Joy Christian

Joy Christian, Einstein Centre for Local-Realistic Physics, Oxford OX2 6LB,
United Kingdom

# Local origins of quantum correlations rooted in geometric algebra

Joy Christian  Einstein Centre for Local-Realistic Physics, Oxford OX2 6LB,
United Kingdom<EMAIL_ADDRESS>
(<day> <Month>, <year>; <day> <Month>, <year>; <day> <Month>, <year>)

###### Abstract

[Abstract]In previous publications I have proposed a geometrical framework
underpinning the local, realistic, and deterministic origins of the strong
quantum correlations observed in Nature, without resorting to
superdeterminism, retrocausality, or other conspiracy loopholes usually
employed to circumvent Bell’s argument against such a possibility. The
geometrical framework I have proposed is based on a Clifford-algebraic
interplay between the quaternionic 3-sphere, or $S^{3}$, which I have taken to
model the geometry of the three-dimensional physical space in which we are
confined to perform all our physical experiments, and an octonion-like
7-sphere, or $S^{7}$, which arises as an algebraic representation space of
this quaternionic 3-sphere. In this paper I first review the above geometrical
framework, then strengthen its Clifford-algebraic foundations employing the
language of geometric algebra, and finally refute some of its critiques.

###### keywords:

Clifford algebra, quantum correlations, geometric algebra, norm division
algebras, local causalityM S C Classification: 11R52; 17C60

††articletype: Research Article

##  1 Introduction

Unlike Einstein’s theory of gravity, our other most fundamental theory of
physics, namely quantum mechanics, is not a locally causal theory. This was
recognized in 1935 by Einstein, Podolsky, and Rosen 1. They hoped, however,
that quantum mechanics can be completed into a locally causal theory
compatible with realism, by adding "hidden variables" to its equations. Today
such a hope of restoring local causality is widely believed to have been
undermined by an argument proposed by Bell 2, which claims that no locally
causal theory can reproduce all of the statistical predictions of quantum
mechanics, such as those of the stronger-than-classical correlations observed
in Nature 3. However, I have argued elsewhere that Bell’s argument is of
limited validity 4, 5, 6, 7. It depends on several implicit and explicit
assumptions, which can be relaxed to achieve what Einstein had hoped for 8.
Following Einstein’s vision, I have proposed a geometrical framework
underpinning the local, realistic, and deterministic origins of quantum
correlations 4, 5, 6, 7, without resorting to superdeterminism,
retrocausality, or any other conspiracy loopholes usually employed to overcome
Bell’s argument. The geometrical framework I have proposed is based on a
Clifford-algebraic interplay between the quaternionic 3-sphere, or $S^{3}$,
which is assumed to model the geometry of the three-dimensional physical space
in which we are confined to perform our physical experiments, and an octonion-
like 7-sphere, or $S^{7}$, which arises as an algebraic representation space
of this quaternionic 3-sphere. In what follows, I first review the above
geometrical framework, then strengthen its Clifford-algebraic foundations
using the language of geometric algebra 9, and finally refute some of its
critiques.

##  2 Review of the quaternionic 3-sphere model of strong correlations

It is well known that the best locally causal theory in physics is Einstein’s
theory of gravity, also known as the general theory of relativity. Indeed, one
of the motivations of Einstein behind constructing the theory was to overcome
the action-at-a-distance intrinsic to Newton’s theory of gravity 10. It is
therefore natural to consider one of the well known cosmological solutions of
Einstein’s field equations of general relativity, namely, the Friedmann-
Lemaître-Robertson-Walker solution, as our starting point:

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

Here ${a(t)}$ is the scale factor, ${\boldsymbol{\Sigma}}$ is a spacelike
hypersurface, ${\rho}$ is the radial coordinate within
${\boldsymbol{\Sigma}}$, ${\kappa}$ is the “normalized” curvature of
${\boldsymbol{\Sigma}}$, and ${\boldsymbol{\Omega}}$ is a solid angle within
${\boldsymbol{\Sigma}}$ 11. Since we are primarily concerned with terrestrial
experiments 3 designed to verify quantum correlations, in what follows,
without loss of generality, I will restrict considerations to the current
epoch of the cosmos by setting the cosmological scale factor to unity in the
above line element: ${a(t)=1}$. It still allows three possible geometries for
spacetime with the product topology
${\mathrm{I\\!R}\times{\boldsymbol{\Sigma}}}$, so that the corresponding
spacelike hypersurfaces ${\boldsymbol{\Sigma}}$ can be isomorphic to
${\mathrm{I\\!R}^{3}}$, $S^{3}$, or $H^{3}$, with $H^{3}$ being a hyperboloid
of negative curvature. Among these possible geometries, only $S^{3}$
represents a closed universe with compact geometry and constant positive
curvature. Moreover, observationally the cosmic microwave background spectra
mapped by the space observatory Planck now prefers a positive curvature or
$S^{3}$ at more than 99% confidence level 12, 13.

Incidentally, Bell seems to have anticipated using gravity to negate his
argument. In Chapter 7 of his book 14, while exploring possible escape routes
to do so, he wrote: “The space time structure has been taken as given here.
How then about gravitation?”

###  2.1 Special Theorem

Considering the above properties of $S^{3}$, in Christian 4, 5, 6, 7 I have
proposed the following experimentally falsifiable 4 hypothesis: The strong
correlations we observe in Nature 3 can be understood as local, realistic, and
deterministic correlations if we model the three-dimensional physical space
${\boldsymbol{\Sigma}}$ as a closed and compact quaternionic 3-sphere using
geometric algebra 9 rather than as a flat and open space
${\mathrm{I\\!R}^{3}}$ using “vector algebra.” This hypothesis has far
reaching consequences 4, 5, 6, 7. To explore them, let us begin by defining
the quaternionic 3-sphere as

$S^{3}:=\left\\{\,{\bf
q}(\theta,\,{\mathbf{r}})=\varrho_{r}\left[\cos\left(\frac{\theta}{2}\right)+{\mathbf{J}}({\mathbf{r}})\,\sin\left(\frac{\theta}{2}\right)\right]\Bigg{|}\;\left|\left|\,{\bf
q}\left(\theta,\,{\mathbf{r}}\right)\,\right|\right|=\varrho_{r}\right\\}\\!,$
(2)

where ${{\mathbf{J}}({\mathbf{r}})=I_{3}{\mathbf{r}}}$ is a unit bivector (or
pure quaternion) rotating about an axis vector ${{\bf
r}\in{\mathrm{I\\!R}}^{3}}$ with rotation angle ${0\leq\theta<4\pi}$,
$\varrho_{r}$ is the radius of the corresponding 3-sphere, and
$I_{3}={\mathbf{e}}_{x}{\mathbf{e}}_{y}{\mathbf{e}}_{z}$ is the standard
trivector. Now, for our purposes it is convenient to express the unit
quaternions in (2) as products ${\bf D}({\bf n})\,{\bf L}({\bf s})$ of two
bivectors, with ${\bf D}({\bf n})$ representing detectors used in Bell-test
experiments 3 and ${\bf L}({\bf s})$ representing the spins being detected. To
that end, consider the even subalgebra of the algebra $\mathrm{Cl}_{3,0}$ of
orthogonal directions in ${\mathrm{I\\!R}^{3}}$, also referred to as a
bivector subalgebra 9. Since the spins ${{\mathbf{L}}({\mathbf{s}}_{1})}$ and
${{\mathbf{L}}({\mathbf{s}}_{2})}$ and the detectors
${{\mathbf{D}}({\mathbf{a}})}$ and ${{\mathbf{D}}({\mathbf{b}})}$ are
different physical systems, the experimenters, traditionally referred to as
Alice and Bob, have no knowledge of the handedness of the spins until their
measurements. It is therefore important to represent them using two different
basis bivectors:

$L_{i}\,L_{j}=-\,\delta_{ij}\,-\sum_{k}\epsilon_{ijk}\,L_{k}$ (3)

and

$D_{i}\,D_{j}=-\,\delta_{ij}\,-\sum_{k}\epsilon_{ijk}\,D_{k}\,.$ (4)

These bases are then related by the orientation (or handedness) $\lambda=\pm
1$ of $S^{3}$ as

$L_{i}=\lambda\,D_{i}\;\;\Longleftrightarrow\;\;D_{i}=\lambda\,L_{i}\,,$ (5)

so that, about any unit vector $\mathbf{n}$, the rotations of the spin and
detector bivectors are related only by the orientation $\lambda$ of $S^{3}$ as

${\mathbf{L}}({\mathbf{n}})\,=\,\lambda\,{\mathbf{D}}({\mathbf{n}})\,\,\Longleftrightarrow\,\,{\mathbf{D}}({\mathbf{n}})\,=\,\lambda\,{\mathbf{L}}({\mathbf{n}})\,.$
(6)

The orientation $\lambda$, with $\lambda^{2}=1$, thus plays a role of a random
“hidden variable” in the model, with 50/50 chance of $\lambda=+1$ or
$\lambda=-1$ during the detection processes at the two ends of a Bell-test
experiment. Since the spins — which originate with the common cause $\lambda$
in the overlap of the backward light-cones of Alice and Bob — are being
measured with respect to the detectors at a later time, we may write them as
functions of the orientation $\lambda$ to remind us of this fact:
${{\mathbf{L}}({\mathbf{s}}_{1},\lambda)}$ and
${{\mathbf{L}}({\mathbf{s}}_{2},\lambda)}$. In the manner of Bell 2, the
detection processes of Alice and Bob can then be specified as limiting scalar
points of two quaternions within $S^{3}$:

$\displaystyle S^{3}\ni{\mathscr{A}}({\mathbf{a}},{\lambda})\,$
$\displaystyle=\lim_{{\mathbf{s}}_{1}\,\rightarrow\,{\mathbf{a}}}\left\\{-\,{\mathbf{D}}({\mathbf{a}})\,{\mathbf{L}}({\mathbf{s}}_{1},\,\lambda)\right\\}=\lim_{{\mathbf{s}}_{1}\,\rightarrow\,{\mathbf{a}}}\left\\{\,+\,{\mathbf{q}}(\eta_{{\mathbf{a}}{\mathbf{s}}_{1}},\,{\mathbf{r}}_{1})\right\\}$
(7)
$\displaystyle=+\lambda\lim_{{\mathbf{s}}_{1}\,\rightarrow\,{\mathbf{a}}}\left\\{\cos(\eta_{{\mathbf{a}}{\mathbf{s}}_{1}})+(I_{3}{\mathbf{r}}_{1})\sin(\eta_{{\mathbf{a}}{\mathbf{s}}_{1}})\right\\},$
(8)

where $\eta_{{\mathbf{a}}{\mathbf{s}}_{1}}$ is the angle between the spin
direction ${\mathbf{s}}_{1}$ and the detector direction ${\mathbf{a}}$ chosen
by Alice at a later time (cf. Figure 1), and

${\mathbf{r}}_{1}=\frac{{\mathbf{a}}\times{\mathbf{s}}_{1}}{||{\mathbf{a}}\times{\mathbf{s}}_{1}||}$
(9)

is the rotation axis of the quaternion
${\mathbf{q}}(\eta_{{\mathbf{a}}{\mathbf{s}}_{1}},\,{\mathbf{r}}_{1})$ in
$S^{3}$. Thus, as ${\mathbf{s}}_{1}\,\rightarrow\,{\mathbf{a}}$ during the
detection process so that the angle
$\eta_{{\mathbf{a}}{\mathbf{s}}_{1}}\rightarrow\,0$,

${\mathscr{A}}({\mathbf{a}},{\lambda})\longrightarrow\pm 1,\,\;\text{with its
average}\;\Bigl{\langle}{\mathscr{A}}({\mathbf{a}},{\lambda})\Bigr{\rangle}\longrightarrow
0,$ (10)

because $\lambda=\pm 1$ is a fair coin. Note that the rotation angle
${\theta}$ of the quaternion is twice the angle
${\eta_{{\mathbf{a}}{\mathbf{s}}_{1}}}$ between ${\mathbf{a}}$ and
${\mathbf{s}}_{1}$. Similarly,

$\displaystyle S^{3}\ni{\mathscr{B}}$
$\displaystyle({\mathbf{b}},\,{\lambda})\,=\lim_{{\mathbf{s}}_{2}\,\rightarrow\,{\mathbf{b}}}\left\\{+\,{\mathbf{L}}({\mathbf{s}}_{2},\,\lambda)\,{\mathbf{D}}({\mathbf{b}})\right\\}=\lim_{{\mathbf{s}}_{2}\,\rightarrow\,{\mathbf{b}}}\left\\{\,-\,{\mathbf{q}}(\eta_{{\mathbf{s}}_{2}{\mathbf{b}}},\,{\mathbf{r}}_{2})\right\\}$
(11)
$\displaystyle=-\lambda\lim_{{\mathbf{s}}_{2}\,\rightarrow\,{\mathbf{b}}}\left\\{\cos(\eta_{{\mathbf{s}}_{2}{\mathbf{b}}})+(I_{3}{\mathbf{r}}_{2})\sin(\eta_{{\mathbf{s}}_{2}{\mathbf{b}}})\right\\},$
(12)

where $\eta_{{\mathbf{s}}_{2}{\mathbf{b}}}$ is the angle between the spin
direction ${\mathbf{s}}_{2}$ and the detector direction ${\mathbf{b}}$ chosen
by Bob at a later time (cf. Figure 1), and

${\mathbf{r}}_{2}=\frac{{\mathbf{s}}_{2}\times{\mathbf{b}}}{||{\mathbf{s}}_{2}\times{\mathbf{b}}||}$
(13)

is the rotation axis of the quaternion
${\mathbf{q}}(\eta_{{\mathbf{s}}_{2}{\mathbf{b}}},\,{\mathbf{r}}_{2})$ in
$S^{3}$. Thus, as ${\mathbf{s}}_{2}\,\rightarrow\,{\mathbf{b}}$ during the
detection process so that the angle
$\eta_{{\mathbf{s}}_{2}{\mathbf{b}}}\rightarrow\,0$,

${\mathscr{B}}({\mathbf{b}},{\lambda})\longrightarrow\mp 1,\,\;\text{with its
average}\;\Bigl{\langle}{\mathscr{B}}({\mathbf{b}},{\lambda})\Bigr{\rangle}\longrightarrow
0.$ (14)

It is important to note that the functions (7) and (11) specify measurement
results obtained in two separate set of experiments, independently carried out
by Alice and Bob regardless of each other. The strong correlations between
their results, however, are obtained when they jointly observe their results
in “coincident counts”, at a later time, within a spacelike hypersurface in
spacetime, which we have assumed to be $S^{3}$. This requires a different set
of experiments and leads to the following question: What will be the average
of the product
${{\mathscr{A}}{\mathscr{B}}({\mathbf{a}},{\mathbf{b}},\lambda)}$ 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 within $S^{3}$? The answer to this question is
given by the following special theorem.

${{\bf s}_{2}}$${{\bf s}_{1}}$${\bf 1}$${\bf 2}$${\bf a}$${\bf
a^{\prime}}$${\bf b}$${\bf
b^{\prime}}$source${\pi^{0}\longrightarrow\,e^{-}+\,e^{+}\,}$total spin = 0
Figure 1: A spin-less neutral pion decays into an electron-positron pair.
Measurements of spin components of each separated fermion are performed at
remote stations ${\mathbf{1}}$ and ${\mathbf{2}}$, obtaining binary results
$\mathscr{A}=\pm 1$ and $\mathscr{B}=\pm 1$ along directions ${\mathbf{a}}$
and ${\mathbf{b}}$. The conservation of spin momentum dictates that the total
spin of the system remains zero during its free evolution. After Christian 6.

######  Theorem 2.1.

The strong 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 ${{\mathscr{A}}({\bf
a},\,{\lambda})=\pm 1}$ and ${{\mathscr{B}}({\bf b},\,{\lambda})=\pm 1}$ of a
quaternionic 3-sphere, or $S^{3}$, taken as a model of the three-dimensional
physical space.

