# Reply to “Comment on ‘Dr. Bertlmann’s Socks in a Quaternionic World of
Ambidextral Reality”’

Joy Christian<EMAIL_ADDRESS>Einstein Centre for Local-Realistic Physics, 15
Thackley End, Oxford OX2 6LB, United Kingdom

###### Abstract

In this paper, I respond to a critique of one of my papers previously
published in this journal, entitled ‘Dr. Bertlmann’s socks in a quaternionic
world of ambidextral reality.’ The geometrical framework presented in my paper
is based on a quaternionic 3-sphere, or $S^{3}$, taken as a model of the
physical space in which we are inescapably confined to perform all our
experiments. The framework intrinsically circumvents Bell’s theorem by
reproducing the singlet correlations local-realistically, without resorting to
backward causation, superdeterminism, or any other conspiracy loophole. In
this response, I demonstrate point by point that, contrary to its claims, the
critique has not found any mistakes in my paper, either in the analytical
model of the singlet correlations or in its event-by-event numerical
simulation based on Geometric Algebra.

##  I Introduction

Mathematical constructions are often obfuscated rather than enlightened by
their misplaced criticisms, and the recent critique in Gill-Ieee  of the
geometrical 3-sphere framework for local-realistically underpinning quantum
correlations I have presented in Disproof ; IJTP ; RSOS ; IEEE-1 ; IEEE-2  is
no exception. Therefore, in Section III below I first summarize the 3-sphere
framework in some detail before responding in Section IV to the critique in
Gill-Ieee . While the critique is focused on the 3-sphere model presented in
my latest paper IEEE-2 , it does not raise any new questions. The issues
raised therein have been extensively addressed by me already, for example in
Reply-Gill ; Reply-Gill-IJTP ; Scott . It is unfortunate that the critique in
Gill-Ieee  does not directly address any of my previous responses. In this
regard, my response Reply-Lasenby  to a separate critique is also of interest.
In addition to these responses, the critique in Gill-Ieee  has also overlooked
the large number of specific questions that are answered in the appendices of
IEEE-1  and IEEE-2 .

The motivations behind the critique in Gill-Ieee  seem to stem from the fact
that the geometrical framework I have presented in Disproof ; IJTP ; RSOS ;
IEEE-1 ; IEEE-2  appears to contradict the well known mathematical theorem
proposed by Bell Bell-1964 . But as we shall soon see, the facts are not as
black and white as the critique has claimed. The proof of Bell’s theorem
relies on the strong correlations predicted by the entangled quantum states,
such as the singlet state. The claim of the theorem is that no local and
realistic theory of the kind espoused by Einstein can reproduce all of the
statistical predictions of quantum mechanics, especially those that are
predicted using quantum entanglement. But in Disproof ; IJTP ; RSOS ; IEEE-1 ;
IEEE-2 , I have demonstrated that the strong correlations observed in Nature,
including those predicted by the singlet state, have little to do with quantum
entanglement per se. In general, they are local and realistic correlations
among the limiting scalar points of an octonion-like 7-sphere RSOS ; Eight ,
which is an algebraic representation space of a quaternionic 3-sphere, taken
as the physical space. Thus, the assumptions underlying Disproof ; IJTP ; RSOS
; IEEE-1 ; IEEE-2  and those of Bell’s theorem are different.

By contrast, in its analysis the critique in Gill-Ieee  begins with an
incorrect version of the above model by writing some of its equations
incorrectly within a flat Euclidean space ${\mathrm{I\\!R}^{3}}$. It then
derives a constant value of the singlet correlations, ${\cal
E}({\mathbf{a}},\,{\mathbf{b}})=-1$ for all detector directions ${\mathbf{a}}$
and ${\mathbf{b}}$, by failing to respect the geometrical properties of the
3-sphere, such as the spinorial sign changes in the quaternions, criticizes
this incorrect value, and concludes that it has thereby criticized the
3-sphere model of Disproof ; IJTP ; RSOS ; IEEE-1 ; IEEE-2 . In the process,
the critique also violates the conservation of zero spin angular momentum.

But before I bring out such oversights from the critique Gill-Ieee , in the
next section I first review the status of Bell’s theorem in order to highlight
in which sense it is circumvented by the 3-sphere model of quantum
correlations presented in Disproof ; IJTP ; RSOS ; IEEE-1 ; IEEE-2 .

##  II Boole’s inequality versus Bell’s theorem

Much is made in the critique Gill-Ieee  of the so-called “theorem” of Bell
that claims that the models such as the one presented in Disproof ; IJTP ;
RSOS ; IEEE-1 ; IEEE-2  are impossible. But mathematically a theorem with
loopholes loopholes  is an oxymoron, while physically we know that the bounds
on Bell inequalities are not respected by Nature. The consequent conclusion
that therefore Nature must be non-local, non-realistic, or conspiratorial is
not justified. For Bell’s theorem depends on a number of assumptions RSOS , in
addition to those of locality and realism. And, in fact, Bell inequalities can
be derived without assuming either locality or realism, as shown, for example,
in Section 4.2 of RSOS .

That is not to say that Bell’s theorem Bell-1964  does not have a sound
mathematical core. When stated as a mathematical theorem in probability
theory, there can be no doubt about its validity. But my work on the subject
Disproof ; IJTP ; RSOS ; IEEE-1 ; IEEE-2  does not challenge this mathematical
core, if it is viewed as a piece of mathematics. What it challenges are the
metaphysical conclusions regarding locality and realism derived from that
mathematical core. My work thus draws a sharp distinction between the
mathematical core of Bell’s theorem and the metaphysical conclusions derived
from it. Let me unpack these remarks to explain in what sense the local-
realistic framework presented in Disproof ; IJTP ; RSOS ; IEEE-1 ; IEEE-2
circumvents Bell’s theorem.

As acknowledged in the critique in Gill-Ieee , the mathematical core of Bell’s
theorem goes back to Boole’s derivation of an inequality within probability
theory, one hundred and eleven years before the publication of Bell’s theorem
Boole-1 ; Boole-2 . In the modern form, it is the famous Bell-CHSH Clauser
inequality

$-2\,\leqslant\,{\cal E}({\mathbf{a}},\,{\mathbf{b}})\,+\,{\cal
E}({\mathbf{a}},\,{\mathbf{b}^{\prime}})\,+\,{\cal
E}({\mathbf{a}^{\prime}},\,{\mathbf{b}})\,-\,{\cal
E}({\mathbf{a}^{\prime}},\,{\mathbf{b}^{\prime}})\,\leqslant\,+2\,,$ (1)

where each of the expectation values is defined as the average

${\cal
E}({\mathbf{a}},\,{\mathbf{b}})\,=\int_{\Lambda}{\mathscr{A}}({\mathbf{a}},\,\lambda)\,{\mathscr{B}}({\mathbf{b}},\,\lambda)\;\,d\rho(\lambda)$
(2)

that satisfies the EPR’s condition of perfect anti-correlation:

${\cal E}({\mathbf{n}},\,{\mathbf{n}})=-1,\;\;\;\forall\;{\mathbf{n}}\in
S^{2}\subset{\mathrm{I\\!R}^{3}}.$ (3)

Here $\lambda\in\Lambda$ denotes a complete specification of the physical
state of the singlet system at a suitable instant, ${\rho(\lambda)}$ denotes
the normalized probability measure on the space ${\Lambda}$ of the complete
states, and ${\mathscr{A}({\mathbf{a}},\,{\lambda})=\pm 1}$ and
${\mathscr{B}({\mathbf{b}},\,{\lambda})=\pm 1}$ are the measurement functions
specifying the results observed by Alice and Bob for a given run of the
experiment. These functions satisfy the following conditions of local
causality:

> Local Causality: Apart from the initial state or a hidden variable
> ${\lambda}$, the result ${{\mathscr{A}}=\pm 1}$ of Alice depends only on the
> measurement direction ${\mathbf{a}}$, chosen freely by Alice, regardless of
> Bob’s actions. And similarly, apart from the initial state ${\lambda}$, the
> result ${{\mathscr{B}}=\pm 1}$ of Bob depends only on the measurement
> direction ${\mathbf{b}}$, chosen freely by Bob, regardless of Alice’s
> actions. In particular, the function
> ${{\mathscr{A}}({\mathbf{a}},\,\lambda)}$ does not depend on ${\mathbf{b}}$
> or ${\mathscr{B}}$, the function ${{\mathscr{B}}({\mathbf{b}},\,\lambda)}$
> does not depend on ${\mathbf{a}}$ or ${\mathscr{A}}$, and, moreover, the
> initial state ${\lambda}$ does not depend on ${\mathbf{a}}$, ${\mathbf{b}}$,
> ${\mathscr{A}}$, or ${\mathscr{B}}$.

In addition, the probability measure ${\rho(\lambda)}$ in (2) is required to
depend only on ${\lambda}$, and not on either ${\mathbf{a}}$ or
${\mathbf{b}}$, which are the freely chosen detector settings at the time of
measurements.

Although I have introduced quite a bit of physics in the foregoing, the
mathematical core of Bell’s theorem is rather simple. It can be stated simply
as the claim that, for the product
${\mathscr{A}({\mathbf{a}},\,{\lambda}){\mathscr{B}({\mathbf{b}},\,{\lambda})}=\pm
1}$ of any local-realistic functions
${\mathscr{A}({\mathbf{a}},\,{\lambda})=\pm 1}$ and
${\mathscr{B}({\mathbf{b}},\,{\lambda})=\pm 1}$, the expectation value

$\displaystyle{\cal E}({\mathbf{a}},\,{\mathbf{b}})\,$
$\displaystyle=\lim_{\,n\,\gg\,1}\left[\frac{1}{n}\sum_{k\,=\,1}^{n}\,{\mathscr{A}}({\mathbf{a}},\,{\lambda}^{k})\;{\mathscr{B}}({\mathbf{b}},\,{\lambda}^{k})\right]$
$\displaystyle\equiv\,\Bigl{\langle}\,{\mathscr{A}}_{k}({\mathbf{a}})\,{\mathscr{B}}_{k}({\mathbf{b}})\,\Bigr{\rangle}$
$\displaystyle=-{\mathbf{a}}\cdot{\mathbf{b}}$ (4)

is impossible to achieve, because it would lead to “violations” of the bounds
of $\pm 2$ claimed in (1) on the CHSH correlator. The expectation values are
therefore constrained to be within

$\displaystyle{\cal
E}({\mathbf{a}},\,{\mathbf{b}})=\,\begin{cases}-\,1\,+\,\frac{2}{\pi}\,\eta_{{\mathbf{a}}{\mathbf{b}}}\;\;\;\text{if}&\\!\\!0\leqslant\eta_{{\mathbf{a}}{\mathbf{b}}}\leqslant\pi\\\
\\\
+\,3\,-\,\frac{2}{\pi}\,\eta_{{\mathbf{a}}{\mathbf{b}}}\;\;\;\text{if}&\\!\\!\pi\leqslant\eta_{{\mathbf{a}}{\mathbf{b}}}\leqslant
2\pi\,,\end{cases}$ (5)

where $\eta_{{\mathbf{a}}{\mathbf{b}}}$ is the angle between the detector
directions ${\mathbf{a}}$ and ${\mathbf{b}}$. Here I have rewritten the
integral in (2) as a discrete sum because that is what is observed in the
experiments, with ${\lambda^{k}}$ being an initial state for the
${k^{\mathrm{th}}}$ run of the experiment.

