# Reply to “Comment on ‘Bell’s Theorem Versus Local Realism in a Quaternionic
Model of Physical Space”’

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 “Bell’s Theorem Versus Local Realism in a
Quaternionic Model of Physical Space.” That paper presents a local-realistic
model of quantum correlations based on a quaternionic 3-sphere, taken as a
physical space in which we are confined to perform all our experiments. The
critique, on the other hand, considers two entirely different models within a
flat Euclidean space, neither related to my quaternionic 3-sphere model. It
then criticizes its own flat space models and claims that it has thereby
criticized the model presented in my paper. Along the way, without providing
evidence or proof, it claims that the results in my paper are based on
mistakes. I demonstrate that there are no mistakes of any kind in my paper. On
the contrary, I bring out a number of elementary mathematical and conceptual
mistakes from the critique and the critiques it relies on.

##  I Introduction

This reply paper should be read as a continuation of my previous reply paper
IEEE-3 , which is a reply published in this journal to a previous critique of
one of my papers by the same author Gill-IEEE-1 . I have also responded to
earliercritiques by the same author elsewhere Reply-Gill ; Reply-Gill-IJTP ;
Scott . It is unfortunate that the critique in Gill-IEEE-2  has repeated many
arguments that I have already addressed in IEEE-3 ; Reply-Gill ; Reply-Gill-
IJTP ; Scott , as well as in Appendix B of IEEE-1  and Appendix B of IEEE-2 .
Sections I, II, and III of my previous reply IEEE-3  provide the background
for understanding the current reply to the critique in Gill-IEEE-2 . In what
follows, I have tried to avoid repetitions of my refutations of the issues
raised in Gill-IEEE-2  that are already addressed by me in IEEE-3 . Needless
to say, prior reading of my original paper IEEE-1  and its pedagogical
exposition IEEE-2  is recommended for understanding this reply.

The central result I have presented in IEEE-1  and IEEE-2 , as well as in my
earlier papers Disproof ; IJTP ; RSOS , is this special theorem:

Theorem 1: Quantum mechanical correlations predicted by the entangled singlet
state can be understood as classical, local, realistic, and deterministic
correlations among the pairs of limiting scalar points of a quaternionic
3-sphere, or $S^{3}$, taken as a model of the three-dimensional physical
space.

I have proved this theorem in IEEE-1 ; IEEE-2 ; Disproof ; IJTP ; RSOS  in
several different ways. In fact, its proof can be presented on a single page
disproof . Its more complete proof can be found in Section III of IEEE-3 .

Now $S^{3}$ turns out to be sufficient for understanding the singlet
correlations local-realistically, but not sufficient for understanding more
general correlations. What is required is an algebraic representation space of
$S^{3}$, which turns out to be an octonion-like 7-sphere, leading to this
general theorem:

Theorem 2: Quantum mechanical correlations predicted by any arbitrary quantum
state can be understood as classical, local, realistic, and deterministic
correlations among the pairs of limiting scalar points of an octonion-like
7-sphere, as an algebraic representation space of the quaternionic 3-sphere.

I have proved this general theorem in RSOS . For a summary, see also slides .
According to these two theorems, the geometrical and topological properties of
the quaternionic 3-sphere, or $S^{3}$, are the raison d’être of the origins
and strengths of all quantum correlations, not quantum entanglement per se.
The latter is merely a placeholder for quantum correlations, similar to how
phlogiston was for combustion until Lavoisier proved it to be a rapid
oxidation, and action-at-a-distance was for Newton’s gravity until Einstein
recognized it to be an effect of the curvature of spacetime. Tsirel’son’s
bounds

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

turn out to be consequences of the above hypothesis IEEE-1 ; RSOS . Although
in my view the strong correlations observed in Nature is evidence enough in
favor of this hypothesis, in IJTP  I have proposed an experiment set in a
macroscopic domain, devoid of any quantum features, that may be able to
falsify it.

In this paper, I demonstrate, point by point, that none of the claims made in
the critique Gill-IEEE-2 , or in the critiques mentioned therein, undermine
the above theorems. Contrary to its claims, the critique has not found any
kind of mistakes in my papers, or in the numerical simulations of the
quaternionic 3-sphere model cited therein. On the contrary, in what follows I
point out a number of mathematical and conceptual mistakes in the critique
Gill-IEEE-2  and the critiques Gill-IEEE-1 ; Gill-Entropy ; Lasenby-AACA  it
relies on.

The common defect in the critiques Gill-IEEE-1 , Gill-IEEE-2 , and Gill-
Entropy  is that they consider and criticize models that are entirely
different from the one presented in IEEE-1  and IEEE-2 , and then claim that
they have thereby criticized the model presented in my papers. This has been
an effective strategy for the critiques in Gill-IEEE-1 , Gill-IEEE-2 , and
Gill-Entropy  despite its manifest logical fallacy. For example, it can be
easily verified by searching the PDF file of the critique Gill-IEEE-2  that in
its text the word “quaternion” appears only once, and that only in a
dismissive sentence: ”But Christian’s speculations about a quaternionic space
seem completely irrelevant.” And the word “3-sphere” also appears only once,
and that only within a sentence quoted from my simulation code rpubs . The
critique thus entirely avoids engaging with the details of the quaternionic
3-sphere model, overlooking many of its vital features and their broader
physical significance.

The minimum prerequisite for understanding the model of quantum correlations
presented in IEEE-1  and IEEE-2  (as well as in Disproof ; IJTP ; RSOS ) is at
least some appreciation that it is based on a quaternionic 3-sphere, or
$S^{3}$, hypothesized to be the physical space in which we are inescapably
confined to perform our Bell-test experiments. The origins and strengths of
quantum correlations are then inevitable consequences of the Clifford algebra
and geometry of this physical space, as we will see in the next Section for
the singlet state. By contrast, the flatland or ${\mathrm{I\\!R}^{3}}$
perspective and the use of vector algebra adhered to in the critiques Gill-
IEEE-1 ; Gill-IEEE-2  leads one to the traditional interpretation of Bell’s
theorem, as I have demonstrated in Section X of IEEE-1 .

##  II Résumé of the quaternionic 3-sphere model

Within the quaternionic 3-sphere model presented in IEEE-1 ; IEEE-2 , the
important question to be answered is this: What will be the value of the joint
result ${{\mathscr{A}}{\mathscr{B}}({\mathbf{a}},{\mathbf{b}},\lambda)=\pm 1}$
when the individual results ${\mathscr{A}}({\mathbf{a}},\lambda)=\pm 1$ and
${\mathscr{B}}({\mathbf{b}},\lambda)=\pm 1$ are observed by Alice and Bob
separately but simultaneously, in “coincidence counts”, within a space-like
hypersurface $S^{3}$ within spacetime? Here ${\mathbf{a}}$ and ${\mathbf{b}}$
are the detector directions, chosen by Alice and Bob, and $\lambda$ is a
common cause, so that

${\mathscr{A}}{\mathscr{B}}({\mathbf{a}},{\mathbf{b}},\lambda)={\mathscr{A}}({\mathbf{a}},\lambda)\,{\mathscr{B}}({\mathbf{b}},\lambda),$
(2)

which is the factorizability condition for the results required by local
causality. To answer the above question, recall that there are three different
sets of experiments involved in any Bell-test experiment IEEE-3 . In the first
two sets of experiments Alice and Bob make spin measurements at their
respective 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 independent sets of experiments in which Alice performs her
experiments regardless of Bob’s existence, and vice versa. Their results,
${\mathscr{A}}=\pm 1$, $\langle{\mathscr{A}}\rangle=0$ and ${\mathscr{B}}=\pm
1$, $\langle{\mathscr{B}}\rangle=0$, are what the measurement functions (66)
and (67) defined in IEEE-1  predict. They describe the detection processes of
Alice and Bob as limiting scalar points of two separate quaternions within
$S^{3}$:

$\displaystyle S^{3}\ni{\mathscr{A}}$
$\displaystyle({\mathbf{a}},{\lambda})\,=\lim_{{\mathbf{s}}_{1}\,\rightarrow\,{\mathbf{a}}}\left\\{-\,{\mathbf{D}}({\mathbf{a}})\,{\mathbf{L}}({\mathbf{s}}_{1},\,\lambda)\right\\}$
(3)
$\displaystyle=\lim_{{\mathbf{s}}_{1}\,\rightarrow\,{\mathbf{a}}}\left\\{\,+\,{\mathbf{q}}(\eta_{{\mathbf{a}}{\mathbf{s}}_{1}},\,{\mathbf{r}}_{1})\right\\}$
(4)
$\displaystyle=+\lambda\lim_{{\mathbf{s}}_{1}\,\rightarrow\,{\mathbf{a}}}\left\\{\cos(\eta_{{\mathbf{a}}{\mathbf{s}}_{1}})+(I\cdot{\mathbf{r}}_{1})\sin(\eta_{{\mathbf{a}}{\mathbf{s}}_{1}})\right\\}$
(5)

implying that, as the angle
$\eta_{{\mathbf{a}}{\mathbf{s}}_{1}}\rightarrow\,0$,

${\mathscr{A}}({\mathbf{a}},{\lambda})\rightarrow\pm
1\;\;\;\text{with}\;\;\;\langle\,{\mathscr{A}}({\mathbf{a}},{\lambda})\rangle=0,$
(6)

and

$\displaystyle S^{3}\ni{\mathscr{B}}$
$\displaystyle({\mathbf{b}},\,{\lambda})\,=\lim_{{\mathbf{s}}_{2}\,\rightarrow\,{\mathbf{b}}}\left\\{+\,{\mathbf{L}}({\mathbf{s}}_{2},\,\lambda)\,{\mathbf{D}}({\mathbf{b}})\right\\}$
(7)
$\displaystyle=\lim_{{\mathbf{s}}_{2}\,\rightarrow\,{\mathbf{b}}}\left\\{\,-\,{\mathbf{q}}(\eta_{{\mathbf{s}}_{2}{\mathbf{b}}},\,{\mathbf{r}}_{2})\right\\}$
(8)
$\displaystyle=-\lambda\lim_{{\mathbf{s}}_{2}\,\rightarrow\,{\mathbf{b}}}\left\\{\cos(\eta_{{\mathbf{s}}_{2}{\mathbf{b}}})+(I\cdot{\mathbf{r}}_{2})\sin(\eta_{{\mathbf{s}}_{2}{\mathbf{b}}})\right\\}$
(9)

implying that, as the angle
$\eta_{{\mathbf{s}}_{2}{\mathbf{b}}}\rightarrow\,0$,

${\mathscr{B}}({\mathbf{b}},{\lambda})\longrightarrow\mp
1\;\;\;\text{with}\;\;\;\langle\,{\mathscr{B}}({\mathbf{b}},{\lambda})\rangle=0,$
(10)

where ${\mathbf{L}}({\mathbf{s}}_{1},\,\lambda)$ and
${\mathbf{L}}({\mathbf{s}}_{2},\,\lambda)$ are spin bivectors and
${\mathbf{D}}({\mathbf{a}})$ and ${\mathbf{D}}({\mathbf{b}})$ are detector
bivectors of Alice and Bob, whose orientations are defined relative to those
of the spin bivectors,

${\mathbf{L}}({\mathbf{n}},\,\lambda)\,=\,\lambda\,{\mathbf{D}}({\mathbf{n}})\,\,\Longleftrightarrow\,\,{\mathbf{D}}({\mathbf{n}})\,=\,\lambda\,{\mathbf{L}}({\mathbf{n}},\,\lambda)\,,$
(11)

$\eta_{{\mathbf{a}}{\mathbf{s}}_{1}}$ and
$\eta_{{\mathbf{s}}_{2}{\mathbf{b}}}$ are angles between the detectors and
spins, and

${\mathbf{r}}_{1}=\frac{{\mathbf{a}}\times{\mathbf{s}}_{1}}{||{\mathbf{a}}\times{\mathbf{s}}_{1}||}\;\;\;\text{and}\;\;\;{\mathbf{r}}_{2}=\frac{{\mathbf{s}}_{2}\times{\mathbf{b}}}{||{\mathbf{s}}_{2}\times{\mathbf{b}}||}.$
(12)

On the other hand, the above equations (6) and (10) (or, equivalently,
equations (66) and (67) of IEEE-1 ) do not tell us anything about the third
set of experiments carried out by Alice and Bob in which they jointly and
simultaneously make measurements within the quaternionic 3-sphere, taken as
the physical space. This is because the geometrical properties of the
quaternionic 3-sphere are highly nontrivial and they must be respected to
respect the hypothesis of the model. Unlike the $\mathrm{I\\!R}^{3}$ model of
the physical space, $S^{3}$ is a compact space that remains closed under
multiplications of the quaternions that constitute it. It is defined as a set
of all unit quaternions,

$S^{3}:=\left\\{\,{\mathbf{q}}(\psi,\,{\mathbf{r}}):=\exp\left[\,{\mathbf{J}}({\mathbf{r}})\,\frac{\psi}{2}\,\right]\Bigg{|}\;||\,{\mathbf{q}}(\psi,\,{\mathbf{r}})\,||^{2}=1\right\\}\\!,$
(13)

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 conventions of Geometric
Algebra are used Clifford .

The correct values of the joint result
${{\mathscr{A}}{\mathscr{B}}({\mathbf{a}},{\mathbf{b}},\lambda)}$ can thus be
inferred only by respecting the geometrical properties of $S^{3}$. They can be
preserved only by following the correct rules of quaternionic multiplication.
But since the above concepts and the next steps they lead to are not well
understood in the critiques Gill-IEEE-1 ; Gill-IEEE-2 ; Gill-Entropy , let me
explain them here step by step.

The crucial question is: What will be the value of the joint result
${{\mathscr{A}}{\mathscr{B}}({\mathbf{a}},{\mathbf{b}},\lambda)}$ within
$S^{3}$? Can it simply be the product of the values seen in (6) and (10)? No,
that cannot be a meaningful value within $S^{3}$. The values seen in (6) and
(10) are scalar numbers, not quaternions. Scalar numbers do not respect the
multiplication rules of quaternions. But, by definition, $S^{3}$ in (12),
taken as the physical space, is the set of all unit quaternions. Therefore the
value of the joint result
${{\mathscr{A}}{\mathscr{B}}({\mathbf{a}},{\mathbf{b}},\lambda)}$ must be a
limiting scalar point of a quaternion within $S^{3}$ as depicted in Fig. 1,
just as the individual results ${{\mathscr{A}}({\mathbf{a}},\lambda)}$ and
${{\mathscr{B}}({\mathbf{b}},\lambda)}$ are the limiting scalar points of two
quaternions, as specified in the equations (4) and (8) above.

Figure 1: The results ${\mathscr{A}}$ and ${\mathscr{B}}$ are scalar 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 of
$S^{3}$. After IEEE-3 .

The next question is: Which quaternion within $S^{3}$ would give the correct
value of the joint result
${{\mathscr{A}}{\mathscr{B}}({\mathbf{a}},{\mathbf{b}},\lambda)=\pm 1}$ as its
limiting scalar point while the results
${{\mathscr{A}}({\mathbf{a}},\lambda)}$ and
${{\mathscr{B}}({\mathbf{b}},\lambda)}$ are observed by Alice and Bob
simultaneously in coincidence counts? The answer to this question can be found
from the condition (2) and the definitions (4) and (8) of the individual
results ${{\mathscr{A}}({\mathbf{a}},\lambda)}$ and
${{\mathscr{B}}({\mathbf{b}},\lambda)}$ within $S^{3}$, giving

$\displaystyle S^{3}\ni\;$
$\displaystyle{\mathscr{A}}{\mathscr{B}}({\mathbf{a}},\,{\mathbf{b}},\,{\lambda})={\mathscr{A}}({\mathbf{a}},\,{\lambda})\,{\mathscr{B}}({\mathbf{b}},\,{\lambda})$
(14)
$\displaystyle\\!=\\!\left[\lim_{{\mathbf{s}}_{1}\,\rightarrow\,{\mathbf{a}}}\left\\{\,+\,{\mathbf{q}}(\eta_{{\mathbf{a}}{\mathbf{s}}_{1}},\,{\mathbf{r}}_{1})\right\\}\right]\\!\left[\lim_{{\mathbf{s}}_{2}\,\rightarrow\,{\mathbf{b}}}\left\\{\,-\,{\mathbf{q}}(\eta_{{\mathbf{s}}_{2}{\mathbf{b}}},\,{\mathbf{r}}_{2})\right\\}\right]\\!.$
(15)

The question now is: How must we evaluate the last product without
compromising the properties of $S^{3}$ that requires us to respect the
quaternionic rules of multiplication when the results
${{\mathscr{A}}({\mathbf{a}},\lambda)}$ and
${{\mathscr{B}}({\mathbf{b}},\lambda)}$ are observed simultaneously? If the
conservation of zero spin angular momentum is satisfied, as it has to be,
which requires us to set ${\mathbf{s}}_{1}={\mathbf{s}}_{2}$ (cf. Fig. 2),
then the correct answer to the above question is: By using the “product of
limits equal to the limits of product” rule, giving

Figure 2: A spin-less neutral pion decays into an electron-positron pair.
Measurements of spin components on each separated fermion are performed by
Alice and Bob at remote stations ${\mathbf{1}}$ and ${\mathbf{2}}$, providing
binary outcomes along freely chosen directions ${\mathbf{a}}$ and
${\mathbf{b}}$. The conservation of spin momentum dictates that the net spin
of the pair remains zero during the free evolution. After IEEE-1 .

