase change.  In the non-laminar phase the vectors $\{V^\mu,V^\mu_+,V^\mu_-\}$ will all have to be defined individually so that $Q\neq0$.  In section I.2 we described how hyperspacetime can be constructed from spacetime by taking some vector in $\mathcal{H}$ and then defining $\{V^\mu,V^\mu_+,V^\mu_-\}$ accordingly.  This is the $Q\neq0$ decoherence mode is where the $V$ are defined from some local qubits instead of the ontological qubit $\hat \Phi$.

Additionally we might explore a situation in which decoherence does always occur but the qubit remains in some local minimum that is preserved through brief geometric flow during the brief period between repeated measurements.  If we associate one qubit with one local minimum in the flow then the eigenstate $|\psi;\hat\pi_0\rangle$ at the beginning of the flow will be the same one at the end of the flow $|\psi;\hat\pi_1\rangle$ as long as the valley of stability representing the eigenstate is preserved.  If the duration of the flow is very short then we expect the local minimum will not merge with some other basin of attraction where the quantum state changes from an eigenstate to a superposition of eigenstates.  This notion might be the long sought after quantum determinism.

Dirac was able to predict antimatter on the basis of sign reversal symmetry in quantum mechanics.  If we extend that principle into another sector in QED we would expect that the perturbation theory that agrees with Nature would be a Laurent series expansion in the fine structure constant but instead the one that describes Nature is only a Taylor series.  QED uses series whose terms are proportional to $\alpha_{Q\!E\!D}^n$ but a continuation of the symmetry argument made by Dirac might lead one to conclude that terms of order $\alpha_{Q\!E\!D}^{\pm n}$ should contribute as in a Laurent series.  Why are the terms associated with $\alpha_{Q\!E\!D}^{-n}$ not present in Nature?  Of course those terms would make the probability interpretation of the theory explode but we could add some non-trivial non-unitary component to the $\sqrt{i}$ channel which would allow the Laurent terms and not just the Taylor terms.  If we put the observer on the $\hat \Phi^0$ level of $\aleph$ it is easy to associate all qubits from the past levels $\hat\Phi^{\{n<0\}}$ with the Taylor series terms and all the future qubits on levels $\hat\Phi^{\{n>0\}}$ with the terms that would add to the Taylor terms to make a complete Laurent series in the fine structure constant.  The critical distinction to note in this regard is that the past terms get encoded on $\hat \varphi$ but the future terms do not.

To examine the full Laurent series representation we can take a Laurent series and split it in half.  Put the positively and negatively exponentiated terms in the chronological sectors and chirological sectors respectively.  It is natural to say that the terms in the chronological sector are QED's Taylor series terms $\alpha_{Q\!E\!D}^n$ and the other terms $\alpha_{Q\!E\!D}^{-n}$ are not encoded in the $\hat \varphi$ sector because initial conditions like $\psi_{\text{initial}}\hat\varphi$ describe the past only.  For now we leave the future terms as a logical remainder but perhaps the continuation of Dirac's symmetry argument will yield observable consequences of the Laurent representation.  We may be able to use the transfinite analytical tools of hypercomplexity to directly write the causality violating advanced potential effects from the future levels of $\aleph$ directly into the well known Taylor series representation of the past levels.  Furthermore, we have struggled with a reason to invert $\alpha_{M\!C\!M}$ so that it becomes to $1/137$ instead of just 137 but now that matters less as we have freedom to choose positive or negative exponents in each sector respectively.\footnote{In reference \cite{IC} we did finally come up with a good way to get $137$ into the denominator.}  Further note that the terms in the chirological sector are assigned to all future levels of $\aleph$ so they are much more complicated than the chronological terms that exist only one the level of $\aleph$.  This distinction between a single level of $\aleph$ or many levels could be a clue regarding how one sector could be so well understood for so many decades while the other parallel sector has remained mysterious.


\begin{figure}[t]
	\makebox[\textwidth][c]{
		\includegraphics[scale=.5]{Stereographic.png}}
	\captionsetup{format=hang}\caption{The ``volume integral'' over the fundamental element of the abstract psychological space is the fine structure constant $\alpha_{M\!C\!M}$ when the volume of the ball is $(\Phi\pi)^3$ and the two vectors each contribute $\pi$ as non-trivial embedded objects corresponding the two $\hat\pi$-sites at the beginning and end of every MCM unit cell.  We might obtain the volume element $(\Phi\pi)^3$ through the inflation of 3-space where the arrow of space is $\hat \pi$.  Recall that inflation of 3-space is the default scenario in $\Lambda$CDM cosmology where the Hubble parameter increases with time.  This figure first appeared in reference \cite{ZETA} where we argued against the Riemann hypothesis.  Reference \cite{ZETA} is a good introduction to the idea of infinite complexity.}
	\label{fig:stereo}
\end{figure}


We might motivate the Taylor/Laurent asymmetry in perturbation theory by noting that we have previously expected to define the past of some moment $\hat \Phi^n$ on the $\hat \Phi^{n-m}$-sites with $m\geq1$ but now we see that $\hat \varphi$ is a better place to define the past.  The analytical expression of the past has to exist in the present with the observer so we may encode the information from the past as $\hat \varphi$ on the leftmost boundary of the cosmological unit cell.\footnote{$\mathcal{H}_1$ and $\Sigma^+$ are on the left side of the unit cell, $\Sigma^-$ and $\mathcal{H}_2$ are on the right.}  Then, instead of a series of past terms in the hypercosmos like $\{\hat \Phi^{n-1},\hat \Phi^{n-2},...\}$, we have a series of terms in perturbation theory like $\{\alpha_{M\!C\!M},\alpha_{M\!C\!M}^2,...\}\approx\{137^{-1},137^{-2},...\}$ which all coexist on the same level of $\aleph$.

To say a little more about the $\alpha_{M\!C\!M}$ note that we still have not decided which of the ontological basis is the arrow of time.  The vector $\hat \pi$ is a good candidate, but it does not have to be the arrow of time and could just as easily be the ``arrow of space'' when $\hat i$ or another one is the arrow of time, as in equation (\ref{eq:timearrowspace}).  The motivation in proposing an arrow of space is the factor of $\pi^3$ that appears in $\alpha_{M\!C\!M}=2\pi+(\Phi\pi)^3$.  While we have made some good definitions for the numerical origin of $\alpha_{M\!C\!M}$ in reference \cite{IC}, if there is a $\hat \pi$ pointing in each of the three dimensions of space where the $\hat \varphi$ object encodes an input for $\hat M^3$, and then space is inflated by $\Phi$ when $\hat \Phi$ acts on $\hat \varphi$ (perhaps generating the unitary sector $\hat 1$), that would be another very natural origin for $(\Phi\pi)^3$.  If that 3-space is on the interior of the 2-sphere in figure \ref{fig:stereo}, and the vectors on its surface of are qubits that return $\pi$ when integrated over, likely pointing to $\hat \pi_0$ and $\hat \pi_1$ on two different levels of $\aleph$ $\hat\Phi^1$ and $\hat\Phi^2$, then the volume integral over the object in the figure will be $\alpha_{M\!C\!M}=2\pi+(\Phi\pi)^3$.



\subsection{Twistors and Spinors}
The mystery of unification in physics is to find a scheme by which quantum particles dynamically warp spacetime meaning that the spacetime is not just a background.  In the previous section we suggested to send qubits into the geometry by encoding them in the torsion.  In this section we show that another option leads to an object in twistor theory.  One obvious way to define a reciprocal mechanism for sending perturbations between QM and GR will be through

\begin{equation}\label{eq:bdiyegwi}
\psi\quad\longleftrightarrow\quad h_{\mu\nu}~~,
\end{equation}\newline

\noindent where $\psi$  is a disturbance of the vacuum and $h_{\mu\nu}$ is the perturbation on the Minkowski metric.  In reference \cite{GC} we showed how an electromagnetic boundary condition on $ \mathcal{H}$ could induce gravitation in the bulk hyperspacetime and vice versa.  This reciprocity between the EM and GR sectors implies the existence of more than one configuration for perturbative information and we are drawn to consider many possible definitions for perturbations.

Equation (\ref{eq:bdiyegwi}) describes a framework where the mass and energy associated with $\psi$ directly perturb spacetime without going through the torsion in hyperspacetime.  Note that both methods for writing qubits as perturbative tensors, through the torsion earlier and now directly, are simply ansatzes taken for the purposes of discussion.  To that end, one is inclined to consider perturbations of the form

\begin{equation}
h_{\mu\nu}=\psi\left(\begin{matrix}
-\dfrac{i}{4} & 0 & 0 & 0 \\
0 & -\dfrac{\varphi}{4} & 0 & 0 \\
0 & 0 & \,\dfrac{1}{8}\, & 0 \\
0 & 0 & 0 & \dfrac{1}{4\pi} \\
\end{matrix}\right)~~,
\end{equation}\newline

\noindent where the reader should recognize the ontological coefficients.  Here the state causes a perturbation in ontological form having two timelike dimensions and two spacelike.  Such objects have been treated as a problem in string theoretical twistor theory by Witten.  He writes the following in reference \cite{WITTEN}.

\begin{quotation}
    ``Before considering scattering amplitudes, we will review some kinematics in four dimensions.  We start out in signature $+\,-\,-\,-$, but we sometimes generalize to other signatures.  Indeed, [\textit{in perturbation theory}] the signature is largely irrelevant as the scattering amplitudes are holomorphic functions of the kinematic variables.  Some things will be simpler with other signatures or for complex momenta with no signature specified.

    ``First we recall that the Lorentz group in four dimensions, upon \textbf{complexification} [\textit{emphasis added}], is locally isomorphic to $SL(2)~\times~SL(2)$, and thus the finite-dimensional representations are classified as $(p,q)$, where $p$ and $q$ are integers or half-integers.  The negative and positive chirality spinors transform in the $(1/2,0)$ and $(0,1/2)$ representations, respectively.  We write generally $\lambda_a$, $a=1,2$, for a spinor transforming as $(1/2,0)$, and $\widetilde \lambda_{\dot a}$, $\dot a=1,2$, for a spinor transforming as $(0,1/2)$.

    ``Spinor indices of type $(1/2,0)$ are raised and lowered with the antisymmetric tensor $\epsilon_{ab}$ and its inverse $\epsilon^{ab}$ (obeying $\epsilon^{ab}\epsilon_{ab}=-\delta^a_c$): $\lambda_a=\epsilon_{ab}\lambda^b$, $\lambda^b=\epsilon^{bc}\lambda_c$.  Given two spinors $\lambda_1$, $\lambda_2$ both of positive chirality, we can form the Lorentz invariant $\langle\lambda_1,\lambda_2\rangle=\epsilon_{ab}\lambda^a_1\lambda^b_2$. From the definitions, it follows that $\langle\lambda_1,\lambda_2\rangle=-\langle\lambda_2,\lambda_1\rangle=-\epsilon^{ab}\lambda_{1a}\lambda_{2b}$.

    ``Similarly, we raise and lower indices of type $(0,1/2)$ with the antisymmetric tensor $\epsilon_{\dot a\dot b}$ and its inverse $\epsilon^{\dot a\dot b}$, again imposing $\epsilon^{\dot a\dot b}\epsilon_{\dot b\dot c}=\delta^{\dot a}_{\dot b}$.  For two spinors $\widetilde{\lambda_1},\widetilde{\lambda_2}$, both of negative chirality, we define $[\widetilde{\lambda_1},\widetilde{\lambda_2}]=\epsilon_{\dot a\dot b}\widetilde \lambda_1^{\dot a}\widetilde \lambda_2^{\dot b}$.

    ``The vector representation of $SO(3,1)$ is the $(1/2,1/2)$ representation. Thus, a momentum vector $p_\mu$, $\mu=0,...,3$, can be represented as a `bi-spinor'  $p_{a\dot a}$ with one spinor index $a$ or $\dot a$ of each chirality.  The explicit mapping from $p_\mu$ to $p_{a\dot a}$ can be made using the chiral part of the Dirac matrices.  With signature $+\,-\,-\,-$, one can take the Dirac matrices to be

    \begin{equation}\label{eq:DIRACMATRICES}
    \gamma^\mu=\left(\begin{matrix}
    0&\sigma^\mu\\
    \bar \sigma^\mu&0\\
    \end{matrix}\right)~~,
    \end{equation}

    \noindent where we take $\sigma^\mu=(1,\vec \sigma)$, $\bar \sigma^\mu=(-1,\vec \sigma)$, with $\vec \sigma$ being the $2~\times~2$ Pauli spin matrices.  In particular, the upper right hand block of $\gamma^\mu$ is a $2~\times~2$ matrix $\sigma^\mu_{a\dot a}$ that maps spinors of one chirality to the other.  For any spinor $p_\mu$, define

    \begin{equation}
    p_{a\dot a}=\sigma^\mu_{a\dot a}p_\mu~~.
    \end{equation}

    \noindent Thus, with the above representation of $\sigma^\mu$, we have $p_{a\dot a}=p_0+\vec \sigma\cdot \vec p$ (where $p_0$ and $\vec p$ are the ``time'' and ``space'' parts of $p^\mu$), from which it follows that

    \begin{equation}
    p_\mu p^\mu=\det(p_{a\dot a})~~.
    \end{equation}

    \noindent Thus a vector $p^\mu$ is lightlike if and only if the corresponding matrix $p_{a\dot a}$ has determinant zero.

    ``Any $2~\times~2$ matrix $p_{a\dot a}$ has rank at most two, so it can be written $p_{a\dot a}=\lambda_a\widetilde\lambda_{\dot a}+\mu_a\widetilde\mu_{\dot a}$ for some spinors $\lambda, \mu$, and $\widetilde\lambda,\widetilde\mu$.  The rank of a $2~\times~2$ matrix is less than two if and only if its determinant vanishes.  So the lightlike vectors $p^\mu$ are precisely those for which

    \begin{equation}\label{eq:WITT1}
    p_{a\dot a}=\lambda_a\widetilde\lambda_{\dot a},
    \end{equation}

    \noindent for some spinors $\lambda_a$ and $\widetilde \lambda_{\dot a}$.

    ``If we wish $p_{a\dot a}$ to be real with Lorentz signature, we must take $\widetilde{\lambda}=\pm\bar{\lambda}$ (where $\bar \lambda$ is the complex conjugate of $\lambda$).  The sign determines whether $p^\mu$ has positive energy or negative energy.

    ``It will also be convenient to consider other signatures.  In \textbf{signature} $\boldsymbol{+\,+\,-\,-}$ [\textit{emphasis added}], $\lambda$ and $\widetilde \lambda$ are independent, real, two-component objects.  Indeed, with signature $+\,+\,-\,-$, the Lorentz group $SO(2,2)$ is, without any complexification, locally isomorphic to $SL(2,\mathbb{R})\times SL(2,\mathbb{R})$, so the spinor representation is real.  With Euclidean signature $+\,+\,+\,+$, the Lorentz group is locally isomorphic to $SU(2)\times SU(2)$; the spinor representations are pseudoreal.  A lightlike vector cannot be real with Euclidean signature.

    ``Obviously, if $\lambda$ and $\widetilde \lambda$ are given, a corresponding lightlike vector $p$ is determined, via (\ref{eq:WITT1}).  It is equally clear that if a lightlike vector $p$ is given, this does not suffice to determine $\lambda$ and $\widetilde \lambda$.  They can be determined only modulo the scaling

    \begin{equation}
    \lambda\to u\lambda,~\widetilde \lambda\to u^{-1}\widetilde \lambda
    \end{equation}

    \noindent for $u\in\mathbb{C}^*$, that is, $u$ is a nonzero complex number.  (In signature $+\,-\,-\,-$, if $p$ is real, we can restrict to $|u|=1$.  In signature $+\,+\,-\,-$, if $\lambda$ and $\tilde \lambda$ are real, we can restrict to real $u$.)  Not only is there no natural way to determine $\lambda$ as a function of $p$; there is in fact no continuous way to do so, as there is a topological obstruction to this.  Consider, for example, massless particles of unit energy; the energy-momentum of such a particle is specified by the momentum three-vector $\vec p$, a unit vector which determines a point [\textit{on a sphere}] $\mathbb{S}^2$.  Once $\vec p$ is given, the space of possible $\lambda$'s is a non-trivial complex line bundle over $\mathbb{S}^2$ that is known as the Hopf line bundle [(\textit{figure \ref{fig:HOPF}})]; non-triviality\footnote{Perhaps it can be argued that the non-triviality of the Hopf fiber bundle is the ultimate source of the requirement for the non-trivial parameter $\chi^5$.  We have proposed that this parameter threads a periodic orbit through the Hopf fiber bundle in each traversal of an MCM unit cell.} of this bundle means that one cannot pick $\lambda$ as a continuously varying function of $\vec p$.

    ``Once $p$ is given, the additional information that is involved in specifying $\lambda$ (and hence $\widetilde \lambda$) is equivalent to a choice of wavefunction for a spin one-half particle with momentum vector $p$.  In fact, the chiral Dirac equation for a spinor $\psi^a$ is

    \begin{equation}\label{eq:WITTfinal}
    i\sigma^\mu_{a\dot a}\dfrac{\partial\psi^a}{\partial x^\mu}=0~~.\mathrm{"}
    \end{equation}  \newline
\end{quotation}

\begin{figure}[p!]
	\makebox[\textwidth][c]{
		\includegraphics[scale=1]{Hopf.png}}
	\captionsetup{format=hang}\caption{The Hopf fiber bundle describes a 4D shape drawn with circles so it offers an alternative viewpoint on hyperdimensional geometry from rectangular representations such as the tesseract.  The bundle looks like the stereographic projection of the 3-sphere onto parallels, meridians, and hypermeridians, as in figure \ref{fig:3-sphere}.  In reference \cite{DE} we assigned the topologies flat, hyperbolic, and spherical to those projections and later they were associated to $\{\aleph,\mathcal{H},\Omega\}$.  Now we are likely proposing to better understand the Hopf fibration by modeling it as the complete cosmological unit cell including $\Sigma^\varnothing$.  The arrows at the top and bottom can be thought of as the input and output of $\hat M^3$.  The Wikipedia caption of the image to the right is, ``The Hopf fibration can be visualized using a stereographic projection of $\mathbb{S}^3$ to $\mathbb{R}^3$ and then compressing $\mathbb{R}^3$ to a ball. This image shows points on $\mathbb{S}^2$ and their corresponding fibers with the same color.''}
	\label{fig:HOPF}
	\makebox[\textwidth][c]{
		\includegraphics[scale=.55]{3-sphere.png}}
	\captionsetup{format=hang}\caption{In the stereographic projection of the 3-sphere onto parallels, meridians, and hypermeridians, Every parallel, meridian, and hypermeridian is a circle.  To get straight lines we simply consider a circles of infinite radius.  }
	\label{fig:3-sphere}
\end{figure}


Equation (\ref{eq:WITTfinal}) is exactly the kind of mechanism wherein some hypercosmological vector $V^\mu$ is derived from the qubit $\psi$ in the lab frame.  Witten writes, ``The explicit mapping from $p_\mu$ to $p_{a\dot a}$ can be made using the chiral part of the Dirac matrices,'' and that calls attention to the fact that we have still not defined the explicit mapping of $\hat M^3$.  We have a preliminary definition that $\hat M\equiv\partial_t$ but the Dirac operator

\begin{equation}
\mathcal{\widehat D}\quad\equiv\quad i\sigma^\mu_{a\dot a}\partial_\mu~~,
\end{equation}\newline

\noindent suggests that we might add complexity to $\hat M^3$ by adding matrix-valued or other complex coefficients to each $\hat M$.

We will not be going too much into twistor theory and it will suffice to say that twistors satisfy the twistor equation.  Wikipedia says that, ``For Minkowski space [\textit{sic}] the solutions to the twistor equation are of the form

\begin{equation}
\Omega^a(x)=\omega^a-ix^{a\dot a}\pi_{\dot a}~~,
\end{equation}\newline

\noindent where $\omega^a$ and $\pi_{\dot a}$ are two constant Weyl spinors and $x^{a\dot a}=\sigma^{a\dot a}_\mu x^\mu$ is a point in Minkowski space.''  Among the primary utilities of twistor theory is to a change the number of indices of objects ($x^\mu\mapsto x^{a\dot a}$ or $p_\mu\mapsto p_{a\dot a}$) which is exactly what is needed for the maps like

\begin{align}
f^3\big|\psi;\hat\pi\big\rangle&~~\mapsto~~ T_{\mu\nu}~~,\label{eq:GR001}\\
&~\nonumber\\
\big|\psi;\hat\Phi\big\rangle&~~\mapsto~~ Rg_{\mu\nu}~~,\label{eq:GR002}\\
&~\nonumber\\
\frac{\Phi}{4}\big|\psi;\hat2\big\rangle&~~\mapsto~~ R_{\mu\nu}~~,\label{eq:GR003}\\
&~\nonumber\\
i\big|\psi;\hat i\big\rangle&~~\mapsto~~ g_{\mu\nu}\Lambda~~,\label{eq:GR004}
\end{align}\newline

\noindent wherein rank one state vectors\footnote{We have been working with non-relativistic spinless rank zero states under the assumption that they will generalize to rank-1 Dirac vectors $\psi^a$ in the usual way.} become rank two tensors in GR.  It follows that we can rewrite equation (\ref{eq:bdiyegwi}) as

\begin{equation}
\big|\psi;\hat \varphi\big\rangle~~\mapsto~~ h_{\mu\nu}~~.
\end{equation}\newline

An exemplary feature of twistor theory is the mapping of the null interval $ds^2=0$ in Minkowski space to its corresponding object in twistor space.   In twistor theory the light ray is a point in twistor space\footnote{See video reference \cite{PENTALK}.  In this talk Penrose explains that his main interest in twistor theory is to discover a spacetime calculus specific to 3+1 dimensions, and perhaps we have done that with $\{\Phi,2,\pi\}$$\,\in\,$$^*\mathbb{R}$ and $\{i\}$$\,\in$$\, ^*\mathbb{C}$.  He states that 4D twistor theory is the best variety of twistor theory and he casts erudite dispersions on the arrangement shown in figure \ref{fig:twistors} which shows the abutment of two 5D twistor spaces $\Sigma^\pm$.} and points in spacetime become Riemann spheres in twistor space.  Using these definitions we can further clarify the inversion operation that we have referred to very many times \cite{TER}.  This operation takes a Riemann sphere tangentially situated between two branes and inverts it so that the origin of coordinates moves to the null point on of $\mathcal{S}^2$ that is not included in the Reimann sphere.  If the map swaps the Riemann sphere's pole with its null point at the other pole, then that is a map between the entire Riemann sphere and a point, \textit{i.e.}, it is the inverse of the map to twistor space.