Now, the proof of Bell’s claim follows from the additivity of expectation
values, which allows us to equate the sum of four separate averages of numbers
$+1$ and $-1$ appearing in the CHSH inequality (1) with a single average of
their sum:

$\displaystyle{\cal E}({\mathbf{a}},\,$ $\displaystyle{\mathbf{b}})\,+\,{\cal
E}({\mathbf{a}},\,{\mathbf{b^{\prime}}})\,+\,{\cal
E}({\mathbf{a^{\prime}}},\,{\mathbf{b}})\,-\,{\cal
E}({\mathbf{a^{\prime}}},\,{\mathbf{b^{\prime}}})$
$\displaystyle=\Bigl{\langle}\,{\mathscr{A}}_{k}({\mathbf{a}})\,{\mathscr{B}}_{k}({\mathbf{b}})\,\Bigr{\rangle}+\Bigl{\langle}\,{\mathscr{A}}_{k}({\mathbf{a}})\,{\mathscr{B}}_{k}({\mathbf{b^{\prime}}})\,\Bigr{\rangle}$
$\displaystyle\;\;\,\;\;\;\;\;\;+\Bigl{\langle}\,{\mathscr{A}}_{k}({\mathbf{a^{\prime}}})\,{\mathscr{B}}_{k}({\mathbf{b}})\,\Bigr{\rangle}-\Bigl{\langle}\,{\mathscr{A}}_{k}({\mathbf{a^{\prime}}})\,{\mathscr{B}}_{k}({\mathbf{b^{\prime}}})\,\Bigr{\rangle}$
(6)
$\displaystyle=\Bigl{\langle}{\mathscr{A}}_{k}({\mathbf{a}}){\mathscr{B}}_{k}({\mathbf{b}})+{\mathscr{A}}_{k}({\mathbf{a}}){\mathscr{B}}_{k}({\mathbf{b^{\prime}}})$
$\displaystyle\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;+{\mathscr{A}}_{k}({\mathbf{a^{\prime}}}){\mathscr{B}}_{k}({\mathbf{b}})-{\mathscr{A}}_{k}({\mathbf{a^{\prime}}}){\mathscr{B}}_{k}({\mathbf{b^{\prime}}})\Bigr{\rangle}.$
(7)

This immediately reduces the sum (6) of four averages to

$\Bigl{\langle}{\mathscr{A}}_{k}({\mathbf{a}})\big{\\{}{\mathscr{B}}_{k}({\mathbf{b}})+{\mathscr{B}}_{k}({\mathbf{b^{\prime}}})\big{\\}}\,+\,{\mathscr{A}}_{k}({\mathbf{a^{\prime}}})\big{\\{}{\mathscr{B}}_{k}({\mathbf{b}})-{\mathscr{B}}_{k}({\mathbf{b^{\prime}}})\big{\\}}\Bigr{\rangle}.$
(8)

And because ${{\mathscr{B}}_{k}({\mathbf{b}})=\pm 1}$, if
${|{\mathscr{B}}_{k}({\mathbf{b}})+{\mathscr{B}}_{k}({\mathbf{b^{\prime}}})|=2}$,
then
${|{\mathscr{B}}_{k}({\mathbf{b}})-{\mathscr{B}}_{k}({\mathbf{b^{\prime}}})|=0}$,
and vice versa. Consequently, using ${{\mathscr{A}}_{k}({\mathbf{a}})=\pm 1}$,
it is easy to see that the absolute value of the above average cannot exceed
2, just as Bell concluded Clauser :

$\displaystyle-\,2\,\leqslant\,\Bigl{\langle}\,$
$\displaystyle{\mathscr{A}}_{k}({\mathbf{a}})\,{\mathscr{B}}_{k}({\mathbf{b}})\,+\,{\mathscr{A}}_{k}({\mathbf{a}})\,{\mathscr{B}}_{k}({\mathbf{b^{\prime}}})$
$\displaystyle+\,{\mathscr{A}}_{k}({\mathbf{a^{\prime}}})\,{\mathscr{B}}_{k}({\mathbf{b}})\,-\,{\mathscr{A}}_{k}({\mathbf{a^{\prime}}})\,{\mathscr{B}}_{k}({\mathbf{b^{\prime}}})\,\Bigr{\rangle}\,\leqslant\,+\,2\,.$
(9)

On the other hand, if we substitute ${\cal
E}({\mathbf{a}},\,{\mathbf{b}})=-\,{\mathbf{a}}\cdot{\mathbf{b}}$, etc., into
(6), then it is easy to demonstrate that the bounds of $\pm 2$ in (1) can be
exceeded for some detector directions, giving

$-2\sqrt{2}\leqslant{\cal E}({\mathbf{a}},\,{\mathbf{b}})+{\cal
E}({\mathbf{a}},\,{\mathbf{b^{\prime}}})+{\cal
E}({\mathbf{a^{\prime}}},\,{\mathbf{b}})-{\cal
E}({\mathbf{a^{\prime}}},\,{\mathbf{b^{\prime}}})\leqslant 2\sqrt{2}.$ (10)

Consequently, according to Bell’s theorem, the quantum mechanical correlations
${\cal E}({\mathbf{a}},\,{\mathbf{b}})=-\,{\mathbf{a}}\cdot{\mathbf{b}}$ are
impossible to achieve within the local-realistic framework specified above.

The above proof, however, while mathematically sound, harbors a profound
physical mistake. The mistake is hidden in the assumption (7) of the
additivity of expectation values, and it is the same mistake that von
Neumann’s ex-theorem against the general hidden variable theories was based
on, as I have explained in Oversight . Both von Neumann’s theorem and Bell’s
theorem unjustifiably assume the additivity of expectation values within
hidden variable theories to derive their respective conclusions. However, for
observables that are not simultaneously measurable, such as those involved at
one of the stations in the Bell-test experiments, the equivalence of the sum
of expectation values and the expectation value of the sum of corresponding
measurement results, although respected within quantum mechanics, need not
hold within hidden variable theories, as noted by Einstein Oversight . Once
this oversight is removed from Bell’s argument and local realism is
implemented correctly, the bounds on the CHSH correlator derived in Eq. (9)
above work out to be $\pm 2\sqrt{2}$ instead of $\pm 2$, thereby mitigating
the conclusion of Bell’s theorem Oversight . Consequently, what is ruled out
by Bell-test experiments is not local realism but the additivity (7) of
expectation values.

##  III Review of the $S^{3}$ Model of correlations

Let us now review the quaternionic 3-sphere model for the strong correlations
proposed in Disproof ; IJTP ; RSOS ; IEEE-1 ; IEEE-2 . As far as the
correlations predicted by the singlet state are concerned (which are the focus
of Gill-Ieee  and IEEE-2 ), it will be sufficient to restrict to the 3-sphere
instead of its algebraic representation space $S^{7}$ considered in RSOS . A
quaternionic 3-sphere can be defined as

$S^{3}:=\left\\{\,{\mathbf{q}}(\psi,\,{\mathbf{r}}):=\exp\left[\,{\mathbf{J}}({\mathbf{r}})\,\frac{\psi}{2}\,\right]\Bigg{|}\;||\,{\mathbf{q}}(\psi,\,{\mathbf{r}})\,||^{2}=1\right\\}\\!,$
(11)

where ${{\mathbf{J}}({\mathbf{r}})}$ is a bivector (or pure quaternion)
rotating about ${{\mathbf{r}}\in{\mathrm{I\\!R}}^{3}}$ with the rotation angle
${\psi}$ in the range ${0\leq\psi<4\pi}$. Here the notations and conventions
of Geometric Algebra are used Clifford . Now, the central hypothesis I have
put forward in IEEE-1  and IEEE-2  is that the strong correlations we observe
in the Bell-test experiments are consequences of the fact that three-
dimensional physical space is best modeled as a closed and compact
quaternionic 3-sphere, $S^{3}$, using Geometric Algebra, as in (11), rather
than as a flat and open space ${\mathrm{I\\!R}^{3}}$ using “vector algebra.”
This is by no means an ad hoc hypothesis. Note that ${S^{3}}$ happens to be
isomorphic to the spatial part of one of the well known cosmological solutions
of Einstein’s field equations of general relativity, representing a closed
universe with positive curvature IEEE-1 ; IEEE-2 . It is universally accepted
that the spacetime geometries of our universe are described by the Friedmann-
Robertson-Walker line element

$ds^{2}=dt^{2}-a^{2}(t)\,d\Sigma^{2},\;\;\,d\Sigma^{2}=\left[\frac{d\rho^{2}}{1-\kappa\,\rho^{2}}+\rho^{2}d\Omega^{2}\right]\\!,$
(12)

where ${a(t)}$ is the scale factor, ${\Sigma}$ is a spacelike hypersurface,
${\rho}$ is the radial coordinate within ${\Sigma}$, ${\kappa}$ is the
“normalized” curvature of ${\Sigma}$, and ${\Omega}$ is a solid angle within
${\Sigma}$ d'Inverno . Since we are primarily concerned with a galactic,
solar, or terrestrial scenario, in what follows, without loss of generality,
we will restrict our attention to the current epoch of the cosmos by setting
the scale factor ${a(t)=1}$ in the above line element. It then allows three
possible geometries for the spacetime with the product topology
${\mathrm{I\\!R}\times\Sigma}$ so that the corresponding spacelike
hypersurfaces $\Sigma$ can be isomorphic only to ${\mathrm{I\\!R}^{3}}$,
$S^{3}$, or $H^{3}$, with $H^{3}$ being a hyperboloid of negative curvature.
Among these possible three-geometries, only $S^{3}$ represents a closed
universe with compact geometry and constant positive curvature. Moreover, the
cosmic microwave background spectra recently mapped by the space observatory,
Planck, now prefers a positive curvature at more than 99% confidence level
closed ; Handley . And yet, topologically $S^{3}$ can be constructed by adding
only a single mathematical point to ${{\mathrm{I\\!R}}^{3}}$ at infinity:

$S^{3}=\,{\mathrm{I\\!R}}^{3}\cup\left\\{\infty\right\\}.$ (13)

What is more, unlike ${\mathrm{I\\!R}^{3}}$ and $H^{3}$, $S^{3}$ is
parallelizable using quaternions. That is to say, given three linearly-
independent vector fields forming a basis of the tangent space at some point
of $S^{3}$, using quaternions it is possible to find three linearly-
independent vector fields forming a basis of the tangent space at every other
point of $S^{3}$. In other words, it is possible to set all of the points of a
quaternionic $S^{3}$ in a smooth flowing motion at the same time, in any
direction, without a fixed point, or a pole, or a singularity, or a cowlick.
On the other hand, the tangent bundle of ${S^{3}}$ happens to be trivial:
${{\rm T}S^{3}=S^{3}\times{\mathrm{I\\!R}}^{3}}$. This renders the tangent
space at each point of ${S^{3}}$ to be isomorphic to ${{\mathrm{I\\!R}}^{3}}$.
Consequently, local experiences of the experimenters within ${S^{3}}$ are no
different from those of their counterparts within ${{\mathrm{I\\!R}}^{3}}$.
The global topology of ${S^{3}}$, however, is clearly different from that of
${{\mathrm{I\\!R}}^{3}}$ Disproof ; IJTP . In particular, the triviality of
the bundle ${{\rm T}S^{3}}$ again means that ${S^{3}}$ is parallelizable. As a
result, a global anholonomic frame can be defined on ${S^{3}}$ that fixes each
of its points uniquely. Such a frame renders ${S^{3}}$ diffeomorphic to the
set of all unit quaternions, as in (11). The properties of $S^{3}$ are thus
uniquely captured by the properties of quaternions.