$\displaystyle S^{3}\ni\;$
$\displaystyle{\mathscr{A}}({\mathbf{a}},\,{\lambda})\,{\mathscr{B}}({\mathbf{b}},\,{\lambda})$
$\displaystyle\\!=\left[\lim_{{\mathbf{s}}_{1}\,\rightarrow\,{\mathbf{a}}}\left\\{\,+\,{\mathbf{q}}(\eta_{{\mathbf{a}}{\mathbf{s}}_{1}},\,{\mathbf{r}}_{1})\right\\}\right]\\!\left[\lim_{{\mathbf{s}}_{2}\,\rightarrow\,{\mathbf{b}}}\left\\{\,-\,{\mathbf{q}}(\eta_{{\mathbf{s}}_{2}{\mathbf{b}}},\,{\mathbf{r}}_{2})\right\\}\right]$
(16)
$\displaystyle\\!=\lim_{\begin{subarray}{c}{\mathbf{s}}_{1}\,\rightarrow\,{\mathbf{a}}\\\
{\mathbf{s}}_{2}\,\rightarrow\,{\mathbf{b}}\end{subarray}}\left\\{\,-\,{\mathbf{q}}(\eta_{{\mathbf{a}}{\mathbf{s}}_{1}},\,{\mathbf{r}}_{1})\,{\mathbf{q}}(\eta_{{\mathbf{s}}_{2}{\mathbf{b}}},\,{\mathbf{r}}_{2})\right\\}$
(17)
$\displaystyle\\!=\lim_{\begin{subarray}{c}{\mathbf{s}}_{1}\,\rightarrow\,{\mathbf{a}}\\\
{\mathbf{s}}_{2}\,\rightarrow\,{\mathbf{b}}\end{subarray}}\left\\{\,-\,{\mathbf{q}}(\eta_{{\mathbf{u}}{\mathbf{v}}},\,{\mathbf{r}}_{0})\right\\},$
(18)

where
${\mathbf{q}}(\eta_{{\mathbf{u}}{\mathbf{v}}},\,{\mathbf{r}}_{0})={\mathbf{q}}(\eta_{{\mathbf{a}}{\mathbf{s}}_{1}},\,{\mathbf{r}}_{1})\,{\mathbf{q}}(\eta_{{\mathbf{s}}_{2}{\mathbf{b}}},\,{\mathbf{r}}_{2})$
is necessarily another unit quaternion within $S^{3}$ because the latter
remains closed under multiplication, with half of its rotation angle

$\displaystyle\eta_{{\mathbf{u}}{\mathbf{v}}}=\cos^{-1}$
$\displaystyle\big{\\{}({\mathbf{a}}\cdot{\mathbf{s}}_{1})({\mathbf{s}}_{2}\cdot{\mathbf{b}})$
$\displaystyle\;\;\;-({\mathbf{a}}\cdot{\mathbf{s}}_{2})({\mathbf{s}}_{1}\cdot{\mathbf{b}})+({\mathbf{a}}\cdot{\mathbf{b}})({\mathbf{s}}_{1}\cdot{\mathbf{s}}_{2})\big{\\}}$
(19)

and the axis of its rotation

$\displaystyle{\mathbf{r}}_{0}$
$\displaystyle=\frac{({\mathbf{a}}\cdot{\mathbf{s}}_{1})({\mathbf{s}}_{2}\\!\times\\!{\mathbf{b}})\\!+\\!({\mathbf{s}}_{2}\cdot{\mathbf{b}})({\mathbf{a}}\\!\times\\!{\mathbf{s}}_{1})\\!-\\!({\mathbf{a}}\\!\times\\!{\mathbf{s}}_{1})\\!\times\\!({\mathbf{s}}_{2}\\!\times\\!{\mathbf{b}})}{\sin\left(\,\eta_{{\mathbf{u}}{\mathbf{v}}}\right)}.$
(20)

Note, however, that the above results do not depend on using the “product of
limits equal to the limits of product” rule, as long as the conservation of
angular momentum is respected.

Now the conservation of spin angular momentum requires
${\mathbf{s}}_{1}={\mathbf{s}}_{2}$ (cf. Fig. 2). And from (19) we note that
for ${\mathbf{s}}_{1}={\mathbf{s}}_{2}$ the half angle
$\eta_{{\mathbf{u}}{\mathbf{v}}}$ reduces to
$\eta_{{\mathbf{a}}{\mathbf{b}}}$, reducing
${\mathbf{q}}(\eta_{{\mathbf{u}}{\mathbf{v}}},\,{\mathbf{r}}_{0})$ to

${\mathbf{q}}(\eta_{{\mathbf{a}}{\mathbf{b}}},\,{\mathbf{r}}_{0})=\cos(\eta_{{\mathbf{a}}{\mathbf{b}}})+(I\cdot{\mathbf{r}}_{0})\,\sin(\eta_{{\mathbf{a}}{\mathbf{b}}}).$
(21)

On the other hand, regardless of whether ${\mathbf{s}}_{1}={\mathbf{s}}_{2}$
or ${\mathbf{s}}_{1}\not={\mathbf{s}}_{2}$,

$\lim_{\begin{subarray}{c}{\mathbf{s}}_{1}\,\rightarrow\,{\mathbf{a}}\\\
{\mathbf{s}}_{2}\,\rightarrow\,{\mathbf{b}}\end{subarray}}{\mathbf{r}}_{0}({\mathbf{s}}_{1},\,{\mathbf{s}}_{2})=\vec{\,\mathbf{0}}$
(22)

during the detection process of the joint result
${\mathscr{A}}{\mathscr{B}}({\mathbf{a}},\,{\mathbf{b}},\,{\lambda})$. The
value ${\mathscr{A}}{\mathscr{B}}$ of the product
${\mathscr{A}}({\mathbf{a}},\lambda)\,{\mathscr{B}}({\mathbf{b}},\lambda)$
thus tends to

$\displaystyle{\mathscr{A}}$
$\displaystyle({\mathbf{a}},\lambda)\,{\mathscr{B}}({\mathbf{b}},\lambda)={\mathscr{A}}{\mathscr{B}}({\mathbf{a}},\,{\mathbf{b}},\,{\lambda})$
$\displaystyle=\\!\lim_{\begin{subarray}{c}{\mathbf{s}}_{1}\,\rightarrow\,{\mathbf{a}}\\\
{\mathbf{s}}_{2}\,\rightarrow\,{\mathbf{b}}\end{subarray}}\left\\{\,-\,{\mathbf{q}}(\eta_{{\mathbf{a}}{\mathbf{b}}},\,{\mathbf{r}}_{0})\right\\}\rightarrow\begin{cases}\\!\\!-1&\\!\\!\text{if}\;\,{\mathbf{s}}_{1}\not={\mathbf{s}}_{2}\\\
\\!\\!-\,{\mathbf{a}}\cdot{\mathbf{b}}&\\!\\!\text{if}\;\,{\mathbf{s}}_{1}={\mathbf{s}}_{2}.\end{cases}$
(23)

Given the equality ${\mathbf{s}}_{1}={\mathbf{s}}_{2}$ required by the
conservation of zero spin angular momentum (cf. Fig. 2), this tendency (23)
for the value ${\mathscr{A}}{\mathscr{B}}$ holds for each run of the
experiment. As a result, the correlation between the results
${{\mathscr{A}}({\mathbf{a}},\,{\lambda})}$ and
${{\mathscr{B}}({\mathbf{b}},\,{\lambda})}$ observed by Alice and Bob within
$S^{3}$ works out as follows. Since $\lambda$ is a fair coin, we can assume
uniform averaging over $\lambda$ with probability density
$p(\lambda)=\frac{1}{n}$, where $n$ is the total number of experimental
trials. This allows us to reduce the expectation function ${\cal
E}({\mathbf{a}},\,{\mathbf{b}})$ to a discrete version:

$\displaystyle{\cal E}_{\mathrm{L.R.}}({\mathbf{a}},\,{\mathbf{b}})$
$\displaystyle=\\!\int{\mathscr{A}}({\mathbf{a}},\,\lambda)\,{\mathscr{B}}({\mathbf{b}},\,\lambda)\;p(\lambda)\,d\lambda$
(24)
$\displaystyle\approx\\!\lim_{\,n\,\gg\,1}\left[\frac{1}{n}\sum_{k\,=\,1}^{n}\,{\mathscr{A}}({\mathbf{a}},\,{\lambda}^{k})\;{\mathscr{B}}({\mathbf{b}},\,{\lambda}^{k})\right]$
(25)
$\displaystyle=\lim_{\,n\,\gg\,1}\left[\frac{1}{n}\sum_{k\,=\,1}^{n}\,\lim_{\begin{subarray}{c}{\mathbf{s}}_{1}\,\rightarrow\,{\mathbf{a}}\\\
{\mathbf{s}}_{2}\,\rightarrow\,{\mathbf{b}}\end{subarray}}\left\\{\,-\,{\mathbf{q}}(\eta_{{\mathbf{a}}{\mathbf{b}}},\,{\mathbf{r}}_{0})\right\\}\right]$
(26) $\displaystyle=\,-\cos(\,\eta_{{\mathbf{a}}{\mathbf{b}}}).$ (27)

This corroborates the hypothesis put forward in Disproof ; IJTP ; RSOS ;
IEEE-1 ; IEEE-2  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. I have derived the correlation (27) in Disproof ; IJTP ; RSOS ;
IEEE-1 ; IEEE-2  in several different ways, proving the Theorem 1 stated above
in the Introduction. See also its derivation in equations (34) to (41e) of
Section III in my previous reply IEEE-3 .

It is evident from the above derivation that the quaternionic 3-sphere model
is not a model of after-the-events procedure followed by experimenters within
${\mathrm{I\\!R}^{3}}$ as implicitly assumed in the critique, but a
theoretical model that predicts the values of the simultaneous events
${\mathscr{A}}$, ${\mathscr{B}}$, and ${{\mathscr{A}}{\mathscr{B}}}$ occurring
within $S^{3}$, dictated by its geometrical and topological properties. Thus
the critique Gill-IEEE-2  confuses epistemology with ontology, committing what
philosophers call “a category error.”