We have not used spinors at all in the MCM because we have not carried out any quantum mechanical analysis requiring the specification a qubit as a fermion.  A spinor is just a multiplex introduced to get the right eigenvalue algebra needed to describe spin in QM.  In fact we can even take the Dirac matrices $\sigma^\mu_{a\dot a}$ as defining Dirac spinors so if we add some coefficients to the derivatives in $\hat M^3$ then that will also be a way to probe the MCM for multiplectic spinor analogues.  A long-standing problem throughout the history of quantum theory has been the search for how to represent the fermionic eigenfunctions of the spin-1/2 operator without spinors.  For a boson we can say that, perhaps $\psi(x)=A\sin(kx)e^{-|x|}$ but it is always impossible to write the wavefunction of a fermion as an ordinary function.  Fermionic wavefunctions always require a multiplectic component.  Witten mentions the problem of the non-existence of any regular representations for the spinors; he points out that there is no continuous way to represent them as a function which would be the ordinary, non-multiplectic way to represent it.  Everything in physics can be represented in this way except for spin.  We propose that stacking $\hat \Phi$ vectors allows us to create a smooth parameter across hypercomplex tiers of infinitude that will go straight through the Hopf fibration pseudo-trivially so that we can define continuous functions that give a regular representation of half-integer spin.  The distinction will be simply that our new functions are functions of hypercomplex rather than simply complex variables.



\begin{figure}[t]
	\makebox[\textwidth][c]{
		\includegraphics[scale=.9]{twistors.png}}
	\captionsetup{format=hang}\caption{Perhaps the 4D twistor theory lives inside the shared 4D boundary of $\Sigma^\pm$ and chiral legs extending outward describe the topology of the boundary conditions of the twistor space, one for the future and one for the past, or two for $\{\mathcal{H},\varnothing\}$.}
	\label{fig:twistors}
\end{figure}


We propose that by shifting certain complexity into the topology it will be possible to define spin algebras without spinors.  In lieu of spinors, we want to use the topological twisting mechanism from figure \ref{fig:twist}.  Where a wavefunction on one arc of $\pi$ radians (in figure \ref{fig:twist}) would be symmetric we can create antisymmetric fermionic wavefunctions by twisting semicircles around the bounce point which is the origin of coordinates.  Where the final pane of figure \ref{fig:twist} shows sinusoidal $ct$ and $-ct$, if we invert only the semicircle on one side, or simply select the complete upper or lower path, then the resultant sinusoidal or piecewise sinusoidal wavefunction that propagates on either $ct$ or $-ct$ will be antisymmetric.  Witten proposes to use a ``bi-spinor'' and that sounds like exactly the kind of thing one might need to twist the topology.  However, where Witten builds on the spinor, we aim to never introduce the spinor except as a limit of some ontological considerations.  In figure \ref{fig:twistors} we can see the general outline of how the chirality of twistor space relates the interface of $\Sigma^\pm$.  The are joined on the projective null space but have chiral legs extending outward.

Note that Witten's spinors $\lambda,\widetilde \lambda$ are only determined up to an inverse scale relationship between $u$ and $u^{-1}$ exactly like the one satisfied by $\Phi$ and $\varphi$.  Perhaps it is specifically when we fix this as the \textbf{golden ratio} with $\{u,u^{-1}\}\equiv\{\Phi,-\varphi\}$ that we are able to set the \textbf{new boundary condition} that lets us \textbf{solve for new physics}.  Regarding new physics, the inverse scale relationship is exactly the relationship between the two charts that cover $\mathbb{S}^2$; they are related by the canonical inversion map.  Where Witten is not able to uniquely determine his spinors $\lambda$ and $\widetilde \lambda$, in our argument against the Riemann hypothesis \cite{ZETA} we used a single chart on $\mathbb{S}^2$ but were unable to uniquely determine which of the two possible charts it was.  Furthermore since any point on $\mathbb{S}^2$ can be the origin of a chart, we were actually unable to determine which of the infinite number of possible charts we were working with.  Furthermore, our argument against the Riemann hypothesis relied heavily on the relationship between a point and the Riemann sphere, and if a spacetime point is the Riemann sphere in twistor space then there likely exists an important connection between that argument, the Riemann zeta function, and twistor theory.  Of course we will not be going in that direction in this book because such nuance lies far beyond the general relevance of the MCM.


\begin{figure}[t]
 	\makebox[\textwidth][c]{
 		\includegraphics[scale=1]{KNBH.png}}
 	\captionsetup{format=hang}\caption{This figure shows the multifaceted region on the interior of a Kerr--Newman black hole.}
 	\label{fig:KNBH}
\end{figure}



Previously we described how the complete geometry of the bulk, the fiber bundle of all the computed geodesics, was very complicated.  Witten suggests that this is the complexity of the famous Hopf fibration so we will have another new result if the ontological basis provides a new understanding of Hopf's famous mathematical representation of geometric complexity.  In reference \cite{KN} we described the topology of $\hat M^3$ as being isomorphic to that of the Kerr--Newman geometry surrounding a charged, rotating black hole.  The topology of the singularity in the Kerr--Newman black hole is $\mathbb{S}^1$, not the normal pointlike singularity, which is actually only one half of $\mathbb{S}^0$ which is two points.  In the natural universe it is unlikely that there exist any uncharged, non-rotating black holes, and therefore there will be pointlike singularities.  One might speculate that, at a very fundamental level, the problems with Hawking's derivation of radiation near a static Schwarzschild black hole cited in reference \cite{ICT} are related to the oversimplification of the Kerr--Newman event horizon as a simple mathematical surface.  The Kerr--Newman horizon shown in figure \ref{fig:KNBH} is multilayered whereas the the Schwarzschild event horizon is a simple surface.

To close this section in which we only lightly touch on twistor theory, and then only with the tip of a long polearm, consider what Misner, Thorne, and Wheeler wrote about Penrose in reference \cite{BBB}.  Penrose's name appeared in the title of the first paper \cite{MOD} this writer submitted to arXiv in 2009.  His name also appeared in the abstract of the second such paper \cite{DE} submitted in 2011 where one finds a statement, ``Following the program of Penrose...''

\begin{quotation}
	``Roger Penrose started out as an algebraic geometer.  [\textit{sic}]  Because if his pure mathematical background, \textbf{his approach to the subject was different} [\textit{emphasis added}] from those which had been adopted hitherto.  He was particularly interested in the global light-come structure of spacetime and in the equations of zero rest-mass fields both of which are preserved under conformal transformations.  He exploited this conformal invariance to give an elegant and powerful treatment of gravitational radiation in terms of a null surface $\mathcal{I}^+$ at infinity.  More recently this interest has led him to develop the theory of twistors, which are the spinors corresponding to the conformal group of Minkowski space.''\newline
\end{quotation}



\subsection{Dyads and Quaternions}

Whereas we have considered a great many things, now we will generate complexity by considering pairs of ontological basis vectors.  There is an intuitive picture wherein one needs to compute arbitrarily long strings of such vectors to define the addresses of arbitrarily many unique lattice sites in the hypercosmos.  It is with these long addresses that we propose to generate the small numbers needed to solve the hierarchy problem.  A first step in that direction is to consider pairs exhibiting the behavior we proposed in reference \cite{ICT}

\begin{align}\label{eq:pipiDYAD}
\hat \pi\hat \pi~~\quad\equiv\quad&~~\pi^2\\
&~\nonumber\\
\hat \Phi\hat \Phi~~\quad\equiv\quad&~~\Phi+1\\
&~\nonumber\\
\hat 2\hat 2~~\quad\equiv\quad&~~1+1+1+1\\
&~\nonumber\\
\hat i\hat i~~\quad\equiv\quad&~~-1~~\label{eq:iiDYAD}.
\end{align}\newline

\noindent A dual vector is a map from vectors to real numbers so these four definitions mean that the ontological basis vectors are their own dual vectors.  The ontological numbers need to have these properties if they are to retain their everyday numerical properties.

Just because Hilbert space vectors are their own dual vectors, that does not mean the ontological vectors live in Hilbert space.  The definition for vectors in $\mathcal{H}'\equiv\mathcal{L}^2$ is

\begin{equation}\label{eq:L2}
\psi\big(x\big)\in\mathcal{L}^2\quad\Longleftrightarrow\quad\int_{-\infty}^{\infty} \psi^*\big(x\big)\psi\big(x\big)\,dx<\infty~~,
\end{equation}\newline

\noindent and that they have an inner product

\begin{equation}\label{eq:HILB01}
\big\langle\vartheta\big|\psi\big\rangle \quad\equiv\quad \int\vartheta^*\psi \,dx~~,
\end{equation}\newline

\noindent which is symmetric under complex conjugation as

\begin{equation}\label{eq:HILB02}
\big\langle\vartheta\big|\psi\big\rangle^* = \big\langle\psi\big|\vartheta\big\rangle~~.
\end{equation}\newline

\noindent We want to impose a non-commutative component where equation (\ref{eq:HILB01}) holds but equation (\ref{eq:HILB02}) does not strictly hold.  We will not discuss every property of $\mathcal{H}'\equiv\mathcal{L}^2$, but only those we intend to systematically change with modular complexification.  We propose to alter equation (\ref{eq:HILB02}) by introducing a new kind of duality along the lines of

\begin{equation}\label{eq:9876tgh}
\big|\psi\in\Omega_1'\big\rangle^*=\big\langle\psi\in\aleph_2'\big|~~,\quad\qquad\text{and}\qquad\quad\big|\psi\in\aleph_1'\big\rangle^*=\big\langle\psi\in\Omega_1'\big|~~.
\end{equation}\newline

\noindent The dual space of either $\aleph'$ or $\Omega'$ will be the next forward iteration of the space so that, following a general process $\mathcal{H}_1\mapsto\Omega_1\mapsto\aleph_2\mapsto\mathcal{H}_2$, the dual of an $\Omega'$ state is in the same unit cell but the dual of an $\aleph'$ state is in the next unit cell (using the convention that the iterator on the object specifies the cell it belongs to).  In slightly more familiar notation equation (\ref{eq:9876tgh}) can be written

\begin{equation}\label{eq:9876tg9h}
\left(\big|\psi;\hat\Phi\big\rangle\,\hat\Phi^1\right)^*=\big\langle\psi;\hat i\big|\,\hat\Phi^2~~,\quad\qquad\text{and}\qquad\quad\left(\big|\psi;\hat i\big\rangle\,\hat\Phi^1\right)^*=\big\langle\psi;\hat\Phi\big|,\hat\Phi^1~~.
\end{equation}\newline



Using these definitions we can contradict the property of $\mathcal{L}^2$ shown in equation (\ref{eq:HILB02}).  The contradictory statement is

\begin{equation}
\big\langle\vartheta\in\aleph_1'\big|\psi\in\Omega_1'\big\rangle^* = \big\langle\psi\in\aleph_2'\big|\vartheta\in\Omega_1'\big\rangle\quad\not\equiv\quad\big\langle\psi\in\aleph_1'\big|\vartheta\in\Omega_1'\big\rangle~~,
\end{equation}\newline

\noindent while still preserving that behavior as a special case like

\begin{equation}
\big\langle\vartheta\in\mathcal{H}'\big|\psi\in\mathcal{H}'\big\rangle^* = \big\langle\psi\in\mathcal{H}'\big|\vartheta\in\mathcal{H}'\big\rangle~~.
\end{equation}\newline

\noindent Another standard property of vectors in Hilbert space that we aim to alter is

\begin{equation}\label{eq:LtwoHILB}
\big\langle\vartheta|\psi\big\rangle\in\mathbb{R}\geq0~~.
\end{equation}\newline

\noindent   The $\sqrt{i}$ channel exists specifically to violate this constraint.  If the three real ontological basis vectors are such that


\begin{equation}
\big\langle\psi;\{\hat \Phi,\hat 2,\hat \pi\}\big|\psi;\{\hat \Phi,\hat 2,\hat \pi\}\big\rangle\geq0~~,
\end{equation}\newline

\noindent then a direct consequence will be that


\begin{equation}
\big\langle\psi;\hat i\big|\psi;\hat i\big\rangle\leq0~~.
\end{equation}\newline

\noindent We aim to make a lot of modifications to the existing theory but there is no guarantee that everything about $\hat M^3$ can be represented in Dirac notation.  Therefore all of the objects presented are, following the style of this book, defined tentatively pending a better follow on analysis.

We have not yet begun to address the products defined by arbitrarily long strings of different lattice vectors that are the addresses of different lattice sites.  If we were to write the cosmological address of a hypothetical lattice site such as the $\hat \pi$-component of the $\hat i$-component of the $\hat \pi$-component of the $\hat 2 $-component of the $\hat \pi $-component of the $\hat \pi $-component of the $\hat 2 $-component of the $\hat i$-component of the $\hat \pi$-component on $\hat\Phi^{11}$ it could look like

\begin{equation}\label{eq:LONGPRODUCT}
\prod_{1}^{N}\hat e^j=\hat \pi\hat i \hat 2 \hat\Phi^7\hat \pi\hat \pi\hat \Phi\hat2\hat \pi\hat\Phi^3\hat i\hat \pi~~.
\end{equation}\newline

The order of the eleven $\hat \Phi$ in equation (\ref{eq:LONGPRODUCT}) does not matter.  The 1D topological flatness condition on chiros means that no matter how many $\hat \Phi^n$'s appear in a string, we will combine them into one hatted object $\hat \Phi^N$, and then we will consider the full nested structure of $\{\hat 2,\hat \pi,\hat i \}$ on that single level of $\aleph$.  By ``full nested structure'' we mean that the order of the non-$\hat \Phi$ objects in the string does matter when specifying an address.  We point this out to demonstrate that $\hat \Phi$ is completely different than $\{\hat 2,\hat \pi,\hat i\}$.  Here we consider a single observer's timeline but point out that nested structure of $\hat \Phi$ would imply divergent timelines.  By ``nested structure of $\hat \Phi$'' we mean the case when the order of $\hat \Phi$ objects in equation (\ref{eq:LONGPRODUCT}) does matter but we are not yet to that level of complexity.  Instead we use only $\{\hat 2,\hat \pi,\hat i \}$ to build an easily understood 3D lattice on one level of $\aleph$, and then we will consider many such levels.  In this book we have proposed to avoid any reliance on the addresses of such sites in the past by encoding the history on $\hat \varphi$ but when we want to compute the amplitudes for the evolutionary futures of the qubit on $\hat \varphi$, it is likely we will need to consider future cosmological lattice sites identified in a manner roughly like that shown in equation (\ref{eq:LONGPRODUCT}).

We have considered pairs of like basis vectors acting on each other outside of the Dirac product but we have not yet considered mixed products of different kinds of ontological basis vectors.  We will make the rule be to take the outer dyadic product when the pairs of basis vectors are mixed.  The cross product is an outer product that only exists in three dimensions but the dyadic product is another outer product that is good for any two basis vectors of consistent dimension.  The dyadic tensor $\hat a \hat b$, also called the dyadic tensor product, is defined by its action on vectors.  It is a map from one vector to another vector, or a map from two vectors to numbers via

\begin{equation}
\big(\hat a\hat b\big)\cdot \hat c = \hat a\big(\hat b\cdot\hat c\big)~~,\quad\qquad\mathrm{and}\quad\qquad\hat d\cdot\big(\hat a\hat b\big)\cdot \hat c = \big(\hat d\cdot\hat a\big)\big(\hat b\cdot\hat c\big)~~.
\end{equation}\newline

\noindent  This is an important property because it allows us to construct both types of objects needed to replicate the Dirac formalism with

\begin{equation}
|\psi\rangle\hat a  \quad\equiv\quad |\psi;\hat a\rangle ~~,
\end{equation}\newline

\noindent so that we can write

\begin{equation}\label{fig:BISP11}
\langle\vartheta;\hat b|\psi;\hat a\rangle \quad\equiv\quad \langle\vartheta|\psi\rangle\,\hat a\hat b=\hat a\hat b\,\langle\vartheta|\psi\rangle~~.
\end{equation}\newline

\noindent The dyadic outer product is not commutative.  For example, using the totally arbitrary assignments for $\hat e^\mu$ let

\begin{equation}\label{eq:ontologiVEC}
\hat \pi = \left(\begin{matrix}
\pi \\ 0 \\ 0 \\ 0 \\
\end{matrix}\right)~,\quad\hat \Phi = \left(\begin{matrix}
0 \\ \Phi \\ 0 \\ 0 \\
\end{matrix}\right)~,\quad\hat 2 = \left(\begin{matrix}
0 \\ 0 \\ 2 \\ 0 \\
\end{matrix}\right)~,\quad\qquad\mathrm{and}\qquad\quad\hat i = \left(\begin{matrix}
0 \\ 0 \\ 0 \\ i \\
\end{matrix}\right)~~.
\end{equation}\newline

\noindent Then we have objects like

\begin{equation}\label{eq:piphiDYAD}
\hat \pi\hat \Phi=\left(\begin{matrix}
0 & \pi\Phi & 0 & 0 \\
0 & 0 & 0 & 0 \\
0 & 0 & 0 & 0 \\
0 & 0 & 0 & 0 \\
\end{matrix}\right)~~,
\end{equation}\newline

\noindent along with another product

\begin{equation}\label{eq:piphiDYAD2}
\hat \Phi\hat \pi=\left(\begin{matrix}
0 & 0 & 0 & 0 \\
\pi\Phi & 0 & 0 & 0 \\
0 & 0 & 0 & 0 \\
0 & 0 & 0 & 0 \\
\end{matrix}\right)~~,
\end{equation}\newline

\noindent that demonstrate the non-commutativity with $\hat\pi\hat\Phi\neq\hat\Phi\hat\pi$.

Equations (\ref{eq:ontologiVEC}-\ref{eq:piphiDYAD2}) all demonstrate objects natural to quaternion operations since they are 4-vectors and $4\times4$ matrices.  Now that we have discovered hyperquaternions in conference proceedings from 1984 \cite{MANN}, and even they have not proven sufficient to finally develop a unified field theory, perhaps $\hat 2$ can act on the hyperquaternions to give an ``ultraquaternion'' solution using the principles outlines in reference \cite{ICT}.  With a fifth component in the quaternion algebra derived from $\hat2x\mapsto x+x$, we may design the additional complexity needed to accommodate O(1,4)$\leftrightarrow$O(2,3) on two quaternion algebras.  When using the ontological objects as pseudo-quaternions (or ultraquaternions) $\mathbb{H}'$ \cite{ICT}, they do not technically meet the definition of quaternions because the coefficient $-i/4$ of $\hat i$ is not real and the quaternions $\mathbb{H}$ all have real coefficients.  Also, the identity is not in the ontological basis but it is a member of the quaternions as formulated by Hamilton.  Therefore let us solve this problem about the inconsistency of $\hat i$ with $\mathbb{H}$ by assigning our ontological pseudo-quaternions as $\{\hat 1,\hat \Phi,\hat\pi,\hat i\}$.   Then if we add a fifth component $\hat 2$ to $\mathbb{H}$, and thereby make two copies of a quaternion algebra (on two co-$\hat \pi$'s), we can take within each instance separately a rigorously perfect set of quaternions $\{\hat 1,\hat \Phi,\hat 2,\hat \pi\}$.

The ordinary quaternions $\{\textbf{1},\textbf{i},\textbf{j},\textbf{k}\}$ have the property that $\textbf{ijk}=-1$ and we can almost certainly replicate this with $ \hat2\hat \pi\hat \Phi=2\pi\Phi$.  We have stated that in each application of $\hat M^3$ the phase in the gauge theory is constrained by the ontological gauge to evolve only across one single cycle of $2\pi$ radians so we can use the periodicity to get rid of $2\pi$ from $2\pi\Phi$ be resetting the final phase $\Delta=2\pi$ to the initial phase required by the MCM: $\Delta=0$.  We also know that when the level of $\aleph$ increases objects like $\hat \Phi^n$ will become like $\hat \Phi^{n-1}$.  For the specific case of $\hat \Phi^1$ this gives $\hat \Phi^0=\hat 1$.  We can attribute the missing negative sign to the $\sqrt{i}$.  In a fairly reasonable and direct manner we have engineered $2\pi\Phi\to-1$.

Now we have considered pairs of basis vectors.  What shall be the general case for the consideration of triples?  This is exactly the problem Hamilton was considering when he invented the quaternions.  We want to give rigorous definitions to long strings of unit vectors that define each lattice site but the dyadic only involves pairs.  It makes a lot of sense to let the self product of two like basis vectors be the inner product and let the mixed product of two dissimilar basis vectors be the outer product, but what would be the interpretation for the long product that keeps track of strings like that in equation (\ref{eq:LONGPRODUCT})?  We have run into a roadblock on the way to complexification after only considering the second order lattice sites.  However, it looks like simple pairs will be enough for now.  If the Dirac matrices can be combined to make exactly 16 independent objects, then it is critically important to note that there are exactly sixteen independent objects when we consider all iterations of dyadics like equations (\ref{eq:piphiDYAD}) and (\ref{eq:piphiDYAD2}).  In this way perhaps it will be simpler to only use the dyadic product, even for like pairs, and replace equations (\ref{eq:pipiDYAD}-\ref{eq:iiDYAD}) with

\begin{align}
\hat \pi\hat \pi&=\left(\begin{matrix}
\pi^2 & 0 & 0 & 0 \\
0 &0  & 0 & 0 \\
0 & 0 &0  & 0 \\
0 & 0 &0  & 0 \\
\end{matrix}\right)\label{eq:sameDYAD2}\\
&~\nonumber\\
\hat \Phi\hat \Phi&=\left(\begin{matrix}
0 & 0 &0  & 0 \\
0& \Phi^2 & 0 &  0\\
0 & 0 &0  & 0 \\
0 & 0 &0  & 0 \\
\end{matrix}\right)\\
&~\nonumber\\
\hat 2\hat 2&=\left(\begin{matrix}
0 & 0 &0  & 0 \\
0 & 0 &0  & 0 \\
0& 0 &  4&0 \\
0 & 0 &0  & 0 \\
\end{matrix}\right)\label{eq:sameDYAD3}\\
&~\nonumber\\
\hat i\hat i&=\left(\begin{matrix}
0 & 0 &0  & 0 \\
0 & 0 &0  & 0 \\
0 & 0 &0  & 0 \\
0& 0 & 0 & -1 \\
\end{matrix}\right)~~.
\end{align}\newline


\noindent We left a lot of unanswered questions when we considered the quaternions in reference \cite{ICT} whose main goal was to raise those questions.  We won't propose to treat those questions now but will refer to one of the main takeaways reported after examining the quaternions in reference \cite{ICT}: the 4D structure is a lot like Dirac's theory (and Einstein's theory.)  We have now shown another set of 16 objects which is different than yet exceedingly similar to the quaternion argument made in reference \cite{ICT}.

After noting how the dyads look like the Clifford algebra, also consider a general Lorentz transformation in matrix form

\begin{equation}\label{eq:Ltransform}
\left(\begin{matrix}
ct'\\
x'\\
y'\\
z'\\
\end{matrix}\right)=\left(\begin{matrix}
\gamma&-\beta\gamma&0&0\\
-\beta\gamma&\gamma&0&0\\
0&0&1&0\\
0&0&0&1\\
\end{matrix}\right)\left(\begin{matrix}
ct\\
x\\
y\\
z\\
\end{matrix}\right)~.
\end{equation}\newline

\noindent  Evidently the ontological dyads can build Lorentz transform matrices as well.  Every Lorentz transformation can be written as the sum of a rotation and a boost so even the symmetries of the Lorentz group might be motivated by the ontological basis.  We know the Lorentz group has six parameters and it will take six dyadic products $\hat \pi\hat \Phi$, $\hat \Phi\hat \pi$, $\hat \pi\hat \pi$, $\hat \Phi\hat \Phi$, $\hat 2\hat 2$, and $\hat i\hat i$ to build equation (\ref{eq:Ltransform})'s typical example of velocity in one dimension.  More complex Lorentz transform matrices will be built with more than six dyads so they are not the Lorentz parameters themselves but it is good that the small set of dyads we used to demonstrate complexity can be used to demonstrate the Lorentz transform in special relativity.