Now, given two unit vectors ${\mathbf{x}}$ and ${\mathbf{y}}$ in
${\mathrm{I\\!R}^{3}}$ and a rotation axis ${\mathbf{r}}$, each element of the
set ${S^{3}}$ can be factorized into a product of corresponding bivectors
${{\mathbf{J}}({\mathbf{x}})}$ and ${{\mathbf{J}}({\mathbf{y}})}$ as follows:

$\displaystyle{\mathbf{q}}(\eta_{{\mathbf{x}}{\mathbf{y}}},\,{\mathbf{r}})$
$\displaystyle=-\,{\mathbf{J}}({\mathbf{x}})\,{\mathbf{J}}({\mathbf{y}})$ (14)
$\displaystyle=-\,(I\,{\mathbf{x}})\,(I\,{\mathbf{y}})$ (15)
$\displaystyle=-I^{2}\,{\mathbf{x}}\,{\mathbf{y}}$ (16)
$\displaystyle={\mathbf{x}}\,{\mathbf{y}}$ (17)
$\displaystyle=\,{\mathbf{x}}\cdot{\mathbf{y}}\,+\,{\mathbf{x}}\wedge{\mathbf{y}}$
(18)
$\displaystyle=\cos(\,\eta_{{\mathbf{x}}{\mathbf{y}}})\,+\,{\mathbf{J}}({\mathbf{r}})\,\sin(\,\eta_{{\mathbf{x}}{\mathbf{y}}})\,,$
(19)

where $I:={\mathbf{e}}_{1}\,{\mathbf{e}}_{2}\,{\mathbf{e}}_{3}$, with
$I^{2}=-1$, is the standard trivector, ${\eta_{{\mathbf{x}}{\mathbf{y}}}}$ is
the angle between ${\mathbf{x}}$ and ${\mathbf{y}}$,
${{\mathbf{x}}\,{\mathbf{y}}}$ is the geometric product between ${\mathbf{x}}$
and ${\mathbf{y}}$, ${{\mathbf{x}}\wedge{\mathbf{y}}}$ is the wedge product
between ${\mathbf{x}}$ and ${\mathbf{y}}$, and ${{\mathbf{J}}({\mathbf{r}})}$
is identified with
${\frac{{\mathbf{x}}\wedge{\mathbf{y}}}{||{\mathbf{x}}\wedge{\mathbf{y}}||}}$.
Comparing (11) and (19), we recognize that the rotation angle ${\psi}$ of the
quaternion is twice the angle between the vectors ${\mathbf{x}}$ and
${\mathbf{y}}$:

$\psi\,=\,2\,\eta_{{\mathbf{x}}{\mathbf{y}}}.$ (20)

As a result, the characteristic spinorial sign changes exhibited by the
quaternions constituting the $S^{3}$ can be expressed as

${\mathbf{q}}(\eta_{{\mathbf{x}}{\mathbf{y}}}+\kappa\pi,\,{\mathbf{r}})\,=\,(-1)^{\kappa}\,{\mathbf{q}}(\eta_{{\mathbf{x}}{\mathbf{y}}},\,{\mathbf{r}})\,\;\;\text{for}\;\,\kappa=0,1,2,3,\dots$
(21)

This equation expresses a key relation that reduces the singlet correlations
we observe in Nature to Dr. Bertlmann’s socks type classical correlations
Bell-1987 , because, as we shall soon see, it forces the product
${{\mathscr{A}}{\mathscr{B}}}$ of the measurement results ${\mathscr{A}}=\pm$
and ${\mathscr{B}}=\pm$ observed, respectively, by Alice and Bob to fluctuate
between the values ${{\mathscr{A}}{\mathscr{B}}=-1}$ and
${{\mathscr{A}}{\mathscr{B}}=+1}$ and vice versa. It thereby allows all four
combinations of the results, ${{\mathscr{A}}{\mathscr{B}}=+\,+}$, ${+\,-}$,
${-\,+}$, and ${-\,-}$, necessary to produce the observed strong correlations
between them.

Incidentally, in the algebra $\mathrm{Cl}_{(3,0)}$ of a three-dimensional
space the four-dimensional object $I\wedge{\mathbf{v}}$ is necessarily zero by
definition. The vectors ${\mathbf{v}}$ in $\mathrm{Cl}_{(3,0)}$ are thus
defined as the solutions of the equation $I\wedge{\mathbf{v}}=0$. Therefore,
we can write

$I{\mathbf{v}}=I\cdot{\mathbf{v}}+I\wedge{\mathbf{v}}=I\cdot{\mathbf{v}}.$
(22)

This choice is explained in Question 8 of Appendix B in IEEE-1 .

Next, let us recall that the measurement results $\mathscr{A}$ and
$\mathscr{B}$ observed by Alice and Bob are events in spacetime. Within the
spacetime defined by the line element (12), they are thus events in
${\mathrm{I\\!R}\times\Sigma}$. Now, traditionally, in Bell-test experiments
$\Sigma$ is implicitly identified with $\mathrm{I\\!R}^{3}$. In other words,
the three-dimensional physical space is implicitly modeled as
$\mathrm{I\\!R}^{3}$. Thus, traditionally, the results $\mathscr{A}$ and
$\mathscr{B}$ observed by Alice and Bob are assumed to be events in
${\mathrm{I\\!R}\times{\mathrm{I\\!R}^{3}}}$. But in my model
${\mathrm{I\\!R}^{3}}$ in ${\mathrm{I\\!R}\times\mathrm{I\\!R}^{3}}$ is
replaced with $S^{3}$, and thus $S^{3}$ is taken to be a spacelike
hypersurface in spacetime, and hence a surface of simultaneity. In other
words, the results $\mathscr{A}$ and $\mathscr{B}$ are viewed as events in
${\mathrm{I\\!R}\times S^{3}}$. But in the EPR-Bohm type experiments the
observed results are necessarily equal-time events, otherwise called
“coincidence counts.” Thus, in my model they are points in $S^{3}$ at the time
of simultaneous measurements by Alice and Bob. These considerations, thanks to
the decomposition (14), lead us to the following theorem.

Theorem 1: The quantum mechanical correlation predicted by the entangled
singlet state can be understood as a classical, local, realistic, and
deterministic correlation among the pairs of limiting scalar points of a
quaternionic 3-sphere defined in (11), with the limiting scalar points defined
by the functions

$\displaystyle S^{3}\ni\,{\mathscr{A}}({\mathbf{a}},\,{\lambda^{k}})\,$
$\displaystyle:=\lim_{{\mathbf{s}}_{1}\,\rightarrow\,{\mathbf{a}}}\left\\{\,+\,{\mathbf{q}}(\eta_{{\mathbf{a}}{\mathbf{s}}_{1}},\,{\mathbf{r}}_{1})\right\\}$
$\displaystyle\equiv\lim_{{\mathbf{s}}_{1}\,\rightarrow\,{\mathbf{a}}}\left\\{-\,{\mathbf{D}}({\mathbf{a}})\,{\mathbf{L}}({\mathbf{s}}_{1},\,\lambda^{k})\right\\}$
$\displaystyle\xrightarrow[{\mathbf{s}}_{1}\,\to\,{\mathbf{a}}]{\,}\begin{cases}+\,1\;\;\;\;\text{if}&\lambda^{k}\,=\,+\,1\\\
-\,1\;\;\;\;\text{if}&\lambda^{k}\,=\,-\,1\end{cases}\Bigg{\\}}$ (23)

and

$\displaystyle S^{3}\ni\,{\mathscr{B}}({\mathbf{b}},\,{\lambda^{k}})\,$
$\displaystyle:=\lim_{{\mathbf{s}}_{2}\,\rightarrow\,{\mathbf{b}}}\left\\{\,-\,{\mathbf{q}}(\eta_{{\mathbf{s}}_{2}{\mathbf{b}}},\,{\mathbf{r}}_{2})\right\\}$
$\displaystyle\equiv\lim_{{\mathbf{s}}_{2}\,\rightarrow\,{\mathbf{b}}}\left\\{+\,{\mathbf{L}}({\mathbf{s}}_{2},\,\lambda^{k})\,{\mathbf{D}}({\mathbf{b}})\right\\}$
$\displaystyle\xrightarrow[{\mathbf{s}}_{2}\,\to\,{\mathbf{b}}]{\,}\begin{cases}-\,1\;\;\;\;\text{if}&\lambda^{k}\,=\,+\,1\\\
+\,1\;\;\;\;\text{if}&\lambda^{k}\,=\,-\,1\end{cases}\Bigg{\\}},$ (24)

where the bivectors ${-\,{\mathbf{L}}({\mathbf{s}}_{1},\,\lambda^{k})}$ and
${+\,{\mathbf{L}}({\mathbf{s}}_{2},\,\lambda^{k})}$ represent the two
fermionic spins emerging from a common source that are subsequently detected
(possibly at a space-like distance from each other) by two detector bivectors
${{\mathbf{D}}({\mathbf{a}})=I\cdot{\mathbf{a}}}$ and
${{\mathbf{D}}({\mathbf{b}})=I\cdot{\mathbf{b}}}$, freely chosen by Alice and
Bob. I have also assumed the handedness ${\lambda^{k}}$ of ${S^{3}}$ to be a
fair coin with 50/50 chance of being ${+1}$ or ${-\,1}$ at the moment of pair-
creation, making the spinning bivector
${{\mathbf{L}}({\mathbf{n}},\,\lambda^{k})}$ a random variable relative to any
given detector bivector such as
${{\mathbf{D}}({\mathbf{n}})=I\cdot{\mathbf{n}}}$,

${\mathbf{L}}({\mathbf{n}},\,\lambda^{k})\,=\,\lambda^{k}\,{\mathbf{D}}({\mathbf{n}})\,\,\Longleftrightarrow\,\,{\mathbf{D}}({\mathbf{n}})\,=\,\lambda^{k}\,{\mathbf{L}}({\mathbf{n}},\,\lambda^{k})\,.$
(25)

The next question is: What will be the value of the product
${{\mathscr{A}}{\mathscr{B}}}$ of these results within $S^{3}$? In other
words, what will be the value of the product ${{\mathscr{A}}{\mathscr{B}}}$
when the results ${\mathscr{A}}$ and ${\mathscr{B}}$ are observed by Alice and
Bob separately but simultaneously in “coincidence counts” RSOS ? We can work
out the value of the product ${{\mathscr{A}}{\mathscr{B}}}$ within $S^{3}$
from the definitions (23) and (24) and the “product of limits equal to limits
of product” rule:

$\displaystyle S^{3}$
$\displaystyle\ni\,{\mathscr{A}}{\mathscr{B}}({\mathbf{a}},\,{\mathbf{b}},\,{\lambda^{k}})\,=\,{\mathscr{A}}({\mathbf{a}},\,{\lambda^{k}})\,{\mathscr{B}}({\mathbf{b}},\,{\lambda^{k}})$
(26)
$\displaystyle=\,\left[\lim_{{\mathbf{s}}_{1}\,\rightarrow\,{\mathbf{a}}}\left\\{\,+\,{\mathbf{q}}(\eta_{{\mathbf{a}}{\mathbf{s}}_{1}},\,{\mathbf{r}}_{1})\right\\}\right]\left[\lim_{{\mathbf{s}}_{2}\,\rightarrow\,{\mathbf{b}}}\left\\{\,-\,{\mathbf{q}}(\eta_{{\mathbf{s}}_{2}{\mathbf{b}}},\,{\mathbf{r}}_{2})\right\\}\right]$
(27)
$\displaystyle=\,\lim_{\begin{subarray}{c}{\mathbf{s}}_{1}\,\rightarrow\,{\mathbf{a}}\\\
{\mathbf{s}}_{2}\,\rightarrow\,{\mathbf{b}}\end{subarray}}\left\\{\,-\,{\mathbf{q}}(\eta_{{\mathbf{a}}{\mathbf{s}}_{1}},\,{\mathbf{r}}_{1})\,{\mathbf{q}}(\eta_{{\mathbf{s}}_{2}{\mathbf{b}}},\,{\mathbf{r}}_{2})\right\\}$
(28)
$\displaystyle=\,\lim_{\begin{subarray}{c}{\mathbf{s}}_{1}\,\rightarrow\,{\mathbf{a}}\\\
{\mathbf{s}}_{2}\,\rightarrow\,{\mathbf{b}}\end{subarray}}\left\\{\,-\,{\mathbf{q}}(\eta_{{\mathbf{u}}{\mathbf{v}}},\,{\mathbf{r}}_{0})\right\\}$
(29) $\displaystyle=\,-1,$ (30)

where, for the moment, I have assumed that
${\mathbf{s}}_{1}\not={\mathbf{s}}_{2}$, giving