##  III Point by point response to the critique

With the above summary of the 3-sphere model, let us now proceed to address
the issues raised in the critique Gill-IEEE-2 . While there are many incorrect
and unproven claims throughout the critique, I focus in this Section on those
that are significant.

###  III.1 Concerning the status of Bell’s theorem

In its Introduction, the critique in Gill-IEEE-2  begins with a serious
misconception that “Bell’s Theorem” is a proven theorem in the mathematical
sense and therefore any critique of it must be flawed. But while even proven
mathematical theorems may not be immune to refutations by counterexamples as
so lucidly explained by Lakatos Lakatos , Bell’s theorem is not a theorem in
the mathematical sense to begin with, as I have pointed out in Answer 1 of
Appendix B in IEEE-1 . It is a physical argument, based on a number of
implicit and explicit physical assumptions, which can be and have been
questioned before, not only by me IEEE-1 ; IEEE-2 ; Disproof ; IJTP ; RSOS
but also by many others (cf. footnote 1 in IEEE-1 ). If it were a proven
theorem in the mathematical sense, then it would not require physical
experiments for its validity and any loophole (or ”gap”) would render it
invalid.

Moreover, as I have discussed in detail in Section II of IEEE-3 , the
mathematical part of Bell’s theorem is nothing more than a reworking of an
inequality in probability theory proven by Boole one hundred and eleven years
before the publication in 1964 of Bell’s physical argument based on it Boole-1
; Boole-2 ; Bell-1964 . The critique in Gill-IEEE-2  seems to be aware of the
historical precedence of Boole’s mathematical inequality, but fails to
appreciate the distinction between a purely mathematical inequality and the
radical metaphysical claims regarding the nature of physical reality made by
Bell based on that mathematical inequality. This is unfortunate, because it is
quite well known that Boole’s inequality can be derived without assuming
locality, as, for example, I have derived in Section 4.2 of RSOS , or without
assuming realism, as, for example, Eberhard has derived in Eberhard , or
without assuming either locality or realism, as, for example, Boole has
derived in Boole-1 ; Boole-2 . By now it is well known that, when viewed as a
mathematical theorem about a probability distribution and its marginals, the
theorem part of Bell’s argument (i.e., Boole’s inequality) can be derived
without any reference to local realism at all.

Bell’s argument, on the other hand, starts off by assuming local realism. It
then proceeds to re-derive Boole’s inequality without referring to Boole.
Since this inequality is not consistent with quantum mechanical probabilities,
Bell argues that quantum mechanics is inconsistent with local realism. But
given the different derivations of Boole’s inequality in Section 4.2 of RSOS ,
Eberhard , and Boole-1 ; Boole-2  mentioned in the previous paragraph, we are
equally justified in concluding that quantum mechanics is inconsistent with
non-local realism, or local non-realism, or non-local non-realism,
respectively. Indeed, one can start off by explicitly assuming non-locality
and non-realism and still derive Boole’s inequality, requiring only the
additivity of expectation values Oversight  and that results of incompatible
experiments along mutually exclusive measurement directions can occur in
nature simultaneously RSOS .

In the light of the above results, the orthodox interpretation of Bell’s
argument adhered to in the critique Gill-IEEE-2  is obsolete.

###  III.2 Concerning my critique of Bell’s theorem

It turns out, however, that even the orthodox interpretation of Bell’s
argument harbors a serious physical mistake Oversight . In Section II of the
critique Gill-IEEE-2 , the standard derivation of the Bell-CHSH inequality is
reviewed. This derivation is well known for at least fifty years. The critique
thus adds nothing new to the existing literature. Towards the end of that
section the critique then quotes the following from my paper IEEE-1 :

> As innocuous as the step [that replaces the four separate averages to a
> single average] in the proof [of Bell-CHSH] may seem mathematically, it is,
> in fact, an illegitimate step physically, because what is being averaged on
> its right-hand are unobservable and unphysical quantities. Indeed, the pairs
> of measurement directions ${({\mathbf{a}},\,{\mathbf{b}})}$,
> ${({\mathbf{a}},\,{\mathbf{b^{\prime}}})}$,
> ${({\mathbf{a^{\prime}}},\,{\mathbf{b}})}$, and
> ${({\mathbf{a^{\prime}}},\,{\mathbf{b^{\prime}}})}$ are mutually exclusive
> measurement directions, corresponding to incompatible experiments which
> cannot be performed simultaneously.

The “step” here concerns the assumption of the additivity of expectation
values. Namely, the replacement of the sum of four separate averages with the
single average of their sum:

$\displaystyle\Bigl{\langle}\,{\mathscr{A}}_{k_{1}}({\mathbf{a}})\,{\mathscr{B}}_{k_{1}}({\mathbf{b}})\,\Bigr{\rangle}+\Bigl{\langle}\,{\mathscr{A}}_{k_{2}}({\mathbf{a}})\,{\mathscr{B}}_{k_{2}}({\mathbf{b^{\prime}}})\,\Bigr{\rangle}$
$\displaystyle\;\;\,\;\;\;\;\;\;+\Bigl{\langle}\,{\mathscr{A}}_{k_{3}}({\mathbf{a^{\prime}}})\,{\mathscr{B}}_{k_{3}}({\mathbf{b}})\,\Bigr{\rangle}-\Bigl{\langle}\,{\mathscr{A}}_{k_{4}}({\mathbf{a^{\prime}}})\,{\mathscr{B}}_{k_{4}}({\mathbf{b^{\prime}}})\,\Bigr{\rangle}$
(28)
$\displaystyle\longrightarrow\Bigl{\langle}{\mathscr{A}}_{k}({\mathbf{a}})\Big{\\{}{\mathscr{B}}_{k}({\mathbf{b}})+{\mathscr{B}}_{k}({\mathbf{b^{\prime}}})\Big{\\}}$
$\displaystyle\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;+{\mathscr{A}}_{k}({\mathbf{a^{\prime}}})\Big{\\{}{\mathscr{B}}_{k}({\mathbf{b}})-{\mathscr{B}}_{k}({\mathbf{b^{\prime}}})\Big{\\}}\Bigr{\rangle}.$
(29)

In Section II of critique Gill-IEEE-2 , it is the following specious step,

> … let us take a look at the following expression
> $Z:=X_{1}Y_{1}-X_{1}Y_{2}-X_{2}Y_{1}-X_{2}Y_{2}$. One can rewrite it as
> $X_{1}(Y_{1}-Y_{2})-X_{2}(Y_{1}+Y_{2})$. … ,

that hides the assumption of the additivity of expectation values. Without
this assumption Bell-CHSH inequalities cannot be derived Oversight . I have
discussed this briefly in Section II of IEEE-3 , and extensively in Oversight
. While mathematically correct, this assumption is physically meaningless. It
amounts to assuming the bound of $2$ on the CHSH correlator instead of
deriving it Oversight . Contrary to the claim made in Gill-IEEE-2 , it is an
assumption over and above the assumptions of locality and realism. In fact, as
noted, it is a profound physical mistake.

It is the same mistake that von Neumann’s former theorem against general
hidden variable theories harbored, as I have explained in Oversight . For
observables that are not simultaneously measurable, such as the observables
involved in the fermionic spin measurements in Bell-test experiments, the
replacement of the sum of expectation values with the expectation value of the
sum, although respected within quantum mechanics, does not hold within hidden
variable theories. This was pointed out by Einstein and Grete Hermann in the
1930s within the context of von Neumann’s theorem, and again some thirty years
later by Bell and others, as I have explained in Oversight .

The problem is that, while the sum of expectation values is mathematically the
same as the expectation value of the sum, as in the assumption that allows us
to mathematically replace (28) with (29) above, and while this assumption is
valid in quantum mechanics because any linear sum of operators represents
another operator in Hilbert space, it is not valid for any hidden variable
theory based on dispersion-free states, because the eigenvalue of a sum of
operators is not the sum of individual eigenvalues (which is what
${\mathscr{A}}_{k}(\mathbf{a})$, ${\mathscr{A}}_{k}(\mathbf{a^{\prime}})$,
${\mathscr{B}}_{k}(\mathbf{b})$, and ${\mathscr{B}}_{k}(\mathbf{b^{\prime}})$
in fact are) when the constituent operators are non-commuting, as in the case
of Bell-test experiments. This makes the replacement of (28) with (29)
physically invalid. But without this replacement the absolute upper bound of 2
on the CHSH correlator cannot be derived.

Once this extraordinary oversight is removed from Bell’s argument and local
realism is implemented correctly Oversight , the bounds on the CHSH correlator
work out to be $\pm 2\sqrt{2}$ instead of $\pm 2$, thereby mitigating the
radical conclusions of Bell’s theorem Oversight . Consequently, what is ruled
out by the Bell-test experiments is not local realism as widely believed, but
the assumption of the additivity of expectation values.

Considering the above mistake, the claim in the Introduction of the critique
Gill-IEEE-2  that “Bell’s theorem has not been disproved…” is rather ironic.
For Bell’s theorem has never been proven in the first place. It has only been
assumed.