I have proved this theorem in several different ways in previous publications
4, 5, 6, 7. The proof amounts to computing the singlet correlations while
preserving the geometric properties of the 3-sphere, using the following
standard formula provided by Bell 2:

${\cal
E}_{\mathrm{L.R.}}({\mathbf{a}},\,{\mathbf{b}})=\int{\mathscr{A}}({\mathbf{a}},\,\lambda)\,{\mathscr{B}}({\mathbf{b}},\,\lambda)\;p(\lambda)\,d\lambda\approx\\!\lim_{\,n\,\gg\,1}\left[\frac{1}{n}\sum_{k\,=\,1}^{n}\,{\mathscr{A}}({\mathbf{a}},\,{\lambda}^{k})\;{\mathscr{B}}({\mathbf{b}},\,{\lambda}^{k})\right]=\,-\cos(\,\eta_{{\mathbf{a}}{\mathbf{b}}}).$
(15)

Since $\lambda$ is a fair coin with only possible values $+1$ or $-1$, the
averaging over $\lambda$ can be assumed to be uniform with probability
distribution $p(\lambda)=\frac{1}{n}$, where $n$ is the total number of
trials. This allows us to reduce the above expectation function to the
discrete version. In Section 5 below, a new proof of Theorem 2.1 is presented
that is immune to a critique of its proofs in Christian 4, 5, 6, 7.

###  2.2 General Theorem

The above special case allows us to reproduce the predictions of simple
quantum states such as the two-particle singlet state and the Hardy state 15.
However, for reproducing more general quantum correlations the quaternionic
3-sphere by itself is not sufficient. What is required is an algebraic
representation space of the quaternionic 3-sphere defined in (2). It turns out
to be a 7-sphere constructed from the eight-dimensional even sub-algebra
(which I have christened ${\cal K}^{\lambda}$) of the ${2^{4}=16}$-dimensional
Clifford algebra ${\mathrm{Cl}_{4,0}\,}$, to be discussed in the next section.
The algebra ${\cal K}^{\lambda}$ happens to be a conformal counterpart of the
algebra ${\mathrm{Cl}_{3,0}}$ of orthogonal directions in
$\mathrm{I\\!R}^{3}$. The general elements of ${\cal K}^{\lambda}$ can be
written as a sum involving two quaternions as follows:

${\cal K}^{\lambda}:=\left\\{{\mathbb{Q}}_{z}:=\,{\bf q}_{r}+{\bf
q}_{d}\,\varepsilon\;\big{|}\;\left|\left|{\mathbb{Q}}_{z}\right|\right|^{2}={\mathbb{Q}}_{z}{\mathbb{Q}}^{\dagger}_{z}={\varrho}^{2}_{c}+{\sigma_{c}^{2}}\,\varepsilon\right\\},$
(16)

where $\dagger$ signifies the conjugate or reverse operation,
${\varrho}^{2}_{c}={\bf q}_{r}\,{\bf q}^{\dagger}_{r}+{\bf q}_{d}\,{\bf
q}^{\dagger}_{d}=\varrho_{r}^{2}+\varrho_{d}^{2}$ and ${\sigma}_{c}^{2}={\bf
q}_{r}\,{\bf q}^{\dagger}_{d}+{\bf q}_{d}\,{\bf q}^{\dagger}_{r}$ are scalar
quantities, and $\varepsilon$ is a pseudoscalar that satisfies
$\varepsilon^{2}=+1$ and $\varepsilon^{\dagger}\\!=\varepsilon$. The required
7-sphere can thus be easily constructed by setting $\sigma_{c}^{2}=0$:

${\cal K}^{\lambda}\hookleftarrow S^{7}:=\,\left\\{{\mathbb{Q}}_{z}:=\,{\bf
q}_{r}+{\bf q}_{d}\,\varepsilon\;\Big{|}\;\sigma_{c}^{2}={\bf q}_{r}\,{\bf
q}^{\dagger}_{d}+{\bf q}_{d}\,{\bf
q}^{\dagger}_{r}=0\Longleftrightarrow||{\mathbb{Q}}_{z}||=\varrho_{c}\right\\},$
(17)

where $\lambda=\pm 1$ specifies the orientation or handedness of ${\cal
K}^{\lambda}$ and $S^{7}$. This leads to the following generalization of
Theorem 2.1:

######  Theorem 2.2.

The 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 ${{\mathscr{A}}({\bf
a},\,{\lambda})=\pm 1}$ and ${{\mathscr{B}}({\bf b},\,{\lambda})=\pm 1}$ of
the octonion-like 7-sphere (17), viewed as an algebraic representation space
of the quaternionic 3-sphere considered in Theorem 2.1.

Here, in analogy with (7) and (11), the measurement results
${{\mathscr{A}}({\bf a},\,{\lambda})=\pm 1}$, ${{\mathscr{B}}({\bf
b},\,{\lambda})=\pm 1}$, etc., are specified by the functions

$\displaystyle S^{7}\ni{\mathscr{A}}({\bf a}\,,\,\lambda):=$
$\displaystyle\lim_{\,\begin{subarray}{c}{\bf s}_{r1}\,\rightarrow\;{\bf
a}_{r}\\\ \,{\bf s}_{d1}\,\rightarrow\;{\bf
a}_{d}\end{subarray}}\left\\{\,\pm\,{\bf D}({\bf a}_{r},\,{\bf
a}_{d},\,0)\,{\bf L}({\bf s}_{r1},\,{\bf
s}_{d1},\,0,\,\lambda)\,\right\\}=\,\begin{cases}\,\mp\,1\;\;\;\;\;{\rm
if}&\lambda\,=\,+\,1\\\ \,\pm\,1\;\;\;\;\;{\rm
if}&\lambda\,=\,-\,1\end{cases}\Bigg{\\}},$ $\displaystyle\;\,\text{together
with}\,\;\Bigl{\langle}\,{\mathscr{A}}({\bf
a}\,,\,\lambda)\,\Bigr{\rangle}\,=\,0\,,$ (18) $\displaystyle
S^{7}\ni{\mathscr{B}}({\bf b}\,,\,\lambda):=$
$\displaystyle\lim_{\,\begin{subarray}{c}{\bf s}_{r2}\,\rightarrow\;{\bf
b}_{r}\\\ \,{\bf s}_{d2}\,\rightarrow\;{\bf
b}_{d}\end{subarray}}\left\\{\,\mp\,{\bf D}({\bf b}_{r},\,{\bf
b}_{d},\,0)\,{\bf L}({\bf s}_{r2},\,{\bf
s}_{d2},\,0,\,\lambda)\,\right\\}=\,\begin{cases}\,\pm\,1\;\;\;\;\;{\rm
if}&\lambda\,=\,+\,1\\\ \,\mp\,1\;\;\;\;\;{\rm
if}&\lambda\,=\,-\,1\end{cases}\Bigg{\\}},$ $\displaystyle\;\,\text{together
with}\,\;\Bigl{\langle}\,{\mathscr{B}}({\bf
b}\,,\,\lambda)\,\Bigr{\rangle}\,=\,0\,,$ (19) $\displaystyle
S^{7}\ni{\mathscr{C}}({\bf c}\,,\,\lambda):=$
$\displaystyle\lim_{\,\begin{subarray}{c}{\bf s}_{r3}\,\rightarrow\;{\bf
c}_{r}\\\ \,{\bf s}_{d3}\,\rightarrow\;{\bf
c}_{d}\end{subarray}}\left\\{\,\pm\,{\bf D}({\bf c}_{r},\,{\bf
c}_{d},\,0)\,{\bf L}({\bf s}_{r3},\,{\bf
s}_{d3},\,0,\,\lambda)\,\right\\}=\,\begin{cases}\,\mp\,1\;\;\;\;\;{\rm
if}&\lambda\,=\,+\,1\\\ \,\pm\,1\;\;\;\;\;{\rm
if}&\lambda\,=\,-\,1\end{cases}\Bigg{\\}},$ $\displaystyle\;\,\text{together
with}\,\;\Bigl{\langle}\,{\mathscr{C}}({\bf
c}\,,\,\lambda)\,\Bigr{\rangle}\,=\,0\,,$ (20)

etc., representing arbitrary number of measurement results observed by
arbitrary number of experimenters. In Christian 5 I have proved this theorem
for arbitrary quantum states by formally computing correlations in analogy
with those computed in (15):

${\cal E}_{\mathrm{L.R.}}({\bf a},\,{\bf b},\,{\bf c},\,{\bf d},\,{\bf
e},\,{\bf
f},\,\dots)\,=\\!\lim_{\,n\,\gg\,1}\left[\frac{1}{n}\sum_{k\,=\,1}^{n}\,{\mathscr{A}}({\bf
a},\,\lambda^{k})\;{\mathscr{B}}({\bf b},\,\lambda^{k})\;{\mathscr{C}}({\bf
c},\,\lambda^{k})\;{\mathscr{D}}({\bf d},\,\lambda^{k})\;{\mathscr{E}}({\bf
e},\,\lambda^{k})\;{\mathscr{F}}({\bf f},\,\lambda^{k})\dots\right].$ (21)

Moreover, as an example I have also reproduced the correlations quantum
mechanically predicted by the four-particle GHZ state:

${\cal E}^{\mathrm{GHZ}}_{\mathrm{L.R.}}({\bf a},\,{\bf b},\,{\bf c},\,{\bf
d})=\,\cos\theta_{\bf a}\,\cos\theta_{\bf b}\,\cos\theta_{\bf
c}\,\cos\theta_{\bf d}\,-\,\sin\theta_{\bf a}\,\sin\theta_{\bf
b}\,\sin\theta_{\bf c}\,\sin\theta_{\bf d}\,\cos\left(\,\phi_{\bf
a}\,+\,\phi_{\bf b}\,-\,\phi_{\bf c}\,-\,\phi_{\bf d}\,\right),$ (22)

where ${\theta_{\bf a}}$, ${\phi_{\bf b}}$, etc., represent the polar and
azimuthal angles, respectively, of the detector directions ${\bf a}$, ${\bf
b}$, etc. This is exactly the prediction of four-particle GHZ state 5. But I
have derived it as a geometrical effect within the locally causal 7-sphere
framework.

In fact, the Tsirel’son’s bounds on a CHSH-type correlator characterizing the
strength of arbitrary quantum correlations, namely

$-2\sqrt{2}\leqslant{\cal E}({\mathbf{a}},\,{\mathbf{b}})+{\cal
E}({\mathbf{a}},\,{\mathbf{b^{\prime}}})+{\cal
E}({\mathbf{a^{\prime}}},\,{\mathbf{b}})-{\cal
E}({\mathbf{a^{\prime}}},\,{\mathbf{b^{\prime}}})\leqslant 2\sqrt{2}\,,$ (23)

also turns out to be a consequence of the quaternionic 3-sphere hypothesis
2.1, and can be reproduced within the octonion-like 7-sphere framework
constructed in Christian 5. According to this hypothesis, the strong
correlations observed in Nature, such as the singlet correlations 3, is a
compelling evidence that we live in a quaternionic 3-sphere and not in
${\mathrm{I\\!R}^{3}}$. However, singlet correlations are usually understood
as due to quantum entanglement. I have therefore proposed an experiment in
Christian 4, set in a macroscopic domain, so that concepts from quantum theory
cannot explain the strong correlations if they are observed in that
experiment. Thus the 3-sphere hypothesis is falsifiable. Considering its
significance, I now turn to its mathematical foundations.

##  3 Clifford-algebraic foundations of the octonion-like 7-sphere

My goal in this section is to establish the Clifford-algebraic foundations of
the 7-sphere constructed in (17) by starting with the quaternionic 3-sphere
defined in (2). I have explained these foundations adequately in Christian 5.
However, some parts of the 7-sphere framework have been challenged in Lasenby
16, making several incorrect claims. In the next section, I address those
claims. But for that purpose, it will be instructive to first understand the
foundations of the 7-sphere framework in some detail.

###  3.1 From even subalgebra of $\mathrm{Cl_{3,0}}$ to even subalgebra of
$\mathrm{Cl_{4,0}}$

In Section 2, we saw how Bell’s impossibility claim 2 can be overcome by
modeling the physical space as quaternionic 3-sphere and using geometric
algebra, instead of modeling it as $\mathrm{I\\!R}^{3}$ and using vector
algebra. Now it is well known that topologically $S^{3}$ can be constructed by
adding a point to ${{\mathrm{I\\!R}}^{3}}$ at infinity, by means of one-point
or Alexandroff compactification (cf. Figure 2) 17:

$S^{3}=\,{\mathrm{I\\!R}}^{3}\cup\left\\{\infty\right\\}.$ (24)

The resulting manifold is then closed and compact space, with remarkable
geometrical properties 4, 18. On the other hand, the algebraic properties of
${{\mathrm{I\\!R}}^{3}}$ are best captured by the Clifford algebra
$\mathrm{Cl_{3,0}}$ of orthogonal directions in ${{\mathrm{I\\!R}}^{3}}$. Thus
the algebra $\mathrm{Cl_{3,0}\,}$, as an eight-dimensional linear vector space
spanned by graded basis, can be viewed as an algebraic representation space of
${{\mathrm{I\\!R}}^{3}}$:

$\mathrm{Cl}_{3,0}={\rm span}\\!\left\\{\,1,\;{\bf e}_{x},\,{\bf e}_{y},\,{\bf
e}_{z},\;{\bf e}_{x}{\bf e}_{y},\,{\bf e}_{z}{\bf e}_{x},\,{\bf e}_{y}{\bf
e}_{z},\;{\bf e}_{x}{\bf e}_{y}{\bf e}_{z}=:I_{3}\,\right\\}\\!.$ (25)

Here ${{\bf e}^{2}_{i}=1}$ is an identity element of grade-0, ${{\bf e}_{i}}$
are three orthonormal vectors of grade-1, ${{\bf e}_{j}{\bf e}_{k}}$ are three
orthonormal bivectors of grade-2, and ${{\bf e}_{i}{\bf e}_{j}{\bf e}_{k}}$ is
a trivector of grade-3. Respectively, they represent points, lines, planes,
and volumes in ${{\rm I\\!R}^{3}}$. There are thus ${2^{3}=8}$ ways to combine
the vectors ${{\bf e}_{i}}$ so that no two products are linearly dependent.
This characterizes ${{\rm I\\!R}^{3}}$ by the bijection

${\cal F}:{\rm I\\!R}^{3}:={\rm span}\\!\left\\{\,{\bf e}_{x},\,{\bf
e}_{y},\,{\bf e}_{z}\right\\}\longrightarrow\,{\rm I\\!R}^{8}:={\rm
span}\\!\left\\{\,1,\;{\bf e}_{x},\,{\bf e}_{y},\,{\bf e}_{z},\;{\bf
e}_{x}{\bf e}_{y},\,{\bf e}_{z}{\bf e}_{x},\,{\bf e}_{y}{\bf
e}_{z},\;I_{3}\,\right\\}=\mathrm{Cl}_{3,0}.$ (26)

The natural question for us then is: What is the analogous algebraic
representation space of the closed and compact space $S^{3}$ defined in (2)
using the even subalgebra (3) of $\mathrm{Cl_{3,0}}$? For the purposes of
local-realistically reproducing quantum correlations more general than those
predicted by the singlet state within the 3-sphere model, this is the most
important question one can ask. To answer this question, let us take a closer
look at the one-point compactification of $\mathrm{I\\!R}^{3}$ depicted in
Figure 2. What we see in this figure is an inverse stereographic projection
of $\mathrm{I\\!R}^{3}$ onto a unit 3-sphere, ${S^{3}}$, by the embedding map
${\vec{\phi}:{\mathrm{I\\!R}}^{3}\rightarrow S^{3}}$ given by

$\vec{\phi}\left({\bf x}\in{\mathrm{I\\!R}}^{3}\right)=\left(\frac{2}{||{\bf
x}||^{2}+1}\right){\bf x}+\left(\frac{2\,||{\bf x}||^{2}}{||{\bf
x}||^{2}+1}\right){{\hat{x}}_{4}}\,.$ (27)

Note that two of the dimensions of ${{\mathrm{I\\!R}}^{3}}$ are suppressed in
Figure 2, and ${{\hat{x}}_{4}}$ represents the fourth dimension of the
embedding space ${{\rm I\\!R}^{4}}$. The important observation in this inverse
stereographic projection is that, as an arbitrary vector ${{\bf
x}\in{\mathrm{I\\!R}}^{3}}$ from the origin approaches infinity, it is mapped
to the same point ${{\bf e}_{\infty}}$ located at ${(0,2)}$, thus closing the
non-compact space ${{\mathrm{I\\!R}}^{3}}$ into a 3-sphere. Now, by shifting
the origin to ${(0,1)}$, the above set of points can be inscribed also by a
radial 4-vector originating from ${(0,1)}$ as

$\vec{\psi}\left({\bf x}\in{\mathrm{I\\!R}}^{3}\right)=\left(\frac{2}{||{\bf
x}||^{2}+1}\right){\bf x}+\left(\frac{2\,||{\bf x}||^{2}}{||{\bf
x}||^{2}+1}-1\right){{\hat{x}}_{4}}\,.$ (28)

The magnitude of this vector then confirms the unity of the radius of our
conformally embedded 3-sphere within the space ${{\rm I\\!R}^{4}}$:

$1=\left|\left|\,\vec{\psi}\left({\bf
x}\in{\mathrm{I\\!R}}^{3}\right)\right|\right|=\text{radius
of}\;S^{3}\hookrightarrow{\rm I\\!R}^{4}.$ (29)

The map ${\vec{\psi}({\bf x})}$ thus transforms the space
${{\mathrm{I\\!R}}^{3}}$ into a unit 3-sphere within ${{\rm I\\!R}^{4}}$,
accomplishing the one-point compactification of ${{\mathrm{I\\!R}}^{3}}$.