$\displaystyle\eta_{{\mathbf{u}}{\mathbf{v}}}:=\cos^{-1}$
$\displaystyle\big{\\{}({\mathbf{a}}\cdot{\mathbf{s}}_{1})({\mathbf{s}}_{2}\cdot{\mathbf{b}})$
$\displaystyle\;\;\;-({\mathbf{a}}\cdot{\mathbf{s}}_{2})({\mathbf{s}}_{1}\cdot{\mathbf{b}})+({\mathbf{a}}\cdot{\mathbf{b}})({\mathbf{s}}_{1}\cdot{\mathbf{s}}_{2})\big{\\}}$
(31)

and

${\mathbf{r}}_{0}=\frac{\scriptstyle{({\mathbf{a}}\cdot{\mathbf{s}}_{1})({\mathbf{s}}_{2}\times{\mathbf{b}})\,+\,({\mathbf{s}}_{2}\cdot{\mathbf{b}})({\mathbf{a}}\times{\mathbf{s}}_{1})\,-\,({\mathbf{a}}\times{\mathbf{s}}_{1})\times({\mathbf{s}}_{2}\times{\mathbf{b}})}}{\scriptstyle{||\,({\mathbf{a}}\cdot{\mathbf{s}}_{1})({\mathbf{s}}_{2}\times{\mathbf{b}})\,+\,({\mathbf{s}}_{2}\cdot{\mathbf{b}})({\mathbf{a}}\times{\mathbf{s}}_{1})\,-\,({\mathbf{a}}\times{\mathbf{s}}_{1})\times({\mathbf{s}}_{2}\times{\mathbf{b}})\,||}}.$
(32)

That the product of the two remote quaternions
${{\mathbf{q}}(\eta_{{\mathbf{a}}{\mathbf{s}}_{1}},\,{\mathbf{r}}_{1})}$ and
${{\mathbf{q}}(\eta_{{\mathbf{s}}_{2}{\mathbf{b}}},\,{\mathbf{r}}_{2})}$ is
yet another quaternion
${{\mathbf{q}}(\eta_{{\mathbf{u}}{\mathbf{v}}},\,{\mathbf{r}}_{0})}$ is not
surprising, because the set ${S^{3}}$ defined in (11) is known to remain
closed under multiplication. A product of any number of quaternions will
result in yet another quaternion belonging to ${S^{3}}$. More importantly for
our hypothesis, the product
${{\mathscr{A}}{\mathscr{B}}({\mathbf{a}},\,{\mathbf{b}},\,{\lambda^{k}})}$ is
again a limiting scalar point, ${-1}$ in this case, of the quaternion
${-\,{\mathbf{q}}(\eta_{{\mathbf{u}}{\mathbf{v}}},\,{\mathbf{r}}_{0})}$ that
also belongs to ${S^{3}}$.

The result (30), namely ${{\mathscr{A}}{\mathscr{B}}=-1}$, suggests that if
Alice finds spin to be “up” at her station, then Bob is guaranteed to find
spin to be “down” at his station, precisely mimicking the perfect anti-
correlation observed in Dr. Bertlmann’s socks type correlations. This may give
the wrong impression that the product ${{\mathscr{A}}{\mathscr{B}}}$ of the
results observed by Alice and Bob will always remain at the fixed value of
${-1}$. But because of the spinorial sign changes described in (21), which any
quaternion in ${S^{3}}$ — including the quaternion
${-\,{\mathbf{q}}(\eta_{{\mathbf{u}}{\mathbf{v}}},\,{\mathbf{r}}_{0})}$
appearing in Eq. (29) as well as those appearing in the definitions (23) and
(24) of the individual measurement results ${\mathscr{A}}$ and ${\mathscr{B}}$
— must respect, the value of the product ${{\mathscr{A}}{\mathscr{B}}}$ will
be altered. Moreover, variations in the detector directions ${\mathbf{a}}$ and
${\mathbf{b}}$ will induce variations in the angle
${\eta_{{\mathbf{u}}{\mathbf{v}}}}$ defined in (31), which can be expressed as
${\eta_{{\mathbf{u}}{\mathbf{v}}}\rightarrow\eta_{{\mathbf{u}}{\mathbf{v}}}+\delta}$.
For variation ${\delta=\kappa\pi}$, the quaternion
${-\,{\mathbf{q}}(\eta_{{\mathbf{u}}{\mathbf{v}}},\,{\mathbf{r}}_{0})}$
appearing in Eq. (29) will then change its sign from
${-\,{\mathbf{q}}(\eta_{{\mathbf{u}}{\mathbf{v}}},\,{\mathbf{r}}_{0})}$ to
${+\,{\mathbf{q}}(\eta_{{\mathbf{u}}{\mathbf{v}}},\,{\mathbf{r}}_{0})}$ for
odd ${\kappa}$. As a result, the value of the product
${{\mathscr{A}}{\mathscr{B}}}$ will change from ${-1}$ to ${+1}$ for odd
${\kappa}$. We would thus have our cake (i.e., Dr. Bertlmann’s socks type
local-realistic interpretation of the correlations) and eat it too — i.e.,
have the value of the product ${{\mathscr{A}}{\mathscr{B}}}$ fluctuate between
${-1}$ and ${+1}$:

${\mathscr{A}}{\mathscr{B}}\in\\{-1,\,+1\\}.$ (33)

In other words, all four possible combinations of outcomes, ${+\,+}$,
${+\,-}$, ${-\,+}$, and ${-\,-}$, will be observed by Alice and Bob despite
the correlations being Dr. Bertlmann’s socks type.

But that is only a necessary part of the singlet correlations. In deriving the
value $-1$ of the product ${{\mathscr{A}}{\mathscr{B}}}$ in (30) we have
assumed ${\mathbf{s}}_{1}\not={\mathbf{s}}_{2}$. That assumption, however,
violates the conservation of zero spin angular momentum of the singlet,

$\displaystyle-\,{\mathbf{L}}({\mathbf{s}}_{1},\,\lambda^{k})+{\mathbf{L}}({\mathbf{s}}_{2},\,\lambda^{k})=0$
$\displaystyle\Longleftrightarrow{\mathbf{L}}({\mathbf{s}}_{1},\,\lambda^{k})={\mathbf{L}}({\mathbf{s}}_{2},\,\lambda^{k})$
$\displaystyle\Longleftrightarrow{\mathbf{s}}_{1}=\,{\mathbf{s}}_{2}\,\equiv\,{\mathbf{s}}\,,$
(34)

which necessarily holds during the free evolution of the spins
${-\,{\mathbf{L}}({\mathbf{s}}_{1},\,\lambda^{k})}$ and
${+\,{\mathbf{L}}({\mathbf{s}}_{2},\,\lambda^{k})}$ from the source $\pi^{0}$
until their detections by ${{\mathbf{D}}({\mathbf{a}})}$ and
${{\mathbf{D}}({\mathbf{b}})}$, as specified in (23) and (24). This condition,
in the light of the analog (48) of the Pauli identity discussed below, is also
equivalent to the condition

${\mathbf{L}}({\mathbf{s}}_{1},\,\lambda^{k})\,{\mathbf{L}}({\mathbf{s}}_{2},\,\lambda^{k})=\left\\{\,{\mathbf{L}}({\mathbf{s}},\,\lambda^{k})\right\\}^{2}=\,{\mathbf{L}}^{2}({\mathbf{s}},\,\lambda^{k})=-1\,.$
(35)

As proved in Section VIII of IEEE-1  and Appendix A of IEEE-2 , in the context
of EPR-Bohm experiments this algebraic condition for the conservation spin
angular momentum is equivalent to the Möbius-like twists in the fiber geometry
of ${S^{3}}$. Thus, the condition ${{\mathbf{s}}_{1}={\mathbf{s}}_{2}}$ is a
part of the very geometry of the physical space $S^{3}$ within which we are
inescapably confined to perform our experiments. Therefore, the result (30),
which can be valid within the physical space modeled as
${\mathrm{I\\!R}^{3}}$, cannot possibly be valid within the physical space
modeled as $S^{3}$.

It is important to note that, even though the conservation of zero spin
angular momentum of the singlet implies ${\mathbf{s}}_{1}={\mathbf{s}}_{2}$,
the physical sense of ${\mathbf{s}}_{1}$ proceeding towards Alice’s detection
process (or measurement interaction) defined in (23) and ${\mathbf{s}}_{2}$
proceeding towards Bob’s detection process (or measurement interaction)
defined in (24) remain unchanged. Moreover, for
${{\mathbf{s}}_{1}={\mathbf{s}}_{2}}$ the angle
${\eta_{{\mathbf{u}}{\mathbf{v}}}}$ defined in (31) reduces to the angle
${\eta_{{\mathbf{a}}{\mathbf{b}}}}$ between the detector directions
${\mathbf{a}}$ and ${\mathbf{b}}$. As a result, for
${{\mathbf{s}}_{1}={\mathbf{s}}_{2}}$ the product of the measurement results
reduces to

$\displaystyle{\mathscr{A}}({\mathbf{a}},\,{\lambda^{k}})\,$
$\displaystyle{\mathscr{B}}({\mathbf{b}},\,{\lambda^{k}})\,\longrightarrow\lim_{\begin{subarray}{c}{\mathbf{s}}_{1}\,\rightarrow\,{\mathbf{a}}\\\
{\mathbf{s}}_{2}\,\rightarrow\,{\mathbf{b}}\end{subarray}}\left\\{\,-\,{\mathbf{q}}(\eta_{{\mathbf{a}}{\mathbf{b}}},\,{\mathbf{r}}_{0})\right\\}$
$\displaystyle=\lim_{\begin{subarray}{c}{\mathbf{s}}_{1}\,\rightarrow\,{\mathbf{a}}\\\
{\mathbf{s}}_{2}\,\rightarrow\,{\mathbf{b}}\end{subarray}}\left\\{-\cos(\,\eta_{{\mathbf{a}}{\mathbf{b}}})-{\mathbf{L}}({\mathbf{r}}_{0},\,\lambda^{k})\,\sin(\,\eta_{{\mathbf{a}}{\mathbf{b}}})\right\\}.$
(36)

Note that, so far, we have not taken the limits
${\mathbf{s}}_{1}\to{\mathbf{a}}$ and ${\mathbf{s}}_{2}\to{\mathbf{b}}$. But
for ${{\mathbf{s}}_{1}={\mathbf{s}}_{2}}$ the only quantity in (36) that still
depends on the spin directions ${\mathbf{s}}_{1}$ and ${\mathbf{s}}_{2}$ is
the rotation axis vector ${\mathbf{r}}_{0}$, which, for
${{\mathbf{s}}_{1}={\mathbf{s}}_{2}}$, using (32), works out to be

${\mathbf{r}}_{0}=\frac{\scriptstyle{({\mathbf{a}}\cdot{\mathbf{s}}_{1})({\mathbf{s}}_{2}\times{\mathbf{b}})\,+\,({\mathbf{s}}_{2}\cdot{\mathbf{b}})({\mathbf{a}}\times{\mathbf{s}}_{1})\,-\,({\mathbf{a}}\times{\mathbf{s}}_{1})\times({\mathbf{s}}_{2}\times{\mathbf{b}})}}{\scriptstyle{\sin\left(\,\eta_{{\mathbf{a}}{\mathbf{b}}}\right)}}.$
(37)