###  III.3 Concerning the Bell inequality violations

Much is made in the critique of the so-called new generation of loophole-free
Bell-test experiments and how they exhibit violations of the Bell-CHSH
inequality. By CHSH inequality the critique essentially means the inequality
discovered by Boole one hundred and eleven years before Bell’s famous paper of
1964 Boole-1 ; Boole-2 . It is a mathematical inequality involving four binary
numbers: ${\mathscr{A}}_{k}(\mathbf{a})=\pm 1$,
${\mathscr{A}}_{k}(\mathbf{a^{\prime}})=\pm 1$,
${\mathscr{B}}_{k}(\mathbf{b})=\pm 1$, and
${\mathscr{B}}_{k}(\mathbf{b^{\prime}})=\pm 1$. As such, the absolute value of
the expression (29) is mathematically bounded by 2. It is impossible to
violate the bound of 2 on (29) with binary scalar numbers $\pm 1$ that
represent the results observed in the Bell-test experiments. And yet,
experimenters are routinely reporting violations of the absolute bound of 2 on
(29) set by application of elementary arithmetic. How is that possible?

It is certainly possible to exceed the bound of 2 on (29) if we can switch to
four separate experiments corresponding to expression (28), the absolute value
of which is bounded by 4, not 2. After all, four separate experiments
corresponding to expression (28) is all we can hope to achieve in practice. So
that is what the experimenters do. In other words, there is extraordinary
bait-and-switch happening (albeit unwittingly) in every experiment that claims
to have violated the absolute bound of 2 on expression (29), rationalized with
statistics.

###  III.4 Concerning a denial of Bell’s assumption

In their Introductions, the critiques Gill-IEEE-1  and Gill-IEEE-2  claim 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.

This claim, however, is not correct. I have already explained why it is not
correct in Answer 2 of Appendix B of IEEE-1  and in Subsection IV B of IEEE-3
. And yet, the claim has been repeated in Gill-IEEE-2  again. But nowhere in
his writings has Bell stated that his theorem holds independently of the
geometry of physical space. On the contrary, in Section 8 of Chapter 7 of his
book Speakable , while exploring possible strategies that can negate his
theorem, he writes: “The space time structure has been taken as given here.
How then about gravitation?” Thus Bell seems to have anticipated the use of a
solution of Einstein’s field equations of general relativity to overcome his
theorem.

Moreover, the derivations of the Bell-CHSH inequalities explicitly require
different detector settings corresponding to different directions in the
physical space, which are then represented – both theoretically and in
configuring Bell-test experiments – by ordinary vectors in
${\mathrm{I\\!R}}^{3}$. It is unfortunate that this implicit assumption is
usually not made explicit in the literature on Bell’s theorem. But why must we
assume such vectors to be within ${\mathrm{I\\!R}^{3}}$? If, instead, they are
embedded within quaternions constituting $S^{3}$ that model the physical
space, then the correlations between measurement events are inevitably
sinusoidal, as I have proved many times in IEEE-1 ; IEEE-2 ; Disproof ; IJTP ;
RSOS , in many different ways. Thus to claim that Bell’s argument “does not
take account of the geometry of space” is to reveal a weakness of Bell’s
argument, not its strength.

The respective topologies of the three-dimensional spaces
${\mathrm{I\\!R}}^{3}$ and $S^{3}$ are dramatically different, even though the
latter differs from the former by a single point added to it at infinity:

$S^{3}=\,{\mathrm{I\\!R}}^{3}\cup\left\\{\infty\right\\}.$ (30)

While ${\mathrm{I\\!R}}^{3}$ is an open space stretching out from the origin
to infinity, $S^{3}$ is both a closed and compact manifold. Unlike
${\mathrm{I\\!R}}^{3}$, the space $S^{3}$ has many special properties. For
example, although locally it is a product space, $S^{3}\approx S^{2}\times
S^{1}$, globally the fiber bundle of $S^{3}$ has no cross-section at all.
Moreover, it is both a connected and simply connected manifold without
boundary, so that any loop or circular path in it can be shrunk continuously
to a point without leaving the manifold Bloch .

Now measurement results, such as the clicks of the detectors, are events in
spacetime. In the EPR-Bohm or Bell-test experiments one is interested in
spacelike separated coincident events in spacetime. In other words, one is
interested in the spacelike separated equal-time points on a spacelike
hypersurface in spacetime, such as those in ${\mathrm{I\\!R}}^{3}$ or $S^{3}$.
And as I have explained in IEEE-1 ; IEEE-2 ; IEEE-3 , both
${\mathrm{I\\!R}}^{3}$ and $S^{3}$ are admissible spatial parts of a well
known solution of Einstein’s field equations of general relativity. There is
thus no escape from the geometries of ${\mathrm{I\\!R}}^{3}$ and $S^{3}$. But
since the topological properties of ${\mathrm{I\\!R}}^{3}$ and $S^{3}$ are so
dramatically different, the correlations between their respective points would
also be dramatically different. And yet, the critique claims that Bell’s
argument is insensitive to this difference. If so, then that is clearly yet
another of several defects in Bell’s argument.

###  III.5 Concerning Equations (66) and (67) in IEEE-1

Another incorrect claim in the critique Gill-IEEE-2  concerns what the
measurement functions (66) and (67) defined in IEEE-1  predict:

> Christian’s explicit definition of measurement functions results in
> measurement outcomes which are equal and opposite with probability one,
> whatever the measurement settings. [Appendix of Gill-IEEE-2 .]

In its Section IV, the critique again claims that, for whatever detector
settings ${\mathbf{a}}$ and ${\mathbf{b}}$ are chosen by Alice and Bob, the
measurement functions (66) and (67) defined in IEEE-1  predict that the
observed results ${\mathscr{A}}({\mathbf{a}},\,\lambda)$ and
${\mathscr{B}}({\mathbf{b}},\,\lambda)$ will satisfy

${\mathscr{A}}=-{\mathscr{B}}=\pm 1=\lambda,$ (31)

with $\lambda=\pm 1$ being a fair coin. But no such equation exists in IEEE-1
, or in any of my other publications IEEE-2 ; Disproof ; IJTP ; RSOS . In
fact, as I explained in Section II above, within the quaternionic 3-sphere
model the above equation is quite meaningless. Moreover, it is very easy to
verify that the measurement functions (66) and (67) defined in IEEE-1  do not
predict perfect anti-correlation ${\mathscr{A}}{\mathscr{B}}=-1$ for
${\mathbf{a}}\not={\mathbf{b}}$ without violating the conservation of zero
spin angular momentum. I have demonstrated this in considerable detail towards
the end of Section VIII in the original paper IEEE-1 . Thus the claim made in
the critique is manifestly wrong. The above equation (31) is simply made up in
the critique Gill-IEEE-2 . As I have explained in Subsection IV E of IEEE-3
and Section III of IEEE-2 , the perfect anti-correlation
${\mathscr{A}}{\mathscr{B}}=-1$ between the measurement results can hold
within $S^{3}$ only for the special case when ${\mathbf{a}}={\mathbf{b}}$. In
general, for ${\mathbf{a}}\not={\mathbf{b}}$, the product
${\mathscr{A}}{\mathscr{B}}$ of the observed results ${\mathscr{A}}$ and
${\mathscr{B}}$ would necessarily fluctuate between $-1$ and $+1$,

${\mathscr{A}}{\mathscr{B}}\in\\{-1,\,+1\\},$ (32)

because of the sign changes in quaternions that constitute $S^{3}$:

${\mathbf{q}}(\eta_{{\mathbf{a}}{\mathbf{b}}}+\kappa\pi,\,{\mathbf{r}})\,=\,(-1)^{\kappa}\,{\mathbf{q}}(\eta_{{\mathbf{a}}{\mathbf{b}}},\,{\mathbf{r}})\,\;\;\text{for}\;\,\kappa=0,1,2,3,\dots$
(33)

In other words, all four possible combinations of the results, ${+\,+}$,
${+\,-}$, ${-\,+}$, and ${-\,-}$, would be seen by Alice and Bob.

The violation of the conservation of spin angular momentum in the critique’s
claim is not difficult to detect. For ${{\mathbf{a}}\not={\mathbf{b}}}$, Eq.
(31) can hold only for ${\mathbf{s}}_{1}\not={\mathbf{s}}_{2}$, where
${\mathbf{s}}_{1}$ and ${\mathbf{s}}_{2}$ are unit vectors about which the
spin bivectors ${\mathbf{L}}({\mathbf{s}}_{1},\,\lambda)$ and
${\mathbf{L}}({\mathbf{s}}_{2},\,\lambda)$ are spinning after emerging from
the source (cf. Fig. 2). This can be verified from the measurement functions
(66) and (67) defined in IEEE-1 . Thus, the critique’s made up equation holds
in general for all choices of ${\mathbf{a}}$ and ${\mathbf{b}}$ only if the
conservation of zero 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 ${{\mathbf{a}}\not={\mathbf{b}}}$, Eq. (31) holds only if the 3-sphere
model is abandoned and one relapses back to the flat geometry of
${\mathrm{I\\!R}^{3}}$, as the critiques in Gill-IEEE-1  and Gill-IEEE-2  tend
to do reflexively.