${{\bf e}_{\infty}}$${\sim(0,2)}$${{\mathrm{I\\!R}}^{3}}$${\rm o}$${{\rm
I\\!R}^{4}}$${\hat{x}_{4}}$${\bf x}$${\bf x}$${\vec{\phi}({\bf
x})}$${S^{3}\\!\sim\vec{\phi}\left({\mathrm{I\\!R}}^{3}\right)}$${\infty}$${\infty}$${\infty}$${\infty}$${(0,1)}$
Figure 2: One-point compactification of ${{\mathrm{I\\!R}}^{3}}$ by means of a
stereographic projection onto ${S^{3}\\!\in{\rm I\\!R}^{4}}$. After Christian
5.

The one-point compactification of $S^{3}$ has thus led us to the embedding
space ${{\mathrm{I\\!R}}^{4}}$ spanned by the vector basis ${\left\\{\,{\bf
e}_{x},\,{\bf e}_{y},\,{\bf e}_{z},\,{\bf e}_{\infty}\right\\}}$, where the
fourth unit vector ${\bf e}_{\infty}$ is the algebraic counterpart of the
dimensionless point ${\\{\infty\\}}$ added to ${{\rm I\\!R}^{3}}$ at infinity
to construct $S^{3}$. The algebra of orthogonal directions in ${{\rm
I\\!R}^{4}}$ is then the ${2^{4}=16}$-dimensional Clifford algebra
${\mathrm{Cl}_{4,0}}$. Consequently, in analogy with how the even subalgebra
(3) of $\mathrm{Cl_{3,0}}$ is used to define $S^{3}$ in (2), the algebraic
representation space of $S^{3}$ can be defined using the even subalgebra of
$\mathrm{Cl_{4,0}}$, which turns out to be the graded vector space we defined
in Eq. (16) of the previous section:

${\cal K}^{\lambda}:=\,{\rm span}\\!\left\\{\,1,\,\lambda\,{\bf e}_{x}{\bf
e}_{y},\,\lambda\,{\bf e}_{z}{\bf e}_{x},\,\lambda\,{\bf e}_{y}{\bf
e}_{z},\,\lambda\,{\bf e}_{x}{\bf e}_{\infty},\,\lambda\,{\bf e}_{y}{\bf
e}_{\infty},\,\lambda\,{\bf e}_{z}{\bf e}_{\infty},\,\lambda\,I_{3}{\bf
e}_{\infty}\,\right\\}\\!.$ (30)

Here $\lambda=\pm$ specifies the orientation or handedness of this vector
space (cf. Milnor 19). For convenience, the multiplication rules satisfied by
the graded basis elements of ${\cal K}^{\lambda}$ are reproduced in Table
1. In Subsection 2.1 of Christian 5, this algebra is arrived at by closing
the volume form $I_{3}={\bf e}_{x}{\bf e}_{y}{\bf e}_{z}$ (which is open with
the topology of $\mathrm{I\\!R}^{3}$) with the unit vector ${\bf e}_{\infty}$,
as shown in Figure 2:

$\mathrm{I\\!R}^{3}\;\Longleftrightarrow\;{\bf e}_{x}{\bf e}_{y}{\bf
e}_{z}\equiv I_{3}\;\longrightarrow\;I_{3}{\bf e}_{\infty}\equiv{\bf
e}_{x}{\bf e}_{y}{\bf e}_{z}{\bf
e}_{\infty}\;\Longleftrightarrow\;\mathrm{I\\!R}^{4}\hookleftarrow S^{3}.$
(31)

Comparing the basis elements of ${\cal K}^{\lambda}$ in (30) with those of
$\mathrm{Cl_{3,0}}$ in (25), we see that each of the odd basis element of
$\mathrm{Cl_{3,0}}$ is now closed with ${\bf e}_{\infty}$ and rendered even,
as a result of the compactification implicit in (31), thus making all basis
elements of ${\cal K}^{\lambda}$ even.

rowsep=10pt  ${*}$ ${1}$ ${\lambda\,{\bf e}_{x}{\bf e}_{y}}$ ${\lambda\,{\bf
e}_{z}{\bf e}_{x}}$ ${\lambda\,{\bf e}_{y}{\bf e}_{z}}$ ${\lambda\,{\bf
e}_{x}{\bf e}_{\infty}}$ ${\lambda\,{\bf e}_{y}{\bf e}_{\infty}}$
${\lambda\,{\bf e}_{z}{\bf e}_{\infty}}$ ${\lambda\,I_{3}{\bf e}_{\infty}}$
[2pt] ${1}$ ${1}$ ${\lambda\,{\bf e}_{x}{\bf e}_{y}}$ ${\lambda\,{\bf
e}_{z}{\bf e}_{x}}$ ${\lambda\,{\bf e}_{y}{\bf e}_{z}}$ ${\lambda\,{\bf
e}_{x}{\bf e}_{\infty}}$ ${\lambda\,{\bf e}_{y}{\bf e}_{\infty}}$
${\lambda\,{\bf e}_{z}{\bf e}_{\infty}}$ ${\lambda\,I_{3}{\bf e}_{\infty}}$
${\lambda\,{\bf e}_{x}{\bf e}_{y}}$ ${\lambda\,{\bf e}_{x}{\bf e}_{y}}$ ${-1}$
${{\bf e}_{y}{\bf e}_{z}}$ ${-{\bf e}_{z}{\bf e}_{x}}$ ${-{\bf e}_{y}{\bf
e}_{\infty}}$ ${{\bf e}_{x}{\bf e}_{\infty}}$ ${I_{3}{\bf e}_{\infty}}$
${-{\bf e}_{z}{\bf e}_{\infty}}$ ${\lambda\,{\bf e}_{z}{\bf e}_{x}}$
${\lambda\,{\bf e}_{z}{\bf e}_{x}}$ ${-{\bf e}_{y}{\bf e}_{z}}$ ${-1}$ ${{\bf
e}_{x}{\bf e}_{y}}$ ${{\bf e}_{z}{\bf e}_{\infty}}$ ${I_{3}{\bf e}_{\infty}}$
${-{\bf e}_{x}{\bf e}_{\infty}}$ ${-{\bf e}_{y}{\bf e}_{\infty}}$
${\lambda\,{\bf e}_{y}{\bf e}_{z}}$ ${\lambda\,{\bf e}_{y}{\bf e}_{z}}$ ${{\bf
e}_{z}{\bf e}_{x}}$ ${-{\bf e}_{x}{\bf e}_{y}}$ ${-1}$ ${I_{3}{\bf
e}_{\infty}}$ ${-{\bf e}_{z}{\bf e}_{\infty}}$ ${{\bf e}_{y}{\bf e}_{\infty}}$
${-{\bf e}_{x}{\bf e}_{\infty}}$ ${\lambda\,{\bf e}_{x}{\bf e}_{\infty}}$
${\lambda\,{\bf e}_{x}{\bf e}_{\infty}}$ ${{\bf e}_{y}{\bf e}_{\infty}}$
${-{\bf e}_{z}{\bf e}_{\infty}}$ ${I_{3}{\bf e}_{\infty}}$ ${-1}$ ${-{\bf
e}_{x}{\bf e}_{y}}$ ${{\bf e}_{z}{\bf e}_{x}}$ ${-{\bf e}_{y}{\bf e}_{z}}$
${\lambda\,{\bf e}_{y}{\bf e}_{\infty}}$ ${\lambda\,{\bf e}_{y}{\bf
e}_{\infty}}$ ${-{\bf e}_{x}{\bf e}_{\infty}}$ ${I_{3}{\bf e}_{\infty}}$
${{\bf e}_{z}{\bf e}_{\infty}}$ ${{\bf e}_{x}{\bf e}_{y}}$ ${-1}$ ${-{\bf
e}_{y}{\bf e}_{z}}$ ${-{\bf e}_{z}{\bf e}_{x}}$ ${\lambda\,{\bf e}_{z}{\bf
e}_{\infty}}$ ${\lambda\,{\bf e}_{z}{\bf e}_{\infty}}$ ${I_{3}{\bf
e}_{\infty}}$ ${{\bf e}_{x}{\bf e}_{\infty}}$ ${-{\bf e}_{y}{\bf e}_{\infty}}$
${-{\bf e}_{z}{\bf e}_{x}}$ ${{\bf e}_{y}{\bf e}_{z}}$ ${-1}$ ${-{\bf
e}_{x}{\bf e}_{y}}$ ${\lambda\,I_{3}{\bf e}_{\infty}}$ ${\lambda\,I_{3}{\bf
e}_{\infty}}$ ${-{\bf e}_{z}{\bf e}_{\infty}}$ ${-{\bf e}_{y}{\bf
e}_{\infty}}$ ${-{\bf e}_{x}{\bf e}_{\infty}}$ ${-{\bf e}_{y}{\bf e}_{z}}$
${-{\bf e}_{z}{\bf e}_{x}}$ ${-{\bf e}_{x}{\bf e}_{y}}$ ${1}$

Table 1: Multiplication table for the geometric algebra ${\cal K}^{\lambda}$
defined in (30). Here ${I_{3}={\bf e}_{x}{\bf e}_{y}{\bf e}_{z}}$, ${{\bf
e}_{\infty}^{2}=+1}$, and ${\lambda=\pm 1}$.

###  3.2 Setting up notations

Next, we seek to close ${\cal K}^{\lambda}$ by normalizing its elements in
analogy with $S^{3}$ defined in (2). Since $S^{3}$ is a closed and compact
space with spherical topology, it is natural to expect its algebraic
representation space to have the same topological properties. To that end,
consider an arbitrary multivector in the vector space ${\cal K}^{\lambda}$
defined in (30) with the coefficients
$\\{q_{0},\,q_{1},\,q_{2},\,q_{3},\,q_{4},\,q_{5},\,q_{6},\,q_{7}\\}$:

${\mathbb{Q}}_{z}=\,q_{0}+q_{1}\,\lambda{\bf e}_{x}{\bf
e}_{y}+q_{2}\,\lambda{\bf e}_{z}{\bf e}_{x}+q_{3}\,\lambda{\bf e}_{y}{\bf
e}_{z}+q_{4}\,\lambda{\bf e}_{x}{\bf e}_{\infty}+q_{5}\,\lambda{\bf e}_{y}{\bf
e}_{\infty}+q_{6}\,\lambda{\bf e}_{z}{\bf e}_{\infty}+q_{7}\,\lambda I_{3}{\bf
e}_{\infty}\,.$ (32)

For our purposes, it is convenient to write this general element of ${\cal
K}^{\lambda}$ in terms of two independent quaternions defined in (2) as

${\mathbb{Q}}_{z}=\,{\bf q}_{r}+{\bf q}_{d}\,\varepsilon\,,$ (33)

so that

${\cal K}^{\lambda}=\left\\{{\mathbb{Q}}_{z}=\,{\bf q}_{r}+{\bf
q}_{d}\,\varepsilon\right\\},$ (34)

with

${\bf q}_{r}:=\,q_{0}+q_{1}\,\lambda\,{\bf e}_{x}{\bf
e}_{y}+q_{2}\,\lambda\,{\bf e}_{z}{\bf e}_{x}+q_{3}\,\lambda\,{\bf e}_{y}{\bf
e}_{z}\,,$ (35) ${\bf q}_{d}:=\,-q_{7}+q_{6}\,{\bf e}_{x}{\bf
e}_{y}+q_{5}\,{\bf e}_{z}{\bf e}_{x}+q_{4}\,{\bf e}_{y}{\bf e}_{z}\,,$ (36)

and pseudoscalar ${\varepsilon:=-\lambda\,I_{3}{\bf e}_{\infty}}$ with its
square ${\,\varepsilon^{2}={\bf e}^{2}_{\infty}=+1}$ and reverse (or
conjugate) $\varepsilon^{\dagger}=\varepsilon$. The quaternions ${\bf q}_{r}$
and ${\bf q}_{d}$ can be normalized, respectively, to radii ${\varrho_{r}}$
and $\varrho_{d}$, defining two copies of a 3-sphere embedded in
${\mathrm{I\\!R}}^{4}$ similar to that defined in (2):

$S_{r}^{3}=\left\\{\,{\bf q}_{r}=\,q_{0}+q_{1}\,\lambda\,{\bf e}_{x}{\bf
e}_{y}+q_{2}\,\lambda\,{\bf e}_{z}{\bf e}_{x}+q_{3}\,\lambda\,{\bf e}_{y}{\bf
e}_{z}\;\Big{|}\;||{\bf q}_{r}||=\sqrt{{\bf q}_{r}\,{\bf
q}_{r}^{\dagger}\,}=\varrho_{r}\,\right\\}\hookrightarrow{\mathrm{I\\!R}}^{4}$
(37)

and

$S_{d}^{3}=\left\\{\,{\bf q}_{d}=-q_{7}+q_{6}\,{\bf e}_{x}{\bf
e}_{y}+q_{5}\,{\bf e}_{z}{\bf e}_{x}+q_{4}\,{\bf e}_{y}{\bf
e}_{z}\;\Big{|}\;||{\bf q}_{d}||=\sqrt{{\bf q}_{d}\,{\bf
q}_{d}^{\dagger}\,}=\varrho_{d}\,\right\\}\hookrightarrow{\mathrm{I\\!R}}^{4}.$
(38)

This amounts to constraining the coefficients
${q_{0},q_{1},q_{2},q_{3},q_{4},q_{5},q_{6},}$ and $q_{7}$ of the vector
${\mathbb{Q}}_{z}$ with a scalar radius $\varrho_{c}$ as follows:

$||{\mathbb{Q}}_{z}||^{2}={\mathbb{Q}}_{z}\cdot{\mathbb{Q}}^{\dagger}_{z}=\,{\bf
q}_{r}\,{\bf q}^{\dagger}_{r}\,+\,{\bf q}_{d}\,{\bf
q}^{\dagger}_{d}\,=\,\varrho^{2}_{r}+\varrho^{2}_{d}\,=\sum_{k\,=\,0}^{7}q_{k}^{2}\,=\,\varrho^{2}_{c}.$
(39)

On the other hand, it is easy to work out that the analogous quantity involved
in the “cross term” also gives a scalar quantity:

$\sigma_{c}^{2}:={\bf q}_{r}\,{\bf q}^{\dagger}_{d}\,+\,{\bf q}_{d}\,{\bf
q}^{\dagger}_{r}=-2\,q_{0}q_{7}+2\,\lambda\,q_{1}q_{6}+2\,\lambda\,q_{2}q_{5}+2\,\lambda\,q_{3}q_{4}\,.$
(40)

###  3.3 Uniformly applicable definition of norm

Unfortunately, because the scalar product
${\mathbb{Q}}_{z}\cdot{\mathbb{Q}}^{\dagger}_{z}$ used in (39) to evaluate
norms is not a fundamental product in geometric algebra and therefore does not
preserve the Clifford algebra structure of ${\cal K}^{\lambda}$, the
prescription
$||{\mathbb{Q}}_{z}||=\sqrt{{\mathbb{Q}}_{z}\cdot{\mathbb{Q}}^{\dagger}_{z}\,}$
leads to internal inconsistency when ${\mathbb{Q}}_{z}$ happens to be a
product of two or more multivectors within ${\cal K}^{\lambda}$, such as
${\mathbb{Q}}_{z1}{\mathbb{Q}}_{z2}$ in (51) below. As we will see in
Subsection 4.3, this internal inconsistency comes to light when the left-hand
side and right-hand side of the composition law (51) ends up requiring two
incompatible products — a combination of geometric product and scalar product
on the left-hand side of (51) but only scalar products on the right-hand side
of (51). The prescription
$||{\mathbb{Q}}_{z}||=\sqrt{{\mathbb{Q}}_{z}\cdot{\mathbb{Q}}^{\dagger}_{z}\,}$
is also inconsistent with how the norms of the quaternions within ${\cal
K}^{\lambda}$ are evaluated using geometric products: $||{\bf
q}_{r}||:=\sqrt{{\bf q}_{r}\,{\bf q}^{\dagger}_{r}\,}$. We will therefore
employ a definition of norm that is uniformly applicable and remains
consistent when products of multivectors are involved:

######  Definition 3.1.

Norm $||{\mathbb{Q}}_{z}||$ :=
$\sqrt{{\mathbb{Q}}_{z}\,{\mathbb{Q}}^{\dagger}_{z}\,}$ with the coefficient
of the pseudoscalar part of the geometric product
${{\mathbb{Q}}_{z}\,{\mathbb{Q}}^{\dagger}_{z}}$ set to zero.