Consequently, in the simultaneous limits ${\mathbf{s}}_{1}\to{\mathbf{a}}$ and
${\mathbf{s}}_{2}\to{\mathbf{b}}$ characterising the measurement interactions
defined in (23) and (24) (during which spin angular momentum is no longer
conserved and therefore the condition ${{\mathbf{s}}_{1}={\mathbf{s}}_{2}}$
does not hold), the rotation axis vector ${\mathbf{r}}_{0}$ in (36) reduces to
a null vector:

$\lim_{\begin{subarray}{c}{\mathbf{s}}_{1}\,\rightarrow\,{\mathbf{a}}\\\
{\mathbf{s}}_{2}\,\rightarrow\,{\mathbf{b}}\end{subarray}}\left\\{\,{\mathbf{r}}_{0}\,\right\\}=\vec{\,0}.$
(38)

As a result, the bivector in (36) also reduces to a null bivector:

$\displaystyle\lim_{\begin{subarray}{c}{\mathbf{s}}_{1}\,\rightarrow\,{\mathbf{a}}\\\
{\mathbf{s}}_{2}\,\rightarrow\,{\mathbf{b}}\end{subarray}}\left\\{{\mathbf{L}}({\mathbf{r}}_{0},\,\lambda^{k})\,\sin(\,\eta_{{\mathbf{a}}{\mathbf{b}}})\right\\}$
$\displaystyle={\mathbf{L}}(\vec{\mathbf{\,}0},\,\lambda^{k})\,\sin(\,\eta_{{\mathbf{a}}{\mathbf{b}}})$
$\displaystyle=(J\cdot\vec{\mathbf{\,}0}\,)\,\sin(\,\eta_{{\mathbf{a}}{\mathbf{b}}}),$
(39)

where $J$ is a trivector representing the volume form on $S^{3}$.
Consequently, the product ${{\mathscr{A}}{\mathscr{B}}}$ in (36) tends to
$-\cos(\,\eta_{{\mathbf{a}}{\mathbf{b}}})$:

$\displaystyle{\mathscr{A}}({\mathbf{a}},\,{\lambda^{k}})$
$\displaystyle\,{\mathscr{B}}({\mathbf{b}},\,{\lambda^{k}})\,\longrightarrow\lim_{\begin{subarray}{c}{\mathbf{s}}_{1}\,\rightarrow\,{\mathbf{a}}\\\
{\mathbf{s}}_{2}\,\rightarrow\,{\mathbf{b}}\end{subarray}}\left\\{\,-\,{\mathbf{q}}(\eta_{{\mathbf{a}}{\mathbf{b}}},\,{\mathbf{r}}_{0})\right\\}$
$\displaystyle=\lim_{\begin{subarray}{c}{\mathbf{s}}_{1}\,\rightarrow\,{\mathbf{a}}\\\
{\mathbf{s}}_{2}\,\rightarrow\,{\mathbf{b}}\end{subarray}}\left\\{-\cos(\,\eta_{{\mathbf{a}}{\mathbf{b}}})-{\mathbf{L}}({\mathbf{r}}_{0},\,\lambda^{k})\,\sin(\,\eta_{{\mathbf{a}}{\mathbf{b}}})\right\\},$
$\displaystyle=\,-\cos(\,\eta_{{\mathbf{a}}{\mathbf{b}}})-{\mathbf{L}}(\vec{\mathbf{\,}0},\,\lambda^{k})\,\sin(\,\eta_{{\mathbf{a}}{\mathbf{b}}}),$
$\displaystyle=\,-\cos(\,\eta_{{\mathbf{a}}{\mathbf{b}}})\,-\,0\,.$ (40)

Evidently, this tendency of ${{\mathscr{A}}{\mathscr{B}}}$ holds for each run
of the experiment. Consequently, using the universally accepted definition of
the correlations function used in the Bell-test experiments, the correlation
between the results ${{\mathscr{A}}({\mathbf{a}},\,{\lambda^{k}})}$ and
${{\mathscr{B}}({\mathbf{b}},\,{\lambda^{k}})}$ observed by Alice and Bob work
out to give

$\displaystyle{\cal E}_{\rm
L.R.}({\mathbf{a}},\,{\mathbf{b}})=\\!\lim_{\,n\,\gg\,1}\left[\frac{1}{n}\sum_{k\,=\,1}^{n}\,{\mathscr{A}}({\mathbf{a}},\,{\lambda}^{k})\;{\mathscr{B}}({\mathbf{b}},\,{\lambda}^{k})\right]$
(41a)
$\displaystyle\;\;\;\;=\lim_{\,n\,\gg\,1}\left[\frac{1}{n}\sum_{k\,=\,1}^{n}\,\lim_{\begin{subarray}{c}{\mathbf{s}}_{1}\,\rightarrow\,{\mathbf{a}}\\\
{\mathbf{s}}_{2}\,\rightarrow\,{\mathbf{b}}\end{subarray}}\left\\{\,-\,{\mathbf{q}}(\eta_{{\mathbf{a}}{\mathbf{b}}},\,{\mathbf{r}}_{0})\right\\}\right]$
(41b)
$\displaystyle\;\;\;\;=\lim_{\,n\,\gg\,1}\left[\frac{1}{n}\sum_{k\,=\,1}^{n}\\{-\cos(\,\eta_{{\mathbf{a}}{\mathbf{b}}})-{\mathbf{L}}(\vec{\mathbf{\,}0},\,\lambda^{k})\,\sin(\,\eta_{{\mathbf{a}}{\mathbf{b}}})\\}\right]$
(41c)
$\displaystyle\;\;\;\;=\,-\cos(\,\eta_{{\mathbf{a}}{\mathbf{b}}})\,-\\!\lim_{\,n\,\gg\,1}\left[\frac{1}{n}\sum_{k\,=\,1}^{n}\,{\mathbf{L}}(\vec{\mathbf{\,}0},\,\lambda^{k})\,\sin(\,\eta_{{\mathbf{a}}{\mathbf{b}}})\,\right]$
(41d)
$\displaystyle\;\;\;\;=\,-\cos(\,\eta_{{\mathbf{a}}{\mathbf{b}}})\,-\,0\,.$
(41e)

This corroborates my hypothesis that the observed singlet correlations are
correlations among the limiting scalar points
${{\mathscr{A}}({\mathbf{a}},\,{\lambda})=\pm 1}$ and
${{\mathscr{B}}({\mathbf{b}},\,{\lambda})\pm 1}$ of a quaternionic 3-sphere.
It is also worth noting that the above derivation of (41e) is just one of
several different ways Theorem 1 is proved in Disproof ; IJTP ; RSOS ; IEEE-1
; IEEE-2 .

Despite this inevitable result, replacing the global topology of the physical
space from $\mathrm{I\\!R}^{3}$ to $S^{3}$ may seem pointless, because
locally, in the topological sense, $S^{3}$ is isomorphic to
$\mathrm{I\\!R}^{3}$, similarly to how Earth ($S^{2}$) is isomorphic to
${\mathrm{I\\!R}^{2}}$. At each point of $S^{3}$ the tangent space is simply
$\mathrm{I\\!R}^{3}$. Consequently, in local experiments the normalized
directions such as ${\mathbf{a}}$ and ${\mathbf{b}}$ can certainly be taken to
be from $S^{2}\in\mathrm{I\\!R}^{3}$. But that does not rule out the
possibility that they are, in fact, embedded in a quaternionic 3-sphere, as I
have proposed in IEEE-2 . An a priori denial of the possibility of a global
$S^{3}$ nature of space would be analogous to a denial of the spherical nature
of Earth.

##  IV Point-by-point response to the critique

Unfortunately, the critique in Gill-Ieee  ignores the above 3-sphere model and
its physical significance entirely. Instead of appreciating that the model is
based on the orientation $\lambda=\pm 1$ of a closed and compact physical
space $S^{3}$, it insists on reinterpreting it as a hidden variable model
based on a detached binary number $\pm 1$ within a flat and non-compact space
${\mathrm{I\\!R}}^{3}$.

While there are too many incorrect claims throughout the critique, in this
Section I focus on those that are significant.

###  IV.1 Concerning one of Bell’s assumptions

Ironically, the critique itself makes one of the implicit and unjustifiable
assumptions of Bell’s theorem quite explicit in its statement of the theorem
Gill-Ieee :

> In the language of probability theory, the mathematical core of Bell’s
> original proof of his theorem is the assertion that one cannot find a single
> probability space on which are defined random variables $X_{\mathbf{a}}$ and
> $Y_{\mathbf{b}}$ taking values in the set $\\{-1,+1\\}$, for all
> ${\mathbf{a}}$, ${\mathbf{b}}$, unit vectors in $\mathbb{R}^{3}$, and such
> that
>
>
> ${\mathbb{E}}(X_{\mathbf{a}}Y_{\mathbf{b}})~{}=~{}-{\mathbf{a}}\cdot{\mathbf{b}}$
> $None$
>
> for all $\mathbf{a}$, $\mathbf{b}$. Moreover, the expectation values of
> $X_{\mathbf{a}}$ and $Y_{\mathbf{b}}$ are all zero.

But why must we assume unit vectors ${\mathbf{a}}$, ${\mathbf{b}}$ to be in
${\mathrm{I\\!R}^{3}}$? As I have summarized in Section III, there are both
theoretical and observational reasons that compel us to model physical space
as a closed and compact quaternionic 3-sphere, or $S^{3}$, instead of a flat
Euclidean space ${\mathrm{I\\!R}^{3}}$, both being admissible spatial parts of
one of the most well known cosmological solutions of Einstein’s field
equations of general relativity. Moreover, as explained in several of my
papers since 2007 Disproof ; IJTP ; RSOS ; IEEE-1 ; IEEE-2 , the correct
language to model $S^{3}$ as physical space is Geometric Algebra, not vector
“algebra.” This implies, in particular, that the directions ${\mathbf{a}}$ and
${\mathbf{b}}$ freely chosen by Alice and Bob to perform their experiments
must be solutions of the equation

$I\wedge{\mathbf{v}}=0,$ (42)

where $I$ is a volume form on the physical space $S^{3}$ and ${\mathbf{v}}$ is
any vector built from the orthogonal directions
$\\{{\mathbf{e}}_{x},\,{\mathbf{e}}_{y},\,{\mathbf{e}}_{z}\\}$ that
characterize the Clifford algebra ${\mathrm{Cl}_{(3,0)}}$. Once the physical
space is modeled as $S^{3}$ instead of $\mathrm{I\\!R}^{3}$ in the manner
explained in Section III and characterized using the powerful language of
Geometric Algebra, the correlations between the measurement results
${\mathscr{A}_{\mathbf{a}}}$ and ${\mathscr{B}_{\mathbf{b}}}$ observed by
Alice and Bob inevitably turn out to be ${\cal
E}({\mathscr{A}_{\mathbf{a}}}{\mathscr{B}_{\mathbf{b}}})=-{\mathbf{a}}\cdot{\mathbf{b}}$
as proven in IEEE-2 .

On the other hand, the standard interpretation of Bell’s theorem adhered to in
Gill-Ieee  is recovered in the flat geometry of ${{\mathrm{I\\!R}^{3}}}$. The
${S^{3}}$ model becomes conducive to the traditional interpretation of the
theorem when the algebraic, geometrical and topological properties of the
compactified physical space ${S^{3}}$ are ignored. In that case the upper
bound of 2 on the Bell-CHSH inequalities is respected. Thus, the results
presented in Disproof ; IJTP ; RSOS ; IEEE-1 ; IEEE-2  do not conflict with
the standard interpretation of Bell’s theorem outright but rather reproduces
that interpretation as a special case in the flat geometry
${{\mathrm{I\\!R}}^{3}}$ of the physical space. This is discussed in more
detail in Section X of IEEE-1 .