\subsection{Unification}
\addtocontents{toc}{\protect\vspace{22pt}}

One of the major things that remains to be clarified in the precise form of the maps

\begin{align}
f^3\big|\psi;\hat\pi\big\rangle\quad&\mapsto\quad T_{\mu\nu}\label{eq:233maps}\\
&~\nonumber\\
i \big|\psi;\hat\Phi\big\rangle\quad&\mapsto\quad G_{\mu\nu}\\
&~\nonumber\\
\big|\psi;\hat i\big\rangle \quad&\mapsto\quad g_{\mu\nu}\Lambda~~,\label{eq:233maps2}
\end{align}\newline

\noindent through which we propose to unify gravitation and quantum theory.  This does not appear to be low hanging fruit and we have not yet reached for it.  Certainly detractors will say that we have not yet unified the theories since we have not yet defined the maps with some specific definition that would have to be confirmed or denied by experimentalists.  This is not an invalid criticism.  To date work on the MCM has only discovered the mechanism of unification.  We have not yet demonstrated it by writing the maps as specific functionals of wavefunctions.  However, the discovery of a new mechanism for unification, the only one that exists,\footnote{The reader should be very careful to note that no one has ever proposed a mechanism such as the MCM mechanism and there should be no insistence to treat it is another blip on a radar screen full of many such blips.  It is the only blip on the screen and it is the only blip that has ever appeared on the screen.} is independently a great accomplishment.  Now it is possible for other researchers to tinker with the mechanism but that was not possible before it was discovered by this writer in 2012 \cite{TER}.  Note well, if no one ever discovers a mechanism of unification then the theories of gravitation and quanta will never be unified.  When the theories are eventually unified according to the specifications of detractors, that unification will unequivocally follow the discovery of the mechanism of that unification.  Even then, detractors will not desist from their detractions until the experimentalists confirm or deny any specific functionals proposed and furthermore detractors seem to be in control of scientific funding so that is unlikely to happen on a satisfactory timescale.  Instead of searching for the specific functionals preferred by everyone who was unable to figure out a mechanism for unification we have wisely pursued the course of research that led to the result in section I.3 that is the main new result presented in this book.  The absence of a set of specifically defined functions for some problem in applied physics takes away nothing from our discovery of the maximum action path hidden along the Cauchy curve around infinity.  It takes away nothing from the many independently valid results derived in the MCM and yet detractors continue to insist that one incomplete problem means that all of the problems are incomplete.

Once the mechanism is discovered then it does become possible to unify the theories of gravitation and quanta.  This is clearly a two step process and accomplishing either step is a big deal.  Thus the identification of a mechanism of unification, the first one ever, is clearly an important discovery worthy of praise despite the protestations of people who know nothing about the scientific method or who do know but are blinded by their hubris.  If the specific maps are found in the next 90 years or so then history will show that finding the mechanism of unification was the more difficult problem between finding the mechanism and finding the functionals.  In the time after unification is achieved according to the semantic definition preferred by detractors and not the one preferred by this writer which would mean that unification has already been achieved, the thing that will be first demonstrated to the physics students of the future will be the mechanism of the unification.  Physics professors will say, ``This mechanism unifies the disparate mathematical frameworks of differential geometry and quantum field theory,'' and the specific forms of the maps will likely be left as a homework problem in advanced graduate coursework.

In addition to the mechanism of unification in maps (\ref{eq:233maps}-\ref{eq:233maps2}) we have demonstrated so many other successful aspects of the theory that any reasonable person should assume that the maps exist and that the two theories will be properly unified through the mechanism exposed in this research program.  When the MCM's other myriad results exist it is highly irregular that funded researchers have not taken up the search for functionals while we remain focused on more important issues dealing with the fundamentals of infinite complexity (such as those in reference \cite{ZETA}.)  Thus are the prerogatives of those researchers and if they choose not do it first then we will get to it when we get to it.  Since it will be impossible to know if any derived maps are correct without an experimental confirmation, all detractors that currently detract are most likely continue to do so even if we did propose a specific set of functions.  If the physics community at large has pretended to ignore our other experimental predictions and other mathematical results that are already fully complete then it is likely that specific analytical relationships of the form

\begin{align}
T_{\mu\nu}&=f_T[\psi;\hat\pi]\\
&~\nonumber\\
G_{\mu\nu}&=f_G[\psi;\hat\Phi]\\
&~\nonumber\\
g_{\mu\nu}\Lambda&=f_{g\Lambda}[\psi;\hat i]~~,
\end{align}\newline

\noindent would also be ignored.\footnote{This statement is highly conditional.  They would only be ignored if they were correct.  If they were demonstrably incorrect, as they likely would be if the endeavors to find them were undertaken by this writer prematurely, then they would likely not be ignored.  In that case detractors would probably pounce upon that erroneous result to rhetorically assault other results that are logically unassailable.}  Recall that we have solved very many problems besides the problem of unification and solving one more is unlikely the change the sentiments of the cretins in the anti-science conspiracy.  The primary of example of that to which we refer is the non-publication of the full spin analysis for the Higgslike particle \cite{RALSTON}.  Why has it not been carried out and published if not to spite this writer by denying him the experimental confirmation that detractors will always refer to regardless of how specific our theoretical results eventually become?

The aspects of the theory that provide such strong evidence of its correctness include the production of three numbers which are unambiguously the most important dimensionless constants in physics.  The leading coefficient $1/4\pi$ of the ontological resolution of the identity

\begin{equation}
\hat 1=\dfrac{1}{4\pi}\,\hat\pi-\dfrac{\varphi}{4}\,\hat\Phi+\dfrac{1}{8}\,\hat2-\dfrac{i}{4}\,\hat i~~,
\end{equation}\newline



\noindent is the dimensionless electromagnetic coupling constant.  Electromagnetism is a cornerstone of the physics that we more or less understand and we have taken $\hat \pi$ as the sector of known physics in $\mathcal{H}$ before adding $\{\hat2,\hat \Phi,\hat i\}$ to construct an ontological basis for new physics in the hypercosmos.\footnote{The hypercosmological lattice structure is not unlike the concept of an akashic record.  Measurements get made at $\hat\pi$-sites and there they remain there when the observer makes another one at the next $\hat\pi$-site.}  It is a good demonstration of the MCM's consistent, non-contradicting validity\footnote{See also reference \cite{QG} for further discussion of the MCM's consistent, non-contradicting validity.} that the coefficient associated with $\hat \pi$ is the dimensionless electromagnetic coupling constant.  We have uncovered consistency in every corner of the MCM but never once have we uncovered any inconsistency and none has been demonstrated by detractors.  Secondly we have generated the dimensionless coupling constant from Einstein's equation $8\pi$ and will discuss it in the next chapter.  Thirdly we arrive at the first important dimensionless constant identified in the MCM: the fine structure constant $\alpha_{M\!C\!M}=2\pi+(\Phi\pi)^3$.  This differs from the currently accepted value $\alpha_{Q\!E\!D}$ by about 0.4\% and it is totally obvious that quantum theory can be reformulated with $\alpha_{M\!C\!M}$ by moving the 0.4\% discrepancy into the quantum uncertainty found in some other corner of the theory.

Another very strong indicator for the overwhelming likelihood of the existence of a set of functions that will clarify and define maps (\ref{eq:233maps}-\ref{eq:233maps2}) with specific, correct, useful definitions is the MCM's modified model of particle physics.  In the standard model of particle physics, there are four fundamental gaugeons plus the spinless scalar Higgs boson.  In the MCM scheme, there are no fundamental scalar particles and we predict two spin-1 gaugeons $G^\pm$ and $\zeta^0$ where the standard model only predicts the spin-0 scalar particle $H^0$.  At a very fundamental level it is likely that we have been able to make so much progress through diagrammatic representations alone because the conformal invariance of the MCM gauge theory is not broken by the existence of a fundamental scalar particle.  All of the pieces of the MCM are conformally invariant in the gauge theory but if we tried to include $H^0$ then there would be a breakdown in the global conformal invariance.

As an aside about particle physics we mention the neutrino mass.  Neutrino masses violate the predictions of the standard model but they have never been measured directly.  The existence of massive neutrinos is inferred from neutrino flavor oscillation; a neutrino of one flavor at a source can show up in a detector as any of the three flavors: electron, muon, or tau.  The argument goes that if neutrinos were massless then they would move at the speed of light along the null interval in spacetime.  When $ds^2=0$ for every neutrino trajectory it means that neutrinos do not experience time and therefore should not be able to change flavors.  Change is uniquely a property associated with time; timeless things don't change.  However, now that we have introduced a second kind of time into the MCM we should re-evaluate whether or not flavor oscillation strictly implies mass.  It might be that massless neutrinos do move on the null interval as predicted by the standard model but there is some oscillation allowed due to the passage of chirological time which is orthogonal to the spacetime of only spacelike, timelike and null intervals.


\begin{figure}[t]
	\makebox[\textwidth][c]{\includegraphics[scale=0.8]{particles.png}}
	\captionsetup{format=hang}\caption{The $4\times4$ symmetry of the standard model is broken by the spin-0 $H$ scalar boson but the $4\times4$ symmetry of the modified model is completed with the spin-1 $G$ and $\zeta$ gauge bosons.  See reference \cite{QS} for details.}\label{fig:particles}
\end{figure}



The symmetry of the MCM particle scheme with respect to the asymmetry of the standard scheme (figure \ref{fig:particles}) makes a good case for the overall correctness of the MCM but it still remains for the political agents at the LHC to report the spin of the particle they discovered in 2012.  The official source of what is and is not known in particle physics is the Particle Data Group.  They publish a yearly volume and the entry for the Higgs boson in the 2016 volume\cite{PDG} we find the following wherein the reader should note the deliberate uncertainty in the choice of words and how that tone contrasts the ordinary style of concise language one finds in that group's publications.

\begin{quotation}
	``$H^0$ refers to the signal that has been discovered in the Higgs searches.  Whereas the observed signal is labeled as a spin 0 particle and called a Higgs Boson, the detailed properties of $H^0$ and its role in the context of electroweak symmetry breaking need to be further clarified.''\newline
\end{quotation}

Indeed they do need to be clarified because no one knows if the particle announced by CERN four years prior to the publication of reference \cite{PDG} is the particle predicted by Higgs and his contemporaries or if it is the particle predicted via the MCM.  If they report that the particle has spin-1 then that will make an even stronger case for the existence of the specific, physical functions that will replace the arrows in maps (\ref{eq:233maps}-\ref{eq:233maps2}).  There are very many things that have worked out surprisingly well in the MCM such as the new mechanism for AdS/CFT \cite{GC} and altogether there is an irrefutable likelihood that specific physical forms of the maps do exist and can be written down.  There have been so many such discoveries in this research program, each making a small argument for the overall physical correctness of the MCM, that altogether they do form a convincing argument. However, the fact remains that we still have not identified the specific functionals of wavefunctions or their inverses and that is on the to do list.

To find the requisite functionals we should look for a connection between the quantum state vectors and the general relativistic equation of state.  Recall that Einstein's equation alone, much like maps (\ref{eq:233maps}-\ref{eq:233maps2}), does not determine physics.  Einstein's equation conserves the energy of a universe but we also need to know the universe's equation of state before we can do physics properly.  There are very many common equations of state for a model universe and reference \cite{BENOIT} contains a comprehensive survey of the most common and useful ones.  One thing that all general relativistic equations of state have in common is a pair of thermodynamics parameters: the pressure $p$ and the energy density $\rho$.  Since any two parameters are defined by their ratio $w$ we can expect that it will be productive to study ontological equations of state wherein the ratio of these two parameters is the golden ratio.  Perhaps we can set them relative to each other as $w=\Phi$ in $\Sigma^+$ and $w=\varphi$ in $\Sigma^-$ to set a non-equilibrium condition across $\mathcal{H}$ or $\varnothing$ which leads to the laws of thermodynamics: entropy increases and time goes forward.  Davies has discovered the golden ratio in an independent study of black hole thermodynamics \cite{DAVIES}.  The title of Davies' paper is \textit{Thermodynamic Phase Transitions of Kerr--Newman Black Holes in de Sitter Space} so here we have the following MCM elements already bundled together in 1989: the golden ratio, Kerr--Newman black holes, phase transitions, and de Sitter space.  To repeat what we have stated about the MCM already, we expect that the topological compartmentalization of the MCM unit cell is exactly that of the Kerr--Newman black hole \cite{KN} and obviously half of that cell is de Sitter space (the other half is anti-de Sitter space.)

Regarding maps (\ref{eq:233maps}-\ref{eq:233maps2}), we have made them such that they point to both sides of Einstein's equation but we could make the four maps on $|\psi;\{\hat i,\hat \Phi,\hat 2,\hat \pi\}\rangle$ point to the 16 places in the matrix representation of or the Einstein tensor $G_{\mu\nu}$ or the stress energy tensor $T_{\mu\nu}$, or there are very many combinations.\footnote{Under ordinary conditions this writer's fellow researchers would jump at the chance put their names on arbitrarily long catalogs of the implications of the finitely many intuitive guesses at specific functionals.  The reader is invited to recall that the quantum hall effect was finally solved by guessing and Laughlin shared the 1998 Nobel Prize in physics for his guess.  However, something about the conditions surrounding the MCM is extraordinary.}  Since one of $T_{\mu\nu}$ or $G_{\mu\nu}$ determines the other through Einstein's equation (given $g_{\mu\nu}\Lambda$) there is no need for for us to send the qubit into both of $T_{\mu\nu}$ and $G_{\mu\nu}$.  However, if we do not send the information to both sides of Einstein's equation then we lose the unique connection to the all-important coefficient $8\pi$.  Another thing to consider is that a single quanta should not determine the state of the entire universe.  In that case we are drawn to maps of the form

\begin{equation}
f^3|\psi;\hat\pi\rangle\quad\mapsto\quad T_{\mu\nu}=T^0_{\mu\nu}+T^\psi_{\mu\nu}
\end{equation}\newline

\noindent or



\begin{equation}
i|\psi;\hat\Phi\rangle\quad\mapsto\quad G_{\mu\nu}=G^0_{\mu\nu}+h_{\mu\nu}\big[\psi(x^\mu_+)\big]
\end{equation}\newline

\noindent  Furthermore when mapping qubits to both sides of Einstein's equation quantum weirdness may imply that quantum states affect the curvature differently than would be expected if we simply put the energy of the qubit into the stress-energy tensor.




\clearpage

\renewcommand\thesection{\Roman{section}}
\section{{\textsf{\LARGE Maximal Symmetry}}}
\renewcommand\thesection{\arabic{section}}

The first section in this chapter restates the MCM hypothesis and points out a few obvious criticisms which are mitigated with arguments in later sections.  The second section recounts some drama related to the development and publication status of the documents that characterize this research program.  Section three contains a few orphaned comments and in section five we make a definition for a multiplex.  Sections four and six are dedicated to addressing the criticisms from section one.  In section four we also propose to create small perturbations $h_{\mu\nu}$ on the Minkowski metric by taking the difference of large perturbations in $\Sigma^\pm$.  Additionally in section six we strengthen and clarify the loose definition $\Psi=0$ that was reported in reference \cite{IC}.  In section seven we discuss the properties of eponymous maximally symmetric spacetimes.  In section eight we treat the geodesics of the MCM unit cell but do not calculate them.  The main result of this chapter is in section nine where we show how dark energy and expanding space are both expected properties of the piecewise metric used in the MCM.  Section ten is a brief summary of relevant aspects of the advanced and retarded potentials.

\subsection{The MCM Hypothesis}


The first step in building the MCM is to assume a hypothesis that momentum is always conserved and thereby pursue a different model than most of the big bang theorists.  Taking a shortcut to something that can be analyzed specifically, we have hypothesized that the third derivative with respect to chronological time should be equal to the third derivative with respect to chirological time.

\begin{equation}\label{eq:onefreq1}
\hat M^3\big|\psi;\hat \pi\big\rangle=\hat M^3\big|\psi;\hat \pi\big\rangle  \quad\implies\quad \partial^{\,3}_0\big|\psi;\hat \pi\big\rangle:=\partial^{\,3}_5|\psi;\hat \pi\big\rangle~~.
\end{equation}\newline

\noindent The operator that appeared as $\partial_4$ in earlier work now appears as $\partial_5$ because we have changed the counting convention on the Latin indices $A$.  The law says that the third derivative of a state with respect to chronological time is defined according to its third derivative with respect to chirological time.  The next step is to choose $\psi$ so that the hypothesis becomes

\begin{equation}\label{eq:onefreq}
\omega^3\big|\psi;\hat \pi\big\rangle=i\pi\Phi^2|\psi;\hat \pi\big\rangle~~,
\end{equation}\newline

\noindent and we will say a lot about how to accomplish that in this chapter (after first referring the reader to reference \cite{IC} for full details.)  We reduce the equation (\ref{eq:onefreq}) which is quadratic in $\Phi$ to a linear one with $\Phi^2=\Phi+1$ to achieve

\begin{equation}\label{eq:ALG1}
\omega^3\big|\psi;\hat \pi\big\rangle=i\pi\Phi|\psi;\hat \pi\big\rangle+i\pi|\psi;\hat \pi\big\rangle~~,
\end{equation}\newline

\noindent and this gives some general idea about how one might quantize third order equations that are ordinarily taken as unquantizable.  We can take an equation of almost any order in $\Phi$ and reduce the order through an operation like $\Phi^3=\Phi^2+\Phi=2\Phi+1$.  We have still not identified all the problems we hope to solve through some method of achieving equations in a given order.  However, since the theory of infinite complexity uses the advanced potential that is third order in $\partial_t$, we will almost certainly be required to quantize a third order potential as part of the quantization process for the classical MCM Hamiltonian of the EM field because it depends on the advanced and retarded potentials.

After writing equation (\ref{eq:ALG1}) we use $\omega=2\pi f$, insert the identities $\hat \pi=-\varphi\pi\hat \Phi$ and $\hat \pi=-i\pi\hat i$, and then shuffle the hats to write

\begin{equation}\label{eq:ALG2}
8\pi^3f^3\big|\psi;\hat \pi\big\rangle=i\pi^2|\psi;\hat \Phi\big\rangle+i\pi^2|\psi;\hat \pi\big\rangle~~.
\end{equation}\newline

\noindent Then we normalize by $\pi^2$ and say that

\begin{equation}
8\pi f^3\big|\psi;\hat \pi\big\rangle=i|\psi;\hat \Phi\big\rangle+|\psi;\hat i\big\rangle\label{eq:MCMalgebra}~~,
\end{equation}\newline

\noindent is Einstein's equation via

\begin{align}
f^3\big|\psi;\hat\pi\big\rangle \quad\equiv\quad &\psi\big(x^\mu\big) \quad\mapsto\quad T_{\mu\nu}\label{eq:33maps}\\
&~\nonumber\\
i \big|\psi;\hat\Phi\big\rangle\quad\equiv\quad &\psi\big(x^\mu_+\big) \quad\mapsto\quad G_{\mu\nu}\\
&~\nonumber\\
\big|\psi;\hat i\big\rangle\quad\equiv\quad &\psi\big(x^\mu_-\big) \quad\mapsto\quad g_{\mu\nu}\Lambda~~,\label{eq:33maps2}
\end{align}\newline

\noindent and we continue to ignore some minus signs.  The central column in equations (\ref{eq:33maps}-\ref{eq:33maps2}) shows another new trick introduced for this research.  Even in exotic studies where the Gelfand triple is considered, the position space representations of the vectors in $\{\aleph',\mathcal{H}',\Omega'\}$ are taken in the coordinates of $\mathcal{H}$ but we will take their position space representations in the coordinates of the similarly named manifolds $\{\aleph,\mathcal{H},\Omega\}$.  We have already shown in the previous chapter how to accommodate $\hat 2$ in the scheme of maps (\ref{eq:33maps}-\ref{eq:33maps2}) with the decomposed Einstein tensor and the remaining definition to compliment the above will be

\begin{equation}\label{eq:eq22222}
\big|\psi;\hat 2\big\rangle \quad\equiv\quad \psi(x^\mu_\varnothing)~~.
\end{equation}\newline

\noindent For the purposes of discussing the logical development of the hypothesis, we will use the original convention given by maps (\ref{eq:33maps}-\ref{eq:33maps2}) without including $\hat 2$.  There are so many possible ways to insert $\hat2$ it is unlikely that we would guess the correct one at this point.

We say equation (\ref{eq:MCMalgebra}) is Einstein's equation but what does that mean?  Here we will be more careful with our notation because there are a few things listed below that are not perfect in the sense of the $=$ symbol that appears in equation (\ref{eq:MCMalgebra}).  However, that takes away little from the magnitude of the discovery because we can swap in the $:=$ symbol and it is clear that a previously undocumented mechanism has been documented and now appears in the literature.  Rather than a strict equality, we can more freely use the $:=$ symbol which means ``is defined according to.''  In section III.5 we will give a formal mathematical definition for \textbf{multiplex} although we will also continue to use the word informally.   Where the $=$ sign denotes an equality, the $:=$ sign shall denote a multiplex.

Before embarking upon an interesting, maximally symmetric aside about the history of the Earth in the years around 2012 A.D. we will list the most obvious criticisms of the hypothesis, and then after discussing the history we will return to the problems pointing out how the multiplectic formalism is superior, in many ways, to the ordinary ideas of formalism for quantized determinism with arbitrary spinors.  The problems with the hypothesis are as follows.

\begin{enumerate}
    \item Equation (\ref{eq:onefreq}) implies that only one frequency $\omega=\sqrt[3]{i\pi\Phi^2}$ is allowed.
    \item When everything is proportional to $\psi$, as in equations (\ref{eq:onefreq1}-\ref{eq:MCMalgebra}), there is no room for complexity-generating double orthogonality in the maps (\ref{eq:33maps}-\ref{eq:33maps2}).  To build double orthogonality we must first start with single orthogonality but $\psi$ is not orthogonal to itself.  When we use maps (\ref{eq:33maps}-\ref{eq:33maps2}) to convert to tensors, all of the tensors will be linearly dependent on each other through $\psi$.  In general, to model an arbitrary cosmos with perturbations, there needs to be at least two independent tensors in Einstein's equation.
    \item In reference \cite{IC} we proposed to make the left and right sides of equation (\ref{eq:onefreq1}) independent by switching to a law of the form $\partial^{\,3}_0|\psi(x^\mu)\rangle=\partial^{\,3}_5|\Psi(\chi^A)\rangle$ but then we have a problem of how a 4D object on one side could be equal to a 5D object on the other side.  Recall from reference \cite{OP} that not only is $\Psi$ a function of the 5D coordinates, it is a five component vector where $\psi$ is only a four component vector.
    \item Even if we solve problem two and accomplish a workaround for the dimensionality issue in problem three, we still have the same problem of linear dependence as in problem two, only to a lesser degree.  When we decompose the Einstein tensor into the Ricci tensor and the metric with

    \begin{equation}\label{eq:decomp762}
    i\big|\Psi;\hat \Phi\big\rangle \quad\equiv\quad G_{\mu\nu} \quad\equiv\quad R_{\mu\nu}-\dfrac{1}{2}Rg_{\mu\nu}~~,
    \end{equation}

    both objects are, by construction, defined by $\Psi$ but we know that fully dynamical spacetime requires $R_{\mu\nu}$ and $g_{\mu\nu}$ to be, at least sometimes, independent.
\end{enumerate}

Since we have introduced the concept of double orthogonality, we could try to solve the problem of the uniform dependence on $\psi$ strictly in the double orthogonal channel for ontological basis vectors (without introducing $\Psi$ as in problem three) but even then the Ricci tensor and the metric tensor are neither single nor double orthogonal because of problem four.  In equation (\ref{eq:decomp762}) $R_{\mu\nu}$ and $g_{\mu\nu}$  both live inside $i|\Psi;\hat{\Phi}\rangle$ so even in the second orthogonal channel they are the same: they are both proportional to $\Psi$ and they both live on $\hat \Phi$.  This is problematic because there is no infinite complexity; there are just four extra complexes $\{\hat i,\hat \Phi,\hat 2,\hat \pi\}$ so if we try to get single and double orthogonality from just four numbers we will be in for a burdensome and likely fruitless task.  Even if we use $\hat 2$ as in equations (\ref{eq:Rconst1}) and (\ref{eq:Rconst2}) which were

\begin{equation}
\frac{1}{2}\big|\Psi;\hat\Phi\big\rangle~~\mapsto~~ R_{\mu\nu}~~,
\qquad\quad\mathrm{and}\qquad\quad
\frac{\Phi}{4}\big|\Psi;\hat2\big\rangle~~\mapsto~~ \frac{1}{2}Rg_{\mu\nu}~~,
\end{equation}\newline

\noindent and

\begin{equation}
\frac{1}{2}\big|\Psi;\hat\Phi\big\rangle~~\mapsto~~ \frac{1}{2}Rg_{\mu\nu}~~,
\qquad\quad\mathrm{and}\qquad\quad
\frac{\Phi}{4}\big|\Psi;\hat2\big\rangle~~\mapsto~~ R_{\mu\nu}~~,
\end{equation}\newline


\noindent that will only achieve single orthogonality, albeit in the second orthogonal channel.  The ontological basis is hardly infinite; we want to generate complexity by applying the basis to the already infinite dimensional Hilbert space of $\psi$ and, presumably, $\Psi$.