I have already explained the above flaws in the critique at great length in
Subsection IV E of my previous reply IEEE-3 , as well as in Scott , in
Appendix C of Reply-Gill , in Answers 9 and 14 in Appendix B of IEEE-1 , and
in Answers 6 and 7 in Appendix B of IEEE-2 . And yet, this frequently refuted
claim is repeated in Gill-IEEE-2 .

###  III.6 Concerning the claim of different models

In its Abstract, the critique Gill-IEEE-2  claims that my paper IEEE-1  “…
contains several conflicting models.” And again in its Sections III and IV it
refers to the 3-sphere model presented in IEEE-1  as my “first model” and
“second model”, implying that I have proposed two different models. But
throughout my work published in IEEE-1 ; IEEE-2 ; Disproof ; IJTP ; RSOS , and
especially in my paper IEEE-1 , I have proposed only one model for the singlet
correlations, and that model is the quaternionic 3-sphere model summarized in
Section III of IEEE-3 . This suggests that the critique Gill-IEEE-2  is based
on quite a mistaken understanding of what is presented in IEEE-1 .

In reality, what is presented in Sections IV and VII of IEEE-1  are not
different models but two different representations of one and the same
quaternionic 3-sphere model. And there is no “conflict” of any kind between
these two representations. They complement each other perfectly, and prove
that, no matter which representation of the quaternionic 3-sphere is used to
perform calculations, the correlations predicted by the quantum mechanical
singlet state can be understood as local, realistic, and deterministic
correlations among the limiting scalar points ${\mathscr{A}}=\pm 1$ and
${\mathscr{B}}=\pm 1$ of the 3-sphere, provided it is taken to model the
three-dimensional physical space in which we are confined to perform our Bell-
test experiments.

It is also important to note that no argument or proof is provided in Gill-
IEEE-2  (or anywhere else) in support of its incorrect claim. The critique
alleges “conflicting models”, but does not provide any evidence or
demonstration of the alleged conflict.

It is well known that mathematically a 3-sphere can be represented in several
different ways. In Sections IV and V of my paper IEEE-1  it is represented
without using the full arsenal of Geometric Algebra in a manner that brings
out an unforeseen link between the quaternionic 3-sphere model and Pearle’s
local hidden variable model for the singlet correlations Pearle . On the other
hand, in Sections VII, VIII, and IX of IEEE-1  the 3-sphere is represented
using the elegant and powerful language of Geometric Algebra. It is the
strength of my analysis that two entirely different representations of the
3-sphere are shown to reproduce the same singlet correlations predicted by
quantum mechanics. What is more, pedagogically each representation illuminates
the quaternionic 3-sphere model in a different way. The critique Gill-IEEE-2 ,
however, claims the two representations of the model to be two different
models. That is analogous to claiming that Heisenberg and Schrödinger
representations of quantum theory are two different theories.

###  III.7 Concerning Pearle’s calculation within $\boldsymbol{S^{3}}$

In its Abstract, critique Gill-IEEE-2  claims that “Most of [my] paper is
devoted to a model based on the detection loophole due to Pearle (1970).” But
only a few formulae from Pearle’s paper are used in only two out of twelve
sections and two appendices of my paper IEEE-1 , and neither of the two
sections are exclusively concerned with his detection loophole analysis. This
suggests that the critique’s reading of IEEE-1  is mistaken.

In fact, far from being “devoted” to Pearle’s detection loophole model Pearle
, the discussion in Section V of IEEE-1  is about what has been missed in
Pearle’s analysis in Pearle , and how the singlet correlation can be recovered
once that defect in Pearle  is rectified. The state space for the singlet spin
system used by Pearle is a well known representation of the rotation group
SO(3), viewed as a unit ball within ${\mathrm{I\\!R}^{3}}$, with each point of
it representing a rotation about a vector ${\mathbf{r}}$ of length $0\leqslant
r\leqslant 1$ from the origin, and the product ${\pi r}$ representing a
rotation by angle $\psi=\pi r$. By contrast, the state space used in the
quaternionic 3-sphere model of the singlet correlations presented in IEEE-1
uses a representation of the double covering group of SO(3). Namely, the group
${\mathrm{SU(2)}}$, which is homeomorphic to ${S^{3}}$, and can be constructed
by identifying the boundaries of two SO(3) balls, providing the double
covering of the group SO(3). The state space ${\mathrm{SU}(2)\cong S^{3}}$ is
thus topologically nontrivial, and sensitive to the spinorial sign changes in
the quaternions that constitute $S^{3}$. By contrast, Pearle’s sate space
SO(3) is insensitive to such sign changes. Thus, what is overlooked in
Pearle’s analysis is the spinorial sign changes in the quaternions described
in Eq. (33) above.

To appreciate this better, recall that, in general, the group
${\mathrm{SU}(2)\cong S^{3}}$ represents rotations of objects relative to
other objects. Such rotations are sensitive to spinorial sign changes and thus
fermionic IJTP . On the other hand, local observations of rotations that do
not refer to external objects are not sensitive to sign changes in the
quaternions that constitute the group ${\mathrm{SU}(2)\cong S^{3}}$. In other
words, the quaternions ${-\,{\mathbf{q}}(\psi,\,{\mathbf{r}})}$ and
${+\,{\mathbf{q}}(\psi,\,{\mathbf{r}})}$ in $S^{3}$ represent one and the same
rotation in the physical space ${\mathrm{I\\!R}^{3}}$. There are thus twice as
many elements in the set ${S^{3}}$ of all unit quaternions than there are
points in the configuration space SO(3) of all possible rotations in the
physical space. This is because every pair of quaternions constituting the
antipodal points of ${S^{3}}$ represent one and the same rotation in
${\mathrm{I\\!R}^{3}}$ (cf. Fig. 3). This can be verified by recalling how a
quaternion and its antipode can rotate a bivector
${{\mathbf{J}}({\mathbf{r}})}$ about ${\mathbf{r}}$ to a bivector
${{\mathbf{J}}({\mathbf{r}^{\prime}})}$ about ${\mathbf{r}^{\prime}}$:

${\mathbf{J}}({\mathbf{r}^{\prime}})\,=\,(+\,{\mathbf{q}})\,{\mathbf{J}}({\mathbf{r}})\,(+\,{\mathbf{q}})^{\dagger}\,=\,(-\,{\mathbf{q}})\,{\mathbf{J}}({\mathbf{r}})\,(-\,{\mathbf{q}})^{\dagger}.$
(34)

Thus ${S^{3}}$, or more precisely the group SU(2) homeomorphic to $S^{3}$,
represents a universal double covering of the rotation group SO(3) so that the
quaternions ${-\,{\mathbf{q}}(\psi,\,{\mathbf{r}})}$ and
${+\,{\mathbf{q}}(\psi,\,{\mathbf{r}})}$ represent one and the same rotation
in the physical space ${\mathrm{I\\!R}^{3}}$. SO(3) is thus a group of
observable rotations in physical space and can be constructed by identifying
each quaternion ${-\,{\mathbf{q}}(\psi,\,{\mathbf{r}})}$ in ${S^{3}}$ with its
antipodal quaternion ${+\,{\mathbf{q}}(\psi,\,{\mathbf{r}})}$ in
${S^{3}\hookrightarrow\mathrm{I\\!R}^{4}}$. The space that results is the real
projective space

${\mathrm{I\\!R}}{\mathrm{P}}^{3}\cong{\mathrm{SO}}(3)\approx\left({\mathrm{SU}}(2)\cong
S^{3}\right)/\\{-1,\,+1\\},$ (35)

which is simply the set of all lines through the origin of
${{\mathrm{I\\!R}}^{4}}$.

Figure 3: The configuration space SO(3) of all possible rotations in physical
space is obtained by identifying the antipodal points of $S^{3}$ — i.e., by
identifying every quaternion ${+{\mathbf{q}}}$ in ${S^{3}}$ with its antipodal
quaternion ${-{\mathbf{q}}}$ in ${S^{3}}$. After IJTP .

There are thus two preimages in ${S^{3}}$, namely ${+\,{\mathbf{q}}}$ and
${-\,{\mathbf{q}}}$ for each rotation within SO(3). In other words, the
projective space ${{\mathrm{I\\!R}}{\mathrm{P}}^{3}}$ is the quotient of
${S^{3}}$ by the map ${{\mathbf{q}}\mapsto-\,{\mathbf{q}}}$, which can be
expressed as ${\varphi:S^{3}\rightarrow{\mathrm{I\\!R}}{\mathrm{P}}^{3}}$.
This quotient map renders the topologies of the spaces ${S^{3}}$ and
${{\mathrm{I\\!R}}{\mathrm{P}}^{3}}$ quite distinct from one another. For
instance, the space ${S^{3}}$ turns out to be simply-connected, whereas the
space ${{\mathrm{I\\!R}}{\mathrm{P}}^{3}}$ is connected but not simply-
connected. Consequently, the geodesic distances
${{\mathscr{D}}({\mathbf{a}},\,{\mathbf{b}})}$ between two quaternions
${{\mathbf{q}}(\psi_{\mathbf{a}},\,{\mathbf{a}})}$ and
${{\mathbf{q}}(\psi_{\mathbf{b}},\,{\mathbf{b}})}$ representing two different
rotations within ${\mathrm{I\\!R}^{3}}$ can be measurably different in the
manifolds ${\mathrm{SU}(2)\cong S^{3}}$ and
${\mathrm{SO}(3)\cong\mathrm{I\\!R}{\mathrm{P}}^{3}}$, providing a signature
of spinorial sign changes between
${{\mathbf{q}}(\psi_{\mathbf{a}},\,{\mathbf{a}})}$ and
${{\mathbf{q}}(\psi_{\mathbf{b}},\,{\mathbf{b}})}$. This signature is shown in
Fig. 4, and discussed in considerable detail in IJTP .