Note that this definition of norm is equivalent to the usual prescription
$||{\mathbb{Q}}_{z}||=\sqrt{{\mathbb{Q}}_{z}\cdot{\mathbb{Q}}^{\dagger}_{z}\,}$
for any individual multivector within ${\cal K}^{\lambda}$ when geometric
products are not involved. The two definitions give one and the same scalar
value for the norm. To see this, let us evaluate the geometric product
${\mathbb{Q}}_{z}{\mathbb{Q}}^{\dagger}_{z}$ using (33), (39), and (40). In
general, it turns out to take the following form:

$\displaystyle{\mathbb{Q}}_{z}{\mathbb{Q}}^{\dagger}_{z}\,$
$\displaystyle=\left({\bf q}_{r}+{\bf q}_{d}\,\varepsilon\right)\left({\bf
q}_{r}+{\bf q}_{d}\,\varepsilon\right)^{\dagger}$ (41)
$\displaystyle=\left({\bf q}_{r}+{\bf q}_{d}\,\varepsilon\right)\left({\bf
q}_{r}^{\dagger}+{\bf q}_{d}^{\dagger}\,\varepsilon\right)$ (42)
$\displaystyle=\left({\bf q}_{r}\,{\bf q}^{\dagger}_{r}\,+\,{\bf q}_{d}\,{\bf
q}^{\dagger}_{d}\right)+\left({\bf q}_{r}\,{\bf q}^{\dagger}_{d}\,+\,{\bf
q}_{d}\,{\bf q}^{\dagger}_{r}\right)\varepsilon$ (43)
$\displaystyle=\left(\varrho^{2}_{r}+\varrho^{2}_{d}\right)+\left(-2\,q_{0}q_{7}+2\,\lambda\,q_{1}q_{6}+2\,\lambda\,q_{2}q_{5}+2\,\lambda\,q_{3}q_{4}\right)\times\varepsilon$
(44) $\displaystyle:=\varrho^{2}_{c}+\sigma_{c}^{2}\,\varepsilon$ (45)
$\displaystyle=\,\text{(a scalar)}\,+\,\text{(a scalar)}\times\varepsilon$
(46) $\displaystyle=\,\text{(a scalar)}\,+\,\text{(a pseudoscalar)}.$ (47)

Thus, the geometric product ${\mathbb{Q}}_{z}{\mathbb{Q}}^{\dagger}_{z}$
resembles a hyperbolic number or split complex number instead of a scalar 20,
21. Definition 3.1 then leads to the following normalization condition for the
elements ${\mathbb{Q}}_{z}$, giving the same value $\varrho_{c}$ for the norm
obtained in (39):

$\sigma_{c}^{2}={\bf q}_{r}{\bf q}^{\dagger}_{d}+{\bf q}_{d}{\bf
q}^{\dagger}_{r}=-2\,q_{0}q_{7}+2\,\lambda\,q_{1}q_{6}+2\,\lambda\,q_{2}q_{5}+2\,\lambda\,q_{3}q_{4}=0.$
(48)

This amounts to setting the coefficient of the pseudoscalar $\varepsilon$ to
zero by assuming that the quaternions ${\bf q}_{r}$ and ${\bf q}_{d}$ are
orthogonal.

###  3.4 $S^{7}$ as an algebraic representation space of $S^{3}$

Equipped with a universally consistent and uniformly applicable definition of
norm, we are ready to prove the following lemmas.

######  Lemma 3.2.

The norms of the elements ${\mathbb{Q}}_{z}\in{\cal K}^{\lambda}$ defined in
(41) by
$||{\mathbb{Q}}_{z}||:=\sqrt{{\mathbb{Q}}_{z}{\mathbb{Q}}^{\dagger}_{z}\,}$
are positive definite, in the sense that

$||{\mathbb{Q}}_{z}||=0\iff{\mathbb{Q}}_{z}=0\,.$ (49)

The proof of Lemma 3.2 is straightforward. Since the norm of any quaternion is
positive definite, the norms of the quaternions ${\bf q}_{r}$ and ${\bf
q}_{d}$ are positive definite: $||{\bf q}_{r}||=0\Longleftrightarrow{\bf
q}_{r}=0$ and $||{\bf q}_{d}||=0\Longleftrightarrow{\bf q}_{d}=0$. From (43)
it is then evident that the scalar part of
$||{\mathbb{Q}}_{z}||:=\sqrt{{\mathbb{Q}}_{z}{\mathbb{Q}}^{\dagger}_{z}}$
cannot vanish unless both ${\bf q}_{r}$ and ${\bf q}_{d}$ are vanishing. But
in that case, ${\mathbb{Q}}_{z}=0$ according to its definition (33). Thus
$||{\mathbb{Q}}_{z}||=0$ if ${\mathbb{Q}}_{z}=0$. Conversely,
${\mathbb{Q}}_{z}$ would be zero only if both ${\bf q}_{r}$ and ${\bf q}_{d}$
are zero. But in that case $||{\mathbb{Q}}_{z}||=0$ using (43).

######  Lemma 3.3.

The norms of the elements ${\mathbb{Q}}_{z}\in{\cal K}^{\lambda}$ defined in
(41) by
$||{\mathbb{Q}}_{z}||:=\sqrt{{\mathbb{Q}}_{z}{\mathbb{Q}}^{\dagger}_{z}\,}$
are multiplicative, in the sense that the norm
$||{\mathbb{Q}}_{z1}{\mathbb{Q}}_{z2}||$ of the product of any two elements
${\mathbb{Q}}_{z1}$ and ${\mathbb{Q}}_{z2}$ in ${\cal K}^{\lambda}$ is equal
to the product of their norms $||{\mathbb{Q}}_{z1}||$ and
$||{\mathbb{Q}}_{z2}||$:

$||{\mathbb{Q}}_{z1}{\mathbb{Q}}_{z2}||=||{\mathbb{Q}}_{z1}||\;||{\mathbb{Q}}_{z2}||.$
(50)

The proof of Lemma 3.3 is given in Appendices A.1 below. It involves
straightforward computation that demonstrates that, for any arbitrary
multivectors in ${\cal K}^{\lambda}$ such as ${\mathbb{Q}}_{z1}$ and
${\mathbb{Q}}_{z2}$, both the left-hand side and the right-hand side of the
composition law 22

$||{\mathbb{Q}}_{z1}{\mathbb{Q}}_{z2}||^{2}\,=\;||{\mathbb{Q}}_{z1}||^{2}\;||{\mathbb{Q}}_{z2}||^{2}$
(51)

are equal to the same quantity

$\displaystyle\left\\{\left(\varrho^{2}_{r1}\,+\,\varrho^{2}_{d1}\right)\left(\varrho^{2}_{r2}\,+\,\varrho^{2}_{d2}\right)+\left({\bf
q}_{r1}\,{\bf q}^{\dagger}_{d1}+{\bf q}_{d1}\,{\bf
q}^{\dagger}_{r1}\right)\left({\bf q}_{r2}\,{\bf q}^{\dagger}_{d2}+{\bf
q}_{d2}\,{\bf q}^{\dagger}_{r2}\right)\right\\}$
$\displaystyle\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;+\left\\{\left(\varrho^{2}_{r1}\,+\,\varrho^{2}_{d1}\right)\left({\bf
q}_{r2}\,{\bf q}^{\dagger}_{d2}+{\bf q}_{d2}\,{\bf
q}^{\dagger}_{r2}\right)+\left(\varrho^{2}_{r2}\,+\,\varrho^{2}_{d2}\right)\left({\bf
q}_{r1}\,{\bf q}^{\dagger}_{d1}+{\bf q}_{d1}\,{\bf
q}^{\dagger}_{r1}\right)\right\\}\,\varepsilon\,.$ (52)

Recalling from (40) that the quantities such as ${\bf q}_{r1}\,{\bf
q}^{\dagger}_{d1}+{\bf q}_{d1}\,{\bf q}^{\dagger}_{r1}$ in (52) are scalar
quantities, we recognize that the above quantity resembles a split complex
number or hyperbolic number 20, 21 similarly to that of the product
${\mathbb{Q}}_{z}{\mathbb{Q}}^{\dagger}_{z}$ in (45). Consequently, as
explained in Appendix A.1, the square-root operation can be meaningfully
applied to (51) by treating the quantity in (52) as a hyperbolic number
$a+b\,\varepsilon$, with $a$ and $b$ being scalar numbers. This allows us to
express the norm relation (51) in square-root form as seen in (50), because a
product of two hyperbolic numbers, say $a_{1}+b_{1}\,\varepsilon$ and
$a_{2}+b_{2}\,\varepsilon$, is also a hyperbolic number.

The norm relation (50) is a general result. It holds for all multivectors
${\mathbb{Q}}_{z1}$ and ${\mathbb{Q}}_{z2}$ in ${\cal K}^{\lambda}$ without
exception, in addition to ${\cal K}^{\lambda}$ being closed under
multiplication. On the other hand, using the normalization condition (48), the
hyperbolic quantity (52) can be reduced to a scalar quantity by setting ${{\bf
q}_{r1}\,{\bf q}^{\dagger}_{d1}+{\bf q}_{d1}\,{\bf q}^{\dagger}_{r1}=0}$ and
${{\bf q}_{r2}\,{\bf q}^{\dagger}_{d2}+{\bf q}_{d2}\,{\bf
q}^{\dagger}_{r2}=0}$. This reduces the norm relation (50) to

$||{\mathbb{Q}}_{z1}\,{\mathbb{Q}}_{z2}||\,=\,\sqrt{\left(\varrho^{2}_{r1}\,+\,\varrho^{2}_{d1}\right)\left(\varrho^{2}_{r2}\,+\,\varrho^{2}_{d2}\right)}\,=\,||{\mathbb{Q}}_{z1}||\;||{\mathbb{Q}}_{z2}||\,.$
(53)

The advantage of this special case is that it reduces the values of the norms
from hyperbolic quantities to scalar quantities as a consequence of the
normalization condition (48), which is rooted in the fundamental nature of the
geometric product. Moreover, (53) allows us to construct the 7-sphere (17)
embedded in ${\mathrm{I\\!R}^{8}}$. This is because the norm relation (53)
ensures that, given two unit multivectors $X$ and $Y$ in ${\cal K}^{\lambda}$,
their product $Z=XY$ would also be a unit multivector in ${\cal K}^{\lambda}$.
This allows us to define $S^{7}$ as

${\cal K}^{\lambda}\hookleftarrow S^{7}:=\,\left\\{\,{\mathbb{Q}}_{z}:=\,{\bf
q}_{r}+{\bf q}_{d}\,\varepsilon\;\Big{|}\;\sigma_{c}^{2}={\bf q}_{r}\,{\bf
q}^{\dagger}_{d}+{\bf q}_{d}\,{\bf
q}^{\dagger}_{r}=0\Longleftrightarrow||{\mathbb{Q}}_{z}||=\sqrt{{\mathbb{Q}}_{z}{\mathbb{Q}}^{\dagger}_{z}\,}=\varrho_{c}\right\\}.$
(54)

This 7-sphere is used in Theorem 2.2 discussed in Subsection 2.2. Its
definition depends on the validity of the following lemma.

######  Lemma 3.4.

The normalization or orthogonality condition ${\sigma_{c}^{2}={\bf
q}_{r}\,{\bf q}^{\dagger}_{d}+{\bf q}_{d}\,{\bf q}^{\dagger}_{r}=0}$ is
preserved under multiplications of ${\mathbb{Q}}_{z}\in{\cal K}^{\lambda}$.

The proof of Lemma 3.4 is given in Appendix A.2. It proves that 7-sphere
defined in (54) remains closed under multiplications of the multivectors
${\mathbb{Q}}_{z}$ in ${\cal K}^{\lambda}$, despite satisfying the nontrivial
normalization condition ${\sigma_{c}^{2}={\bf q}_{r}\,{\bf
q}^{\dagger}_{d}+{\bf q}_{d}\,{\bf q}^{\dagger}_{r}=0}$ defined in (48).

######  Lemma 3.5.

The associative algebra ${{\cal K}}^{\lambda}$ defined in (34) that respects
the normalization condition ${\sigma_{c}^{2}={\bf q}_{r}\,{\bf
q}^{\dagger}_{d}+{\bf q}_{d}\,{\bf q}^{\dagger}_{r}=0}$ defined in (48) so
that the norm relation (53) holds with scalar values of the norms for all
multivectors ${\mathbb{Q}}_{z}\in{\cal K}^{\lambda}$, is a division algebra.

Given the scalar values of the norms that appear in (53), the proof of Lemma
3.5 is straightforward. It is given in Appendix A.3 below. The 7-sphere
defined in (54) is the algebraic representation space of the 3-sphere defined
in (2) we sought to arrive at. Just as the algebra ${\mathrm{Cl}_{3,0}}$ is an
algebraic representation space of ${\mathrm{I\\!R}^{3}}$, the $S^{7}$ defined
in (54) is an algebraic representation space of $S^{3}$.

###  3.5 Comparison with Hurwitz’s theorem

The claim of the original theorem by Hurwitz 23, 24 is that normed division
algebras are possible only in 1, 2, 4, and 8 dimensions. However, the claim of
the modern version of Hurwitz’s theorem is much stronger 25: Every normed
division algebra is isomorphic to either $\mathbb{R}$, $\mathbb{C}$,
$\mathbb{H}$, or $\mathbb{O}$, where $\mathbb{R}$, $\mathbb{C}$, $\mathbb{H}$,
and $\mathbb{O}$ stand, respectively, for the real numbers, complex numbers,
quaternions, and octonions. Consequently, the results proved above appear to
be in conflict with the modern claim of Hurwitz’s theorem. However, the proof
of Hurwitz’s theorem uses inner products to compute the norms whereas I have
used geometric products together with the normalization condition (48),
because using inner products leads to inconsistencies, as explained in
Subsection 4.3 below. Consequently, the underlying coefficient algebra (46) of
${{\cal K}^{\lambda}}$ resembles that of split complex numbers instead of real
numbers. Thus the assumptions underlying the modern claim of Hurwitz’s theorem
are different from those underlying the above results.

##  4 Response to some critiques

Let me now turn to some critiques in Lasenby 16 and Lasenby 26 of the 7-sphere
framework summarized above. My point-by-point response to the critique in
Lasenby 16 is available online 27. However, some of the same claims were
repeated in Lasenby 26, along with some new ones, in a presentation at the
AGACSE 2021 conference in honor of Professor David Hestenes. Unfortunately,
these critiques have preferred to view the algebraic framework I have
presented in Christian 5 as a “1d up” approach to Conformal Geometric Algebra
rather than the octonion-like 7-sphere perspective I have proposed for
reproducing the quantum correlations in Christian 5. This has led to several
misunderstandings. I address them in this section by responding to the main
issues raised.

###  4.1 Concerning the orientation of ${\cal K}^{\lambda}$

I have kept the orientation $\lambda$ unspecified for the vector space ${\cal
K}^{\lambda}$ discussed in the previous section. It plays the role of a
“hidden variable” or "common cause", or an initial state of the physical
system within a Bell-type 2 local-realistic framework for reproducing quantum
correlations. There is nothing unorthodox about this notion, as one can easily
find the definition of orientation of a vector space in any good textbook on
linear algebra. For example, we find the following definition of orientation
in Milnor 19:

######  Definition 4.1.

An orientation of a finite dimensional vector space ${{\cal V}_{n}}$ is an
equivalence class of ordered basis, say
${\left\\{b_{1},\,\dots,\,b_{n}\right\\}}$, which determines the same
orientation of ${\,{\cal V}_{n}}$ as the basis
${\left\\{b^{\prime}_{1},\,\dots,\,b^{\prime}_{n}\right\\}}$ if
${b^{\prime}_{i}=\omega_{ij}\,b_{j}}$ holds with ${{\rm det}(\omega_{ij})>0}$,
and the opposite orientation of ${{\cal V}_{n}}$ as the basis
${\left\\{b^{\prime}_{1},\,\dots,\,b^{\prime}_{n}\right\\}}$ if
${b^{\prime}_{i}=\omega_{ij}\,b_{j}}$ holds with ${{\rm det}(\omega_{ij})<0}$.