If we wanted to, we could even trace this issue about linear dependence all the way back to equation (\ref{eq:onefreq1}) to say that there is no room for any complexity at all: $\psi=\psi$ and there is only one frequency.  However, we will not go that far.  We will guarantee some modicum of complexity simply by defining the $\partial^{\,3}_5$ operator to require a representation of $\psi$ in the chirological coordinates.  Recall from chapter one that there is some complex discrepancy between the $x^\mu$ and $\chi^A$ coordinates due to technical nuance in the dual tangent space.

\subsection{Historical Context}


In this chapter we will show how none of problems one through four are actually problems at all.  Since we are writing a short book rather than a short paper with the intention of using these extra pages to review the basics of that which should have been immediately obvious to any subject matter experts with half the entire field committed to memory, we will condense all previous $:=$ workarounds into formalism that is rigorously correct using the multiplectic formalism.  By ``immediately obvious'' we mean to derisively imply that the more than seven years it has been since people started passing MCM literature around is far too long and the so-called self-correcting mechanism in science is broken as fuck.

This section is dedicated to evidence of the brokenness (non-existence) of the alleged self-correcting mechanism in science and shortly we will return to the problems listed in the previous section.  Reference \cite{MOD} contains the first written description of the MCM.  It was rejected by arXiv in September 2009 and that manuscript is likely the basis for the articles titled ``Is the Universe Inside a Black Hole?,'' that Nikodem Poplawski has been successfully publishing in popular media since 2010.  The MCM phrase ``inverse radial spaghettification'' \cite{DE} is a fancy way to say that the universe is inside a black hole and now, in newer research, we have gone on to show that the observer resides on a singularity at the origin of coordinates marking each level of $\aleph$.\footnote{In fact, even by the time arXiv rejected reference \cite{DE} in 2011, we had already moved the observer from an arbitrary moment to the moment at the apex of a quantum geometric bounce.  Such bounces only occur on the interiors of the event horizons of black holes.  When the entire universe bounces \textit{\`a la} LQC, that is simply occurring inside a super-massive black hole which contains (or is) the entire universe.  The MCM goes far beyond LQC when it identifies a dark energy candidate in the mechanism.}  It is commonly understood that singularities mark the center of black holes so universe-in-a-black-hole is very much a facet of the MCM.  We suggest that Poplawski began writing these articles after he was inspired to do so by the original MCM manuscript \cite{MOD} that he obtained \textit{somehow}.

Similarly at the end of September 2009, Ashtekar, Campiglia, and Henderson published reference \cite{ABHAY1} wherein the first citation is to the Feynman paper \cite{FEYN} that we have considered in chapter one.  This is interesting because Ashtekar had not been citing Feynman's war-era papers from 70 years ago but then he did do so immediately after this writer distributed reference \cite{MOD}.  Reference \cite{MOD} begins with a quote taken from one of Feynman's less famous war era papers where he makes comments about the time ordering of events not being as important as the way they are encoded in his formalism.  A main result of reference \cite{MOD} was an alternative interpretation for the method Feynman described.\footnote{This method is the one in long excerpt from Feynman's reference \cite{FEYN} which appears presently in section I.3.}  This is the method we have proposed to modify by the inclusion of the maximum action path.  arXiv lists the submission date on Ashtekar \textit{et al.}'s paper, reference \cite{ABHAY1}, as about one or two weeks after an anonymous and/or unscrupulous reviewer at arXiv rejected reference \cite{MOD}.\footnote{Unfortunately we have no record of the date of the original submission of reference \cite{MOD} to arXiv, but it was probably around the 15th of September.  After presenting a result to the 2009 meeting of the IceCube Collaboration at Humboldt University in Berlin, this writer enjoyed the gracious patience of the Germans for a few days and then returned to Atlanta to upload the manuscript to arXiv on a Monday or Tuesday expecting it to appear online either Tuesday or Wednesday.}  Since we multiply cited LQC\footnote{(LQC and LQG were not cited directly in 2009 but instead we used the terms ``bouncing'' and ``the repulsive force of quantum geometry'' which were taken from Ashtekar's 2009 talk at Georgia Tech.} within reference \cite{MOD}, a theory whose bottom-liners include Ashtekar,\footnote{The bottom-liners also include Bojowald who declared LQC ``dead'' in 2013.  See video reference \cite{BOJO}.} it is likely that the arXiv reviewer, if it was not Ashtekar himself, sent the manuscript to Ashtekar.  Additionally, Ashtekar may have obtained the manuscript not through arXiv but through another channel.  Just weeks before Ashtekar \textit{et al.} published reference \cite{ABHAY1}, this writer had distributed copies of reference \cite{MOD} in the newly opened Center for Relativistic Astrophysics (CRA) whose founding faculty include two former colleagues of Ashtekar's: Pablo Laguna\footnote{Laguna deserves an honorable mention and thanks for inviting not just Ashtekar to Georgia Tech, but also Penrose, meaning that both of the speakers that inspired the MCM were the invitees of Laguna.} and Dierdre Shoemaker.  The purpose of the email distribution was to advertise that this writer would give a talk on the MCM in the CRA that week.  Shoemaker, who had been working side by side with Ashtekar in Pennsylvania just a year earlier, was in attendance but she was most intently on her phone throughout the talk,\footnote{One wonders how one could pursue a PhD, make it through the academic grinder into a tenure track position, get a promotion as a founding member of a center for relativistic astrophysics, and then show absolutely no interest when some of the most important astrophysical mysteries of the universe are plainly spelled out before one's eyes on a white board.  Affirmative action likely explains the whole thing.} almost intentionally projecting disinterest, or \textit{disrespect}, and is unlikely to have made any effort to help this writer disseminate his research.

The key point in all of this is that although somehow reference \cite{MOD} was not good enough even to be uploaded as a preprint, it seems to have been good enough to prompt an immediate response paper \cite{ABHAY1} from leading names in the field.  Usually eliciting a response paper at all is considered a high achievement in theoretical physics and an immediate response from a leader in the field (Ashtekar) is high praise indeed.  As a counter example, consider that most papers passing the ``very high,'' ``very meaningful,'' ``critically important'' bar of peer-review go on to be completely ignored and accumulate a layer of dust serving as a reminder that paper did at one point pass peer-review meaning that the publishing cartel bestowed a cookie upon the authors who can all add the cookie crumbles to their C.V.'s... which mean nothing weighed against the merit of the research that appeared in the publication.  The cartel's cookie crumbles have become overly important in the modern era where the merit of the research in question is too often non-existent or not significant.

Despite science's alleged self-correcting mechanism, the exact dynamic from 2009 unfolded again in 2011.  Once again arXiv rejected another manuscript, reference \cite{DE}, based on some unpublished set of censorship guidelines.\footnote{If they are unpublished, are they even guidelines?  Or does the uncertainty principle mean that they are always whatever the anonymous reviewer wants them to be?  What does the uncertainty principle tell us about unpublished guidelines in the national security apparatus?}  It seems that after this newer manuscript made the backchannel rounds, negative frequency resonant radiation was immediately discovered \cite{RUBINO} and a team at USC immediately built a working quantum computer \cite{QCOMP}.  Note that since frequency is inverse time, negative frequency resonant radiation is a negative time mode exactly like the $|t_-\rangle$ state we suggested only months earlier in reference \cite{DE}.   In reference \cite{FSC} we suggested to look for correlations with delay and then just a few months later the BaBar collaboration announced that they had decided to reanalyze their old data for correlations with delay and that they did affirmatively find them \cite{BABAR}.

In 2009, reference \cite{MOD} was not even good enough to be allowed as a preprint but it garnered a response, which is very high praise.  In 2011, the paper \cite{DE} still did not meet the bar of arXiv's unpublished censorship criteria and not only did it garner a response paper, it garnered response experiments.  This is outrageously high praise because experiments cost time \textit{and} money whereas papers only cost time.  It means that the ``peers'' of this writer have ``reviewed'' the manuscript and decided to change research direction in favor of the MCM/TOIC.  If the results of the experimental response had been negative, then the praise would be lessened only somewhat because it would still be true that we had presented a new idea which is the primary function of theorists: to theorize new theories.  In this regard one may compare the MCM/TOIC to other very famous theories that are worse yet still manage to reap all the theoretical praise.  However, unlike the experimental tests of very many respected and praise-worthy theories, the results of the experimental response were all positive.  Therefore, although the TOIC has not passed ``peer-review,'' it has been known for an experimental fact, multiple experimental facts actually \cite{RUBINO, QCOMP, BABAR}, that it describes Nature better than any other theory that currently exists.  This was known all throughout 2013, 2014, 2015, and 2016 but there has been no accompanying update to the public understanding of science.

We are essentially accusing Abhay Ashtekar, Nikodem Poplawski, and others of plagiarism but in the technical sense there has been no plagiarism.\footnote{It would be impossible to steal this writer's research because the main intention in carrying it out has been to give it away for free.  It is only this writer's accolades that have been stolen and, God willing, much blood will be spilled over this thievery.}  In the technical sense, the complaints listed here only suggest that the alleged self-correcting mechanism in science ``fucked this writer over big time.''  We pointed out Ashtekar \textit{et al.}'s spurious Feynman citation as evidence of his having viewed reference \cite{MOD} so consider that in reference \cite{ABHAY1} Ashtekar \textit{et al.} wrote that they were being so vague not to avoid writing about the MCM directly, but rather because they will leave ``the detailed derivations and discussions to a longer article.''  Did those derivations exist at the time of the publication of reference \cite{ABHAY1} or had they been first suggested after someone looked at the manuscript  which arXiv rejected \cite{MOD} but not carried out during the hasty preparation and revision of the rough draft that preceded the preprint cited here as reference \cite{ABHAY1}?  One wonders if the promised detailed derivations ever did appear in the literature.  If not, did they ever come into existence?  If not, was reference \cite{ABHAY1} worded so as to mislead readers about the existence of the derivations?

Ashtekar \textit{et al.} write the following in reference \cite{ABHAY1} and one further wonders how Ashtekar \textit{et al.} managed to report a rigorously developed Hamiltonian theory without even reporting a rigorous development of anything.

\begin{quotation}
    ``Because of [\textit{sic}] the Schr\"odinger equation we can now pass to a sum over histories a la Feynman.  [\textit{sic}]  We emphasize that the result was derived from a Hamiltonian theory.  We did not postulate that [\textit{our equation}] is given by a formal path integral.  Rather a rigorously developed Hamiltonian theory guaranteed that [\textit{our equation}] is well-defined.''\newline
\end{quotation}

\noindent

In reference \cite{MOD} we did not include a detailed derivation and we did not claim rigor without derivation which is what Ashtekar \textit{et al.} have done.  The diagrams in reference \cite{MOD} explain an idea much more clearly than Ashtekar \textit{et al.} were able to explain anything with their non-rigorous rigor of math salad.  They included neither diagrams nor derivations but somehow their paper was good and ours was unacceptably terrible.  How have Ashtekar \textit{et al.} ``rigorously developed'' it while leaving the ``detailed derivations'' to a longer article?  Furthermore, the reader should be very careful to note that if the rigor of Ashtekar \textit{et al.}'s result is offloaded elsewhere then references \cite{MOD} and \cite{ABHAY1} are very similar indeed. Ashtekar \textit{et al.}'s murky, imprecise, arguably self-contradictory wording starkly contrasts reference \cite{MOD} where one finds in the abstract a sentence, ``No attempt at quantification is made.''  Instead we pursue a qualitative analysis of the diagrams that guarantee our framework is well-defined.  Again, this sharply contrasts reference \cite{ABHAY1} when the qualitative discussion of diagrams is practical to a degree that is at least an order of magnitude greater than the practicality of qualitative analysis of quantitative equations that don't, when taken all together, form a rigorous derivation of anything.  Generally quantitative analysis is only superior to qualitative analysis when it is rigorous.

As an example of real quantitative rigor consider the unassailable truth of the appearance of the coefficient of Einstein's equation $8\pi$ in the first intuitive manipulations of the MCM/TOIC once the equally unassailable truth of

\begin{equation}
2\pi+\big(\Phi\pi\big)^3\approx 137~~,
\end{equation}\newline

\noindent was established.\footnote{The reader should note that some further confirmation of the validity of the TOIC is seen when the coefficient $1/4\pi$ of the leading term of the ontological resolution of the identity is also a common coupling coefficient for the electromagnetic interaction.}  Somehow some particular individuals have snuck into the halls of power and convinced everyone that Feynman was wrong when he is famously paraphrased as stating that all good physicists have the fine structure constant on the wall in their offices and ask themselves where it comes from, and that no one has a good explanation for it, and that if they did it would ``probably be related to $\pi$ or something.''  This is paraphrased rather than quoted because the original quote, which this writer had understood to be one of Feynman's greatest quotes of all time, does not appear in any internet search results returned to this writer's computer terminal on February 20, 2017 A.D.  Feynman's findable quotes that appear in internet search results at the beginning of the third millennium of \textit{Domini}, include,  ``We know what kind of a dance to do experimentally to measure [\textit{the fine structure constant}] very accurately, but we don't know what kind of dance to do on the computer to make this number come out, without putting it in secretly!''  The specification of this other dance is a great success of the MCM.  Feynman's other quotes include the following.

\begin{quotation}
    ``There was no way, without full understanding, that one could have confidence that conditions the next time might not produce erosion three times more severe than the time before. Nevertheless, officials fooled themselves into thinking they had such understanding and confidence, in spite of the peculiar variations from case to case. A mathematical model was made to calculate erosion. This was a model based not on physical understanding but on empirical curve fitting.''\newline
\end{quotation}

Given that Ashtekar \textit{et al.} were able to produce the inferior analysis that became reference \cite{ABHAY1} \textit{likely within just days} of reading about the MCM, all within the context of their own years or decades long familiarity with the material, that shows exactly how well-defined the MCM already was in 2009.

Ashtekar \textit{et al.} strongly emphasize that their result was derived from a Hamiltonian theory.  They do not say whether or not they were inspired to make that derivation for the first time immediately after viewing the contentious paper that arXiv rejected in 2009: reference \cite{MOD}.  When they write that they did not postulate that their formula is given by a formal path integral, is that to distinguish their paper from reference \cite{MOD} wherein we postulate that the MCM is given by the formal path integral?  Their emphasis on the Hamiltonian theory refers to the type of extra mathematical details presented in this book that are not needed to understand the idea.  In fact, we are still moving steadily toward that eventual Hamiltonian derivation.  In 2011 the purpose of reference \cite{DE} was not to make any precise predictions and this book is only about the theory's general relevance so we will not derive a new Hamiltonian here.  Making precise predictions is clearly the top priority of theoretical physics but it should be clear that there are at least two steps in the process that produces them.  Before one makes a prediction, one must define how to predictions are to be made.  This illustrates the standard distinction between fundamental science and applied science.

The critical reader will notice that ``detailed derivations and discussions'' are left out in both references \cite{MOD} and \cite{ABHAY1} but only one of them appears on arXiv today.  In the acknowledgments section of reference \cite{ABHAY1}, Ashtekar \textit{et al.}'s first thanks are to Jerzy Lewandowski who was the advisor or colleague of Poplawski at the University of Warsaw.  In April 2010, around the time Poplawski began publishing his very, very, very many popular science articles about the universe being in a black hole, he also published reference \cite{POPLAW}.  Note how the title of that paper is evocative of the idea of inverse radial spaghettification:\footnote{The term ``inverse radial spaghettification'' did not appear in the literature until 2012 because arXiv did not allow it to be added to the literature in 2011.  To understand how the title of Poplawski's 2010 paper is evocative of 2009's reference \cite{MOD}, note that radial motion means 1D motion, and together with ``into an Einstein--Rosen bridge,'' it means motion toward a bridge between two distant regions of the universe along the 1D manifold defined by the motion.  The idea presented in reference \cite{MOD} was that dark energy is an expected feature in pairs of worldsheets in the hypercosmos connected in 1D through a bounce.  The connection is 1D because it is along $\chi^5$.} ``Radial Motion into an Einstein--Rosen Bridge.''  Likewise, the title of Lewandowski's October 2009 talk at LSU was evocative: ``Spin foams from loop quantum gravity perspective.''  What was this new perspective that Lewandowski was evangelizing in Louisiana just a month after arXiv rejected reference \cite{MOD}?

While on the topic of the conduct of science in a manner that is other than ethical, consider the following.  At some point in 2011 while preparing a draft of reference \cite{DE} this writer encountered a slideshow from another a talk given at LSU.  The title was something like ``Path Integral Approach to Spin Foams'' and the name on the slides was likely Jonathan Engle who was also a speaker in video reference \cite{BOJO}.  The slides were dated from the end of 2008, but when this writer checked on the seminar schedule at the host university, LSU, the talk was actually given at the end of 2009 and the date from 2008 appears to have been ``a typo.''  This is notable because the path integral formulation of spin foams was not yet conceived in 2008 and a lesser typo might not have changed the year of initial formulation.  Based on the description of a new use for the Feynman path integral in reference \cite{MOD}, and also the fact that Engle was Ashtekar's PhD student, it is likely that the new topic presented and misdated in this talk was inspired by reference \cite{MOD}.  When one views reference \cite{LSU}, which shows the LSU Physics and Astronomy talk schedule archives, one sees all the years 2004--Present except 2009--2012: the window in which Engle presented the misdated slides.  If other researchers were already jockeying in 2009 to position themselves to receive credit for a discovery that was not their own, then whose discovery was it?  A full forensic accounting of the failure of physics to self-correct in this regard is required.

Finally, we wish to point out that Lewandowski is a coauthor on reference \cite{JLEW} which was published in September 2009 around the same time we were proposing to wrap the Minkowski diagram around a cylinder \cite{MOD}.  Therein Kami\'nski \textit{et al.} refer to an unusual cylindrical object Cyl$(\mathcal{A}(\Sigma))$ and one also sees that object in at least one earlier arXiv preprint coauthored by Lewandowski \cite{CYL}.  However, one wonders if perhaps they have contracted with Mossad--Fonseca to do a more professional time stamp alteration job than was suggested above when discussing Engle's ``Path Integral Formulation of Spin Foams'' slides.

For science to self-correct, everyone named in this section will need to consult with the grand inquisitor of the self-correcting mechanism in science.\footnote{In November 2011, as we were putting reference \cite{DE} together with intention to give arXiv a shot at redemption, we discovered all of the above information related to Ashtekar and cohort at Penn State, ranted about it prolifically, and the reader will recall that November 2011 is the month that the FBI swarmed Penn State over ``decades old Sandusky allegations.''}

\subsection{A Few Miscellanea}

Regarding the technical matters at hand we restate problem one as

\begin{equation}\label{q:sh8dfv98y}
\omega^3\big|\psi;\hat \pi\big\rangle=i\pi\Phi^2\big|\psi;\hat \pi\big\rangle\quad\qquad\implies \quad\qquad\omega^3=i\pi\Phi^2~~.
\end{equation}\newline

\noindent The implied value $\omega^3=i\pi\Phi^2$ is a cubic function of $\omega$ so the fundamental theorem of algebra says it will have three roots, not one, and two of them might not even be real numbers.  However, the problem remains: equation (\ref{q:sh8dfv98y}) does not allow the full frequency spectrum of free particle eigenstates available to unbound quanta gravitating on a continuum.

With quadratic equations which would define $\omega^2$ rather than $\omega^3$ one has two solutions.  An example is the map used to generate the Mandelbrot set $f_c(z)=z^2+c$, or even $x^2=x+1$ whose two roots are

\begin{equation}\label{eq:GGGG}
\Phi=\dfrac{1+\sqrt{5}}{2}~~,\qquad\quad\mathrm{and}\qquad\quad\varphi=\dfrac{1-\sqrt{5}}{2}~~.
\end{equation}\newline

\noindent Quadratic equations $ax^2+bx+c=0$ always have two roots given by ``the quadratic equation''

\begin{equation}\label{eq:quadraticEQ}
x=\frac{-b\pm\sqrt{{ b^2-4ac}}}{2a}~~,
\end{equation}\newline

\noindent but for cubic equations we don't even know how to find the roots without guessing because there is no more-complex version of equation (\ref{eq:quadraticEQ}) that we could call ``the cubic equation.''  This is at least important to understand before attempting to understand why one ``can't quantize'' cubic equations of motion despite quadratic second order equations integrating so famously nicely.  About the only thing we do know about cubic equations is that they will never have exactly two real roots.

In the development of the MCM we have previously vacillated concerning the sign of $\varphi$ and, independently, the magnitude of $\aleph$'s parameter of curvature because there were many possible choices.  Among the many choices, here we have a good reason to choose $\varphi<0$ with $\varphi\approx-0.62$ and to take it as the magnitude of the parameter of curvature on $\aleph$.  Therefore the slice $\aleph$ shall be located at $\chi^5_-=\varphi$.  The reader should note that the distance in the $\chi^5_-$ direction toward the past from a given $\mathcal{H}_i$ might be irrelevant and instead the contentious distance may be how far in front of $\Omega$ the next $\aleph$ lies in the $\chi^5_+$ direction (or the $\chi^5_\varnothing$ direction.)  This follows from the introduction of $\hat \varphi$ and the implied separation of the past from the present which contains a record of the past.  Another thing we have vacillated on is the length of $\chi^5$ across the complete MCM unit cell.  If we say that the length of $\chi^5_+$ between $\mathcal{H}$ and $\Omega$ is $\Phi$, that $\chi^5_\varnothing$ has no width between $\Omega$ and $\aleph$, and then that the length of $\chi^5_-$ between $\aleph$ and $\mathcal{H}$ is $\varphi$, then the total chirological distance will be unity via $1=\Phi+0+\varphi$.  This could have a further application to the unitarity constraint that has not been precisely specified other than to say that it exists because we are using quantum theory.  However, rather than exploring unitarity, we will remain focused mostly on the general relevance of the existing model without adding too many bells and whistles.  When unitarity becomes relevant we will explore it then but there are a lot of other things that will need to be covered first.  For instance, the MCM unit cell considered here does not even begin to make accommodations for the compact unit cell described in reference \cite{KN} and it is the opinion of this writer that the ideas presented there are not irrelevant, but rather will be closer to the finalized convention than the convention discussed here.  The reader should take note of the condensed mechanism in reference \cite{KN} because it is probably ``more correct.''  We discuss this other mechanism in section V.1.