The failure of Pearle’s theoretical model in Pearle  to take into account the
spinorial sign changes captured in (33) is rectified in my paper IEEE-1  with
amicable consequences. If we include the spinorial sign changes in Pearle’s
analysis, then the relation between the rotation angle ${{\pi}r}$ in Pearle’s
state space ${\mathrm{SO}(3)\cong{\mathrm{I\\!R}}{\mathrm{P}}^{3}}$ and the
rotation angle ${2\eta_{{\mathbf{z}}{\mathbf{s}}_{o}}}$ in the state space
${{\mathrm{SU}(2)}\cong S^{3}}$ I have used in IEEE-1 ; IEEE-2 ; Disproof ;
IJTP ; RSOS  works out to be

$\displaystyle\cos\left(\frac{\pi}{2}r\right)=$
$\displaystyle-1+\frac{2}{\sqrt{1+3\left(\frac{\eta_{{\mathbf{z}}{\mathbf{s}}_{o}}}{\kappa\pi}\right)}\,}=f(\eta_{{\mathbf{z}}{\mathbf{s}}_{o}}),$
(36) $\displaystyle\cos\left(\frac{\pi}{2}r\right)=$
$\displaystyle-1+\frac{2}{\sqrt{4-3\left(\frac{\eta_{{\mathbf{z}}{\mathbf{s}}_{o}}}{\kappa\pi}\right)}\,}=f(\pi-\eta_{{\mathbf{z}}{\mathbf{s}}_{o}}).$
(37)

There is then a one-to-one correspondence between the initial spin states
emitted by the source and the final spin states detected by the remote
detectors of Alice and Bob. Thus, every initial state that is emitted by the
source is detected by the detectors, and vice versa. Consequently, unlike
Pearle’s model, the model presented in IEEE-1  is not about data rejection or
detection loophole. In the latter model, the fraction
${{\mathrm{g}}(\eta_{{\mathbf{a}}{\mathbf{b}}})}$ of events in which both
particles are detected is exactly one:

${\mathrm{g}}(\eta_{{\mathbf{a}}{\mathbf{b}}})\\!=\frac{P_{12}^{+-}(\eta_{{\mathbf{a}}{\mathbf{b}}})}{\frac{1}{2}\cos^{2}\\!\left(\frac{\eta_{{\mathbf{a}}{\mathbf{b}}}}{2}\right)}=\frac{P_{12}^{++}(\eta_{{\mathbf{a}}{\mathbf{b}}})}{\frac{1}{2}\sin^{2}\\!\left(\frac{\eta_{{\mathbf{a}}{\mathbf{b}}}}{2}\right)}=1\;\forall\;\eta_{{\mathbf{a}}{\mathbf{b}}}\in[0,\,\pi].$
(38)

This is made very clear in IEEE-1 , in the following passage:

> Clearly, a measurement event cannot occur if there does not exist a state
> which can bring about that event. Since the initial state of the system is
> specified by the pair ${({\mathbf{e}}_{o},\,{\mathbf{s}}_{o})}$ and not just
> by the vector ${{\mathbf{e}}_{o}}$, there are no states of the system for
> which
> ${|\cos(\,\eta_{{\mathbf{n}}{\mathbf{e}}_{o}})|\,<\,f(\eta_{{\mathbf{z}}{\mathbf{s}}_{o}})}$
> for any vector ${\mathbf{n}}$. Thus a measurement event cannot occur for
> ${|\cos(\,\eta_{{\mathbf{n}}{\mathbf{e}}_{o}})|\,<\,f(\eta_{{\mathbf{z}}{\mathbf{s}}_{o}})}$,
> no matter what ${\mathbf{n}}$ is. As a result, there is a one-to-one
> correspondence between the initial state
> ${({\mathbf{e}}_{o},\,{\mathbf{s}}_{o})}$ selected from the set (31) and the
> measurement events ${\mathscr{A}}$ and ${\mathscr{B}}$ specified by the Eqs.
> (34) and (35). This means, in particular, that the “fraction”
> ${{\mathrm{g}}(\eta_{{\mathbf{a}}{\mathbf{b}}})}$ in our model is equal to 1
> for all ${\eta_{{\mathbf{a}}{\mathbf{b}}}}$, dictating the vanishing of the
> probabilities
>
>
> $P_{12}^{00}(\eta_{{\mathbf{a}}{\mathbf{b}}})=1+{\mathrm{g}}(\eta_{{\mathbf{a}}{\mathbf{b}}})-2\,{\mathrm{g}}(0)=0,$
> (49)
>
> which follows from Pearle’s Eq. (9). Moreover, from his Eq. (8) we also have
>
>
> ${P_{12}^{+0}(\eta_{{\mathbf{a}}{\mathbf{b}}})=\frac{1}{2}\left[\,{\mathrm{g}}(0)-{\mathrm{g}}(\eta_{{\mathbf{a}}{\mathbf{b}}})\right]},$
>
> giving
>
> $\displaystyle
> P_{12}^{+0}(\eta_{{\mathbf{a}}{\mathbf{b}}})=P_{12}^{-0}(\eta_{{\mathbf{a}}{\mathbf{b}}})$
> $\displaystyle=P_{12}^{0+}(\eta_{{\mathbf{a}}{\mathbf{b}}})$
> $\displaystyle=P_{12}^{0-}(\eta_{{\mathbf{a}}{\mathbf{b}}})=0.$ (50)
>
> Together with the probabilities for individual detections,
>
> $\displaystyle P_{1}^{+}({\mathbf{a}})=P_{1}^{-}({\mathbf{a}})$
> $\displaystyle=P_{2}^{+}({\mathbf{b}})$
> $\displaystyle=P_{2}^{-}({\mathbf{b}})=\frac{1}{2}\,{\mathrm{g}}(0)=\frac{1}{2}\,,$
> (51)
>
> the correlation between ${\mathscr{A}}$ and ${\mathscr{B}}$ then works out
> to be
>
> $\displaystyle{\cal E}$ $\displaystyle({\mathbf{a}},\,{\mathbf{b}})$
> $\displaystyle=\lim_{\,n\,\gg\,1}\left[\frac{1}{n}\sum_{i\,=\,1}^{n}\,{\mathscr{A}}({\mathbf{a}};\,{\mathbf{e}}^{i}_{o},\,{\mathbf{s}}_{o}^{i})\;{\mathscr{B}}({\mathbf{b}};\,{\mathbf{e}}^{i}_{o},\,{\mathbf{s}}_{o}^{i})\right]$
> $\displaystyle=\,\frac{P_{12}^{++}\,+\,P_{12}^{--}\,-\,P_{12}^{+-}\,-\,P_{12}^{-+}}{P_{12}^{++}\,+\,P_{12}^{--}\,+\,P_{12}^{+-}\,+\,P_{12}^{-+}}$
> $\displaystyle=\,-\,\cos\left(\eta_{{\mathbf{a}}{\mathbf{b}}}\right).$ (52)
>
> Since all of the probabilities predicted by our local model in ${S^{3}}$
> match exactly with the corresponding predictions of quantum mechanics, the
> violations of not only the CHSH inequality, but also Clauser-Horne
> inequality follow.

The critique Gill-IEEE-2 , on the other hand, does not mention any of the
foregone reasoning and analysis. That is unfortunate, for it is very clear
from this analysis that the usual interpretation of Pearle’s model as a
detection loophole model is mistaken.

Figure 4: Comparison of the geodesic distances
${{\mathscr{D}}({\mathbf{a}},{\mathbf{b}})}$ on ${\mathrm{SU}(2)\cong S^{3}}$
and ${\mathrm{SO}(3)\cong{\mathrm{I\\!R}}{\mathrm{P}}^{3}}$ as functions of a
half of the rotation angle. The dashed lines depict the geodesic distances on
${\mathrm{SO}(3)\cong{\mathrm{I\\!R}}{\mathrm{P}}^{3}}$, which was taken to be
the state space of the singlet system by Pearle in Pearle , whereas the solid
curve depicts their horizontal lift to the covering group
${\mathrm{SU}(2)\cong S^{3}}$, which is taken to model the three-dimensional
physical space in the quaternionic 3-sphere model of the singlet correlations
I have proposed in IEEE-1 ; IEEE-2 ; Disproof ; IJTP ; RSOS . After IJTP .