###  IV.2 Concerning a denial of Bell’s assumption

In the last sentence of Introduction the critique Gill-Ieee  claims that

> Bell does not take account of the geometry of space because his argument, on
> the side of local realism, does not depend on it in any way whatsoever.

But this claim is immediately contradicted in the critique in the very
paragraph that follows the sentence quoted above. As I just discussed in the
previous subsection, the critique explicitly assumes the geometry of physical
space to be ${\mathrm{I\\!R}^{3}}$ in its statement of Bell’s theorem. Indeed,
it is not possible for us to escape the geometry of physical space while
performing our experiments. For this reason, nowhere in his writings has Bell
stated that his theorem holds independently of the geometry of physical space.
In fact, Bell’s proposed local-realistic framework does assume implicitly that
physical space in which we are confined to perform our experiments is modeled
as $\mathrm{I\\!R}^{3}$, as the critique does in the first paragraph of its
Section II. It is unfortunate that this assumption is usually not made
explicit in the literature on the subject. A deeper reflection on how the
physical space is modeled in analyzing the Bell-test experiments is necessary
to uncover this implicit assumption. In the analyses of such experiments
ordinary vector algebra (which does not, in fact, form an algebra at all)
within $\mathrm{I\\!R}^{3}$ is implicitly assumed. On the other hand, I have
modeled the physical space as a quaternionic 3-sphere using Geometric Algebra
Clifford . The fact that I have been able to reproduce the strong singlet
correlations by modeling the physical space as a quaternionic 3-sphere is a
confirmatory evidence that the strategy to relax the unwarranted implicit
assumption built-in Bell’s theorem has been successful.

###  IV.3 Concerning Local causality of the model

In its Introduction, the critique Gill-Ieee  claims that “Christian’s idea
that quantum correlations are explained by the geometry of space might seem
appealing, but … such an explanation would not be “local” in any meaningful
sense.” However, we are not at liberty to guess what is or is not a meaningful
sense of locality. Einstein and Bell have given us a very precise notion of
local causality and I have strictly adhered to that notion throughout my work
on the subject. As is well known, a violation of relativistic local causality
can be separated into two distinct parts: (1) a signalling non-locality
incompatible with general relativity, and (2) a non-signalling non-locality
compatible with general relativity. These two distinct parts are captured by
Bell in his definitions ${{\mathscr{A}}({\mathbf{a}},\,\lambda)}$ and
${{\mathscr{B}}({\mathbf{b}},\,\lambda)}$ of local measurement functions for
any given initial state ${\lambda}$ of a given physical system. This separates
relativistic local causality into independence of the parameter ${\mathbf{a}}$
from ${\mathbf{b}}$ (and vice versa) preserving signalling locality, and
independence of the outcome ${\mathscr{A}}$ from ${\mathscr{B}}$ (and vice
versa) preserving non-signalling locality, in any EPR-Bohm type experiment.

In the model presented in Disproof ; IJTP ; RSOS ; IEEE-1 ; IEEE-2  and the
previous section the question of signalling non-locality does not arise
because the quaternionic 3-sphere on which it is based is a part of the
solution of Einstein’s field equations of general relativity. And the question
of non-signalling non-locality is also implicitly addressed within the model
by recognizing that the measurement functions (23) and (24) define manifestly
local-realistic functions. Apart from the hidden variable ${\lambda}$, the
result ${{\mathscr{A}}=\pm 1}$ depends only on the measurement direction
${\mathbf{a}}$, chosen freely by Alice, regardless of Bob’s actions. And
similarly, apart from the hidden variable ${\lambda}$, the result
${{\mathscr{B}}=\pm 1}$ depends only on the measurement direction
${\mathbf{b}}$, chosen freely by Bob, regardless of Alice’s actions. In
particular, the function ${{\mathscr{A}}({\mathbf{a}},\,\lambda)}$ does not
depend on ${\mathbf{b}}$ or ${\mathscr{B}}$ and the function
${{\mathscr{B}}({\mathbf{b}},\,\lambda)}$ does not depend on ${\mathbf{a}}$ or
${\mathscr{A}}$. Moreover, the hidden variable ${\lambda}$ does not depend on
either ${\mathbf{a}}$, ${\mathbf{b}}$, ${\mathscr{A}}$, or ${\mathscr{B}}$.

Unimpressed by this, the critique Gill-Ieee  continues its claim:

> Christian seems to see the local spatial coordinate system of Alice being
> the mirror image of Bob’s, the two orientations being determined completely
> at random, again and again! However, in modern accounts of Bell’s theorem,
> angles and orientations play no role whatsoever. The new generation of
> loophole-free Bell experiments … measure correlations between four binary
> variables: two binary inputs and two binary outputs; one input and output at
> each of two distant locations.

These comments provide a clear evidence that the critique in Gill-Ieee  is
based on a mistaken understanding of what is meant by the orientation
$\lambda$ of the 3-sphere. The orientation $\lambda$ of $S^{3}$ does not
concern the local spatial coordinate systems of Alice and Bob. It describes
the handedness of the physical space $S^{3}$ itself. It specifies whether the
3-sphere is inside-out or outside-out111A good analogy of this is an ordinary
hand-glove. If the outside of a hand-glove is right-handed, then pulling it
inside out will make it left-handed. with respect to the detectors in a given
run of the experiment, with 50/50 chance. The value of $\lambda$ is fixed for
the detectors ${{\mathbf{D}}({\mathbf{a}})=I\cdot{\mathbf{a}}}$ and
${{\mathbf{D}}({\mathbf{b}})=I\cdot{\mathbf{b}}}$, chosen freely by Alice and
Bob, for all runs of the experiment. But it is not fixed for the spins
${-\,{\mathbf{L}}({\mathbf{s}}_{1},\,\lambda)}$ and
${+\,{\mathbf{L}}({\mathbf{s}}_{2},\,\lambda)}$ emerging from the source.
Thus, $\lambda$ plays a role of a hidden variable, or an initial state of the
system, relative to the fixed handedness of the detectors. Since it plays the
role of a hidden variable, Alice and Bob need not be concerned about it at
all. All they need to worry about are four binary variables: two binary inputs
“${\mathbf{a}}$” and “${\mathbf{b}}$”, and two binary outputs
“${\mathscr{A}}$” and “${\mathscr{B}}$”; one input and one output at each of
the two distant locations.

###  IV.4 Concerning Equations (32) and (33) of IEEE-2

In Section III of the critique, in the paragraph before the one containing
equation (4), it is stated, incorrectly, that in IEEE-2  the bivectors
${\mathbf{L}}({\mathbf{a}},\lambda)$, ${\mathbf{L}}({\mathbf{b}},\lambda)$,
${\mathbf{D}}({\mathbf{a}})$, and ${\mathbf{D}}({\mathbf{b}})$ are introduced
by equations (32) and (33). This incorrect starting point is the reason behind
much of the confusion manifest in the critique.

In fact, the spin bivectors ${\mathbf{L}}({\mathbf{s}},\lambda)$ in IEEE-2
are introduced by the bivector subalgebra (28), and the detector bivectors
${\mathbf{D}}({\mathbf{n}})$ are introduced by the bivector subalgebra (31);
namely, by

$L_{i}(\lambda)\,L_{j}(\lambda)\,=\,-\,\delta_{ij}\,-\,\sum_{k}\,\epsilon_{ijk}\,L_{k}(\lambda)$
(43)

and

$D_{i}\,D_{j}\,=\,-\,\delta_{ij}\,-\,\sum_{k}\,\epsilon_{ijk}\,D_{k}\,,$ (44)

respectively. In other words, the spin bivectors
${\mathbf{L}}({\mathbf{s}},\lambda)$ and the detector bivectors
${\mathbf{D}}({\mathbf{n}})$ are introduced using two different basis vectors,
because of the experimental prerequisite that the detectors are located at
remote stations at spacelike distance from each other, whereas the spins
originate at the central source, independently of the detectors. Indeed, it is
explicitly stated in IEEE-2  that the basis bivectors for the spins,

$L_{i}(\lambda)=J\cdot{\mathbf{e}}_{i}^{\prime}\,,$ (45)

are defined in terms of the basis vectors
${\\{{\mathbf{e}}_{1}^{\prime},\,{\mathbf{e}}_{2}^{\prime},\,{\mathbf{e}}_{3}^{\prime}\\}}$
and the corresponding trivector
$J={\mathbf{e}}_{1}^{\prime}{\mathbf{e}}_{2}^{\prime}{\mathbf{e}}_{3}^{\prime}\,$;
and the basis bivectors for the detectors,

$D_{i}=I\cdot{\mathbf{e}}_{i}\,,$ (46)

are defined in terms of the basis vectors
${\\{{\mathbf{e}}_{1},\,{\mathbf{e}}_{2},\,{\mathbf{e}}_{3}\\}}$ and the
corresponding trivector $I={\mathbf{e}}_{1}{\mathbf{e}}_{2}{\mathbf{e}}_{3}$.
It is surprising that these explicit definitions of $L_{i}(\lambda)$ and
$D_{i}$ are missed in Gill-Ieee . It is not easy to notice relations (32) and
(33) in IEEE-2  but miss the explicit definitions (28) and (31) written in the
same column.

In any case, once the above definitions (45) and (46) of $L_{i}(\lambda)$ and
$D_{i}$ specifying two different bivector basis are not missed, with $\lambda$
being the orientation of $S^{3}$, then, as elaborated in Question 13 of
Appendix B in IEEE-2  leading up to Eq. (74), the basis bivectors
$L_{i}(\lambda)$ and $D_{i}$ are clearly related by $\lambda$ as

$L_{i}(\lambda)\,=\,\lambda\,D_{i}\;\;\Longleftrightarrow\;\;D_{i}\,=\,\lambda\,L_{i}(\lambda)\,.$
(47)

Contracting on both sides of (47) with the components $n^{i}$ of an arbitrary
unit vector ${\mathbf{n}}$ then gives the relation (25) stated above, and
contracting the bivector subalgebra defined in (43) above on both sides with
the components $a^{i}$ and $b^{j}$ of arbitrary unit vectors ${\mathbf{a}}$
and ${\mathbf{b}}$ gives the Pauli identity,

${\mathbf{L}}({\mathbf{a}},\,\lambda)\,{\mathbf{L}}({\mathbf{b}},\,\lambda)\,=\,-\,{\mathbf{a}}\cdot{\mathbf{b}}\,-\,{\mathbf{L}}({\mathbf{a}}\times{\mathbf{b}},\,\lambda),$
(48)

with unit ${\mathbf{L}(\mathbf{a},\,\lambda):=a^{i}L_{i}(\lambda)}$ and unit
${\mathbf{L}(\mathbf{b},\,\lambda):=b^{j}L_{j}(\lambda)}$.

The above definitions allow the spin bivectors
${\mathbf{L}}({\mathbf{a}},\lambda)$ and ${\mathbf{L}}({\mathbf{b}},\lambda)$
to relate to the detector bivectors ${\mathbf{D}}({\mathbf{a}})$ and
${\mathbf{D}}({\mathbf{b}})$ by the orientation $\lambda$ of $S^{3}$ at the
time of their measurements, as specified by equations (32) and (33) of IEEE-2
. The critique, however, ignores these definitions and insists on interpreting
the contingent relations (32) and (33) between the spin bivectors and detector
bivectors as the definition of the spin bivectors. Moreover, the critique’s
interpretation is also physically incorrect. For when they emerge from the
common source the spins would be spinning about the direction ${\mathbf{s}}$
that would have no prior relation to what directions ${\mathbf{a}}$ and
${\mathbf{b}}$ Alice and Bob may have chosen to perform their measurements.
And even if we go along with the critique’s insistence on focusing on the
contingent relations (32) and (33) as a piece of mathematics, the internal
consistency of the 3-sphere model is robust enough to prevent the critique’s
strategy from succeeding, as I now demonstrate.