One of the big struggles in this research has been to show how to unify second order classical mechanics under Newton's laws with first order, or linear, quantum mechanics governed by the Schr\"odinger equation.  To this end, we have chosen the scalar field $\phi$ in the KK metric

\begin{equation}\label{eq:5Dmetriccc}
\Sigma_{AB}=\left(\begin{matrix}
g_{\mu\nu}+B_\mu B_\nu\phi^2(\chi^5)& &B_\mu\phi^2(\chi^5)\\[8pt]
B_\nu\phi^2(\chi^5) & &\phi^2(\chi^5)
\end{matrix}\right)~~,
\end{equation}\newline

\noindent to be linear in $\chi^5$ with

\begin{equation}
\phi^2(\chi^5) \quad\equiv\quad \chi^5~~.
\end{equation}\newline


\noindent Even when the metric has this linear definition, the double dot operator in $f=m\ddot x$ is not linear.  We want to make everything linear for two reasons: it is easier to compute linear mathematics and if we can linearize quadratic equations, then we can probably quadratify cubic ones and then quantize them canonically.  It would be a big conceptual advance if there was some gauge symmetry associated with the golden ratio that would universally allow such manipulations.  Regarding quadratic equations, and not making an argument about gauge symmetry but rather only outlining an idea at a very high level, we need to split the non-linear $\partial^2$ operator into two parts.  They should be unequal so that they can generate complexity but the only obvious way to split it is through

\begin{equation}\label{eq:partial2222222}
\partial^2=\partial\circ\partial~~.
\end{equation}\newline

\noindent This formulation certainly contains nothing new.

In section II.1 we showed how Hamilton's equations, two first order equations, can give either the geodesics or the field lines of different second order theories but there was no application to quantum theory where the Hamiltonian only appears inside Schr\"odinger's first order equation.  Hamilton's equations, linear or not, are exactly equal to classical mechanics so that formulation gives us little direction regarding a classical description of quantum physics.  In reference \cite{IC} we discussed the normal decomposition of second order equations into pairs of linear equations that are amenable to analysis with numerical methods.  The correlation of numerical $x$ and $v$ to the specific case of Hamiltonian $q$ and $p$ is easy to see with


\begin{equation}\label{eq:numerical}
\ddot x = \frac{F}{m}\quad\mapsto\quad \begin{cases}
\dot x &= v \\
&~\\
\dot v &= \dfrac{F}{m}
\end{cases}~~,
\end{equation}\newline

\noindent but this is not what we refer to as linearization.  The difficult form of linearization we refer to is the one that turns one second order equation into one first order equation.

It will likely be easiest to implement the unification of first and second order equations at the level of numerical algorithms and we will discuss such things in the next chapter.  Before moving on to solve problems one through four we will discuss one possible analytical feature that could motivate later alterations to classical numerical analysis algorithms.  All the physical equations and dynamics that were derived analytically over the years have already been converted into computer ready algorithms so it may be simpler to modify those algorithms and then reverse engineer their analytical underpinnings than to advance solely through the proposition of new underpinnings.

The topological component of double orthogonality uses sums; products should generally be understood as a single object with a single topology.  If this avenue of splitting $\partial^2$ is to pan out we must learn something about how addition on the RHS of

\begin{equation}\label{eq:tempPHI}
\partial^2:=A+B~~,
\end{equation}\newline

\noindent relates to multiplication on the LHS.

In mathematics, multiplication and addition are important operations but they are more important in physics because the physical interpretation assigned to each operation is very well understood.  Almost everything non-trivial in physics depends on some combination of additive and multiplicative operations.  Earlier we quoted Feynman's reference \cite{FEYN} as follows.

\begin{quotation}
    ``We shall see that it is the possibility [\emph{of expressing the action}] $\mathcal{S}$ as a sum, and hence $\Phi$ as a product, of contributions from successive sections of the path, which leads to the possibility of defining a quantity having the properties of a wavefunction.''\newline
\end{quotation}

When Boltzmann was formulating his famous entropy formula $S=k\log\Omega$ he did not randomly pick the logarithm from a very large sea of possible operations.  Boltzmann simply wrote down what he knew about the interpretation of addition and multiplication as they relate to entropy and the logarithm was the only choice that would preserve it.  Therefore, pursuing a very high level concept, to obtain some new representation of $\partial^2$ we should find equation (\ref{eq:tempPHI}) when either $A$ or $B$ has to be the partial derivative operator because we must be able to reconstruct $\partial^2$ from the pieces $A$ and $B$.  For instance if we let $A=\partial$ and $B=\Box^2$ we can reconstruct $\partial^2$ with

\begin{equation}
\partial^2=A^2+0\times B~~,
\end{equation}\newline

\noindent but that is not a valid solution because

\begin{equation}
\partial^2\neq\partial+\Box^2.
\end{equation}\newline

\noindent However, we are not totally constrained by $\partial^2=A+B$ because definition (\ref{eq:tempPHI}) is not an equation and even if it was, it might be irrelevant.  (We suggest an idea only vaguely right now.)  To get the complexity generating representation of $\partial^2$, if it exists (and if it is actually useful), we will likely have to solve an equivalence relationship like definition (\ref{eq:tempPHI}) and we already know an equation that looks like that.  The two roots of

\begin{equation}\label{eq:ROOTS77}
x^2=x+1~~,
\end{equation}\newline

\noindent are the two numbers $\Phi$ and $\varphi$ that are both called the golden ratio.  If the decomposition of the non-linear operator $\partial^2$ into a linear form that unifies physics was simple, it would have been done by now.  To develop a new law in this regard -- for future investigations -- we could suggest that $\partial^2$ has a representation as one full cycle ($2\pi$ radians) which can be decomposed into co-$\hat \pi$'s that undergo different types of gauge transformations.  Note the similar properties of $\pi$ and $\Phi$ such as

\begin{equation}\label{eq:Sec3-2modes}
\pi+\pi=\big(\sqrt{2\pi}\big)^2~~,\quad\qquad\mathrm{and}\quad\qquad\Phi+1=\Phi^2~~.
\end{equation}\newline

In the case of $\partial^2:=\pi+\pi$, $A=B$ but it is quadratic in the mysterious coefficient $\sqrt{2\pi}$ from the Fourier transform and there exists an independent result which proves that $\sqrt{2}$ is an irrational number.  Perhaps the ontological basis is best represented as $\{\hat i, \hat{\sqrt{2}_{\!~~}}\!, \hat \Phi, \hat \pi \}$ and $\hat{\sqrt{2}_{~~}}$ and $\hat i$ combine in the $\sqrt{i}$ channel to give the rational real numbers $\mathbb{Q}$ when the contraction of two ontological objects gives $(\sqrt{2})^2=2$.  We won't go in this direction now, but we remind the reader of the suggested coupling between ``sites'' in the continued fraction forms of the irrational number $\pi$ and the most irrational number $\Phi$


\begin{equation}
\Phi=1+\frac{1}{1+\frac{1}{1+\frac{1}{1+\frac{1}{1+...}}}}~~,\quad\qquad\mathrm{and}\quad\qquad\pi=3+\frac{1}{6+\frac{3^2}{6+\frac{5^2}{6+\frac{7^2}{6+...}}}}~~.
\end{equation}\newline

\noindent A product defined to combine the piecewise elements of continued fractions might be a good avenue for philosophical inquiry when calculating to assemble $2$ from pairs of $\sqrt{2}$.  A result of the contraction of information stored in $\pi$ and $\Phi$ respectively might be to combine copies of $\sqrt{2}$ from each.  In this way there will be no rational real numbers in the ontological basis itself, but they are emergent through operations like $i^2=-1$ or $\sqrt{2}^{\,2}=2$ which give $i^2+\sqrt{2}^2=1$.  Taking these numbers together with addition we can generate the integers $\mathbb{Z}$ and when we add multiplication as well we recover the rationals $\mathbb{Q}$.  Then noting that $\hat i$ is still in there we recover complex numbers $\mathbb{C}$, and then through the rules for $\hat \Phi$ we recover $^\star\mathbb{C}$.  We will say more about this in the next chapter which is in part about mathematical analysis.  Once we have constructed $2$, either from $\hat{\sqrt{2}}$ or $\hat 2$ directly then we will have more pieces for building complex cosmological clockwork with

\begin{equation}
\Phi^{n+1}=\Phi^n+\Phi^{n-1}\quad\qquad\text{and}\quad\qquad2^{n+1}=2^n+2^n~~.
\end{equation}\newline


\subsection{Problems One and Two}

Problem one references a frequency constraint stated as

\begin{equation}\label{eq:onefreq77}
\omega^3\big|\psi;\hat \pi\big\rangle=i\pi\Phi^2|\psi;\hat \pi\big\rangle \quad\qquad\implies\qquad\quad \omega^3=i\pi\Phi^2~~.
\end{equation}\newline

\noindent The kets on both sides are the same so their coefficients $\omega^3$ and $i\pi\Phi^2$ must also be the same.  $i\pi\Phi^2$ is fairly well introduced in reference \cite{QG} which revises and improves the original argument for the criticality of that number as it relates to deriving Einstein's equation \cite{TER}.  All of the TOIC structure is encoded on the value $i\pi\Phi^2$ so we can't change it much.  Instead of changing the scalar coefficient, we proposed in reference \cite{IC} to change the ket on the right side of the fundamental TOIC formula which is recast as

\begin{equation}\label{eq:onefreq7}
\hat M^3\big|\psi;\hat\pi\big\rangle=\hat M^3\big|\Psi;\hat\pi\big\rangle\quad\qquad\implies\quad\qquad\partial^{\,3}_0\big|\psi(x^\mu);\hat \pi\big\rangle=\partial^{\,3}_5|\Psi(\chi^A);\hat \pi\big\rangle~~.
\end{equation}\newline

\noindent Now any frequency is allowed when the degree of freedom for $\omega$ is preserved in the freedom to choose $\Psi\neq\psi$.  From equation (\ref{eq:onefreq7}) all the steps (\ref{eq:onefreq}-\ref{eq:MCMalgebra}) follow directly and we have incidentally solved problem two as well.  Problem two stated that when all of the terms in

\begin{equation}
8\pi f^3\big|\psi;\hat \pi\big\rangle=i|\psi;\hat \Phi\big\rangle+|\psi;\hat i\big\rangle\label{eq:MCMalgebra4546}~~,
\end{equation}\newline

\noindent are proportional to $\psi$ there is no way to achieve double orthogonality.  The introduction of $\Psi$ fixes this.  With $\Psi$, when we convert from the quantum language to the gravitational language, the stress energy tensor $T_{\mu\nu}$ on the LHS, defined by $\psi$, can be different than everything on the other side which will depend on $\Psi$.

Simply capitalizing $\psi$ on the RHS solves both problems one and two.  $\Psi$ is the solution to problem two that we will use moving forward but there is another way to solve problem two that demonstrates complexity and is worth exposing.  We may redefine $\psi$ so that it begins as a third rank tensor

\begin{equation}\label{eq:3rdRNK}
\psi\quad\equiv\quad\psi^\sigma_{\,\mu\nu}~~,
\end{equation}\newline

\noindent and then let the ontological vectors act on $\psi^\sigma_{\,\mu\nu}$ like a Kronecker delta per the usual prescription for basis vectors.  Considering for example only the $\hat e^3\equiv\hat i$ term that we expect to map to $g_{\mu\nu}\Lambda$, we see how one might use the basis vectors to extract independent tensors via


\begin{equation}\label{eq:3rdRNK1}
|\psi;\hat i\rangle \quad\equiv\quad  \psi^\sigma_{\,\mu\nu}\hat e_{3}=\psi^\sigma_{\,\mu\nu}\delta^3_\sigma=\psi^3_{\,\mu\nu}= g_{\mu\nu}\Lambda~~.
\end{equation}\newline


\noindent The other objects of general relativity, $8\pi T_{\mu\nu}$, $R_{\mu\nu}$, and $1/2\,Rg_{\mu\nu}$ can all be extracted from $\psi^\sigma_{\,\mu\nu}$ with the other $\hat e_\mu$ when they are defined accordingly.

Equation (\ref{eq:3rdRNK1}) is reminiscent of the interpretation of the connection coefficients $\Gamma^\sigma_{\mu\nu}$ as a set of four 2D matrices: one $\mu\nu$-matrix for each $\sigma$.
In equation (\ref{eq:3rdRNK1}), $\hat i$ picks out $g_{\mu\nu}\Lambda$ because that is the definition specified by maps (\ref{eq:33maps}-\ref{eq:33maps2}).  $g_{\mu\nu}\Lambda$ would be just one of the four 2D matrices contained in $\psi^\sigma_{\,\mu\nu}$.  This formulation motivates the inclusion of $\hat 2$ as it will have an object defined for it in the counting over $\sigma$ but there would be a leftover element if we tried to do this with only $\hat e_i\in\{\hat \Phi,\hat \pi,\hat i\}$.  However, we will not presently solve problem two with equation (\ref{eq:3rdRNK}) because we are just moving our problem to another sector.  If we choose to use $\psi^\sigma_{\,\mu\nu}$, we would have to add a problem to the list about how to use three index tensors as state vectors in quantum mechanics.  Between the two proposed solutions, the former is better because it is simpler; we will choose to capitalize $\psi$ rather than adding three Greek indices.  However, not only is the three index proposal like the connection, it is also like the torsion.

In reference \cite{CARR} Carroll writes the following.

\begin{quotation}
    ``The first thing to notice is that the difference of two connections is a tensor.  Imagine we have defined two different kinds of covariant derivative, $\nabla_\mu$ and $\widehat{\nabla}_\mu$, with associated connection coefficients $\Gamma^\lambda_{\mu\nu}$ and $\widehat{\Gamma}^\lambda_{\mu\nu}$.  Then the difference

    \begin{equation}
    S^\lambda_{\,\mu\nu}=\Gamma^\lambda_{\mu\nu}-\widehat{\Gamma}^\lambda_{\nu\mu}~~,
    \end{equation}

    \noindent is a (1,2) tensor.  (Notice that we had to choose a convention for the index placement.)  We could show this by brute force, plugging in the transformation laws for the connection coefficients, but let's be a little more slick.  Given an arbitrary vector field $V^\lambda$, we know that both $\nabla_\mu V^\lambda$ and $\widehat{\nabla}_\mu V^\lambda$ are tensors, so their difference must also be.  This difference is simply

    \begin{align}
    \nabla_\mu V^\lambda-\widehat{\nabla_\mu} V^\lambda&=\partial_\mu V^\lambda+\Gamma^\lambda_{\mu\nu}V^\nu-\partial_\mu V^\lambda-\widehat{\Gamma}^\lambda_{\mu\nu}\\
    ~&\nonumber\\
    &=S^\lambda_{\,\mu\nu} V^\nu~~.
    \end{align}

    \noindent Since $V^\lambda$ was arbitrary, and the left hand side is a tensor, $S^\lambda_{\,\mu\nu}$ must be a tensor.  As a trivial consequence, we learn that any set of connection coefficients can be expressed as some fiducial connection plus a tensorial correction,

    \begin{equation}
    \Gamma^\lambda_{\mu\nu}=\widehat{\Gamma}^\lambda_{\mu\nu}+S^\lambda_{\,\mu\nu}~~.
    \end{equation}

    ``Next notice that, given a connection specified by $\Gamma^\lambda_{\mu\nu}$, we can immediately form another connection simply by permuting the lower indices.  That is, the set of coefficients $\Gamma^\lambda_{\nu\mu}$ will also transform according to

    \begin{equation}\label{eq:CarrNonTensor}
    \Gamma^{\lambda'}_{\mu'\nu'}=\dfrac{\partial x^\mu}{\partial x^{\mu'}}\dfrac{\partial x^\nu}{\partial x^{\nu'}}\dfrac{\partial x^\lambda}{\partial x^{\lambda'}}\Gamma^\lambda_{\mu\nu}\dfrac{\partial x^\mu}{\partial x^\mu}\dfrac{\partial x^\nu}{\partial x^\nu}\dfrac{\partial^2 x^{\lambda'}}{\partial x^\mu\partial x^\nu}~~.
    \end{equation}

    \noindent [\textit{Since the partial derivatives appearing in the last term of equation (\ref{eq:CarrNonTensor}) commute}], they determine a distinct connection.  There is thus a tensor we can associate with any given connection, known as the \textbf{torsion tensor}, defined by

    \begin{equation}
    T^\lambda_{\,\mu\nu}=\Gamma^\lambda_{\mu\nu}-\Gamma^\lambda_{\nu\mu}=2\Gamma^\lambda_{\mu\nu}~~.
    \end{equation}

    \noindent It is clear that the torsion is antisymmetric in its lower indices, and a connection that is symmetric in its lower indices is known as `torsion-free.'$\,$''\newline
\end{quotation}

Noting what Carroll has stated above regarding the difference of two connections being a tensor, we can set $B_\mu=0$ in the KK metric, not calculate the bulk geometry, and simply take the difference of the connections $\Gamma^{C~\pm}_{AB}$ in $\Sigma^\pm$ as the perturbation in $\mathcal{H}$.  Even when $\Sigma^\pm$ are empty 5D space, the fact that they have topologies O(1,4) and O(2,3) means that the connections will be different on either side of $\mathcal{H}$.


\subsection{What is a Multiplex?}

Implication (\ref{eq:onefreq7}) solves problem two about all the objects being linearly dependent on $\psi$ but by going the route with $\Psi$ we introduce a new problem: problem three.  How can the 4D object $\psi$ on the left be equal to the 5D object $\Psi$ on the right?\footnote{The objects $\psi$ and $\Psi$ were introduced in reference \cite{IC}}  This cannot be true in general and we don't want to introduce any unnecessary complexity so we should reformulate the rigid equality in implication (\ref{eq:onefreq7}) as

\begin{equation}\label{eq:GROUP}
\partial^{\,3}_0\big|\psi(x^\mu);\hat \pi\big\rangle:=\partial^{\,3}_5|\Psi(\chi^A);\hat \pi\big\rangle~~.
\end{equation}\newline

Equation (\ref{eq:GROUP}) uses the $:=$ symbol because we want to show a new kind of relationship, possibly between a topological object and an algebraic one, or between a geometric object and a group theoretical one, or some such juxtaposition, such as connecting the ontological basis to the ontological group, or perhaps vice versa.  Since it is not clear at this point how the type of object on the RHS of equation (\ref{eq:GROUP}) compares with the type on the LHS we will focus on the symbol $:=$ in the center and going forward we will say ``multiplex (\ref{eq:GROUP})'' instead of ``equation (\ref{eq:GROUP})'' when this symbol appears.

An equation has an LHS and an RHS but a multiplex will have an LHS/RHS pair, as in multiplex (\ref{eq:GROUP}), and also an imaginary hand side (IHS) which consists of all the objects needed to make the multiplectic relationship true.  We can select any two elements of a multiplex as an RHS and LHS which are logically related through an independent IHS.  If there are $N$ objects in a multiplex, then we can always write another ``is defined according to'' statement between any two objects leaving the remainder of $N-2$ objects as an IHS.  Those familiar with group theory will see many obvious parallels, but we will not discuss the likeness here other than to say that the foundation of group theory is that every group has the identity operator as one of its elements so $\{\hat 1,\hat \pi,\hat 2,\hat \Phi,\hat i\}$ is a good set to explore for group structure but the ontological basis $\{\hat \pi,\hat 2,\hat \Phi,\hat i\}$ is not.

We will also use the word multiplex to refer to an IHS from which no LHS/RHS pairs has been selected; the most general multiplex contains all the information about all possible $:=$ pairings.  If we want to make some arbitrary example that shows how all the pieces of a multiplex work, we can consider a multiplex

\begin{equation}
Z_M\quad\equiv\quad\{Z_1,Z_2,Z_3,Z_4,Z_5\}~~.
\end{equation}\newline

\noindent When we choose to consider, perhaps in an effort to understand an equation, two elements of a multiplex taken as an LHS and an RHS, then there is a corresponding IHS that contains the full multiplex minus the two elements that are being used to define a $:=$ relationship.  If we take an LHS $Z_1$ along with an RHS $Z_2$, then that defines an IHS $\bar Z_{\{Z_1,Z_2\}}$.  Together they look like

\begin{equation}
Z_1:=Z_2~~,\quad\qquad\mathrm{and}\quad\qquad \bar Z_{\{Z_1,Z_2\}}=\{Z_3,Z_4,Z_5\}~~.
\end{equation}\newline

\noindent $\bar Z_{\{Z_1,Z_2\}}$ is the set of objects that makes $Z_1:=Z_2$ true.  The IHS contains everything needed to demonstrate how a LHS is defined by an RHS.  Then we also have

\begin{equation}
Z_5:=Z_3~~,\quad\qquad\mathrm{with}\quad\qquad \bar Z_{\{Z_5,Z_3\}}=\{Z_1,Z_2,Z_4\}~~,
\end{equation}\newline

\noindent and so on.

Unlike relationships given with $=$ those defined with $:=$ are not self contained.  The full meaning is contained on the LHS, the RHS, and on the IHS but the IHS does not need to be specified to understand the most important part of the $:=$ relationship.  The LHS/RHS multiplex relationship is statement that a sufficient IHS exists, and there is an implication (or an expectation) that some IHS will connect the LHS and RHS fairly directly, or ``irreducibly,'' and not be some mathematical Rube Goldberg of the type that can be made complicated enough to relate any two things.

\subsection{Problems Three and Four}

Problem three manifests in multiplex (\ref{eq:GROUP}) when the LHS is a 4-vector and the RHS is a 5-vector \cite{IC}, and the two cannot technically satisfy the $=$ relationship.  Here we will rederive $\Psi$ to show that there is yet another undiscussed arrangement for resolving the linear dependence issue.