However, in Lasenby 16 my use of an unspecified orientation $\lambda$ of the
vector space ${\cal K}^{\lambda}$ is not understood and claimed to be a
"problem”, without explanation. In fact, there is no problem with keeping the
orientation of ${\cal K}^{\lambda}$ unspecified, because nowhere in Christian
5 or elsewhere have I used ${\cal K}^{+}$ and ${\cal K}^{-}$ to represent two
different algebras. They both span the same algebra. Moreover, it is evident
from the definition stated above that the notion of orientation is of only
relative significance. If one of the vector spaces (say ${\cal K}^{+}$) is
deemed right-handed, then the other vector space (${\cal K}^{-}$) would be
left-handed, and vice versa. The spaces ${\cal K}^{+}$ and ${\cal K}^{-}$ are
thus mathematically not identical as claimed in Lasenby 16. This is very
clearly explained in Christian 5 as follows:

> “It is easy to verify that the bases of ${{\cal K}^{+}}$ and ${{\cal
> K}^{-}}$ are indeed related by an ${8\times 8}$ diagonal matrix whose
> determinant is ${(-1)^{7}<0}$. Consequently, ${{\cal K}^{+}}$ and ${{\cal
> K}^{-}}$ indeed represent right-oriented and left-oriented vector spaces,
> respectively, in accordance with our definition of orientation. We can
> therefore leave the orientation unspecified and write ${{\cal K}^{\pm}}$ as
>
> ${\cal K}^{\lambda}=\,{\rm span}\\!\left\\{\,1,\,\lambda{\bf e}_{x}{\bf
> e}_{y},\,\lambda{\bf e}_{z}{\bf e}_{x},\,\lambda{\bf e}_{y}{\bf
> e}_{z},\,\lambda{\bf e}_{x}{\bf e}_{\infty},\,\lambda{\bf e}_{y}{\bf
> e}_{\infty},\,\lambda{\bf e}_{z}{\bf e}_{\infty},\,\lambda I_{3}{\bf
> e}_{\infty}\,\right\\}\\!,\,\;\lambda^{2}=1\iff\lambda=\pm 1.\text{''}$

###  4.2 Concerning relative handedness within $S^{3}$

Unfortunately, the above notion of vector space with an unspecified
orientation has led to some confusion in Lasenby 16 and Lasenby 26. This has
led to an instructive conceptual mistake in Lasenby 26. Since I have already
addressed it rather extensively in Questions 13, 14, and 15 in Appendix B of
Christian 7 and in Subsection IV D of Christian 28, it will suffice to be
brief here.

The issue raised in Lasenby 26 concerns the handedness (or orientation)
$\lambda$ of the spin bivector basis (3) with respect to the detector bivector
basis (4) discussed in Subsection 2.1 above. Mathematically, the relative
handedness of the spins with respect to their detectors is specified by the
relations (5) and (6). As I stressed in Subsection 2.1, in the 3-sphere model
the orientation $\lambda$ of $S^{3}$ has only relative meaning, in accordance
with the Definition 4.1 of orientation stated above. For the two bivector
bases $\left\\{\,1,\;L_{1},\;L_{2},\;L_{3}\,\right\\}$ and
${\left\\{\,1,\,D_{1},\,D_{2},\,D_{3}\,\right\\}}$ satisfying, respectively,
the subalgebras (3) and (4), this implies the following relation:

$\left(\begin{array}[]{c}1\\\ L_{1}\\\ L_{2}\\\
L_{3}\end{array}\right)\;=\;\begin{pmatrix}\;1\;\,&\;\,0\;\,&\;\,0\;\,&\;\,0\;\;\,\\\
\,0&\lambda&0&0\,\\\ \,0&0&\lambda&0\,\\\
\,0&0&0&\lambda\end{pmatrix}\;\left(\begin{array}[]{c}1\\\ D_{1}\\\ D_{2}\\\
D_{3}\end{array}\right).$ (55)

Consequently, the relationship between the two bivector frames is the
following:

$L_{1}L_{2}L_{3}=\lambda\,D_{1}D_{2}D_{3}=\pm\,D_{1}D_{2}D_{3}.$ (56)

This equation describes how the perspectives of the spins and the detectors
are related to each other while both objectively respect the same bivector
subalgebra. Moreover, physically this is not only a relative relationship but
also a contingent relationship. Since the spins ${{\mathbf{L}}({\mathbf{s}})}$
and the detectors ${{\mathbf{D}}({\mathbf{n}})}$ are two entirely unrelated
physical systems, Alice and Bob have no prior knowledge of the handedness of
the spins until they are detected by their detectors. This makes the
relationship (6) between the spins and detectors contingent upon the spin
direction ${\mathbf{s}}$ being aligned to the detector direction
${\mathbf{n}}$ during the detection processes specified by the measurement
functions (7) and (11). This is evident not only from these functions but also
from the defining relation (6).

Unfortunately, in Lasenby 26 the above point of view is either overlooked or
misinterpreted. For in Lasenby 26 the contingent relation
$D_{i}=\lambda\,L_{i}$ from Eq. (5) is substituted into Eq. (4) that defines
the bivector subalgebra respected by the detectors to obtain

$L_{i}\,L_{j}\,=\,-\,\delta_{ij}\,-\lambda\sum_{k}\epsilon_{ijk}\,L_{k}.$ (57)

Similarly, by substituting the relation $L_{i}(\lambda)=\lambda\,D_{i}$ from
(5) into (3) that defines the bivector subalgebra for spins we can obtain

$D_{i}\,D_{j}\,=\,-\,\delta_{ij}\,-\lambda\sum_{k}\epsilon_{ijk}\,D_{k}.$ (58)

But such a mathematical substitution has no meaning either within the physics
of the Bell-test experiments or within the quaternionic 3-sphere model 6, 7.
Indeed, comparing the above pair of equations with the defining equations (3)
and (4) of spins and detectors we notice that there is an extra $\lambda$
appearing in (57) and (58) just before the summation. As a result, when
$\lambda=-1$, equations (57) and (58) do not represent the bivector subalgebra
of the algebra ${\mathrm{Cl}}_{(3,0)}$ and therefore do not constitute the
quaternionic 3-sphere defined in (2). Thus the above mathematical step in
Lasenby 26 has taken us out of the quaternionic 3-sphere model altogether.
This can be verified by expressing the spin bivector basis $L_{i}$ in terms of
the vector basis
$\\{\mathbf{e}_{1}^{\prime},\mathbf{e}_{2}^{\prime},\mathbf{e}_{3}^{\prime}\\}$
so that

$L_{1}=\mathbf{e}_{2}^{\prime}\mathbf{e}_{3}^{\prime},\;\;\;L_{2}=\mathbf{e}_{3}^{\prime}\mathbf{e}_{1}^{\prime},\;\;\;\text{and}\;\;\;L_{3}=\mathbf{e}_{1}^{\prime}\mathbf{e}_{2}^{\prime}.$
(59)

Straightforward textbook calculations 9 then give us the bivector subalgebra
(3), not (57). Given the rules of Geometric Algebra and the properties of
$\delta_{ij}$ and $\epsilon_{ijk}$, there is no way to change the sign from
minus to plus appearing before the sum on the right-hand side of (3).
Therefore, equations (57) and (58) are not correct even mathematically, let
alone their physical meaninglessness.

What has been overlooked in Lasenby 26 that led us to the incorrect equations
(57) and (58) with extra $\lambda$ is the fact that orientation of $S^{3}$ is
of only relative significance between spin bivectors and detector bivectors.
Since equation (57) was obtained by substituting the contingent relation
$D_{i}=\lambda\,L_{i}$ from (5) into (4), it is meaningful only with respect
to the detector bivectors defined by (4). Likewise, since equation (58) was
obtained by substituting the contingent relation $L_{i}=\lambda\,D_{i}$ from
(5) into (3), it is meaningful only with respect to the spin bivectors defined
by (3). Neither (57) nor (58) has objective meaning. The only correct
interpretation of the equations (3) to (6) defining the two sets of bases
within $S^{3}$ is how I have interpreted them in Christian 6 and Christian 7.
As demonstrated in Christian 7, the mutual perspectives of the spin and
detector bivectors are related as follows:

${\mathbf{L}}({\mathbf{a}},\,{\lambda}=+1)\;{\mathbf{L}}({\mathbf{b}},\,{\lambda}=+1)={\mathbf{D}}({\mathbf{a}})\;{\mathbf{D}}({\mathbf{b}}),$
(60)

and

${\mathbf{L}}({\mathbf{a}},\,{\lambda}=-1)\;{\mathbf{L}}({\mathbf{b}},\,{\lambda}=-1)={\mathbf{D}}({\mathbf{b}})\;{\mathbf{D}}({\mathbf{a}}).$
(61)

In other words, the order of the product of two spin bivectors and the order
of the product of two detector bivectors is mutually opposite. This is what
the physics of the EPR-Bohm experiments within the quaternionic 3-sphere model
dictates. Accordingly, when the hidden variable or initial state ${\lambda}$
happens to be equal to ${+1}$,
${{\mathbf{L}}({\mathbf{a}},{\lambda}){\mathbf{L}}({\mathbf{b}},{\lambda})={\mathbf{D}}({\mathbf{a}}){\mathbf{D}}({\mathbf{b}})}$,
and when the hidden variable or initial state ${\lambda}$ happens to be equal
to ${-1}$,
${{\mathbf{L}}({\mathbf{a}},{\lambda}){\mathbf{L}}({\mathbf{b}},{\lambda})={\mathbf{D}}({\mathbf{b}}){\mathbf{D}}({\mathbf{a}})}$.
As result, the expectation value (15) reduces at once to

$\displaystyle{\cal E}_{\mathrm{L.R.}}({\mathbf{a}},{\mathbf{b}})$
$\displaystyle=\frac{1}{2}\\{\,{\mathbf{D}}({\mathbf{a}}){\mathbf{D}}({\mathbf{b}})\\}+\frac{1}{2}\\{\,{\mathbf{D}}({\mathbf{b}}){\mathbf{D}}({\mathbf{a}})\\}$
(62)
$\displaystyle=\frac{1}{2}(I\cdot{\mathbf{a}})(I\cdot{\mathbf{b}})+\frac{1}{2}(I\cdot{\mathbf{b}})(I\cdot{\mathbf{a}})$
(63)
$\displaystyle=-\,\frac{1}{2}\left\\{{\mathbf{a}}{\mathbf{b}}+{\mathbf{b}}{\mathbf{a}}\right\\}=-\,{\mathbf{a}}\cdot{\mathbf{b}}\,,$
(64)

because the orientation ${\lambda}$ of ${S^{3}}$ is a fair coin with 50/50
chance. Here the last equality follows from the definition of inner product.

In Section 5 below I have presented an alternative derivation of the singlet
correlations (15) within the quaternionic 3-sphere model that is immune to the
critique in Lasenby 26, because it is based on a fixed orientation,
$\lambda\equiv+1$, between spins and detectors.

###  4.3 Concerning the validity of composition law

In Lasenby 16 and Lasenby 26 another incorrect claim is made. It concerns the
proof in Christian 5 that the norm relation (50) or (53) discussed above holds
for all elements $X$ and $Y$ of ${\cal K}^{\lambda}$. As we saw in Subsection
3.4, the proof of (53) is quite straightforward. It can be found in Subsection
2.5 of Christian 5, and, with greater detail, in Appendix A.1 below. However,
in Lasenby 16 and Lasenby 26 it is claimed that the norm relation (53) is not
correct, and an attempt is made to disprove it with a counterexample.

Unfortunately, the alleged counterexample is based on an internal
inconsistency. Two incompatible product rules are used on the two sides of the
norm relation (50) to demonstrate contradiction. Using the pseudoscalar
$\varepsilon$ involved in the algebra ${\cal K}^{\lambda}$ that satisfies the
properties $\varepsilon^{\dagger}=\varepsilon$ and $\varepsilon^{2}=1$, the
following two multivectors within ${\cal K}^{\lambda}$ are considered to
arrive at a contradiction:

$X=\frac{1}{\sqrt{2}}(1+\varepsilon)\;\;\;\;\text{and}\;\;\;\;Y=\frac{1}{\sqrt{2}}(1-\varepsilon).$
(65)

But, to begin with, no such two-dimensional objects play any role whatsoever
in the 7-sphere framework presented in Christian 5. Thus, even if the above
multivectors are mathematically admissible, they have no relevance for the
physical framework presented in Christian 5. Next, scalar products ${X\cdot
X^{\dagger}}=||X||^{2}$ and ${Y\cdot Y^{\dagger}}=||Y||^{2}$ are used to
evaluate the norms $||X||$ and $||Y||$, giving

$||{X}||=\left|\left|\frac{1}{\sqrt{2}}(1+\varepsilon)\right|\right|=\sqrt{\frac{1}{2}(1+\varepsilon)\cdot(1+\varepsilon)^{\dagger}}=\sqrt{\frac{1}{2}(1+1)}=1$
(66)

and

$||{Y}||=\left|\left|\frac{1}{\sqrt{2}}(1-\varepsilon)\right|\right|=\sqrt{\frac{1}{2}(1-\varepsilon)\cdot(1-\varepsilon)^{\dagger}}=\sqrt{\frac{1}{2}(1+1)}=1.$
(67)

Substituting for these values into the right-hand side of (50) therefore gives
$||X||\,||Y||=1$. On the other hand, to evaluate the left-hand side of (50), a
geometric product between the two multivectors $X$ and $Y$ is used in Lasenby
16 and Lasenby 26 to obtain

$||XY||=\left|\left|\frac{1}{2}(1+\varepsilon)(1-\varepsilon)\right|\right|=\frac{1}{2}\left|\left|(1-\varepsilon^{2})\right|\right|=||(1-1)||=||\,0\,||=0.$
(68)

Indeed, not using geometric product on the left-hand side of (50) is not an
option because that would not result in $XY\in{\cal K}^{\lambda}$. But since
two incompatible product rules are used on the two sides of (50), it is not
surprising that desired contradiction is achieved:

$0=||{X}{Y}||\not=||{X}||\,||{Y}||=1.$ (69)

Since not using the geometric product for $XY$ on the left-hand side of (50)
is not a meaningful operation within ${\cal K}^{\lambda}$, the only way to
avoid the inconsistency in what is considered in Lasenby 16 and Lasenby 26 is
by using geometric products throughout the above calculation, as discussed in
Lemma 3.3 of Subsection 3.4. Then the correct calculation of the norms $||X||$
and $||Y||$ gives

$||X||=\left|\left|\frac{1}{\sqrt{2}}(1+\varepsilon)\right|\right|=\sqrt{\frac{1}{2}(1+\varepsilon)(1+\varepsilon)^{\dagger}\,}=\sqrt{\frac{1}{2}(1+\varepsilon+\varepsilon+\varepsilon^{2})\,}=\sqrt{1+\varepsilon\,}$
(70)

and

$||Y||=\left|\left|\frac{1}{\sqrt{2}}(1-\varepsilon)\right|\right|=\sqrt{\frac{1}{2}(1-\varepsilon)(1-\varepsilon)^{\dagger}\,}=\sqrt{\frac{1}{2}(1-\varepsilon-\varepsilon+\varepsilon^{2})\,}=\sqrt{1-\varepsilon\,},$
(71)

which now gives 0 instead of 1 for the right-hand side of (50):

$||X||\,||Y||=\left(\sqrt{1+\varepsilon}\,\right)\left(\sqrt{1-\varepsilon}\,\right)=\sqrt{(1+\varepsilon)(1-\varepsilon)}=\sqrt{1-\varepsilon^{2}}=0.$
(72)

Comparing (68) and (72) we see that now both the left-hand side $||XY||$ and
the right-hand side $||X||\,||Y||$ of (50) have the same value (namely, 0),
and consequently the norm relation $||XY||=||X||\,||Y||$ is satisfied. As a
result, the counterexample to (50) alleged in Lasenby 16 and Lasenby 26 fails.
This is not surprising because, as discussed in Subsection 3.4 above and
proved explicitly in Appendix A.1 below in full generality, the norm relation
(50) holds for any arbitrary multivectors $X$ and $Y$ in ${\cal K}^{\lambda}$.