###  III.8 Concerning a computer code included in IEEE-1

By now it should be evident that the critique in Gill-IEEE-2  has not
understood the 3-sphere model I have presented in IEEE-1 . It is therefore not
surprising that it has also misunderstood one of the six numerical simulations
of the singlet correlations discussed in my paper. By relapsing back to the
flat geometry of ${\mathrm{I\\!R}^{3}}$, the critique claims that the code of
the simulation cited as [40] in IEEE-1  is a simulation of Pearle’s “data
rejection” model. But in the previous subsection we established that
interpreting the theoretical model proposed by Pearle in Pearle  as a data
rejection or detection loophole model is a mistake.

To begin with, I have discussed, not one, but six different numerical
simulations of the singlet correlations in my paper, cited as references [40],
[41], [45], [46], [47], and [48]. The critique, however, discusses only one of
them, namely, the one cited as reference [40] in IEEE-1 . More importantly, as
I discussed in Subsection III.6 above, it is quite well known that,
mathematically, a 3-sphere can be represented in several different ways. In
Sections IV, V, and VI of my paper IEEE-1  the 3-sphere is represented without
using Geometric Algebra in such a way that it almost aligns with Pearle’s
hidden variable model for the singlet correlations Pearle . This makes it
possible to parallel the detection loophole type simulation analysis but
without failing to detect any initial states emerging form the source in the
EPR-Bohm type experiment. On the other hand, in Sections VII, VIII, and IX of
IEEE-1  the same 3-sphere is represented using the powerful and elegant
language of Geometric Algebra, which facilitates the use of a GAViewer program
for simulating the resulting singlet correlations. It is the strength of my
analysis in IEEE-1  that two entirely different representations of the
3-sphere are shown to reproduce the same singlet correlations predicted by
quantum mechanics.

Now for the Geometric Algebra based simulations [45] to [48] cited in IEEE-1
the issue of data rejection does not arise, and the critique in Gill-IEEE-2
does not claim that it does. On the other hand, the simulations cited in
IEEE-1  as [40] and [41] simulates the singlet correlations using the first of
the two representations of the 3-sphere mentioned in the previous paragraph.
For them, the critique in Gill-IEEE-2  claims that the simulation cited as
[40] in IEEE-1  depends on “the post-selection of data” and thus uses “data
rejection.” But this claim is not correct, as can be easily verified by
examining the simulation in the context of the 3-sphere model discussed in the
Sections IV, V, and VI of IEEE-1 . There is no post-selection of data in this
simulation. There is only pre-selection of only those initial states that can
exist within the 3-sphere. Those are the only initial states that are
meaningful in the model, and none of them are rejected. Consequently, as noted
in the Subsection III.7 above, there is one-to-one correspondence between the
initial states of the system and those states that are detected by Alice and
Bob.

This fact is more evident in the simulation cited as [41] in IEEE-1 , but the
critique in Gill-IEEE-2  has overlooked the analysis in [41]. In that
simulation it is demonstrated how, by setting the distribution function f = 0,
the geometry of $S^{3}$ reduces to that of the flat geometry of
${\mathrm{I\\!R}^{3}}$, and that, in turn, reduces the sinusoidal correlations
to the saw-tooth shaped correlations.

There is another simulation that is worth mentioning here, even though I have
neglected citing it in IEEE-1 ; namely, Ref. Simulation-C  below. The analysis
in this simulation is aimed at countering the incorrect claims of “post-
selection of data”, “use of data rejection”, or “exploiting the detection
loophole” leveled in the critique Gill-IEEE-2  against the simulation cited as
[40] in IEEE-1 . From the detailed numerical analysis presented in the
simulation Simulation-C  cited below it is abundantly clear that any claim of
data rejection is nothing more than wishful thinking.

###  III.9 Concerning Lasenby’s critique of my work

Next, let me briefly comment on the critique Gill-IEEE-2 ’s appeal to
Lasenby’s critique Lasenby-AACA  in support of its own mistaken claims:

> … Lasenby (2020) Lasenby-AACA  shows that a central and purely algebraic
> result in Christian (2018) RSOS  (published in Royal Society Open Science)
> is wrong. … Lasenby identifies exactly the same GA errors as I did in my
> papers Gill-IEEE-1 , Gill-Entropy  …

Lasenby does make such claims in Lasenby-AACA . Unfortunately, while his
critique in Lasenby-AACA  is more cogently presented compared to those in
Gill-IEEE-1 ; Gill-IEEE-2 , and Gill-Entropy , it too succumbs to some of the
same conceptual and mathematical mistakes I have brought out from Gill-IEEE-1
; Gill-IEEE-2 ; Gill-Entropy  in IEEE-3 ; Reply-Gill ; Reply-Gill-IJTP ; Scott
, Appendix B of IEEE-1 , and Appendix B of IEEE-2 . This is not surprising
because they are uncritically borrowed from Gill-IEEE-1  and Gill-Entropy , as
acknowledged in Lasenby-AACA . In particular, the claims made in Lasenby-AACA
are not correct. I have demonstrated them to be incorrect in Reply-Lasenby
and Reply-to-Lasenby .

###  III.10 Concerning the alleged survey of my work

In its Introduction, critique Gill-IEEE-2  claims to have surveyed my work on
Bell’s theorem in the critique Gill-Entropy  (see also Gill-apology ):

> The present author surveyed Christian’s work on Bell’s theorem from 2007 to
> 2019 in Gill-Entropy , … Soon after that, I published a “Comment” Gill-
> IEEE-1  to Christian’s (2020) companion IEEE Access paper IEEE-2 .

Unfortunately, the alleged survey of my work in Gill-Entropy  is anything but
a survey of my work. It contains a large number of incorrect assertions, made
up history, and armchair sociology and psychology, that even a minimal fact
checking would reveal to be unfit for any scientific discourse. It also
contains numerous mathematical and conceptual mistakes, some of which I have
brought out in IEEE-3 ; Reply-Gill ; Reply-Gill-IJTP ; Scott . The mistakes in
Gill-Entropy  begin with its very title, which reads: “Does geometric algebra
provide a loophole to Bell’s theorem?” But my work in IEEE-1 ; IEEE-2 ;
Disproof ; IJTP ; RSOS  on quantum correlations has nothing to do with any
loopholes. Thus, the title reveals a lack of understanding of what my work is
all about. The other mistakes in Gill-Entropy ; Gill-apology  are too numerous
to bring out here, but some of them have been published in the reviewer
reports Gill-reports . Also, the entire presentation in Gill-Entropy  is
marred by the use of matrices and vector ”algebra” (which, of course, does not
form an algebra), thus missing the very point of my use of Geometric Algebra
in IEEE-1 ; IEEE-2 ; Disproof ; IJTP ; RSOS . This is quite a serious
conceptual mistake in Gill-Entropy .

##  IV Conclusion

The common defect in the critiques Gill-IEEE-1 , Gill-IEEE-2 , and Gill-
Entropy  is that, instead of engaging with the original quaternionic 3-sphere
model presented in my papers IEEE-1 ; IEEE-2 ; Disproof ; IJTP ; RSOS ; IEEE-3
using Geometric Algebra, they insist on criticizing entirely unrelated flat
space models based on matrices and vector ”algebra.” This logical fallacy by
itself renders the critiques invalid. Nevertheless, in this paper I have
addressed every claim made in the critique Gill-IEEE-2  and the critiques it
relies on, and demonstrated, point by point, that none of the claims made in
the critiques are correct. I have demonstrated that the claims made in the
critique Gill-IEEE-2  are neither proven nor justified. In particular, I have
demonstrated that, contrary to its claims, critique Gill-IEEE-2  has not found
any mistakes in my paper IEEE-1 , or in my other related papers, either in the
analytical model for the singlet correlations or in its event-by-event
numerical simulations. Moreover, I have brought out a large number of mistakes
and incorrect statements from the critique Gill-IEEE-2  and the critiques it
relies on. Some of these mistakes are surprisingly elementary.

In its Conclusion, the critique Gill-IEEE-2  claims that my “paper IEEE-1
makes no contribution to the ongoing debates concerning Bell’s theorem.”
However, contributing to the ongoing debates concerning Bell’s theorem has
never been my goal. In my view, Bell’s theorem is a deeply flawed argument,
and therefore it is irrelevant for the future of physics. With the problem of
reconciling general relativity with quantum mechanics on the back of my mind
Newton-Cartan , my goal has always been understanding the origins and
strengths of quantum correlations local-realistically, in terms of the
geometry and topology of spacetime, as envisaged by Einstein Pais . In my
view, I have taken the first steps in that direction in IEEE-1 ; IEEE-2 ;
Disproof ; IJTP ; RSOS . Nothing the critique Gill-IEEE-2  and the critiques
it relies on have claimed undermines this work. In IEEE-1 ; IEEE-2 ; Disproof
; IJTP ; RSOS  I have shown that the raison d’être of the observed strong
correlations is not quantum entanglement per se (which is merely a placeholder
for the strong correlations), but the geometry and topology of the physical
space in which we are confined to perform our experiments. I have demonstrated
in IEEE-1 ; IEEE-2 ; Disproof ; IJTP ; RSOS  that the origins and strengths of
strong correlations can be easily understood local-realistically if we model
the physical space as $S^{3}$ instead of ${\mathrm{I\\!R}^{3}}$. There is no
need to succumb to nonlocality, superdeterminism, retrocausality, or
conspiracy.

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