After inserting $\Psi$ into equation (\ref{eq:MCMalgebra}) it becomes

\begin{equation}\label{eq:MCMalgebra3}
8\pi f^3\big|\psi;\hat{\pi}\big\rangle=i\big|\Psi;\hat{\Phi}\big\rangle+\big|\Psi;\hat{i}\big\rangle~~,
\end{equation}\newline

\noindent and this solves problems one and two.  Problem one is solved when the starting point already exhibits single orthogonality between $\psi$ and $\Psi$ and problem two is solved when there is an obvious symmetry that will allow the full frequency spectrum when $\Psi$ always becomes whatever is needed to satisfy equation (\ref{eq:MCMalgebra3}) for a given $f$.  Recalling that we introduced $\Psi$ in reference \cite{IC} as the complete state of the universe, and noting that the complete state is unobservable, there is no constraint on how complicated we can make its wavefunction.  Still, as mentioned above, there is a lingering issue with the $\Psi$ workaround: when we introduced the 5D $\Psi$ vector, we also defined it to contain $\psi$ as a 4D subspace.  This is a severe constraint!  $\Psi$ is almost completely determined by $\psi$ so, in truth, we have just barely solved problem two.  To add more freedom between $\psi$ and whatever we put on the RHS of equation (\ref{eq:MCMalgebra3}), and to resolve the dimensionality discrepancy of problem three, consider a third representation


\begin{equation}\label{eq:MCMalgebra4}
8\pi f^3\big|\psi;\hat{\pi}\big\rangle=i\big|\phi;\hat{\Phi}\big\rangle+\big|\phi;\hat{i}\big\rangle~~.
\end{equation}\newline

To get $\phi$ we take

\begin{align}
\big|\Psi\big\rangle&=0\label{eq:PSI0}\\
~&~\nonumber\\
&=1+(-1)\label{eq:PSI00}\\
~&~\nonumber\\
&=1-\left( \frac{1}{4\pi}\right) \hat\pi+\bigg( \frac{\varphi}{4}\bigg) \hat\Phi-\left( \frac{1}{8}\right) \hat2+\left( \frac{i}{4}\right) \hat i\\
~&~\nonumber\\
&=1-\psi\\
~&~\nonumber\\
&=\phi-\psi\label{eq:PhiPsi77}
\end{align}\newline



\noindent with


\begin{equation}\label{eq:350}
\psi\quad\equiv\quad\big|\psi\big\rangle=\left(\begin{matrix}
-\dfrac{1}{4\pi}\\
~\\[1pt]
\dfrac{\varphi}{4}\\
~\\[1pt]
-\dfrac{1}{8}\\
~\\[1pt]
\dfrac{i}{4}
\end{matrix}\right)~~,\qquad\quad\mathrm{and}\qquad\quad\phi\quad\equiv\quad\big\langle\phi\big|=\left(\begin{matrix}
\dfrac{1}{4\pi}\\
~\\[1pt]
-\dfrac{\varphi}{4}\\
~\\[1pt]
\dfrac{1}{8}\\
~\\[1pt]
-\dfrac{i}{4}
\end{matrix}\right)^{\!T}~~.
\end{equation}\newline

Obviously the constant state vectors presented here (and in reference \cite{IC}) will lead to more problems because the wavefunction should vary over space and time but $\psi$ and $\phi$ do not, and certainly $\Psi=0$ does not.  When this structure was introduced in reference \cite{IC} we did not explicitly state a method by which we could introduce the coordinate dependence into the wavefunctions and the implication of $\Psi=0$ would have to be that $\Psi\neq0$ in some perturbative limit, or some other limit where it is equal to zero only on a certain level of $\aleph$.  Note well, when $\Psi=0$ we can use the algebraic structure that we have built with $\Psi$ to host perturbations in the form of $h_{\mu\nu}$, $h_{\alpha\beta}$, and $h_{AB}$, and then move to the computational frame where the qubit is encoded in the perturbation.  That computational frame is likely what we have previously described as G-space \cite{OP} and perhaps the difference between $\hat M^3$ and $\tilde M^4$ is that $\hat M^3$ takes a qubit in a position or momentum representation and outputs a qubit in an inverted representation but $\tilde M^4$ has an output in the same representation as its input.  Likewise we described this mode between the structure $\{\Psi,\psi,\phi\}$ and the qubit $\{h_{\mu\nu}, h_{\alpha\beta}, h_{AB}\}$ as exhibiting Yangian symmetry in reference \cite{OP}.  Reference \cite{OP} is among the most rigorous mathematical analyses undertaken so far in this research program.\footnote{From this writer's perspective the main result of the work unit whose output was \textit{Ontological Physics} \cite{OP} was that further inquiry in that direction should be carried out with computers.  This writer considered it prudent to continue with the survey of that which is better analyzed without computers, which is sometimes called philosophy, because certainly there are thousands or millions of other people who are already well trained in computerization.}  We will discuss those features in chapter four and specifically how they relate to newer material related to infinite complexity presented in reference \cite{ZETA}.  When we add the coordinate dependence that will make our wavefunctions $\psi$ and $\phi$ functions instead of constants, whatever change we make in $\psi$ must be perfectly balanced in $\phi$ if equation (\ref{eq:PSI0}) is to hold.  However, it is not required that the $\hat \pi$ component of $\psi$ is perfectly offset by the $\hat \pi$ component of $\phi$.  Both $\psi\hat\pi$ and $\phi\hat\pi$ can vary independently when we uses the other components to keep everything balanced.  Finally note that we can solve problem three simply by replacing 5D $\Psi$ with some 4D vector $\phi$ that is otherwise not constrained.  This means we just have to introduce a second 4D wavefunction that doesn't need to have anything to do with equations (\ref{eq:PSI0}-\ref{eq:350}).

In quantum theory the ordinary application of any resolution of the identity is to say that every object is multiplied by one, and now we have the option to also say that every object has zero added to it which allows us to insert two identities as $\hat0=\hat1-\hat1$.  Furthermore, where Feynman has discussed the role of addition in the action and the role of multiplication for wavefunctions, and with these new ``ontological'' resolutions of both one and zero, we can make use of either channel anywhere by adding zero or multiplying by one.  We expect that this will be an important feature of infinite complexity but for now it will suffice to say that there are so many options in this regard, specifically regarding $\phi$, that we can classify problem three as solved.  Regardless of $\phi$'s complete technical specification, it is 4D so problem three goes away.

With three problems out of the way, we come to problem four about how the linear dependence issue lingers inside the Einstein tensor when

\begin{equation}\label{eq:BBBBBB}
\big|\phi;\hat \Phi\big\rangle~~\mapsto~~G_{\mu\nu}\quad\equiv\quad R_{\mu\nu}-\frac{1}{2}Rg_{\mu\nu}~~.
\end{equation}\newline

\noindent $\phi$ resolves the issues discussed earlier but the drawback is that they do not solve the problem of the linear dependence of the objects inside $G_{\mu\nu}$.  As mentioned a few times already, we could use $\hat 2$ so that the four ontological vectors correspond to the four terms of Einstein's equation written without the brevity of the Einstein tensor

\begin{equation}
8\pi T_{\mu\nu}=R_{\mu\nu}-\dfrac{1}{2}\,Rg_{\mu\nu}+g_{\mu\nu}\Lambda~~,
\end{equation}\newline

\noindent but we have avoided this for two reasons.  First, the MCM algebra generates three terms naturally, as in equation (\ref{eq:MCMalgebra3}), and we would have to unnaturally pick one of the three terms to split in half with $\hat 2$ before assigning one of the halves to $\hat 2$.  Second, it is nice when can interpret general relativity as a manifestation of the principle

\begin{equation}\label{eq:LQC77}
\widehat{LQC}\big|\mathrm{bounce} \big\rangle:=\big|t_+\big\rangle+\big|t_-\big\rangle~~,
\end{equation}\newline

\noindent which first appeared in reference \cite{DE} and there is something very interesting we can say about this.  The original intention with equation (\ref{eq:LQC77}) had been to show that the operation of the LQC operator on the bounce state was not just to decompose it into the past and future, but into the past, present, and future via

\begin{equation}
\widehat{LQC}\big|\mathrm{bounce} \big\rangle:=\big|t_+\big\rangle+\big|t_\star\big\rangle+\big|t_-\big\rangle~~.
\end{equation}\newline

\noindent In the course of the development of the idea it became clear that the bounce state should be the present denoted with $|t_\star\rangle$ and that is how we arrived at equation (\ref{eq:LQC77}) in the original paper: reference \cite{DE}.  However, now that we have added $\Sigma^\varnothing$ with the $\varnothing$ coordinates, we are absolutely able to preserve the original interpretation by writing

\begin{equation}\label{eq:LQC762}
\hat M^3\big|\mathrm{bounce} \big\rangle:=\big|t_+\big\rangle+\big|t_\star\big\rangle+\big|t_-\big\rangle~~,
\end{equation}\newline

\noindent with an implication that

\begin{equation}
\big|t_\star\big\rangle:=\Sigma^\varnothing~~.
\end{equation}\newline

\noindent We refer the reader to reference \cite{DE} for a lot of the ``irrelevant'' details regarding the $\widehat{LQC}$ formalism.

The concept illustrated by equation (\ref{eq:LQC762}) has four terms, just like would be needed to write Einstein's equation in long form with four terms corresponding to the four ontological basis vectors.  The only remaining obstacle to using $\hat 2$ from the outset is that the MCM algebra generates three terms and we don't know how to best add a fourth one.  Luckily we can sidestep this issue because dS and AdS belong to a small set of spacetimes that are called \textbf{maximally symmetric}.  The definition of a maximally symmetric space is one whose Ricci tensor is linearly dependent on the metric.

\begin{equation}
R_{\mu\nu}\propto g_{\mu\nu}\quad\implies\quad R_{\mu\nu}:= g_{\mu\nu}~~.
\end{equation}\newline


Since we are only requiring the possibility of constructing a Lorentz frame at the three slices $\{\aleph,\mathcal{H},\Omega\}$ of the cosmological unit cell but not the bulk hypercosmos between slices, we have miraculously generated a constraint that means we don't need to change anything else.  Problem four is not a problem at all because we have already, based on completely independent considerations, chosen $\{\aleph,\mathcal{H},\Omega\}$ to be exactly those unique spaces where the metric and Ricci tensor are \textit{never} independent from each other.  In maximally symmetric spacetime all of the tensors in Einstein's equations are linearly dependent on each other.

If the Ricci tensor and the metric are linearly dependent on each other, then the stress-energy tensor must also be linearly dependent with them.  This means problem two was never a problem to begin with because we had already constructed the MCM to be the special case where complete linear dependence on both sides of Einstein's equation is expected.  Furthermore, as a statement of the non-problematic nature of any of the ``problems,'' note how





\begin{equation}
 \partial^{\,3}_0\big|\psi;\hat \pi\big\rangle=\partial^{\,3}_5|\psi;\hat \pi\big\rangle\quad\rightarrow\quad\partial^{\,3}_0\big|\psi;\hat \pi\big\rangle=\partial^{\,3}_5|\psi';\hat \pi\big\rangle~~,
\end{equation}\newline


\noindent also solves problems one and three.  $\psi$ and $\psi'$ are both 4D and $\psi'$ will vary to accommodate any frequency $\omega$.  Simply adding the tick mark to $\psi$ is sufficient to completely put aside all the issues raised in section III.1.

\subsection{Maximally Symmetric Spacetime}


Noting that flat expanding space in not curved, the Friedman--Lema\^itre--Robertson--Walker (FLRW) metric of flat expanding space is

\begin{equation}
ds^2=-(dx^0)^2+a^2(t) [(dx^1)^2+(dx^2)^2+(dx^3)^2]~~.
\end{equation}\newline

Consider what Misner, Thorne, and Wheeler say about this metric in reference \cite{BBB}.


\begin{quotation}
	``Turn now to the 3-geometry $\gamma_{ij}dx^idx^j$ for the arbitrary initial hypersurface $\mathcal{S}_I$.  This 3-geometry must be homogeneous and isotropic.  A close scrutiny of its three-dimensional Riemann curvature must yield no `handles' to distinguish one point on $\mathcal{S}_I$ from any other, or distinguish one direction at a given point from any other.  `No handles' means that [\textit{the 3D Riemann tensor} $R^l_{\,ijk}$] must be constructed algebraically from pure numbers and from the only `handle-free' tensors that exist: the 3-metric $\gamma_{ij}$ and the three-dimensional Levi-Civita tensor $\epsilon_{ijk}$.  (All other tensors pick out preferred directions or locations.)  One possible expression for $R^l_{\,ijk}$ is

	\begin{equation}\label{eq:BBB1}
	R_{ijkl}=K(\gamma_{ik}\gamma_{jl}-\gamma_{il}\gamma_{jk});~~K=``\mathrm{curvature~ parameter}"=\mathrm{constant}~~.
	\end{equation}

	\noindent Trial and error soon convince one that this is the \textit{only} expression that both has the correct symmetries for a curvature tensor and can be constructed solely from constants, $\gamma_{ij}$ and $\epsilon_{ijk}$.  Hence, this must be the three curvature of $\mathcal{S}_I$.  (One says that any manifold with a curvature tensor of this form is a manifold of `\textit{constant curvature}.')

	``As one might expect, the metric for $\mathcal{S}_I$ is completely determined, up to coordinate transformations, by the form [\textit{equation (\ref{eq:BBB1})}] of its curvature tensor.  [\textit{sic}]  With an appropriate choice of coordinates, the metric reads [\textit{sic}],\footnote{In the original text of this excerpt \cite{BBB} the letter $\chi$ was used where we write $\xi$.}

	\begin{align}
	d\sigma^2=\gamma_{ij}dx^idx^j&=K^{-1}[d\xi^2+\sin^2\xi(d\theta^2+\sin^2\theta ~d\phi^2)]~~\mathrm{if}~~K>0~~,\label{eq:BBB2}\\
	&~\nonumber\\
	d\sigma^2=\gamma_{ij}dx^idx^j&=d\xi^2+\xi^2(d\theta^2+\sin^2\theta ~d\phi^2)~~\mathrm{if}~~K=0~~, \label{eq:BBB3}\\
	&~\nonumber\\
	d\sigma^2=\gamma_{ij}dx^idx^j&= (-K)^{-1}[d\xi^2+\sinh^2\xi(d\theta^2+\sin^2\theta ~d\phi^2)]~~\mathrm{if}~~K<0\label{eq:BBB4}~~.
	\end{align}

	\noindent Absorb\footnote{Here ``absorb'' refers to an area of physics where the mathematical rigor is less than superb with respect to what absorption is.  We have previously chosen in reference \cite {TER} to ``suppress one power of $\pi$ so $\partial_t|\psi\rangle=\Phi m|\psi\rangle$,'' and here the authors refers to the exact same thing as absorption.  There are very many information sinks in the standard model.  Each of the almost twenty empirical parameters are information sinks in addition to all the topological ones relating to open and closed boundaries at infinity and many other anomalous sinks.  We propose to increase the rigor associated with what it means to absorb or suppress a qubit of defining a geometric set of absorptive and adsorptive qubits according to $\{\hat i,\hat \Phi,\hat 2,\hat \pi\}$, and thereby increase how good the best theory we have is.} the $K^{-1/2}$ or $(-K)^{-1/2}$ into the expansion factor $a(t)$ [\textit{sic}], and define the function

	\begin{align}
	&\Sigma~\,\equiv\,~\sin \xi, &&\mathrm{if}~~k~~\equiv~~ K/|K|=+1~~(\mathrm{``positive~ spatial ~curvature"})~,\label{eq:BBB5}\\
	&~&&\nonumber\\
	&\Sigma~\,\equiv~\,\xi, &&\mathrm{if}~~k~~\equiv~~ K=0~~(\mathrm{``zero~ spatial ~curvature"})~,\label{eq:BBB6}\\
	&~&&\nonumber\\
	&\Sigma~\,\equiv~\,\sinh \xi, &&\mathrm{if}~~k~~\equiv~~ K/|K|=-1~~(\mathrm{``negative~ spatial ~curvature"})~.\label{eq:BBB7}
	\end{align}

	\noindent Thus write the full spacetime geometry in the form

	\begin{equation}\label{eq:BBB8}
	ds^2=-dt^2+a^2(t)\gamma_{ij}dx^idx^j~~,
	\end{equation}

	\begin{equation}\label{eq:BBB9}
	\gamma_{ij}\,dx^i\,dx^j=d\xi^2+\Sigma^2(d\theta^2+\sin^2\theta ~d\phi^2),
	\end{equation}

	\noindent and the three-curvatures of the homogeneous hypersurfaces\footnote{This solution in particular, in addition to Ashtekar and Singh's question about emergent time in reference \cite{ABHAY2}, is what motivated the original object $\{\aleph,\mathcal{H},\Omega\}$.  In reference \cite{ABHAY2} Ashtekar and Singh asked, ``Can we extract, from the arguments of the wave function, one variable which can
	serve as emergent time with respect to which the other arguments `evolve'?''   It was the preexisting knowledge of this solution that made this writer immediately recognize the Gelfand triple as the correct algebraic dual to the $K\in\{\pm1,0\}$ geometries in the framework where Ashtekar and Singh's ``emergent time'' is the superposition of the positive and negative time modes.  Brian Kennedy had assigned Chris Isham's book \textit{Lectures on Quantum Theory} as part of my first graduate course in quantum mechanics and within this writer found the above mentioned reference to Gelfand's object.

	Kennedy criticized our first semester graduate course for doing so poorly on an exam.  I was absent for the criticism because he was handing back the test and reviewing student errors but I was sure I had aced the test.  He gave me 5/20 on one of three 20 point problems and left a note asking why I didn't solve it like he showed to do it in class.  I went to him the following week and showed him that if you set the three parameters I had left in my answer to one, then it was the right answer.  He immediate agreed and upped me to 15/20 which made me above the grade he had cited as a threshold in his criticism to the class.  I was really bitter about that because if there has ever been a 19/20 partial credit answer it was mine, it would have made me get the highest grade in the class, I would have almost got 100\% on the heavily weighted test almost ensuring my A after the final but I got docked four points because he had made an error in grading, compounded his error by leaving a note about not doing it his way, and didn't even realize I had aced the test during post-test lecture the previous week.  I was embittered by those four points and after that I was not interested in trying to get an A or being particularly studious in his second semester quantum mechanics course.  However, Kennedy deserves thanks for adding a second textbook to the syllabus for the breadth of his students' expose to the fundamental issues.  Isham's book has very helpful to me, and even inspirational in its direct straightforwardness.} in the form

	\begin{equation}\label{eq:BBB10}
	^{(3)}R_{ijkl}=[k/a^2(t)][\gamma_{ik}\gamma_{jl}-\gamma_{il}\gamma_{jk}]~~.
	\end{equation}

	\noindent The curvature parameter $K$, after this renormalization, is evidently

	\begin{equation}\label{eq:BBB11}
	K=k/a^2(t)~~.
	\end{equation}

	``Why is the word `renormalization' appropriate?  Previously $a(t)$ was a scale factor describing expansion of linear dimensions relative to the linear dimensions as they stood at some arbitrarily chosen epoch; but the choice of that fiducial epoch was a matter of indifference.  Now $a(t)$ has lost that arbitrariness.  It has been normalized so that its value here and now gives the curvature of a spacelike hypersurface of homogeneity here and now.  Previously the factor $a(t)$ was conceived as dimensionless.  Now it has dimensions of a length.  This length is called the `radius of the model universe' when the curvature is positive.  Even when the curvature is negative one sometimes speaks of $a(t)$ as a `radius.'  Only for zero curvature does the normalization of $a(t)$ still retain its former arbitrariness.  Thus, for zero curvature, consider two choices for $a(t)$, one of them $a(t)$ and the other $\bar a(t)=2a(t)$.  Then with $\bar \xi=\tfrac{1}{2}\xi$, one can write proper distances in the three dimensions of interest with perfect indifference in either of two ways:

	\begin{align}
	\left(\begin{matrix}
	\mathrm{proper~distance}\\
	\mathrm{in~ the~ direction}\\
	\mathrm{of~increasing~}\xi\\
	\end{matrix}\right)&=a(t)~d\xi=\bar a(t)~d\bar \xi~~,\label{eq:BBB12}\\
	&~\nonumber\\
	\left(\begin{matrix}
	\mathrm{proper~distance}\\
	\mathrm{in~ the~ direction}\\
	\mathrm{of~increasing~}\theta\\
	\end{matrix}\right)&=a(t)\xi ~d\theta=\bar a(t)\bar\xi~ d\theta~~,\label{eq:BBB13}\\
	&~\nonumber\\
	\left(\begin{matrix}
	\mathrm{proper~distance}\\
	\mathrm{in~ the~ direction}\\
	\mathrm{of~increasing~}\phi\\
	\end{matrix}\right)&=a(t)\xi \sin\theta~d\theta=\bar a(t)\bar\xi\sin\theta~ d\phi~~.\label{eq:BBB14}
	\end{align}

	``No such freedom of choice is possible when the model universe is curved, because then the $\xi$'s in the last two lines are replaced by a function, $\sin \xi$ or $\sinh \xi$, that is not linear in its argument.

	``Despite the feasibility in principle of determining the absolute value of the `radius' $a(t)$ of a curved spacetime, in practice [\textit{1973's}] accuracy falls short of what is required to do so.  Therefore it is appropriate in many contexts to continue to regard $a(t)$ as a factor of relative expansion, the absolute value of which one tries to keep from entering into any equation exactly because it is difficult to determine.''\newline
\end{quotation}

The first thing we will point out is that Misner, the first author of reference \cite{BBB}, is the ``M'' in ADM, whose result \cite{ADM} we have rejected in reference \cite{GC} on the basis of modern CMB data that shows there is a heavenly multipole moment that we can very much ``grasp'' as a ''handle'' in the manifold.  We are not using ``no handles.''  The hats on all four of the ontological numbers are handles and we have formulated $\chi^5$ as another sort of handle.  In the above excerpt the authors state that the implication of no-handles is that the Riemann tensor ``must be constructed algebraically from pure numbers and from the only `handle-free' tensors that exist: the 3-metric $\gamma_{ij}$ and the three-dimensional Levi-Civita tensor $\epsilon_{ijk}$.''  Since the MCM does include handles, we will need to verify whether or not the MCM allows additional pieces to contribute to the Riemann curvature tensor.

We have not given much attention to the 3D metric $\gamma_{ij}$ since the main avenue of complexification is expected along the high-dimensional channels unique to $\Sigma^\pm$.  Of all the ways to deform the flat Minkowski space there is a subset of simpler deformations that only change the geometry of 3-space.  Among that subset, there is a simpler subset of scalar deformations that curve Minkowski space by some constant factor everywhere in the universe.  The spaces $\aleph$ and $\Omega$ have been assigned as de Sitter space and anti-de Sitter space exactly because they of this last variety.  Minkowski space, de Sitter space, and anti-de Sitter space are all maximally symmetric.  $\{\aleph,\mathcal{H},\Omega\}$ is especially maximally symmetric.

It is important to note that while $\mathcal{H}$ has a simple metric

\begin{equation}\label{eq:MMMink}
ds^2_\pm=-(dx^0)^2+\sum\limits_{i=1}^3(dx^i)^2~~,
\end{equation}\newline

\noindent $\aleph$ and $\Omega$ have different metrics that are non-trivial despite their respective conditions of maximal symmetry.  Carroll develops the metrics as follows.

\begin{quotation}
    ``The maximally symmetric spacetime with positive curvature ($\kappa>0$) is called \textbf{de Sitter space}.  Consider a five-dimensional Minkowski space with metric $ds^2_5=-du^2+dx^2+dy^2+dz^2+dw^2$, and embed a hyperboloid given by

    \begin{equation}
    -u^2+x^2+y^2+z^2+w^2=\alpha^2~~.
    \end{equation}

    \noindent Now introduce coordinates $\{t,\chi,\theta,\phi\}$ on the hyperboloid via

    \begin{align}
    u&=\alpha\sinh(t/\alpha)\\
    ~&\nonumber\\
    w&=\alpha\cosh(t/\alpha)\cos\chi\\
    ~&\nonumber\\
    x&=\alpha\cosh(t/\alpha)\sin\chi\cos\theta\\
    ~&\nonumber\\
    y&=\alpha\cosh(t/\alpha)\sin\chi\sin\theta\cos\phi\\
    ~&\nonumber\\
    z&=\alpha\cosh(t/\alpha)\sin\chi\sin\theta\sin\phi~~.
    \end{align}

    \noindent The metric on the hyperboloid is then

    \begin{equation}\label{eq:CarrMet1}
    ds^2=-dt^2+\alpha^2\cosh^2(t\alpha)\left[d\chi^2+\sin^2\chi\left(d\theta^2+\sin^2\theta d\phi^2\right)\right]~~.
    \end{equation}

    \noindent We recognize the expression in round brackets as the metric on a two-sphere, $d\Omega^2_2$, and the expression in square brackets as the metric on a three-sphere, $d\Omega^2_3$.  Thus, de Sitter space describes a spatial three-sphere that initially shrinks, reaching a minimum size at $t=0$, and then re-expands.\footnote{Shrinking, reaching a minimum, and then reexpanding is also known as ``bouncing.''}  Of course this particular description is inherited from a certain coordinate system; we will see that there are equally valid alternative descriptions.