The critique begins by identifying ${{\mathbf{L}}({\mathbf{a}},\,\lambda)}$
with $\lambda\,I\cdot{\mathbf{a}}$ and
${{\mathbf{L}}({\mathbf{b}},\,\lambda)}$ with $\lambda\,I\cdot{\mathbf{b}}$,
but neglects to make the identification

${\mathbf{L}}({\mathbf{a}}\times{\mathbf{b}},\,\lambda)=\lambda\,I\cdot({\mathbf{a}}\times{\mathbf{b}}),$
(49)

which is demanded by the geometrical consistency of vectors within
$S^{2}\subset\mathrm{I\\!R}^{3}$. The critique Gill-Ieee  then claims:

> It follows directly from Christian’s (32) and (33) that
>
>
> $\mathbf{L}(\mathbf{a},\lambda)\mathbf{L}(\mathbf{b},\lambda)=\lambda^{2}I^{2}\mathbf{a}\mathbf{b}=-\mathbf{a}\mathbf{b},$
> (4)
>
> which does not depend on $\lambda$ at all.

But that is not correct. The RHS of the critique’s Eq. (4) is not independent
of $\lambda$. In fact, $\lambda$ is implicit in the product
${-{\mathbf{a}}{\mathbf{b}}=-{\mathbf{a}}\cdot{\mathbf{b}}-{\mathbf{a}}\wedge{\mathbf{b}}}$,
not absent from it. It would be a meaningless equation if its LHS “depended”
on $\lambda$ while its RHS did not. Moreover, to begin with, Eqs. (32) and
(33) of IEEE-2  do not specify what the geometric product of
${{\mathbf{L}}({\mathbf{a}},\,\lambda)}$ with
${{\mathbf{L}}({\mathbf{b}},\,\lambda)}$ is. Eq. (29) of IEEE-2  does. Thus,
Eq. (4) of Gill-Ieee  is rather presumptuous. And why must we stop at the
first step in Eq. (4)? Why not continue the derivation of the product by
recalling from the definition (45) above that, by definition,
${{\mathbf{L}}({\mathbf{a}},\,\lambda)=J\cdot{\mathbf{a}}}$ and therefore
${J\cdot{\mathbf{a}}=\lambda\,I\cdot{\mathbf{a}}}$ if we follow the critique’s
absolute identification, or, equivalently,

$J=\lambda\,I\Longleftrightarrow\,I=\lambda\,J,$ (50)

which then gives

$\displaystyle{\mathbf{L}}({\mathbf{a}},\,\lambda)\,{\mathbf{L}}({\mathbf{b}},\,\lambda)\,$
$\displaystyle=\,-\,{\mathbf{a}}{\mathbf{b}}$ (51)
$\displaystyle=\,-\,{\mathbf{a}}\cdot{\mathbf{b}}\,-\,{\mathbf{a}}\wedge{\mathbf{b}}$
(52)
$\displaystyle=\,-\,{\mathbf{a}}\cdot{\mathbf{b}}\,-\,J\cdot({\mathbf{a}}\times{\mathbf{b}})$
(53)
$\displaystyle=\,-\,{\mathbf{a}}\cdot{\mathbf{b}}\,-\,\lambda\,I\cdot({\mathbf{a}}\times{\mathbf{b}})$
(54)
$\displaystyle=\,-\,{\mathbf{a}}\cdot{\mathbf{b}}\,-\,{\mathbf{L}}({\mathbf{a}}\times{\mathbf{b}},\,\lambda)\,.$
(55)

But this is precisely Eq. (29) of IEEE-2  or (48) above, and its RHS does
“depend” on $\lambda$, contrary to the critique’s claim. In other words, Eqs.
(32) and (33) of IEEE-2  do not contradict its Eq. (29), and consequently the
critique’s argument and strategy fail.

Unfortunately, the critique’s conceptual mistake here is even more serious. It
treats the orientation $\lambda$ of $S^{3}$ as if it were an argument of the
spin bivector ${\mathbf{L}}({\mathbf{s}},\,\lambda)$ by itself without
reference to the detector bivector ${\mathbf{D}}({\mathbf{n}})$. But $\lambda$
is not just a number or an argument of ${\mathbf{L}}({\mathbf{s}},\,\lambda)$
by itself. It represents the handedness of the spin bivector
${\mathbf{L}}({\mathbf{n}},\,\lambda)$ relative to that of the detector
bivector ${\mathbf{D}}({\mathbf{n}})$, and vice versa, as specified in Eq.
(25) above. Thus, between ${\mathbf{L}}({\mathbf{s}},\,\lambda)$ and
${\mathbf{D}}({\mathbf{n}})$, $\lambda$ has only relative significance, and
$\lambda$ for the spin ${\mathbf{L}}({\mathbf{s}},\,\lambda)$ is meaningful
only with respect to the detector ${\mathbf{D}}({\mathbf{n}})$, and vice
versa. Consequently, using the identity (48) and the relations (25), the
products of two spin bivectors can be evaluated as

$\displaystyle{\mathbf{L}}({\mathbf{a}},\,{\lambda}=+1)\;{\mathbf{L}}({\mathbf{b}},\,$
$\displaystyle{\lambda}=+1)$
$\displaystyle=\,-\,{\mathbf{a}}\cdot{\mathbf{b}}\,-\,{\mathbf{L}}({\mathbf{a}}\times{\mathbf{b}},\,\lambda=+1)$
$\displaystyle=-\,{\mathbf{a}}\cdot{\mathbf{b}}\,-\,{\mathbf{D}}({\mathbf{a}}\times{\mathbf{b}})$
$\displaystyle={\mathbf{D}}({\mathbf{a}})\;{\mathbf{D}}({\mathbf{b}})$ (56)

and

$\displaystyle{\mathbf{L}}({\mathbf{a}},\,{\lambda}=-1)\;{\mathbf{L}}({\mathbf{b}},\,$
$\displaystyle{\lambda}=-1)$
$\displaystyle=\,-\,{\mathbf{a}}\cdot{\mathbf{b}}\,-\,{\mathbf{L}}({\mathbf{a}}\times{\mathbf{b}},\,\lambda=-1)$
$\displaystyle=-\,{\mathbf{a}}\cdot{\mathbf{b}}\,+\,{\mathbf{D}}({\mathbf{a}}\times{\mathbf{b}})$
$\displaystyle=-\,{\mathbf{b}}\cdot{\mathbf{a}}\,-\,{\mathbf{D}}({\mathbf{b}}\times{\mathbf{a}})$
$\displaystyle={\mathbf{D}}({\mathbf{b}})\;{\mathbf{D}}({\mathbf{a}}).$ (57)

It is now evident from the above equations that, once the relevant products
are correctly evaluated, the ordering relation between the spin bivectors
${{\mathbf{L}}({\mathbf{a}},\,\lambda)}$ and
${{\mathbf{L}}({\mathbf{b}},\,\lambda)}$ and the detector bivectors
${{\mathbf{D}}({\mathbf{a}}})$ and ${{\mathbf{D}}({\mathbf{b}})}$ is
equivalent to the hypothesis that the orientation $\lambda=\pm 1$ of the
3-sphere is a fair coin. This disproves the conjecture in the critique Gill-
Ieee  that equations (34) and (35) of IEEE-2  could not both be correct.

The rest of the argument in the critique pertaining to the validity of the
identity (48) is now moot. But it is instructive to unpack it to appreciate
the mistake in it. In fact, it is an elementary mistake in ordinary vector
algebra, as pointed out in Appendix C of Reply-Gill . The critique considers
both right-handed and left-handed cross products. But there is no such
suggestion in Disproof ; IJTP ; RSOS ; IEEE-1 ; IEEE-2 . Contrary to what the
critique considers, namely, two cross products $\times_{\\!\lambda}$ with
$\lambda=\pm 1$ and the rules

$\mathbf{a}\times_{\\!+1}\\!\mathbf{b}~{}=~{}\mathbf{a}\times\mathbf{b}\qquad\text{and}\qquad\mathbf{a}\times_{\\!-1}\\!\mathbf{b}~{}=~{}\mathbf{b}\times\mathbf{a},$
(58)

the correct versions of these equations in vector algebra are

$\mathbf{a}\times_{\\!+1}\\!\mathbf{b}~{}=~{}\mathbf{a}\times\mathbf{b}\qquad\text{and}\qquad\mathbf{a}\times_{\\!-1}\\!\mathbf{b}~{}=~{}\mathbf{a}\times\mathbf{b}.$
(59)

As pointed out in Appendix C of Reply-Gill , in vector algebra the cross
product between vectors ${\mathbf{a}}$ and ${\mathbf{b}}$ remains the same in
both right-handed and left-handed coordinates. But by using the incorrect
equations (58), the critique arrives at its Eq. (7),

$\mathbf{L}(\mathbf{a},\lambda)\mathbf{L}(\mathbf{b},\lambda)~{}=~{}-\mathbf{a}\cdot\mathbf{b}-\mathbf{L}(\mathbf{a}\times_{\\!\lambda}\\!\mathbf{b},\lambda),$
(60)

which, according to its definition of
$\mathbf{a}\times_{\\!\lambda}\\!\mathbf{b}$, is equivalent to

$\mathbf{L}(\mathbf{a},\,\lambda)\mathbf{L}(\mathbf{b},\,\lambda)~{}=~{}-\mathbf{a}\cdot\mathbf{b}-\,\lambda\;\mathbf{L}(\mathbf{a}\times\mathbf{b},\,\lambda).$
(61)

Note that there is now a redundant $\lambda$ on the right-hand side of this
equation in the second term, in contrast to the identity (48) above. In terms
of the basis, the equation takes the form

$L_{i}(\lambda)\,L_{j}(\lambda)=-\,\delta_{ij}-\lambda\sum_{k}\epsilon_{ijk}\,L_{k}(\lambda),$
(62)

with the extra $\lambda$ appearing just before the summation. The critique now
claims that, with this equation, it has “restored consistency.” But, in fact,
what it has done is to introduce inconsistency. For the above equation does
not represent the bivector subalgebra of ${\mathrm{Cl}}_{(3,0)}$, and
therefore it cannot constitute a quaternionic 3-sphere. The correct bivector
subalgebra of ${\mathrm{Cl}}_{(3,0)}$ is given by (43) discussed above,
without the extra $\lambda$. Thus, unlike Eq. (28) of IEEE-2 , the critique’s
Eq. (7) with the extra $\lambda$ is both conceptually and mathematically
incorrect.

###  IV.5 Concerning Equations (39) and (40) of IEEE-2

Figure 1: The results ${\mathscr{A}}$ and ${\mathscr{B}}$ are simultaneous
points of a quaternionic 3-sphere, or ${S^{3}}$. Since ${S^{3}}$ remains
closed under multiplication, the product ${{\mathscr{A}}{\mathscr{B}}}$ is
also a point of ${S^{3}}$, with its binary value ${\pm 1}$ dictated by the
geometry.