    ``These coordinates cover the entire manifold.  You can generally check this by, for example, following the behavior of geodesics near the edges of the coordinate system; if the coordinates were incomplete, geodesics would appear to terminate in finite affine parameter.  The topology of de Sitter is thus $\mathbb{R}\times\mathbb{S}^3$.  This makes it very simple to derive the conformal diagram, since the important step in constructing conformal diagrams is to write the metric in a form which is conformally related to the Einstein static universe (a spacetime with topology $\mathbb{R}\times\mathbb{S}^3$, describing a spatial three-sphere with constant radius through time).  Consider the coordinate transformation from $t$ to $t'$ via

    \begin{equation}
    \cosh(t/\alpha)=\dfrac{1}{\cos(t')}~~.
    \end{equation}

    \noindent The metric (\ref{eq:CarrMet1}) now becomes

    \begin{equation}
    ds^2=\dfrac{\alpha^2}{\cos^2(t')}d\bar s^2~~,
    \end{equation}

    \noindent where $d\bar s^2$ represents the metric of the Einstein static universe,

    \begin{equation}\label{Dbar77}
    d\bar s^2=-(dt')^2+d\chi^2+\sin^2\chi d\Omega^2_2~~.
    \end{equation}

    \noindent The range of the new time coordinate is

    \begin{equation}
    -\pi/2<t'<\pi/2~~.
    \end{equation}

    ``The conformal diagram of de Sitter space will simply be a representation of the patch of the Einstein static universe to which the de Sitter is conformally related.  It looks like a square, as shown [\textit{on the left side of figure \ref{fig:conformal}}].  A spacelike slice of constant $t'$ represents a three-sphere; the dashed lines at the left and right edges are the north and south poles of this sphere.  The diagonal lines represent null rays; a photon released at past infinity will get precisely to the antipodal point on the sphere at future infinity.  Keep in mind that the spacetime `ends' to the past and future only through the magic of conformal transformations; the actual de Sitter space extends indefinitely into the future and past.  Note also that two points can have future (or past) light cones that are completely disconnected; this reflects the fact that the spherical spatial sections are expanding so rapidly that light from one point can never come into contact with light from the other.

    ``A similar hyperboloid construction reveals the $\kappa<0$ spacetime of maximal symmetry, known as \textbf{anti-de Sitter} space.  Begin with a fictitious five-dimensional manifold with metric $ds^2_5=-du^2-dv^2+dx^2+dy^2+dz^2$, and embed a hyperboloid given by

    \begin{equation}
    -u^2-v^2+x^2+y^2+z^2=-\alpha^2~~.
    \end{equation}

    \noindent Note all the minus signs.  Then we can induce coordinates $\{t',\rho,\theta,\phi\}$ on the hyperboloid via

    \begin{align}\label{CarrNewCoord77}
    u&=\alpha\sin(t')\cosh(\rho)\\
    ~&\nonumber\\
    v&=\alpha\cos(t')\cosh(\rho)\\
    ~&\nonumber\\
    x&=\alpha\sinh(\rho)\cos\theta\\
    ~&\nonumber\\
    y&=\alpha\sinh(\rho)\sin\theta\cos\phi\\
    ~&\nonumber\\
    z&=\alpha\sinh(\rho)\sin\theta\sin\phi~~,
    \end{align}

    \noindent yielding a metric on this hyperboloid of the form

    \begin{equation}\label{eq:CarrMet762}
    ds^2=\alpha^2\left(-\cosh^2(\rho)dt'^2+d\rho^2+\sinh^2(\rho)d\Omega^2_2\right)~~.
    \end{equation}

    \noindent These coordinates have a strange feature, namely that $t'$ is periodic.\footnote{The MCM proposal to modify spacetime by wrapping time around a cylinder, a.k.a. giving $x^0$ a circular topology, has the same effect of making time periodic.}  From [\textit{equation (\ref{CarrNewCoord77})}], $t'$ and $t'+2\pi$ represent the same place on the hyperboloid.  Since $\partial_{t'}$ is everywhere timelike, a curve with constant $\{\rho,\theta,\phi\}$ as $t'$ increases will be a closed timelike curve.  However, this is not an intrinsic property of the spacetime, merely an artifact of how we have derived the metric from a particular embedding.  We are welcome to consider the `covering space' of this manifold, the spacetime with the metric given by (\ref{eq:CarrMet762}) in which we allow $t'$ to range from $-\infty$ to $\infty$.  There are no closed timelike curves in this space, which we will take to be the definition of anti-de Sitter space.

    ``To derive the conformal diagram, perform a coordinate transformation analogous to that used for de Sitter, but now on the radial coordinate:

    \begin{equation}
    \cosh(\rho)=\dfrac{1}{\cos\chi}~~,
    \end{equation}

    \noindent so that

    \begin{equation}
    ds^2=\dfrac{\alpha^2}{\cos^2\chi}d\bar s^2,
    \end{equation}

    \noindent where $d\bar s^2$ represents the metric on the Einstein static universe (\ref{Dbar77}).  Unlike in de Sitter, the radial coordinate now appears in the conformal factor.  In addition, for anti-de Sitter, the $t'$ coordinate goes from minus infinity to plus infinity, while the range of the radial coordinate is

    \begin{equation}
    0\leq\chi<\frac{\pi}{2}~~.
    \end{equation}

    \noindent Thus, anti-de Sitter space is conformally related to half of the Einstein static universe.  The conformal diagram is shown [\textit{on the right side of figure \ref{fig:conformal}}], which illustrates a few representative timelike and spacelike geodesics passing through the point $t'=0$, $\chi=0$.  Since $\chi$ only goes to $\pi/2$ rather than all the way to $\pi$, a spacelike slice of this spacetime has the topology of the interior of a hemisphere $\mathbb{S}^3$; that is, it is topologically $\mathbb{R}^3$ (and the entire spacetime therefore has the topology $\mathbb{R}^4$).  Note that we have drawn the diagram in polar coordinates, such that a point on the left side represents a point at the spatial origin, while [\textit{the}] one on the right side represents a two-sphere at spatial infinity.  Another popular representation is to draw the spacetime in cross-section, so that the spatial origin lies in the middle and the right and left sides together comprise spatial infinity.\footnote{Note how Carroll's description of the options for drawing the conformal diagram mirror our own options for drawing the MCM unit cell centered on $\mathcal{H}$ or $\varnothing$}


    \begin{figure}[t]
        \makebox[\textwidth][c]{\includegraphics[scale=.5]{Conformal.png}}
        \captionsetup{format=hang}\caption{These conformal diagrams are most appreciatively reproduced from Carroll's textbook, reference \cite{CARR}: Spacetime and Geometry: An Introduction to General Relativity.}\label{fig:conformal}
    \end{figure}


    ``An interesting feature of anti-de Sitter is that infinity takes the form of a timelike hypersurface, defined by $\chi=\pi/2$.  Because infinity is timelike, the space is not globally hyperbolic, we do not have a well-posed initial value problem in terms of information specified on a spacelike slice, since information can always `flow in from infinity.'  Another interesting feature is that the exponential map is not onto the entire spacetime; geodesics, such as those drawn [\textit{on the right side of figure \ref{fig:conformal}}], which leave from a specified point do not cover the whole manifold.  The future-pointing timelike geodesics, as indicated, can initially move radially outward from $t'=0$, $\chi=0$, but eventually refocus to the point $t'=\pi, \chi=0$ and will then move radially outward once again.\footnote{This is ordinarily described as bouncing.}

    ``As an aside it is irresistible to point out that the timelike nature of infinity enables a remarkable feature of string theory, the `AdS/CFT correspondence.'  Here, AdS is of course the anti-de Sitter space we have been discussing, while CFT stands for conformally-invariant field theory defined on the boundary (which is, for an $n$-dimensional AdS, an $(n-1)$-dimensional spacetime in its own right).  The AdS/CFT correspondence suggests that, in a certain limit, there is an equivalence between quantum gravity (or a supersymmetric version thereof) on an AdS background and a conformally-invariant non-gravitational field theory defined on the boundary.  Since we know a lot about non-gravitational quantum field theory that we don't know about quantum gravity, this correspondence (if true, which seems likely but remains unproven\footnote{See reference \cite{GC} for a proof.}) reveals a great deal about what can happen in quantum gravity.''\newline
\end{quotation}

Carroll's example of embedded hyperboloids demonstrates exactly what we have done to embed the chronological dS and AdS metrics onto the chirological slices of $\Sigma^\pm$.  In chapter four we will go into some detail about the exponential map because it is closely related to the winding number on the quantum phase that will be discussed there.  We have included these long excerpts mainly to demonstrate the general relevance of the MCM: almost every type of thing we have imposed as an ``unmotivated'' constraint was already an inherent constraint on the objects from which the MCM is assembled.  Furthermore, Carroll shows how the conformal diagrams of dS and AdS are constructed from co-$\hat \pi$'s or pairs of halves of co-$\hat\pi$'s, as in figure \ref{fig:conformal}.



\begin{figure}[t]
	\makebox[\textwidth][c]{\includegraphics[scale=1]{Principia.png}}
	\captionsetup{format=hang}\caption{Calculation of the the geodesics of the MCM will require a computational facility much greater than the one relied upon by this writer.  That facility is demonstrated in this figure which shows a page excerpted from reference \cite{NEWTON}}.\label{fig:Principia}
\end{figure}

The important takeaway from figure \ref{fig:conformal} is that where dS is constructed naturally on two complete co-$\hat \pi$'s, AdS is only constructed from one and a half co-$\hat \pi$'s.  Through the presence of half co-$\hat \pi$'s we can see that it will be easy to modularize the coordinate systems described by Carroll for immediate application to the construction of complete $\hat\pi$-sites in the MCM.  Any advanced calculation using these coordinates would be impractical to carry out by hand, as is the analytical style of this writer and others, as in figure \ref{fig:Principia}.  On the up side, the interested party should be excited to know that tensor algebra solver software exists and is described in reference \cite{LLWILL}.  Williams' paper can likely be trusted as the source of record for many common formulae that appear in various states of correctness and incorrectness throughout the literature because he has derived them with software.\footnote{Regarding imperfect formulae, even software can have errors and in any case a comprehensive survey of the literature is always in order.  Regarding literature known to contain errata: just as a literate person can understand a sentence with a misspelled word in it, so can a numerate person often recognize what an equation demonstrates even when its formalism is not algorithmically impeccable.  Therefore when the critic is numerate criticisms of the form, ``This is wrong,'' or, ``This is not even wrong,'' can be rebutted with ``Did I demonstrate something there?''  If the answer is yes then the target of the criticism might close his rebuttal,  ``\textit{Q.E.D.}''}


\subsection{Toward Geodesics}

Geodesics are another topic better handled on computers.  Wikipedia says, ``The full Kaluza equations under the cylinder condition are quite complex, and most English-language reviews as well as the English translations of [\textit{foreign language reports}] contain some errors.''  If we did undertake the effort to calculate by hand and then report the geodesics of the MCM our report would almost certainly contain errors of this sort and thereby we would have added little to the literature in our publication of an erroneous or even error-riddled set of geodesics.  Certainly this is an exercise best left to computer experts or until this writer becomes more expert with computers.  We very much refer the reader to reference \cite{LLWILL} wherein Williams has done a great service to mankind putting together a comprehensive and seemingly error-free review of common formulas related to the 5D KK metric which is




\begin{align}\nonumber
\Sigma_{AB}
&=\left(\begin{matrix}
-1+B_1B_1\chi^5 & B_1B_2\chi^5 & B_1B_3\chi^5 & B_1B_4\chi^5 &  B_1\chi^5\\[8pt]
B_2B_1\chi^5 & 1+B_2B_2\chi^5 & B_2B_3\chi^5 & B_2B_4\chi^5 &  B_2\chi^5\\[8pt]
B_3B_1\chi^5 & B_3B_2\chi^5 & 1+B_3B_3\chi^5 & B_3B_4\chi^5 &  B_3\chi^5\\[8pt]
B_4B_1\chi^5 & B_4B_2\chi^5 & B_4B_3\chi^5 & 1+B_4B_4\chi^5 &  B_4\chi^5\\[8pt]
B_1\chi^5 & B_2\chi^5 & B_3\chi^5 & B_4\chi^5 &  \chi^5\\[8pt]
\end{matrix}\right)\\
&~\nonumber\\
&=\left(\begin{matrix}
g_{\alpha\beta}+B_\alpha B_\beta\chi^5& &B_\alpha\chi^5\\[8pt]
B_\beta\chi^5 & &\chi^5
\end{matrix}\right)~~.
\end{align}\newline

We will not complete the calculation here because they are too complicated to do efficiently by hand.  Note that the formulas in reference \cite{LLWILL} use the cylinder condition throughout the 5D space but we are only forced to apply the cylinder condition on the slices $\aleph$ and $\Omega$ but not necessarily on each individual slice of $\Sigma^\pm$ where $\chi^5_\pm\not\in\{\Phi,0,\varphi\}$.  $\mathcal{H}$ and $\varnothing$ are further differentiated from $\aleph$ and $\Omega$ when we note that the cylinder condition is not a prerequisite for $\{\mathcal{H},\varnothing\}$.  $\mathcal{H}$ is a flat 4D brane embedded in a smoothly varying continuum of de Sitter branes and that continuum, including $\mathcal{H}$, is constrained to have a representation in general relativity where $x^4$ is the parameter of curvature.  $\mathcal{H}$ can therefore not be part of a Kaluza--Klein theory with an inherent scalar field $\phi^2$ induced through the cylinder constraint.  $\varnothing$ is a qubit at infinity whose topology is defined as the topology of $\mathcal{H}$ because the only thing we might ever consider using it for is to define a Lorentz frame; $\varnothing$ is so defined and $\Sigma^\varnothing$ is constrained accordingly.  The Lorentz approximation is that the lab frame is Minkowski space devoid of any perturbations due the mass-energy of lab equipment, planets, moons, scientists, \textit{etc}., so therefore we can always define that frame in $\varnothing$.  The MCM condition means that the origin of coordinates in $\varnothing$ would be the point of $\Sigma^\varnothing$ that stitches $\chi^5_\pm$ together in the piecewise definition of geodesics parameterized in $\chi^5$.  If we were going to undertake computerization, a first check on the robust stability of MCM would be to compare its geodesics in the cases where $B_\alpha=0$ and $B_\alpha=1$. Since it is only potential difference that contribute to observables we should get the exact same geodesics for any constant potential $B_\alpha$.

Everything in the geodesic equation

\begin{equation}
\frac{d^2\chi^C}{d\lambda^2}+\Gamma^C_{AB}\frac{d\chi^A}{d\lambda}\frac{d\chi^B}{d\lambda}=0~~,
\end{equation}\newline

\noindent comes from the derivatives of the metric.  The non-tensorial Christoffel symbols (connection coefficients) are

\begin{equation}
\Gamma_{AB}^C=\tfrac{1}{2}\,\Sigma^{CD}\,\big( \partial_A \Sigma_{BD}+\partial_B \Sigma_{DA}-\partial_D \Sigma_{AB}\big)~~,
\end{equation}\newline

\noindent and they appear in the geodesic equation as

\begin{equation}
\frac{d^2\chi^C}{d\lambda^2}+\Gamma^C_{AB}\frac{d\chi^A}{d\lambda}\frac{d\chi^B}{d\lambda}=0~~.
\end{equation}\newline

\noindent  The geodesic equation reflects an impressive amount of complexity already present in Einstein's theory from 100 years ago.  Certainly Einstein carried out more intense manual computations than we have in this research program.  In fact it is due to his efforts that we recognized the dimensionless coefficient of proportionality $8\pi$ as highly relevant in reference \cite{TER}.  It must be noted that Einstein had no hope of using computers to make his calculations and that his contemporaries such as Schwarzschild, Nordstr\"om, and others were the first to make many of the most elementary calculations in general relativity.  Likewise, any of this writer's contemporaries could be the first to make elementary calculations in the MCM/TOIC but it seems science has taken a turn since the olden times.  To find the connection coefficients we need the inverse metric which is


\begin{equation}
\Sigma^{AB}=\left(\begin{matrix}
-1 & 0 & 0 &0 & B_1 \\[8pt]
0 & 1 & 0 &0 & -B_2 \\[8pt]
0 & 0 & 1 &0 & -B_3 \\[8pt]
0 & 0 & 0 &1 & -B_4 \\[8pt]
B_1 &-B_2 &-B_3 &-B_4 &\left(\eta^{\alpha\beta}B_\alpha B_\beta +\dfrac{1}{\chi^5}\right)
\end{matrix}\right)~~.
\end{equation}\newline

In reference \cite{IC} we pointed out that since $\chi^5\equiv\chi^5_+ \oplus\chi^5_\varnothing \oplus\chi^5_-$ is a non-standard pseudo-dimension that means we have a problem understanding $\hat M^3\equiv\partial^3$ in terms of the chirological time.  However, $x^0$ is not a pseudo-dimension so in the chronological sense $\hat M^3\equiv\partial_t^{\,3}$, and this is where Einstein's $8\pi$ comes from but we had to define an alternate convention for chirological $\partial^{\,3}_\chi$ \cite{IC}.  That convention is

\begin{equation}\label{eq:triplederiv}
\partial^{\,3}_5 \quad\equiv\quad \frac{\partial}{\partial\chi^5_-}\frac{\partial}{\partial\chi^5_\varnothing}\frac{\partial}{\partial\chi^5_+}~~.
\end{equation}\newline

Here we will assume a certain set of gauge transformations

\begin{equation}\label{eq:gauge7622}
\psi'=e^{i\Lambda}\psi\quad\qquad\longleftrightarrow \quad\qquad B_\mu '=B_\mu+\dfrac{1}{e}\,\partial_\mu \Lambda~~.
\end{equation}\newline

\noindent We will use gauge transformations to define the operator $\partial_5^{\,3}$ according to the prescription in reference \cite{IC}.  In that reference we showed only how the wavefunction transforms but here we will also consider the transformation of the potential because it drives the transformation of the metric and geodesics.  To derive the fully relativistic gauge transformation we will need to substitute the covariant derivative for the partial derivative that appears in equation (\ref{eq:gauge7622}) but we will proceed in the limit where the covariant derivative is equal to the partial derivative.  Therefore what follows in this section is only an estimate.

To derive a non-trivial ``ontological'' potential that could be used to define a specific metric and therefore a specific set of geodesics, we will start with

\begin{equation}
\psi_0=e^{-i\omega t}~~,\quad\qquad\mathrm{and}\quad\qquad B_\mu=0~~.
\end{equation}\newline

\noindent We will make gauge transformations to accommodate the operator $\partial^{\,3}_5$ according to equation (\ref{eq:triplederiv}).  The first derivative in $\partial^{\,3}_5$ is with respect to $\chi^5_+$ so we make the gauge transformation with

\begin{equation}
\Lambda_1=\Phi\chi^5_++\omega t ~~.
\end{equation}\newline

\noindent This gives the wavefunction

\begin{equation}
\psi_1=e^{i\Lambda_1}\psi_0=e^{i(-\omega t+\Phi\chi^5_++\omega t)}=e^{i\Phi\chi^5_+}~~.
\end{equation}\newline

\noindent Gauge theory requires a corresponding transformation of the potential, as in equation (\ref{eq:gauge7622}).  The transformations employed here do not affect the $B_i$ components of the 4-potential.  Those components as they appear in the gauge transformed potential will be proportional to the $\partial_i$ derivatives of $\Lambda_1$ and it is obvious those will all vanish.  The new potential is

\begin{align}
B_0^{\,\{1\}}&=\dfrac{1}{e} \,\partial_0 \big(\Phi\chi^5_++\omega t \big)=\dfrac{\omega}{e}\\
~&\nonumber\\
B_i^{\,\{1\}}&=B_i=0~~,
\end{align}\newline

To apply $\partial_\varnothing$ we need to apply another gauge transformation

\begin{equation}
\psi_2=e^{i(\Phi\chi^5_++\Lambda_2)}~~,\quad\qquad\mathrm{with}\quad\qquad\Lambda_2=\pi\chi^5_\varnothing-\Phi\chi^5_+~~,
\end{equation}\newline

\noindent which gives

\begin{equation}
\psi_2=e^{i\pi\chi^5_\varnothing}~~.
\end{equation}\newline

\noindent The derivative of $\Lambda''$ with respect to $x^\mu$ vanishes so there is no change in the potential and we see

\begin{align}
B_0^{\,\{2\}}&=B_0^{\,\{1\}}=\dfrac{\omega}{e} \\
&~\nonumber\\
B_i^{\,\{2\}}&=B_i^{\,\{1\}}=B_i=0~~.
\end{align}\newline

\noindent We need to complete taking $\partial^{\,3}_5$ with one more gauge transformation for $\partial_-$ which is

\begin{equation}
\psi_3=e^{i(\pi\chi^5_\varnothing+\Lambda_3)}~~,\quad\qquad\mathrm{with}\quad\qquad\Lambda_3=-\Phi\chi^5_--\pi\chi^5_\varnothing~~,
\end{equation}\newline

\noindent giving

\begin{equation}
\psi'''=e^{-i\Phi\chi^5_-}~~.
\end{equation}\newline

\noindent Again the derivative of $\Lambda_3$ with respect to $x^\mu$ vanishes so

\begin{align}
B_0^{\,\{3\}}&=B_0^{\,\{2\}}=B_0^{\,\{1\}}=\dfrac{\omega}{e} \\
&~\nonumber\\
B_i^{\,\{3\}}&=B_i^{\,\{2\}}=B_i^{\,\{1\}}=B_i=0~~.
\end{align}\newline


\begin{figure}[t]
    \makebox[\textwidth][c]{\includegraphics[scale=1]{bounce762.png}}
    \captionsetup{format=hang}\caption{A plane wave incident on the bounce state, which is generally understood by now as either the endpoints of the MCM unit cell or the central point corresponding to $p\in\Sigma^\varnothing$, can be either transmitted or reflected.}.\label{fig:bbounce}
\end{figure}

The final remaining step is to convert back to a wavefunction of the $x^\mu$ coordinates which we can do in a few separate ways.  This is very interesting and surely contains a lot more nuance than will be discussed here.  We have started with an ordinary wavefunction $e^{-i\omega t}$ and used gauge transformation operations to take derivatives with respect to the three chirological times in the $\hat M^3\equiv\partial^{\,3}_5$ operator.  The final gauge transformation will be applied to convert the wavefunction back into a form that does not depend on chiros.