The critique’s second incorrect claim concerns the definitions (39) and (40)
in IEEE-2  of measurement functions, which I have reproduced in (23) and (24)
above. The critique claims that the functions (39) and (40) predict perfect
anti-correlation

${\cal E}({\mathbf{a}},\,{\mathbf{b}})=-1$ (63)

for all choices of measurement directions ${\mathbf{a}}$ and ${\mathbf{b}}$,
instead of the cosine correlations derived in (41e) for
${{\mathbf{a}}\not={\mathbf{b}}}$. This claim is refuted in Appendix C of
Reply-Gill  and in Scott , as well as in Answers 9 and 14 in Appendix B of
IEEE-1  and Answers 6 and 7 in Appendix B of IEEE-2 . But the claim is made
again in this critique, in Eq. (3) near the end of its Section II, and
repeated in the paragraph that includes Eqs. (9) and (10), which read

$\displaystyle{\mathscr{A}}({\mathbf{a}},\lambda)$ $\displaystyle=+\lambda,$
(64) $\displaystyle{\mathscr{B}}({\mathbf{b}},\lambda)$
$\displaystyle=-\lambda.$ (65)

In other words, the critique identifies the measurement results
${\mathscr{A}}$ and ${\mathscr{B}}$ observed at the space-like separated
stations with the initial state $\lambda$ of the spins that originates from
the source located in the overlap of the backward light-cones of Alice and
Bob. But no such identification is made in the definitions (39) and (40) of
the measurement functions in IEEE-2 . What is encapsulated by these functions
are the measurement interactions. Alice chooses her detector
${\mathbf{D}}({\mathbf{a}})$ about a measurement direction ${\mathbf{a}}$ and
Bob chooses his detector ${\mathbf{D}}({\mathbf{b}})$ about a measurement
direction ${\mathbf{b}}$, at a space-like distance from each other. The spins
${-\,{\mathbf{L}}({\mathbf{s}}_{1},\,\lambda)}$ and
${+\,{\mathbf{L}}({\mathbf{s}}_{2},\,\lambda)}$, on the other hand, originate
from the source located in the overlap of the backward light-cones of Alice
and Bob. And the results ${\mathscr{A}}({\mathbf{a}},\lambda)$ and
${\mathscr{B}}({\mathbf{b}},\lambda)$ defined in (39) and (40) do not come
about until the time of measurements, and even then only via two different
quaternions within $S^{3}$. The identifications (64) and (65), however, allow
the critique to write their product as

${\mathscr{A}}({\mathbf{a}},\lambda)\,{\mathscr{B}}({\mathbf{b}},\lambda)=(+\lambda)(-\lambda)=-\lambda^{2}=-1,$
(66)

for all choices of ${\mathbf{a}}$ and ${\mathbf{b}}$. But for
${{\mathbf{a}}\not={\mathbf{b}}}$, Eqs. (64) and (65) are valid only for
$\mathbf{s}_{1}\not=\mathbf{s}_{2}$. They ignore the conservation of zero spin
angular momentum, which, as we saw in Eq. (34), amounts to setting
${\mathbf{s}}_{1}={\mathbf{s}}_{2}={\mathbf{s}}$. In other word, the
critique’s equations (9) and (10) hold in general for all choices of
${\mathbf{a}}$ and ${\mathbf{b}}$ if and only if the conservation of spin
angular momentum is violated, or, equivalently, the Möbius-like twists in the
Hopf bundle of $S^{3}$ are ignored. That is to say, for
${{\mathbf{a}}\not={\mathbf{b}}}$, equations (64) and (65) hold if and only if
the 3-sphere model is abandoned and one stoops back to the flat geometry of
${\mathrm{I\\!R}^{3}}$.

For convenience, I have reproduced one of the correct derivations of the
singlet correlations above in the paragraphs containing equations (26) to
(41e). This derivation preserves the geometrical properties of the 3-sphere,
without relapsing back to the flat geometry of ${\mathrm{I\\!R}^{3}}$ as the
critique tends to do. It is also worth noting that the derivation in (26) to
(41e) above is just one of several different ways it is demonstrated in IEEE-1
and IEEE-2  that within a quaternionic 3-sphere, taken as a physical space,
the correlations are inevitably ${\cal
E}({\mathbf{a}},\,{\mathbf{b}})=-{\mathbf{a}}\cdot{\mathbf{b}}$.

There is also a related conceptual issue that is important to address here.
Recall that there are, in fact, three different sets of experiments involved
in any EPR-Bohm type experiments. Alice and Bob can independently detect spins
of the particles they receive at their respective observation stations,
obtaining the results $\pm 1$, with 50/50 chance, so that both
$\langle{\mathscr{A}}\rangle=0$ and $\langle{\mathscr{B}}\rangle=0$. These are
two separate sets of experiments, because Alice can perform her experiments
and obtain the same results regardless of Bob’s existence, and vice versa.
Their separate and independent results, ${\mathscr{A}}=\pm 1$,
$\langle{\mathscr{A}}\rangle=0$ and ${\mathscr{B}}=\pm 1$,
$\langle{\mathscr{B}}\rangle=0$, respectively, are exactly what the
measurement functions (39) and (40) defined in IEEE-2  predict.

Then, in a third set of experiments, Alice and Bob jointly and simultaneously,
but again independently, detect spins at their respective stations, regardless
of which spin result the other party has observed. Their recorded results are
then compared later by a third party, say Charlie, and calculated to exhibit
the correlation $\langle{\mathscr{A}}{\mathscr{B}}=\pm
1\rangle=-\cos(\,\eta_{{\mathbf{a}}{\mathbf{b}}})$ between the results
${\mathscr{A}}=\pm 1$ and ${\mathscr{B}}=\pm 1$. In actual experiments this
can be done only by “coincidence counts” of joint and simultaneous detections
of spins by Alice and Bob (cf. Section 4.1 of RSOS ). It is therefore a
mistake to confuse the first two sets of experiments with the third set of
experiments.

Now recall that the results $\mathscr{A}$ and $\mathscr{B}$ observed
simultaneously but independently by Alice and Bob are necessarily events in
spacetime. Within the spacetime defined by the line element (12) together with
$\Sigma=S^{3}$ and $a(t)=1$, they are thus events in ${\mathrm{I\\!R}\times
S^{3}}$. Thus, recalling that $S^{3}$ — which, as defined in (11), is a set of
unit quaternions — remains closed under multiplication (cf. Fig. 1), the
correct question to ask here is the one I have posed just before Eq. (26) in
Section III: What will be the average of the product
${{\mathscr{A}}{\mathscr{B}}}$ of the results ${\mathscr{A}}$ and
${\mathscr{B}}$ within the space-like hypersurface $S^{3}$? In other words,
what will be the average of the product ${{\mathscr{A}}{\mathscr{B}}}$ when
the results ${\mathscr{A}}$ and ${\mathscr{B}}$ are observed by Alice and Bob
separately but simultaneously, in “coincidence counts”, within $S^{3}$? We can
only work out the correct average of ${{\mathscr{A}}{\mathscr{B}}}$ within
$S^{3}$ from the quaternionic definitions (39) and (40), not from their
travesties (64) and (65), and that answer, contrary to the claim in Gill-Ieee
, works out to be ${\cal
E}({\mathbf{a}},\,{\mathbf{b}})=-{\mathbf{a}}\cdot{\mathbf{b}}$.

###  IV.6 Concerning a pair of binary variables

In the paragraph before last in its Section II, the critique Gill-Ieee
claims:

> For $\mathbf{a}\neq\pm\mathbf{b}$, the probability distribution of the pair
> of binary variables $(X_{\mathbf{a}},Y_{\mathbf{b}})$ predicted by quantum
> mechanics gives positive probability to each of four distinct joint outcomes
> $(\pm 1,\pm 1)$. There is no way one can simulate a single draw from a
> probability distribution over four outcomes, each of positive probability,
> as a deterministic function of the outcome of one fair coin toss.
> Christian’s hidden variable $\lambda$, which one may identify with the
> elementary outcome $\omega$ of the alleged probability model on which all
> those random variables are defined, is a fair coin toss, and in his model,
> the results of measurement of spin of the two particles in any two
> directions are functions only of $\lambda$ and of the relevant direction.

This argument is again based on a mistaken reading of what the hidden variable
$\lambda$ is in the $S^{3}$ model. In the $S^{3}$ model ${\lambda}$ is not a
detached fair coin toss in the flat space ${\mathrm{I\\!R}^{3}}$. Instead, it
represents an orientation or handedness of the closed and compact space
$S^{3}$ itself. The alternatives are thus between an inside-out 3-sphere and
outside-out11footnotemark: 1 3-sphere, with respect to the detectors
${\mathbf{D}}({\mathbf{a}})$ and ${\mathbf{D}}({\mathbf{b}})$. This again
illustrates that the critique has misinterpreted what the 3-sphere model
actually is and how it predicts the correlations ${\cal
E}({\mathbf{a}},\,{\mathbf{b}})=-{\mathbf{a}}\cdot{\mathbf{b}}$.

###  IV.7 Concerning the computer Code in IEEE-2

Next, the critique Gill-Ieee  turns to the computer code presented in IEEE-2
and claims that it contains the following “revealing line”:

$\texttt{if(lambda==1) \\{q=A B;\\} else \\{q=B A;\\}},$

“which serves to switch between the geometric product and its transpose
according to the sign of $\lambda$” Gill-Ieee . But it is easy to see that the
above line is a faithful representation of the analytical equations (34) and
(35) in IEEE-2  that switch the order of the detectors with respect to that of
the spins (as annotated in the code), thereby shuffling the alternative
orientations of $S^{3}$, exactly as required by those equations. In other
words, the line in question is necessitated by the model itself. In fact, the
line in question is the very essence of the 3-sphere model. Moreover, by now
the code has been independently translated by several professional programmers
into different computer languages, such as Python, Maple, R, and Mathematica
Fred .

It is worth stressing here that the computer code included in IEEE-2 , by
itself, is not the model, or even a proof of the model. Its purpose is to
demonstrate how the model works. It is thus a pedagogical tool that verifies
the analytical computations presented in IEEE-2 . Needless to say, the
analytical computations stand on their own and do not require a numerical
simulation for their validity. On the other hand, the computer code for an
event-by-event simulation of the singlet correlations does provide additional
support to the analytical computations, for it is both pedagogically and
statistically illuminating.

By contrast, in the paragraph before last in its Section II, and then at the
end of its Section III, the critique Gill-Ieee  claims:

> Christian’s computer simulation program uses a fair coin toss to average
> Geometric Algebra products using the fundamental GA formula
> $\mathbf{a}\cdot\mathbf{b}=\frac{1}{2}\mathbf{a}\mathbf{b}+\frac{1}{2}\mathbf{b}\mathbf{a}$.
> … About half the time, the “product of the measurements” is defined by the
> code as the quaternion $-\mathbf{a}\mathbf{b}$, the other half of the time
> it is the quaternion $-\mathbf{b}\mathbf{a}$.

But these claims are not correct. According to the correct algorithm, which is
annotated in the code, the orientation $\lambda$ of the 3-sphere plays the
role of a fair coin quite independently of the definition of the geometric
product. Moreover, contrary to the claim in Gill-Ieee , the values of the
scalar part of the product ${\mathbf{a}}{\mathbf{b}}$ are not “grouped into
bins.” Instead, what is being plotted is the scalar part of the correlation
directly with respect to its associated angle
$\eta_{{\mathbf{a}}{\mathbf{b}}}$. The purpose of the code is to verify the
analytical calculations, which it does quite successfully.

##  V Concluding remarks

After summarizing the local-realistic 3-sphere model for the singlet
correlations presented in IEEE-2 , I have demonstrated that none of the claims
made against the model in the critique Gill-Ieee  are correct. In particular,
contrary to its claim, the critique has not identified any mistakes in IEEE-2
, either in the analytical model of the singlet correlations or in its event-
by-event numerical simulation. The critique begins with an incorrect version
of the 3-sphere model by writing some of its equations incorrectly. It then
derives a constant value for the singlet correlations by failing to respect
the geometrical properties of the 3-sphere such as the spinorial sign changes
in the quaternions, criticizes this incorrect value, and concludes that it has
thereby criticized the 3-sphere model. This strategy also violates the
conservation of spin angular momentum. In this paper I have addressed the
claims in the critique Gill-Ieee  point by point and shown that they are
neither proven nor justified.

## References

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