These two simple options, $a$ and $b$, for for completing $\psi_0(x^\mu)\mapsto\psi_4(x^\mu)$ are

\begin{equation} \label{eq:psipossible}
\psi_4^{\,a\pm}=e^{\mp i\omega t}~~,\quad\qquad\mathrm{or}\quad\qquad\psi_4^{\,b\pm}=e^{\pm ikx}~~,
\end{equation}\newline


\noindent and there are also non-simple forms where the phase has mixed dependence on both time and space with $\Delta\equiv kx-\omega t$.  Even if we choose the simpler option where the initial and final states both depend on $t$ but not $x$, we still have the option choose the sign that changes the direction of the wavepacket's propagation.  Figure \ref{fig:bbounce} illustrates how this arrangement is very natural to the MCM.  Also note that $\psi_4$ seems logically related to the new process $\tilde M^4$ though it arose purely through consideration of $\hat M^3$.  We must associate the two forms $\psi_4^{\,a}$ and $\psi_4^{\,b}$ with the idea that the topology on either side of $\mathcal{H}$ has different numbers of spacelike and timelike dimensions.  A 5D plane wave incident on $\mathcal{H}$ from $\Sigma^-$, if transmission is possible, must have some component converted from a spacelike domain onto a timelike domain.  Since timelike dimensions use dimensional transposing parameters such as $c$, we can expect that this change of domain significantly alters the character of transmitted wave.  In fact, it is not outlandish to think that the dimensional transposing parameter might even alter the energy of the wave as it passes through $\mathcal{H}$.  However, since this is a new idea, we will not treat it here and instead leave it to a devoted treatment to appear the future $\Omega$.  Here we remain focused on the gauge potential that arises due to the gauge transformation of the wavefunction presented above.

\begin{figure}[t]
    \makebox[\textwidth][c]{\includegraphics[scale=.77]{unitcell_762.png}}
    \captionsetup{format=hang}\caption{The original graphical iteration of the MCM unit cell is, perhaps, the most intuitive representation of its important geometric features.}\label{fig:origunitcell}
\end{figure}

To achieve $\psi_4^{\,a}$ or $\psi_4^{\,b}$ will have to consider two $\Lambda_4$'s

\begin{equation}
\Lambda_4^{\,a\pm}=\mp\omega t+\Phi\chi^5_-~~,\quad\qquad\mathrm{and}\quad\qquad\Lambda_4^{\,b\pm}=\pm kx+\Phi\chi^5_-~~.
\end{equation}\newline

\noindent If we choose to arrive at $\psi_4^{\,a}$ then the $B^{\,\{4a\}}_i$ components will all vanish and the final gauge transformed potential is

\begin{align}
B_{0}^{\,\{4a\mp\}}&=\dfrac{\omega}{e}+\dfrac{1}{e}\,\partial_0\big(\mp\omega t+\Phi\chi^5_-\big)=\dfrac{\omega}{e}\mp\dfrac{\omega}{e}\\
~&\nonumber\\
B_{i}^{\,\{4a\mp\}}&=0~~.
\end{align}\newline

\noindent These values for $B_\mu^{\,\{4\}}$ look very good.  If we select the initial and final wavefunctions as $e^{-i\omega t}$ then the potential at the end is equal to the potential at the start: $B_\mu=0$.  However, at some point (not presently) we will need to study the induced potential

\begin{equation}
B_{0}^{\,\{4a+\}}=\dfrac{2\omega}{e}~~,
\end{equation}\newline

\noindent associated with a reflection of the wavepacket as in figure \ref{fig:bbounce}.

For $\Lambda_4^{\,b}$ we have

\begin{align}
B_{0}^{\,\{4b\pm\}}&=\dfrac{\omega}{e}+\dfrac{1}{e}\partial_0\big(\pm kx+\Phi\chi^5_-\big)=\dfrac{\omega}{e}\\
~&\nonumber\\
B_{i}^{\,\{4b\pm\}}&=0+\dfrac{1}{e}\,\partial_i\big(\pm kx-\Phi\chi^5_-\big)=\pm\dfrac{k}{e}~~,
\end{align}\newline

\noindent which is more complicated than $B_\alpha^{\,4\,a}$ because it involves both $x$ and $t$.  In this case, the final potential will have non-vanishing components $B_i^{\,\{4a\}}$.

Here we will state a motivation for assigning two timelike dimensions to $\Sigma^+$ and only one timelike dimension to $\Sigma^-$.  For $\hat M^1$ we want to evolve some initial state along the chirological time instead of the chronological time.  This is, in general, how the MCM mode of evolution differs from old physics.  If we evolved the qubit with chronological time, then we would be in perfect agreement with what the classical, or ordinary, theory says to do.  In order to construct a second timelike path for new physics, we should assign the O(2,3) topology to $\Sigma^+$.  To do this, we simply redefine the scalar field that appears in the 5D metric as

\begin{equation}\label{eq:changePHIA}
\phi^2\quad\equiv\quad\chi^5\quad\qquad\longrightarrow\quad\qquad\phi^2\quad\equiv\quad-\chi^5~~.
\end{equation}\newline

\noindent Now the signs in $\Sigma^\pm_{AB}$ will show that $\Sigma^+$ has two timelike dimensions and $\Sigma^-$ will have four spacelike dimensions.

Consider the role of the second timelike dimension in $\Sigma^+$ after we formulate an initial condition at $\chi^5=0$.  We know that the qubit will evolve in proper time according to the Schr\"odinger equation but we want to add the chirological mode of evolution along geodesics so we may explain aspects of quantum weirdness with ``interference effects'' between the two timelike evolutionary channels.  We evolve the initial qubit through $\chi^5_+\in(0,\Phi]$ in $\Sigma^+$ and note that the interval $(0,\Phi]$ is just long enough to hold one half of one co-$\hat \pi$, or one half of one $\hat \pi$ vector, just as is required to construct the conformal diagram of AdS shown in figure \ref{fig:conformal}.  This follows because

\begin{equation}
\frac{\pi}{2}\approx1.57~~,\quad\qquad\mathrm{and}\quad\qquad\Phi\approx1.62~~,
\end{equation}\newline

\noindent and it also follows that in the modularization process for AdS and dS on co-$\hat \pi$'s  we can expect characteristic remainder terms on the order of $1.62\!-\!1.57=0.05<<1$ which can be used to make perfect geometry fuzzy.

\subsection{Dark Energy and Expanding Space}

To finish this chapter we will return to the MCM's first result: dark energy.  Unification of the theories of gravitation and quantization is a vastly more primary problem than dark energy because it predates all of what we now call modern physics or post-war physics but dark energy is still very important.  The MCM dark energy result \cite{MOD,DE} was followed by a new numerical formula for the fine structure constant \cite{FSC,TER}, a derivation of Einstein's equation based on the numbers in that formula \cite{TER}, a new geometric explanation for the AdS/CFT correspondence \cite{GC}, an explanation for the structure of the fundamental particles in the standard model as facets of the AdS/CFT geometry \cite{QS}, a prediction for two new spin-1 particles \cite{GC}, and many other interesting results \cite{OP,BELL,KN,QG,ICT,WICK,IC,ZETA}.  The MCM mechanism for dark energy, called inverse radial spaghettification, can be thought of as ordinary spaghettification toward the past (to the left) in the cosmological unit cell where the radial coordinate can be intuitively extrapolated from the rectangular coordinates in the figures.

Using equation (\ref{eq:changePHIA}), consider the KK metric in the limit of vanishing complexity.  When $B_\alpha=0$ we have a pair of ``ground state'' metrics

\begin{equation}\label{eq:KK3}
\Sigma_{AB}^{+~O(2,3)}=\left(\begin{matrix}
-1~& 0& 0& 0&0\\
0& ~1~&0 &0 &0\\
0&0 & ~1~&0&0\\
0&0 &0 &~1~ &0\\
0&0 &0 &0 &~-\chi^5_+
\end{matrix}\right)~~,
\end{equation}\newline

\noindent and

\begin{equation}\label{eq:KK53}
\Sigma_{AB}^{-~O(1,4)}=\left(\begin{matrix}
-1~& 0& 0& 0&0\\
0& ~1~&0 &0 &0\\
0&0 & ~1~&0&0\\
0&0 &0 &~1~ &0\\
0&0 &0 &0 &~-\chi^5_-
\end{matrix}\right)~~.
\end{equation}\newline

\noindent The form of each metric is identical and it is only the sign of $\chi^5_\pm$ that distinguishes the number of timelike dimensions.  The metrics $\Sigma^\pm_{AB}$ are essentially the same so we may follow the ordinary prescription to derive the line element for a generic KK metric $\Sigma_{AB}$.

\begin{align}
ds^2_{MCM}&=\Sigma_{AB}\,d\chi^A\,d\chi^B\\
&~\nonumber\\
&=-\big(d\chi^1\big)^2+\big(d\chi^2\big)^2+\big(d\chi^3\big)^2+\big(d\chi^4\big)^2-\chi^5\big(d\chi^5\big)^2~~.\label{eq:KKFLRW}
\end{align}\newline

On cosmological scales, $B_\alpha=0$ is good approximation for the nearly negligible intergalactic magnetic field so we should consider the implications of equation (\ref{eq:KKFLRW}) as physical implications.  At first glance $ds^2_{MCM}$ is very nearly the FLRW line element

\begin{equation}\label{eq:flrwmetriccc}
ds^2_{FLRW}=-\big(dx^0\big)^2+a^2\big(x^0\big) \left[\big(dx^1\big)^2+\big(dx^2\big)^2+\big(dx^3\big)^2\right]~~.
\end{equation}\newline

\noindent of flat expanding space.  The FLRW metric is flat expanding rather than curved expanding because the scale factor $a^2(x^0)$ only depends on $x^0$ but not the spatial coordinates $x^i$.  Note how the scale factor in the FLRW metric is a quadratic function of a timelike dimension exactly like the scale factor in equation (\ref{eq:KKFLRW}).  Using equation (\ref{eq:changePHIA}) to substitute the scalar field $\phi^2(\chi^5)\equiv-\chi^5$ back into $ds^2_{MCM}$ we see the two line elements are indeed similar with $a^2(x^0)\to\phi^2(\chi^5)$ in

\begin{equation}\label{eq:MCMmetriccc}
ds^2_{MCM}=-\big(d\chi^1\big)^2+\big(d\chi^2\big)^2+\big(d\chi^3\big)^2+\big(d\chi^4\big)^2+\phi^2\big(\chi^5\big)\left[\big(d\chi^5\big)^2\right]~~.
\end{equation}\newline


In equation (\ref{eq:flrwmetriccc}) the scale factor stretches all three spatial dimensions equally but in equation (\ref{eq:MCMmetriccc}) the scale factor only stretches the fifth dimension.  In $\Sigma^+$ where $\chi^5_+$ is a positive valued timelike dimension the metric shows contracting time.  On the other side of $\mathcal{H}$ where $\chi^5_-$ is negative and spacelike, the metric shows the expanding space but not in all four spacelike dimensions.  \textbf{Expanding space} in the past is exactly what astrophysicists observe and contracting time in the future is more or less the mechanism by which we have proposed to generate \textbf{dark energy}.  Furthermore, the expansion of space by a parameter in $[\varphi,0)$ is unlikely to be exactly counterbalanced by a contraction of time proportional to a parameter in $(0,\Phi]$ and that can lead to a pressure gradient which causes the arrow of time to point preferentially in one direction.

The FLRW metric describes flat space because the expansion parameter is a function of a timelike variable.  In $\Sigma^-$ the fifth dimension is spacelike so not only do we get expanding space there, we get expanding curved space exactly like what will be required to embed slices that contain 4D dS and AdS.  Our example of expanding space only shows expanding space in one dimension so there is a natural interpretation related to increasing levels of $\aleph$ along $\chi^5$, and another natural interpretation where the expansion of three spatial dimensions that is properly astrophysical can a reciprocal effect derived from a modularized dependence of the expansion of the fourth spacelike dimension in $\Sigma^-$.


\subsection{Advanced and Retarded Potentials} \addtocontents{toc}{\protect\vspace{22pt}}

When approaching $\mathcal{H}$ from the directions of $\chi^5_\pm$, the discrepancy between the O(1,4) and O(2,3) topologies makes it impossible to model the boundary as a simple 1D transmission problem for the quantum mechanical probability current to pass through $\mathcal{H}$ or ($\varnothing$.)  In chapter four we will model the MCM in 1D (motion along $\chi^5$ only) to derive a hypercomplex potential function $V$ that plugs into the Hamiltonian of the Schr\"odinger equation as

\begin{equation}
\mathbb{\hat H}=\frac{\hat p^2}{2m}+\hat V~~.
\end{equation}\newline

\noindent With a well defined Hamiltonian, we will solve for the transmission and reflection coefficients for the probability current in the hypercosmos.  We mention that in this section because we may be able to identify those coefficients, $T$ and $R$, with the advanced and retarded potentials.  If $T$ is the amplitude of the signal in the forward time direction from $\mathcal{H}_1$, then it reaches the next moment $\mathcal{H}_2$ as a retarded signal from the past.  However, the signal that is reflected from $\mathcal{H}_1$ with amplitude $R$ will move in reverse time meaning that it will be incident on an earlier $\mathcal{H}$-brane (or an earlier $\varnothing$-brane) from the future as is expected of the advanced potential.

Consider the place of the advanced potential as given by reference \cite{ZEH}.\footnote{We substitute $B^\mu$ for the $A^\mu$ that appeared in the original citation.}

\begin{quotation}
    ``Electromagnetic radiation will here be considered as an example for wave phenomena in general.\footnote{Although the Schr\"odinger equation is the heat equation rather than the intuitive wave equation, it does certainly fall under the topic of wave phenomena in general.}  It may be described in terms of the four-potential $B^\mu$, which in the Lorenz gauge obeys the wave equation

    \begin{equation}\label{eq:ZEH1}
    -\partial^\nu\partial_\nu B^\mu(\textbf{r},t)=4\pi j^\mu(\textbf{r},t)~~,\quad\qquad\mathrm{with}\quad\qquad\partial^\nu\partial_\nu = -\partial^2_t+\Delta~~,
    \end{equation}

    \noindent with $c=1$, where the notations\footnote{$\Delta$ is the Laplacian operator: $\Delta\equiv\nabla^2\equiv\nabla\cdot\nabla$.} $\partial_\mu:=\partial/\partial x^\mu$ and $\partial^\mu:=g^{\mu\nu}\partial_{\nu}$ are used together with Einstein's [\emph{summation convention}].  When an appropriate boundary condition is imposed, one may write $A^\mu$ as a functional of the sources $j^\mu$.  For two well known boundary conditions one obtains the $\emph{retarded}$ and the $\emph{advanced}$ potentials,

   \begin{align}
    B^\mu_{\mathrm{ret}}&=\int\frac
	{j^\mu(\textbf{r}, t - |\textbf{r}-\textbf{r}'|)}
	{|\textbf{r}-\textbf{r}'|}d^3\textbf{r}'\\
	~&\nonumber\\
	B^\mu_{\mathrm{adv}}&=\int\frac
	{j^\mu(\textbf{r}, t + |\textbf{r}-\textbf{r}'|)}
	{|\textbf{r}-\textbf{r}'|}d^3\textbf{r}'~~.
   \end{align}

    \noindent These two functionals of $j^\mu(\textbf{r},t)$ are related to one another by a reversal of retardation time $|\textbf{r}-\textbf{r}'|$  [$\emph{sic}$].  Their linear combinations are solutions of  [(\ref{eq:ZEH1}).]''\newline
\end{quotation}

The origin of $B^\mu_{\mathrm{adv}}$ and $B^\mu_{\mathrm{ret}}$ is a tedious subject in classical electromagnetism and will be beyond the scope of this book.  The point of this book is (mostly) to go into greater detail regarding the general relevance of that which has been published already.  However, we do want to emphasize that when we impose the MCM condition that sets $t=0$, the second argument in $j^\mu$ is completely natural to the MCM coordinates.  For $j^\mu$ evaluated at the advanced time we can expect

\begin{equation}
|\mathbf{r}-\mathbf{r}'|=\Phi~~,
\end{equation}\newline

\noindent and for $j^\mu$ evaluated at the retarded time we expect

\begin{equation}
|\mathbf{r}-\mathbf{r}|=-\varphi~~.
\end{equation}\newline

Consider the following from reference \cite{ADVRET}.

\begin{quotation}
	``In 1909 Walter Ritz and Albert Einstein (former classmates at the University of Zurich) debated the question of whether there is a fundamental temporal asymmetry in electrodynamics, and if so, whether Maxwell's equations (as they stand) can justify this asymmetry. As mentioned above, the potential field equation

	\begin{equation}
	\frac{\partial^2\phi}{\partial x^2}+\frac{\partial^2\phi}{\partial y^2}+\frac{\partial^2\phi}{\partial z^2}-\frac{1}{c^2}\frac{\partial^2\phi}{\partial t^2}=-4\pi\rho~~,
	\end{equation}

	\noindent is equally well solved with either of two functions

	\begin{equation}
	\phi_1=\int\frac{\rho(x,y,z,t-r/c)}{r}\,dx\,dy\,dz\qquad\quad \phi_2=\int\frac{\rho(x,y,z,t+r/c)}{r}\,dx\,dy\,dz~~,
	\end{equation}

	\noindent where $\phi_1$ is called the retarded potential and $\phi_2$ the advanced potential. Ritz believed the exclusion of the advanced potentials represents a physically significant restriction on the set of possible phenomena, and yet it could not be justified in the context of Maxwell's equations. From this he concluded that Maxwell's equations were fundamentally flawed, and could not serve as the basis for a valid theory of electrodynamics. Ironically, Einstein too did not believe in Maxwell's equations, at least not when it came to the micro-structure of electromagnetic radiation, as he had written in his 1905 paper on what later came to be called photons. However, Ritz's concern was not related to quantum effects (which he rejected along with special relativity), it was purely classical, and in the classical context Einstein was not troubled by the exclusion of the advanced potentials. He countered Ritz's argument by pointing out (in his 1909 paper ``On the Present State of the Radiation Problem'') that the range of solutions to the field equations is not reduced by restricting ourselves to the retarded potentials, because all the same overall force-interactions can be represented equally well in terms of advanced or retarded potentials (or some combinations of both). He wrote

	\begin{quotation}
	``If $\phi_1$ and $\phi_2$ are [\textit{retarded and advanced}] solutions of the [\textit{potential field}] equation, then $\phi_3 = a_1\phi_1 + a_2\phi_2$ is also a solution if $a_1 + a_2 = 1$. But it is not true that the solution $\phi_3$ is a more general solution than $\phi_1$ and that one specializes the theory by putting $a_1 = 1, a_2 = 0$. Putting $\phi  = \phi_1$ amounts to calculating the electromagnetic effect at the point $x,y,z$ from those motions and configurations of the electric quantities that took place prior to the instant $t$. Putting $\phi  = \phi_2$ we are determining the above electromagnetic effects from the motions that take place after the instant $t$. In the first case the electric field is calculated from the totality of the processes producing it, and in the second case from the totality of the processes absorbing it. If the whole process occurs in a (finite) space bounded on all sides, then it can be represented in the form $\phi  = \phi_1$ as well as in the form $\phi  = \phi_2$. If we consider a field that is emitted from the finite into the infinite, we can naturally use only the form $\phi  = \phi_1$, precisely because the totality of the absorbing processes is not taken into consideration. But here we are dealing with a misleading paradox of the infinite. Both kinds of representations can always be used, regardless of how distant the absorbing bodies are imagined to be. Thus one cannot conclude that the solution $\phi  = \phi_1$ is more special than the solution $\phi  = a_1\phi_1 + a_2\phi_2$ where $a_1 + a_2 = 1$.''
	\end{quotation}

	``Ritz objected to this, pointing out that there is a real observable asymmetry in the propagation of electromagnetic waves, because such waves invariably originate in small regions and expand into larger regions as time increases, whereas we never observe the opposite happening. Einstein replied that a spherical wave-shell converging on a point is possible in principle, it is just extremely improbable that a widely separate set of boundary conditions would be sufficiently coordinated to produce a coherent in-going wave. Essentially the problem is pushed back to one of asymmetric boundary conditions.''\newline
\end{quotation}

A field emitted from the finite into the infinite is exactly the problem of how to begin to propagate fields in $\mathcal{H}_1$ through the unit cell to the higher level of $\aleph$ on $\mathcal{H}_2$.  Recall the transfinite origin of the concept of a level of $\aleph$ \cite{OP}.  We will rely on a general understanding of the coefficients

\begin{equation}
a_1+a_2=1\quad\qquad\longleftrightarrow\quad\qquad\Phi+\varphi=1~~,
\end{equation}\newline

\noindent while noting that complexified solutions of the form


\begin{equation}
\phi=a_1\phi_1+a_2\phi_2+a_3\phi_3+a_4\phi_4~~,\quad\qquad\text{with}\quad\qquad a_1+a_2+a_3+a_4=1~~,
\end{equation}\newline


 \noindent are independently interesting on account of

\begin{equation}
\dfrac{1}{4\pi}\,\hat\pi -\dfrac{\varphi}{4}\,\hat +\dfrac{1}{8}\,\hat2 -\dfrac{i}{4}\,\hat i =\hat1~~.
\end{equation}
\clearpage



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\begin{quotation}
	And he shall speak great words against the most High, and shall wear out the saints of the most High, and think to change times and laws: and they shall be given into his hand until a time [$x^0$] and times [$x^0,\chi^5$] and the dividing of time [$\chi^5_+, \chi^5_\varnothing, \chi^5_-$].

	-- Daniel 7:25
\end{quotation}

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\renewcommand\thesection{\Roman{section}}
\section{{\textsf{\LARGE Computation and Analysis in Quantum Cosmology}}}
\renewcommand\thesection{\arabic{section}}

Section four describes the information current and we compute a toy model of the mass parameters of the universe $\{\Omega_{\text{Matter}},\Omega_{\text{DM}},\Omega_{\text{DE}}\}$ that is in perfectly within the parameter space allowed by $\Lambda$CDM.


\subsection{Tipler Sinusoids}

Consider what Tipler wrote in reference \cite{TIPLER}, a paper containing in its abstract the interesting phrase ``would act as a time machine.''

\begin{quotation}
	``Since the work of Hawking and Penrose, it has become accepted that classical general relativity predicts some sort of pathological behavior.  However, the exact nature of the pathology is under intense debate at present, primarily because solutions to the field equations can be found which exhibit virtually any type of bizarre behavior.  It is thus of utmost importance to know what types of pathologies might be expected to occur in actual physical situations.  One of these pathologies is causality violation, and in [\textit{Tipler's paper Tipler argues}] that if we make the assumptions concerning the behavior of matter and manifold usual in general relativity, then it should be possible in principle to set up an experiment in which this particular pathology could be observed.

	``Because general relativity is a local theory with no \textit{a priori} restrictions on the global topology, causality violation can be introduced into solutions quite easily by injudicious choices of topology;\footnote{Tipler's comment is evocative of an axiom stated in reference \cite{ZETA}, ``If the topology required by the new definitions is not reducible to a [\textit{sufficient form}] then the definitions will amount to what Laithewaite has called `the multiplication of bananas by umbrellas' meaning that the definitions are contrived.''} for example, we could assume that the timelike coordinate in the metric is periodic,\footnote{Obviously this is what we have done in the MCM \cite{MOD}.} or we could make wormhole modifications in Reissner--Nordstr\"om space [(\textit{figure \ref{fig:RN}})].  In both of these cases the causality violation takes the form of closed timelike lines (CTL) which are not homotopic to zero, and these need cause no worries since they can be removed by reinterpreting the