It is instructive here to understand how the normalization condition ${\bf
q}_{r}{\bf q}^{\dagger}_{d}+{\bf q}_{d}{\bf q}^{\dagger}_{r}=0$ is satisfied
in this case for the product $XY$. To that end, using (33) and (65), let us
compare the coefficients of the two multivectors $X$ and $Y$ with those
defined in (33):

$X=\frac{1}{\sqrt{2}}(1+\varepsilon)={\mathbb{Q}}_{z1}=\,{\bf q}_{r1}+{\bf
q}_{d1}\,\varepsilon$ (73)

and

$Y=\frac{1}{\sqrt{2}}(1-\varepsilon)={\mathbb{Q}}_{z2}=\,{\bf q}_{r2}+{\bf
q}_{d2}\,\varepsilon\,.$ (74)

Thus, the coefficients for $X$ and $Y$ chosen in (65) are the following: ${\bf
q}_{r1}=\frac{1}{\sqrt{2}}$, ${\bf q}_{d1}=\frac{1}{\sqrt{2}}$, ${\bf
q}_{r2}=\frac{1}{\sqrt{2}}$, and ${\bf q}_{d2}=\frac{-1}{\sqrt{2}}$. As a
result, we have: ${\varrho^{2}_{r1}={\bf q}_{r1}{\bf
q}^{\dagger}_{r1}=\frac{1}{2}}$, ${\varrho^{2}_{d1}={\bf q}_{d1}{\bf
q}^{\dagger}_{d1}=\frac{1}{2}}$, ${\varrho^{2}_{r2}={\bf q}_{r2}{\bf
q}^{\dagger}_{r2}=\frac{1}{2}}$, ${\varrho^{2}_{d2}={\bf q}_{d2}{\bf
q}^{\dagger}_{d2}=\frac{1}{2}}$, ${{\bf q}_{r1}{\bf
q}^{\dagger}_{d1}=\frac{1}{2}}$, ${{\bf q}_{d1}{\bf
q}^{\dagger}_{r1}=\frac{1}{2}}$, ${{\bf q}_{r2}{\bf
q}^{\dagger}_{d2}=\frac{-1}{2}}$, and ${{\bf q}_{d2}{\bf
q}^{\dagger}_{r2}=\frac{-1}{2}}$. Substituting these into the quantity (52)
that is equal to both left-hand and right-hand sides of (51) reduces (52) to

$\left\\{\left(\frac{1}{2}+\frac{1}{2}\right)\left(\frac{1}{2}+\frac{1}{2}\right)+\left(\frac{1}{2}+\frac{1}{2}\right)\left(-\frac{1}{2}-\frac{1}{2}\right)\right\\}+\left\\{\left(\frac{1}{2}+\frac{1}{2}\right)\left(-\frac{1}{2}-\frac{1}{2}\right)+\left(\frac{1}{2}+\frac{1}{2}\right)\left(\frac{1}{2}+\frac{1}{2}\right)\right\\}\varepsilon=\\{0\\}+\\{0\\}\,\varepsilon=0.$
(75)

This again confirms the results (68) and (72) for the two sides of the norm
relation (51). But more importantly, we see from (75) that the coefficient of
the pseudoscalar $\varepsilon$ vanishes, and consequently the normalization
condition (48) is automatically satisfied.

###  4.4 Concerning a subsequent incorrect claim

In Lasenby 16 a further incorrect claim is made that if $X$ and $Y$ "are unit
vectors, it does not follow that $Z=XY$ is also a unit vector, despite what
Christian says in his Eq. (2.41) [of Christian 5].” But now it is easy to
appreciate that this claim is not correct because it depends on the validity
of the counterexample alleged in Lasenby 16, which I have just shown to be
invalid. Consequently, contrary to the claim in Lasenby 16, Eq. (2.41) of
Christian 5 is quite correct. Since the norm relation (53) holds for any
multivectors $X$ and $Y$ in ${\cal K}^{\lambda}$ that have scalar values for
their norms, if $X$ and $Y$ happen to be unit vectors so that $||X||=1$ and
$||Y||=1$, then (53) necessitates that their product $Z=XY$ will also be a
unit vector: $||Z||=||XY||=||X||\,||Y||=1$.

###  4.5 Concerning the reduction in dimensions

In Lasenby 16 it is claimed that “… while the scalar part of $X\tilde{X}$ sets
up a nice match with $S^{7}$, the extra constraint from
$\langle{X\tilde{X}}\rangle_{4}=0$ reduces the overall dimension down to 6, …”
However, this claim is also incorrect. Here what is meant by
$\langle{X\tilde{X}}\rangle_{4}=0$ is the vanishing of the coefficient of the
pseudoscalar as in the normalization condition (48). But it is easy to see
from this condition that it only reduces the eight dimensions of ${\cal
K}^{\lambda}$ to the seven dimensions of $S^{7}$, precisely as one would
expect from the relationship between the embedding space ${\cal
K}^{\lambda}\sim\mathrm{I\\!R}^{8}$ and the embedded space $S^{7}$. To
appreciate this explicitly, let us express (48) as follows:

${q}_{0}=\frac{\lambda}{{q}_{7}}({q}_{1}{q}_{6}+{q}_{2}{q}_{5}+{q}_{3}{q}_{4}).$
(76)

This shows that the condition (48) can be used to eliminate only one of the
eight coefficients of ${\mathbb{Q}}_{z}\in{\cal K}^{\lambda}$ defined in (32),
say $q_{0}\,$, by writing it in terms of the remaining seven coefficients of
${\mathbb{Q}}_{z}$, as in the above equation. But that still leaves seven
numbers that would be necessary to specify a point in ${\cal K}^{\lambda}$
located at the radius $\varrho_{c}$ from the origin, not six numbers as
claimed in Lasenby 16. Moreover, this reduction of one dimension while
transitioning from the eight dimensions of ${\cal
K}^{\lambda}\sim\mathrm{I\\!R}^{8}$ to the seven dimensions of $S^{7}$ as a
result of the normalization condition (48) or (76) is perfectly consistent
with the one less dimension of the spheres $S^{0}$, $S^{1}$, $S^{3}$, and
$S^{7}$ compared to that of the corresponding normed division algebras
$\mathbb{R}$, $\mathbb{C}$, $\mathbb{H}$, and $\mathbb{O}$ mentioned in
Subsection 3.5.

##  5 Alternative Derivation of the singlet correlations within $S^{3}$

Let me now present an alternative proof of Theorem 2.1 stated in Subsection
2.1 that does not depend on two different orientations of $S^{3}$ for the spin
bivectors and detector bivectors. In other words, in what follows the
orientation of $S^{3}$ is fixed to the value $\lambda=+1$, and kept fixed
throughout the experiment for both the spin bivectors and detector bivectors.
This makes the derivation of the singlet correlations immune to the kind of
misinterpretation of $\lambda=\pm 1$ in Lasenby 26 we discussed in Subsection
4.2 above.

Since the handedness of spin bivectors and detector bivectors is now fixed to
be $\lambda=+1$, they both satisfy the right-handed bivector subalgebra 9.
However, because the spins and detectors are different physical systems,
unrelated to each other until the acts of measurements, we will continue to
represent them using different notations, $L_{i}$ for spin basis and $D_{i}$
for detector basis:

$L_{i}\,L_{j}\,=\,-\,\delta_{ij}\,-\sum_{k}\epsilon_{ijk}\,L_{k}$ (77)

and

$D_{i}\,D_{j}\,=\,-\,\delta_{ij}\,-\sum_{k}\epsilon_{ijk}\,D_{k}.$ (78)

Since the binary freedom of orientation $\lambda$ is no longer available,
instead of (6) let us introduce the following two sign-functions:

$\mu_{1}=\text{sign}(\mathbf{a}\cdot\mathbf{s}^{i}_{1})=\pm
1\;\;\;\text{and}\;\;\;\mu_{2}=\text{sign}(\mathbf{s}^{i}_{2}\cdot\mathbf{b})=\pm
1,$ (79)

where the subscripts 1 and 2 on $\mu$ and ${\mathbf{s}}$ refer to the remote
observation stations of the experimenters Alice and Bob (cf. Figure 1), and
the superscript $i$ on ${\mathbf{s}}$ indicates its initial direction at the
source with respect to the eventually chosen detector directions. Moreover,
since $\lambda$ is no longer available as a hidden variable, let the spin
direction ${\mathbf{s}}={\mathbf{s}}_{1}={\mathbf{s}}_{2}$ now act as a hidden
variable, just as in Bell’s original local model 2, 29. Then the function
${\mu_{1}=\mathrm{sign}({\mathbf{a}}\cdot{\mathbf{s}}^{i}_{1})}$ can be
understood as follows. If, initially (i.e., before the detection process
defined by the measurement functions to be specified below), the two unit
vectors ${\mathbf{a}}$ and ${\mathbf{s}^{i}_{1}}$ happen to be pointing
through the same hemisphere of $S^{2}\hookrightarrow\mathrm{I\\!R}^{3}$
centered at the origin of ${\mathbf{s}^{i}_{1}}$, then
$\mu_{1}=\mathrm{sign}({\mathbf{a}}\cdot{\mathbf{s}}^{i}_{1})$ will be equal
to $+1$, and if the two unit vectors ${\mathbf{a}}$ and ${\mathbf{s}^{i}_{1}}$
happen to be pointing through the opposing hemispheres of $S^{2}$ centered at
the origin of ${\mathbf{s}^{i}_{1}}$, then
$\mu_{1}=\mathrm{sign}({\mathbf{a}}\cdot{\mathbf{s}}^{i}_{1})$ will be equal
to $-1$, provided that ${{\mathbf{a}}\cdot{\mathbf{s}}^{i}_{1}\not=0}$. If
${{\mathbf{a}}\cdot{\mathbf{s}}^{i}_{1}}$ happens to be zero, then
$\mu_{1}=\mathrm{sign}({\mathbf{a}}\cdot{\mathbf{s}}^{i}_{1})$ will be assumed
to be equal to the sign of the first nonzero component from the set
${\\{a_{x},\,a_{y},\,a_{z}\\}}$. And likewise for the function
${\mu_{2}=\mathrm{sign}({\mathbf{s}}^{i}_{2}\cdot{\mathbf{b}})}$.

Next, Theorem 2.1 says that to explain the singlet correlations local-
realistically we should consider correlations among pairs of limiting scalar
points of the 3-sphere defined in (2), which is a set of unit quaternions. We
therefore use products of the spin bivectors
${\mathbf{L}}({\mathbf{s}}_{1})=I_{3}{\mathbf{s}}_{1}$ and
${\mathbf{L}}({\mathbf{s}}_{2})=I_{3}{\mathbf{s}}_{2}$ with detector bivectors
${\mathbf{D}}({\mathbf{a}})=I_{3}{\mathbf{a}}$ and
${\mathbf{D}}({\mathbf{b}})=I_{3}{\mathbf{b}}$ using the trivector $I_{3}={\bf
e}_{x}{\bf e}_{y}{\bf e}_{z}$ to represent the physical interactions that take
place during the detection processes at the two ends of a Bell-test
experiment. Following Bell 2, these can be expressed as measurements functions
${\mathscr{A}}({\mathbf{a}},{\mathbf{s}}_{1})$ and
${\mathscr{B}}({\mathbf{b}},{\mathbf{s}}_{2})$ for the experimenters Alice and
Bob:

$\displaystyle S^{3}\ni{\mathscr{A}}({\mathbf{a}},{\mathbf{s}}_{1})\,$
$\displaystyle=\lim_{{\mathbf{s}}_{1}\,\rightarrow\,\mu_{1}{\mathbf{a}}}\left\\{-\,{\mathbf{D}}({\mathbf{a}})\,{\mathbf{L}}({\mathbf{s}}_{1})\right\\}$
(80)
$\displaystyle=\lim_{{\mathbf{s}}_{1}\,\rightarrow\,\mu_{1}{\mathbf{a}}}\left\\{-(I_{3}{\mathbf{a}})(I_{3}{\mathbf{s}}_{1})\right\\}$
(81)
$\displaystyle=\lim_{{\mathbf{s}}_{1}\,\rightarrow\,\mu_{1}{\mathbf{a}}}\left\\{{\mathbf{a}}\cdot{\mathbf{s}}_{1}+I_{3}({\mathbf{a}}\times{\mathbf{s}}_{1})\right\\}$
(82)
$\displaystyle=\lim_{{\mathbf{s}}_{1}\,\rightarrow\,\mu_{1}{\mathbf{a}}}\left\\{\cos(\eta_{{\mathbf{a}}{\mathbf{s}}_{1}})+(I_{3}{\mathbf{r}}_{1})\sin(\eta_{{\mathbf{a}}{\mathbf{s}}_{1}})\right\\}$
(83)
$\displaystyle=\lim_{{\mathbf{s}}_{1}\,\rightarrow\,\mu_{1}{\mathbf{a}}}\left\\{\,+\,{\mathbf{q}}(\eta_{{\mathbf{a}}{\mathbf{s}}_{1}},\,{\mathbf{r}}_{1})\right\\},$
(84)

where $\eta_{{\mathbf{a}}{\mathbf{s}}_{1}}$ is the angle between the spin
direction ${\mathbf{s}}_{1}$ and the detector direction ${\mathbf{a}}$ chosen
by Alice (cf. Figure 1), and ${\mathbf{r}}_{1}$ is the rotation axis of the
quaternion
${\mathbf{q}}(\eta_{{\mathbf{a}}{\mathbf{s}}_{1}},\,{\mathbf{r}}_{1})$ as in
(9). Thus, given the definition (79) of $\mu_{1}$, the spin direction
${\mathbf{s}}_{1}$ will tend to $+{\mathbf{a}}$ if initially the two unit
vectors ${\mathbf{a}}$ and ${\mathbf{s}_{1}}$ happen to be pointing through
the same hemisphere of $S^{2}$ centered at the origin of ${\mathbf{s}_{1}}$,
and otherwise the spin direction ${\mathbf{s}}_{1}$ will tend to
$-{\mathbf{a}}$. Consequently, as
${\mathbf{s}}_{1}\rightarrow\mu_{1}{\mathbf{a}}=\pm{\mathbf{a}}$ during the
detection process by Alice so that the angle
$\eta_{{\mathbf{a}}{\mathbf{s}}_{1}}\\!\rightarrow 0$ or $\pi$, the binary
value of the observed result by Alice is obtained because
$\cos(\eta_{{\mathbf{a}}{\mathbf{s}}_{1}})\rightarrow\pm 1$ and
$\sin(\eta_{{\mathbf{a}}{\mathbf{s}}_{1}})\rightarrow 0$:

${\mathscr{A}}({\mathbf{a}},{\mathbf{s}}_{1})\longrightarrow+\mu_{1}=\pm 1.$
(85)

Similarly, we can define the measurement function for Bob as

$\displaystyle S^{3}\ni{\mathscr{B}}({\mathbf{b}},{\mathbf{s}}_{2})\,$
$\displaystyle=\lim_{{\mathbf{s}}_{2}\,\rightarrow\,\mu_{2}{\mathbf{b}}}\left\\{+\,{\mathbf{L}}({\mathbf{s}}_{2})\,{\mathbf{D}}({\mathbf{b}})\right\\}$
(86)
$\displaystyle=\lim_{{\mathbf{s}}_{2}\,\rightarrow\,\mu_{2}{\mathbf{b}}}\left\\{+(I_{3}{\mathbf{s}}_{2})(I_{3}{\mathbf{b}})\right\\}$
(87)
$\displaystyle=\lim_{{\mathbf{s}}_{2}\,\rightarrow\,\mu_{2}{\mathbf{b}}}\left\\{-\,{\mathbf{s}}_{2}\cdot{\mathbf{b}}-I_{3}({\mathbf{s}}_{2}\times{\mathbf{b}})\right\\}$
(88)
$\displaystyle=\lim_{{\mathbf{s}}_{2}\,\rightarrow\,\mu_{2}{\mathbf{b}}}\left\\{-\cos(\eta_{{\mathbf{s}}_{2}{\mathbf{b}}})-(I_{3}{\mathbf{r}}_{2})\sin(\eta_{{\mathbf{s}}_{2}{\mathbf{b}}})\right\\}$
(89)
$\displaystyle=\lim_{{\mathbf{s}}_{2}\,\rightarrow\,\mu_{2}{\mathbf{b}}}\left\\{\,-\,{\mathbf{q}}(\eta_{{\mathbf{s}}_{2}{\mathbf{b}}},\,{\mathbf{r}}_{2})\right\\},$
(90)

where $\eta_{{\mathbf{s}}_{2}{\mathbf{b}}}$ is the angle between the spin
direction ${\mathbf{s}}_{2}$ and the detector direction ${\mathbf{b}}$ chosen
by Bob (cf. Figure 1), and ${\mathbf{r}}_{2}$ is the rotation axis of the
quaternion
${\mathbf{q}}(\eta_{{\mathbf{s}}_{2}{\mathbf{b}}},\,{\mathbf{r}}_{2})$ as in
(13). Consequently, as
${\mathbf{s}}_{2}\rightarrow\mu_{2}{\mathbf{b}}=\pm{\mathbf{b}}$ during the
detection process by Bob so that the angle
$\eta_{{\mathbf{s}}_{2}{\mathbf{b}}}\\!\rightarrow 0$ or $\pi$, the binary
value of the observed result by Bob is obtained because
$\cos(\eta_{{\mathbf{s}}_{2}{\mathbf{b}}})\rightarrow\pm 1$ and
$\sin(\eta_{{\mathbf{s}}_{2}{\mathbf{b}}})\rightarrow 0$:

${\mathscr{B}}({\mathbf{b}},{\mathbf{s}}_{2})\longrightarrow-\mu_{2}=\mp 1.$
(91)

Thus $\mu_{1}$ and $\mu_{2}$ play the same role as the orientation $\lambda$
of $S^{3}$ played in the functions (7) and (11). It follows from (85) and (91)
that, in general, for the choices ${\mathbf{a}}\not={\mathbf{b}}$, the product
of the results observed by Alice and Bob will fluctuate between $-1$ and $+1$:

${\mathscr{A}}({\mathbf{a}},{\mathbf{s}}_{1})\,{\mathscr{B}}({\mathbf{b}},{\mathbf{s}}_{2})=\left[\lim_{{\mathbf{s}}_{1}\,\rightarrow\,\mu_{1}{\mathbf{a}}}\left\\{-\,{\mathbf{D}}({\mathbf{a}})\,{\mathbf{L}}({\mathbf{s}}_{1})\right\\}\right]\left[\lim_{{\mathbf{s}}_{2}\,\rightarrow\,\mu_{2}{\mathbf{b}}}\left\\{+\,{\mathbf{L}}({\mathbf{s}}_{2})\,{\mathbf{D}}({\mathbf{b}})\right\\}\right]=-\mu_{1}\mu_{2}=-\,(\pm
1)(\pm 1)=\mp 1.$ (92)

On the other hand, for the choice ${\mathbf{b}}={\mathbf{a}}$, we will have
${\mu}_{2}={\mu}_{1}$, provided ${\mathbf{s}}_{2}=\,{\mathbf{s}}_{1}$, and
perfect anti-correlation will be observed:

${\mathscr{A}}({\mathbf{a}},{\mathbf{s}}_{1})\,{\mathscr{B}}({\mathbf{a}},{\mathbf{s}}_{2})=\left[\lim_{{\mathbf{s}}_{1}\,\rightarrow\,\mu_{1}{\mathbf{a}}}\left\\{-\,{\mathbf{D}}({\mathbf{a}})\,{\mathbf{L}}({\mathbf{s}}_{1})\right\\}\right]\left[\lim_{{\mathbf{s}}_{2}\,\rightarrow\,\mu_{1}{\mathbf{a}}}\left\\{+\,{\mathbf{L}}({\mathbf{s}}_{2})\,{\mathbf{D}}({\mathbf{a}})\right\\}\right]=-(\mu_{1})^{2}=-1.$
(93)

Note that the conservation of zero spin angular momentum requires the equality
${\mathbf{s}}_{1}=\,{\mathbf{s}}_{2}$ to hold during the free evolution of the
spins, but not necessarily during their detection processes. Consequently,
while the physical interactions are taking place during the detection
processes at the two ends of the experiment, ${\mathbf{s}}_{2}$ can tend to
$\mu_{2}{\mathbf{b}}$ jointly as ${\mathbf{s}}_{1}$ tends to
$\mu_{1}{\mathbf{a}}$, despite the fact that
${\mathbf{s}}_{1}=\,{\mathbf{s}}_{2}$ must hold during the free evolution of
the spins. If now the direction
${\mathbf{s}}={\mathbf{s}}_{1}={\mathbf{s}}_{2}$ of the spins originating at
the source is assumed to be uniformly distributed over $S^{2}$ as in Bell’s
local model 2, 29, specified by a normalized probability measure
$p({\mathbf{s}})$,

$\int_{S^{2}}p({\mathbf{s}})\;d{\mathbf{s}}=1,$ (94)

then the results (85) observed by Alice will be equal to $+1$ or $-1$ with
50/50 chance, and likewise for the results (91) observed by Bob. Consequently,
in analogy with the quantum mechanical predictions, the expectation values of
their results will vanish:

${\cal
E}_{\mathrm{L.R.}}(\mathbf{a})=\int_{S^{2}}\mathscr{A}({\mathbf{a}},\mathbf{s}_{1})\,p({\mathbf{s}})\,d{\mathbf{s}}=0\;\;\;\;\text{and}\;\;\;\;{\cal
E}_{\mathrm{L.R.}}(\mathbf{b})=\int_{S^{2}}\mathscr{B}({\mathbf{b}},\mathbf{s}_{2})\,p({\mathbf{s}})\,d{\mathbf{s}}=0,$
(95)

where the subscripts 1 and 2 on the spin directions are retained for clarity
even though they are the same direction (cf. Figure 1).

The question now is: What will be the correlations within $S^{3}$ between the
results ${\mathscr{A}}$ and ${\mathscr{B}}$ observed jointly but independently
by Alice and Bob, in coincident counts, at a space-like distance from each
other? To work out the correlations, we begin with

${\cal
E}_{\mathrm{L.R.}}({\mathbf{a}},{\mathbf{b}})=\int_{S^{2}}{\mathscr{A}}({\mathbf{a}},{\mathbf{s}}_{1})\;{\mathscr{B}}({\mathbf{b}},{\mathbf{s}}_{2})\,p({\mathbf{s}})\,d{\mathbf{s}}=\int_{S^{2}}\left[\lim_{{\mathbf{s}}_{1}\,\rightarrow\,\mu_{1}{\mathbf{a}}}\left\\{+\,{\mathbf{q}}(\eta_{{\mathbf{a}}{\mathbf{s}}_{1}},\,{\mathbf{r}}_{1})\right\\}\right]\left[\lim_{{\mathbf{s}}_{2}\,\rightarrow\,\mu_{2}{\mathbf{b}}}\left\\{-\,{\mathbf{q}}(\eta_{{\mathbf{s}}_{2}{\mathbf{b}}},\,{\mathbf{r}}_{2})\right\\}\right]p({\mathbf{s}})\,d{\mathbf{s}}\,,$
(96)

which follows from the definitions (84) and (90). Next, the “product of limits
equal to limits of product” rule gives the equality

$\left[\lim_{{\mathbf{s}}_{1}\,\rightarrow\,\mu_{1}{\mathbf{a}}}\left\\{+\,{\mathbf{q}}(\eta_{{\mathbf{a}}{\mathbf{s}}_{1}},\,{\mathbf{r}}_{1})\right\\}\right]\left[\lim_{{\mathbf{s}}_{2}\,\rightarrow\,\mu_{2}{\mathbf{b}}}\left\\{-\,{\mathbf{q}}(\eta_{{\mathbf{s}}_{2}{\mathbf{b}}},\,{\mathbf{r}}_{2})\right\\}\right]=\lim_{\begin{subarray}{c}{\mathbf{s}}_{1}\,\rightarrow\,\mu_{1}{\mathbf{a}}\\\
{\mathbf{s}}_{2}\,\rightarrow\,\mu_{2}{\mathbf{b}}\end{subarray}}\Big{\\{}-{\mathbf{q}}(\eta_{{\mathbf{a}}{\mathbf{s}}_{1}},\,{\mathbf{r}}_{1})\,{\mathbf{q}}(\eta_{{\mathbf{s}}_{2}{\mathbf{b}}},\,{\mathbf{r}}_{2})\Big{\\}}.$
(97)

This equality holds for quaternions or bivectors. It is proved explicitly in
Appendix A.4 below. Moreover, in the Appendix A.5 the following equality is
proved, which follows from the fact that $S^{3}$ as a set of unit quaternions
remains closed under multiplication:

${\mathbf{q}}(\eta_{{\mathbf{a}}{\mathbf{s}}_{1}},\,{\mathbf{r}}_{1}){\mathbf{q}}(\eta_{{\mathbf{s}}_{2}{\mathbf{b}}},\,{\mathbf{r}}_{2})={\mathbf{q}}(\eta_{{\mathbf{u}}{\mathbf{v}}},\,{\mathbf{r}}_{0})=\cos(\eta_{{\mathbf{u}}{\mathbf{v}}})+(I_{3}{\mathbf{r}}_{0})\,\sin(\eta_{{\mathbf{u}}{\mathbf{v}}})\,,$
(98)

where

$\eta_{{\mathbf{u}}{\mathbf{v}}}({\mathbf{s}}_{1},\,{\mathbf{s}}_{2})=\cos^{-1}\\!\big{\\{}({\mathbf{a}}\cdot{\mathbf{s}}_{1})({\mathbf{s}}_{2}\cdot{\mathbf{b}})-({\mathbf{a}}\cdot{\mathbf{s}}_{2})({\mathbf{s}}_{1}\cdot{\mathbf{b}})+({\mathbf{a}}\cdot{\mathbf{b}})({\mathbf{s}}_{1}\cdot{\mathbf{s}}_{2})\big{\\}}$
(99)

and

${\mathbf{r}}_{0}({\mathbf{s}}_{1},\,{\mathbf{s}}_{2})=\frac{({\mathbf{a}}\cdot{\mathbf{s}}_{1})({\mathbf{s}}_{2}\times{\mathbf{b}})+({\mathbf{s}}_{2}\cdot{\mathbf{b}})({\mathbf{a}}\times{\mathbf{s}}_{1})-({\mathbf{a}}\times{\mathbf{s}}_{1})\times({\mathbf{s}}_{2}\times{\mathbf{b}})}{\sin\left(\eta_{{\mathbf{a}}{\mathbf{b}}}\right)}.$
(100)

Now, it is easy to see from (100) that during the simultaneous detection
processes by Alice and Bob, which are mathematically represented by the limits
defined in (84) and (90), the axis vector
${\mathbf{r}}_{0}({\mathbf{s}}_{1},\,{\mathbf{s}}_{2})$ of the quaternion (98)
reduces to the zero vector $\vec{\mathbf{0}}$,

$\lim_{\begin{subarray}{c}{\mathbf{s}}_{1}\,\rightarrow\,\mu_{1}{\mathbf{a}}\\\
{\mathbf{s}}_{2}\,\rightarrow\,\mu_{2}{\mathbf{b}}\end{subarray}}{\mathbf{r}}_{0}({\mathbf{s}}_{1},\,{\mathbf{s}}_{2})=\vec{\mathbf{0}},$
(101)

regardless of whether ${\mathbf{s}}_{1}\\!={\mathbf{s}}_{2}$ necessitated by
the conservation of angular momentum holds. Thus the only part of the
quaternion (98) that survives in the simultaneous detection processes is its
scalar part, which reduces to one of the following two possibilities:

$\displaystyle\lim_{\begin{subarray}{c}{\mathbf{s}}_{1}\,\rightarrow\,\mu_{1}{\mathbf{a}}\\\
{\mathbf{s}}_{2}\,\rightarrow\,\mu_{2}{\mathbf{b}}\end{subarray}}\,\left[\cos{\\{\eta_{{\mathbf{u}}{\mathbf{v}}}({\mathbf{s}}_{1},\,{\mathbf{s}}_{2})\\}}\right]=$
$\displaystyle\mu_{1}\mu_{2}=\text{sign}(\mathbf{a}\cdot\mathbf{s}^{i}_{1})\,\text{sign}(\mathbf{b}\cdot\mathbf{s}^{i}_{2})$
$\text{if}\;\,{\mathbf{s}}^{f}_{1}\not=\,{\mathbf{s}}^{f}_{2}$ (102)
$\displaystyle\lim_{\begin{subarray}{c}{\mathbf{s}}_{1}\,\rightarrow\,\mu_{1}{\mathbf{a}}\\\
{\mathbf{s}}_{2}\,\rightarrow\,\mu_{2}{\mathbf{b}}\end{subarray}}\,\left[\cos{\\{\eta_{{\mathbf{u}}{\mathbf{v}}}({\mathbf{s}}_{1},\,{\mathbf{s}}_{2})\\}}\right]=$
$\displaystyle\lim_{\begin{subarray}{c}{\mathbf{s}}_{1}\,\rightarrow\,\mu_{1}{\mathbf{a}}\\\
{\mathbf{s}}_{2}\,\rightarrow\,\mu_{2}{\mathbf{b}}\end{subarray}}\,\left[\cos{\\{\eta_{{\mathbf{u}}{\mathbf{v}}}({\mathbf{s}}_{1},\,{\mathbf{s}}_{2})\\}}\right]=$
$\displaystyle\cos(\eta_{{\mathbf{a}}{\mathbf{b}}})$
$\text{if}\;\,{\mathbf{s}}^{f}_{1}=\,{\mathbf{s}}^{f}_{2},$ (103)

where the superscript $f$ on ${\mathbf{s}}$ indicates its direction just
before detection. It is easy to see that these possibilities follow from (99):

$\lim_{\begin{subarray}{c}{\mathbf{s}}_{1}\,\rightarrow\,\mu_{1}{\mathbf{a}}\\\
{\mathbf{s}}_{2}\,\rightarrow\,\mu_{2}{\mathbf{b}}\end{subarray}}\,\left[\cos{\\{\eta_{{\mathbf{u}}{\mathbf{v}}}({\mathbf{s}}_{1},\,{\mathbf{s}}_{2})\\}}\right]=\lim_{\begin{subarray}{c}{\mathbf{s}}_{1}\,\rightarrow\,\mu_{1}{\mathbf{a}}\\\
{\mathbf{s}}_{2}\,\rightarrow\,\mu_{2}{\mathbf{b}}\end{subarray}}\big{\\{}({\mathbf{a}}\cdot{\mathbf{s}}_{1})({\mathbf{s}}_{2}\cdot{\mathbf{b}})-({\mathbf{a}}\cdot{\mathbf{s}}_{2})({\mathbf{s}}_{1}\cdot{\mathbf{b}})+({\mathbf{a}}\cdot{\mathbf{b}})({\mathbf{s}}_{1}\cdot{\mathbf{s}}_{2})\big{\\}}.$
(104)

Now, from (99) and (103) we see that for ${\mathbf{s}}_{1}={\mathbf{s}}_{2}$
the angle $\eta_{{\mathbf{u}}{\mathbf{v}}}$ reduces to the angle
$\eta_{{\mathbf{a}}{\mathbf{b}}}$ between ${\mathbf{a}}$ and ${\mathbf{b}}$,
and thus (98) reduces to

${\mathbf{q}}(\eta_{{\mathbf{a}}{\mathbf{b}}},\,{\mathbf{r}}_{0})=\cos(\eta_{{\mathbf{a}}{\mathbf{b}}})+(I_{3}{\mathbf{r}}_{0})\,\sin(\eta_{{\mathbf{a}}{\mathbf{b}}}).$
(105)

As we will soon see, this is what reproduces the strong singlet correlations
within $S^{3}$. But first it is instructive to understand the meaning and
significance of (102), which describes the case in which the condition
${\mathbf{s}}^{f}_{1}=\,{\mathbf{s}}^{f}_{2}$ required by the conservation of
spin angular momentum does not hold just before the detection processes begin.
That is possible if some physical interaction external to the singlet system
influences the rotation directions ${\mathbf{s}}_{1}$ and ${\mathbf{s}}_{2}$
sufficiently during their journey from the source to detectors to deviate them
from each other. This destroys the fragile singlet system and therefore the
correlations (96) fail to be sinusoidal:

$\displaystyle{\cal E}_{\mathrm{L.R.}}({\mathbf{a}},{\mathbf{b}})$
$\displaystyle=\int_{S^{2}}{\mathscr{A}}({\mathbf{a}},{\mathbf{s}}_{1})\;{\mathscr{B}}({\mathbf{b}},{\mathbf{s}}_{2})\,p({\mathbf{s}})\,d{\mathbf{s}}$
(106)
$\displaystyle=\int_{S^{2}}\left[\lim_{{\mathbf{s}}_{1}\,\rightarrow\,\mu_{1}{\mathbf{a}}}\left\\{+\,{\mathbf{q}}(\eta_{{\mathbf{a}}{\mathbf{s}}_{1}},\,{\mathbf{r}}_{1})\right\\}\right]\left[\lim_{{\mathbf{s}}_{2}\,\rightarrow\,\mu_{2}{\mathbf{b}}}\left\\{-\,{\mathbf{q}}(\eta_{{\mathbf{s}}_{2}{\mathbf{b}}},\,{\mathbf{r}}_{2})\right\\}\right]p({\mathbf{s}})\,d{\mathbf{s}}$
(107)
$\displaystyle=\int_{S^{2}}\Bigg{[}\lim_{\begin{subarray}{c}{\mathbf{s}}_{1}\,\rightarrow\,\mu_{1}{\mathbf{a}}\\\
{\mathbf{s}}_{2}\,\rightarrow\,\mu_{2}{\mathbf{b}}\end{subarray}}\Big{\\{}-{\mathbf{q}}(\eta_{{\mathbf{a}}{\mathbf{s}}_{1}},\,{\mathbf{r}}_{1})\,{\mathbf{q}}(\eta_{{\mathbf{s}}_{2}{\mathbf{b}}},\,{\mathbf{r}}_{2})\Big{\\}}\Bigg{]}p({\mathbf{s}})\,d{\mathbf{s}}$
(108)
$\displaystyle=\int_{S^{2}}\Bigg{[}\lim_{\begin{subarray}{c}{\mathbf{s}}_{1}\,\rightarrow\,\mu_{1}{\mathbf{a}}\\\
{\mathbf{s}}_{2}\,\rightarrow\,\mu_{2}{\mathbf{b}}\end{subarray}}\Big{\\{}-{\mathbf{q}}(\eta_{{\mathbf{u}}{\mathbf{v}}},\,{\mathbf{r}}_{0})\Big{\\}}\Bigg{]}p({\mathbf{s}})\,d{\mathbf{s}}$
(109)
$\displaystyle=\int_{S^{2}}\Bigg{[}\lim_{\begin{subarray}{c}{\mathbf{s}}_{1}\,\rightarrow\,\mu_{1}{\mathbf{a}}\\\
{\mathbf{s}}_{2}\,\rightarrow\,\mu_{2}{\mathbf{b}}\end{subarray}}\Big{\\{}-\cos(\,\eta_{{\mathbf{u}}{\mathbf{v}}})-\left(I_{3}{\mathbf{r}}_{0}\right)\sin(\,\eta_{{\mathbf{u}}{\mathbf{v}}})\Big{\\}}\\!\Bigg{]}p({\mathbf{s}})\,d{\mathbf{s}}$
(110)
$\displaystyle=\int_{S^{2}}\Bigg{[}\lim_{\begin{subarray}{c}{\mathbf{s}}_{1}\,\rightarrow\,\mu_{1}{\mathbf{a}}\\\
{\mathbf{s}}_{2}\,\rightarrow\,\mu_{2}{\mathbf{b}}\end{subarray}}\Big{\\{}-\cos(\,\eta_{{\mathbf{u}}{\mathbf{v}}})\Big{\\}}-\lim_{\begin{subarray}{c}{\mathbf{s}}_{1}\,\rightarrow\,\mu_{1}{\mathbf{a}}\\\
{\mathbf{s}}_{2}\,\rightarrow\,\mu_{2}{\mathbf{b}}\end{subarray}}\Big{\\{}I_{3}{\mathbf{r}}_{0}\Big{\\}}\times\lim_{\begin{subarray}{c}{\mathbf{s}}_{1}\,\rightarrow\,\mu_{1}{\mathbf{a}}\\\
{\mathbf{s}}_{2}\,\rightarrow\,\mu_{2}{\mathbf{b}}\end{subarray}}\Big{\\{}\sin(\,\eta_{{\mathbf{u}}{\mathbf{v}}})\Big{\\}}\Bigg{]}\,p({\mathbf{s}})\,d{\mathbf{s}}$
(111)
$\displaystyle=\int_{S^{2}}\Bigg{[}\big{\\{}-\mu_{1}\mu_{2}\big{\\}}-\left\\{I_{3}\vec{\mathbf{0}}\right\\}\times\lim_{\begin{subarray}{c}{\mathbf{s}}_{1}\,\rightarrow\,\mu_{1}{\mathbf{a}}\\\
{\mathbf{s}}_{2}\,\rightarrow\,\mu_{2}{\mathbf{b}}\end{subarray}}\Big{\\{}\sin(\,\eta_{{\mathbf{u}}{\mathbf{v}}})\Big{\\}}\Bigg{]}p({\mathbf{s}})\;d{\mathbf{s}}$
(112)
$\displaystyle=\int_{S^{2}}\Bigg{[}\Big{\\{}\\!-\text{sign}(\mathbf{a}\cdot\mathbf{s}^{i}_{1})\,\text{sign}(\mathbf{b}\cdot\mathbf{s}^{i}_{2})\Big{\\}}-0\Bigg{]}p({\mathbf{s}})\;d{\mathbf{s}}\;\;[\text{the
additive identity 0 is the same across all grades in}\;\mathrm{Cl_{3,0}}],$
(113)

where the last equation is well known to produce correlations that are
incapable of exceeding the bounds on Bell inequalities 2, 29:

${\cal
E}_{\mathrm{L.R.}}({\mathbf{a}},{\mathbf{b}})\,=\int_{S^{2}}\\!\left\\{-\,\text{sign}(\mathbf{a}\cdot\mathbf{s}^{i}_{1})\;\text{sign}(\mathbf{b}\cdot\mathbf{s}^{i}_{2})\right\\}\,p({\mathbf{s}})\;d{\mathbf{s}}\,=\,\begin{cases}-\,1\,+\,\frac{2}{\pi}\,\eta_{{\mathbf{a}}{\mathbf{b}}}\;\;\;\text{if}&\\!\\!0\leqslant\eta_{{\mathbf{a}}{\mathbf{b}}}\leqslant\pi,\\\
\\\
+\,3\,-\,\frac{2}{\pi}\,\eta_{{\mathbf{a}}{\mathbf{b}}}\;\;\;\text{if}&\\!\\!\pi\leqslant\eta_{{\mathbf{a}}{\mathbf{b}}}\leqslant
2\pi.\end{cases}$ (114)

[axesstyle=frame,linewidth=0.01mm,tickstyle=full,ticksize=0pt,dx=90Dx=180,dy=1
Dy=+2,Oy=-1](0,0)(180,1.0) ${90}$${270}$${+}$${{\cal
E}_{{\\!}_{\mathrm{L.R.}}}}$${\\!({\bf a},{\bf b})}$${0}$${\eta_{{\bf a}{\bf
b}}}$[linewidth=0.35mm,linecolor=black]0.0180x dup cos exch cos mul 1.0 mul
neg 1 add  Figure 3: Graphs of the correlations (114) and (123). The x-axis
depicts the angle in degrees between the detector directions ${\mathbf{a}}$
and ${\mathbf{b}}$, and the y-axis depicts the corresponding expectation value
or correlation. The dotted straight lines depict the correlations if
${\mathbf{s}}^{f}_{1}\not=\,{\mathbf{s}}^{f}_{2}$ holds, and the solid curve
represents the predictions of the 3-sphere model for which
${\mathbf{s}}^{f}_{1}=\,{\mathbf{s}}^{f}_{2}$ holds. After Christian 7.

In Fig. 3 these weak correlations are depicted by saw-tooth shaped dashed
straight lines. By contrast, the sinusoidal solid curve in Fig. 3 depicts
the strong singlet correlations predicted by the 3-sphere model and quantum
mechanics. They follow for the result (103) in which the spin angular momentum
is conserved up until the detection processes, and can be derived as follows:

$\displaystyle{\cal E}_{\mathrm{L.R.}}({\mathbf{a}},{\mathbf{b}})$
$\displaystyle=\int_{S^{2}}{\mathscr{A}}({\mathbf{a}},{\mathbf{s}}_{1})\;{\mathscr{B}}({\mathbf{b}},{\mathbf{s}}_{2})\,p({\mathbf{s}})\,d{\mathbf{s}}$
(115)
$\displaystyle=\int_{S^{2}}\left[\lim_{{\mathbf{s}}_{1}\,\rightarrow\,\mu_{1}{\mathbf{a}}}\left\\{+\,{\mathbf{q}}(\eta_{{\mathbf{a}}{\mathbf{s}}_{1}},\,{\mathbf{r}}_{1})\right\\}\right]\left[\lim_{{\mathbf{s}}_{2}\,\rightarrow\,\mu_{2}{\mathbf{b}}}\left\\{-\,{\mathbf{q}}(\eta_{{\mathbf{s}}_{2}{\mathbf{b}}},\,{\mathbf{r}}_{2})\right\\}\right]p({\mathbf{s}})\,d{\mathbf{s}}$
(116)
$\displaystyle=\int_{S^{2}}\Bigg{[}\lim_{\begin{subarray}{c}{\mathbf{s}}_{1}\,\rightarrow\,\mu_{1}{\mathbf{a}}\\\
{\mathbf{s}}_{2}\,\rightarrow\,\mu_{2}{\mathbf{b}}\end{subarray}}\Big{\\{}-{\mathbf{q}}(\eta_{{\mathbf{a}}{\mathbf{s}}_{1}},\,{\mathbf{r}}_{1})\,{\mathbf{q}}(\eta_{{\mathbf{s}}_{2}{\mathbf{b}}},\,{\mathbf{r}}_{2})\Big{\\}}\Bigg{]}p({\mathbf{s}})\,d{\mathbf{s}}$
(117)
$\displaystyle=\int_{S^{2}}\Bigg{[}\lim_{\begin{subarray}{c}{\mathbf{s}}_{1}\,\rightarrow\,\mu_{1}{\mathbf{a}}\\\
{\mathbf{s}}_{2}\,\rightarrow\,\mu_{2}{\mathbf{b}}\end{subarray}}\Big{\\{}-{\mathbf{q}}(\eta_{{\mathbf{a}}{\mathbf{b}}},\,{\mathbf{r}}_{0})\Big{\\}}\Bigg{]}p({\mathbf{s}})\,d{\mathbf{s}}$
(118)
$\displaystyle=\int_{S^{2}}\Bigg{[}\lim_{\begin{subarray}{c}{\mathbf{s}}_{1}\,\rightarrow\,\mu_{1}{\mathbf{a}}\\\
{\mathbf{s}}_{2}\,\rightarrow\,\mu_{2}{\mathbf{b}}\end{subarray}}\Big{\\{}-\cos(\,\eta_{{\mathbf{a}}{\mathbf{b}}})-\left(I_{3}{\mathbf{r}}_{0}\right)\sin(\,\eta_{{\mathbf{a}}{\mathbf{b}}})\Big{\\}}\\!\Bigg{]}p({\mathbf{s}})\,d{\mathbf{s}}$
(119)
$\displaystyle=\,-\cos(\,\eta_{{\mathbf{a}}{\mathbf{b}}})-\int_{S^{2}}\Bigg{[}\lim_{\begin{subarray}{c}{\mathbf{s}}_{1}\,\rightarrow\,\mu_{1}{\mathbf{a}}\\\
{\mathbf{s}}_{2}\,\rightarrow\,\mu_{2}{\mathbf{b}}\end{subarray}}\left(I_{3}{\mathbf{r}}_{0}\right)\Bigg{]}\sin(\,\eta_{{\mathbf{a}}{\mathbf{b}}})\,p({\mathbf{s}})\,d{\mathbf{s}}$
(120)
$\displaystyle=\,-\cos(\,\eta_{{\mathbf{a}}{\mathbf{b}}})-\left[I_{3}\vec{\mathbf{0}}\right]\sin(\,\eta_{{\mathbf{a}}{\mathbf{b}}})\int_{S^{2}}\\!p({\mathbf{s}})\;d{\mathbf{s}}$
(121)
$\displaystyle=\,-\cos(\,\eta_{{\mathbf{a}}{\mathbf{b}}})-\left[I_{3}\vec{\mathbf{0}}\right]\sin(\,\eta_{{\mathbf{a}}{\mathbf{b}}})$
(122)
$\displaystyle=-\cos(\,\eta_{{\mathbf{a}}{\mathbf{b}}})-0\;\;\;[\text{the
additive identity 0 is the same across all grades in}\;\mathrm{Cl_{3,0}}]\,,$
(123)

where the equality between the product of two quaternions appearing in (117)
and the one in (118) is proved in Appendix A.5. The only difference between
the derivations of the weak correlations (114) and strong correlations (123)
is the condition ${\mathbf{s}}^{f}_{1}=\,{\mathbf{s}}^{f}_{2}$.

It is important to bear in mind that the above local-realistic derivation of
the singlet correlations is a theoretical prediction of the quaternionic
3-sphere model 6, 7. It should not be confused with after-the-event data
analyses followed by experimenters 30.

Note also that there are only two differences between the above derivation of
the singlet correlations and the one I discussed in Subsection 2.1 above.
First, I have not used the relative handedness $\lambda$ between the spin
bivectors and the detector bivectors. This eliminates one of the concerns
raised in Lasenby 26, even though I demonstrated in Subsection 4.2 that that
concern is unjustified. Second, instead of $\lambda$, I have introduced and
used the limits discussed just below Eq. (79). These limits have quite
transparent meanings. As we see in (92), they unambiguously predict
${\mathscr{A}}{\mathscr{B}}=\pm 1$ in general for the choices
${\mathbf{a}}\not={\mathbf{b}}$, even when
${\mathbf{s}}_{1}\not={\mathbf{s}}_{2}$ holds.

Finally, the derivation presented above is independently verified in Diether31
in a numerical simulation using Mathematica. In Christian4, 32 I have also
proposed a macroscopic experiment that may be able to falsify the 3-sphere
hypothesis. For this purpose, let us note again that when the zero spins
angular momentum is conserved, as it must be, the angular relation (99)
reduces to

$\cos(\eta_{{\mathbf{u}}{\mathbf{v}}})=({\mathbf{a}}\cdot{\mathbf{s}}_{1})({\mathbf{s}}_{2}\cdot{\mathbf{b}})-({\mathbf{a}}\cdot{\mathbf{s}}_{2})({\mathbf{s}}_{1}\cdot{\mathbf{b}})+({\mathbf{a}}\cdot{\mathbf{b}})({\mathbf{s}}_{1}\cdot{\mathbf{s}}_{2})=\cos(\eta_{{\mathbf{a}}{\mathbf{b}}}).$
(124)

Now, the second equality in the above equation can be equivalently written as

$-\int_{S^{2}}\left[\lim_{\begin{subarray}{c}{\mathbf{s}}_{1}\,\rightarrow\,\mu_{1}{\mathbf{a}}\\\
{\mathbf{s}}_{2}\,\rightarrow\,\mu_{2}{\mathbf{b}}\end{subarray}}\left\\{({\mathbf{a}}\cdot{\mathbf{s}}_{1})({\mathbf{s}}_{2}\cdot{\mathbf{b}})-({\mathbf{a}}\cdot{\mathbf{s}}_{2})({\mathbf{s}}_{1}\cdot{\mathbf{b}})+({\mathbf{a}}\cdot{\mathbf{b}})({\mathbf{s}}_{1}\cdot{\mathbf{s}}_{2})\right\\}\right]p({\mathbf{s}})\;d{\mathbf{s}}\;=\,-\int_{S^{2}}\left[\lim_{\begin{subarray}{c}{\mathbf{s}}_{1}\,\rightarrow\,\mu_{1}{\mathbf{a}}\\\
{\mathbf{s}}_{2}\,\rightarrow\,\mu_{2}{\mathbf{b}}\end{subarray}}\left\\{\cos(\eta_{{\mathbf{a}}{\mathbf{b}}})\right\\}\right]p({\mathbf{s}})\;d{\mathbf{s}}\,,$
(125)

which, upon taking limits, simplifies to

$\int_{S^{2}}\\!\left\\{-\mu_{1}\mu_{2}\right\\}\,p({\mathbf{s}})\;d{\mathbf{s}}=-\cos(\eta_{{\mathbf{a}}{\mathbf{b}}}).$
(126)

Using (85), (91), and definitions (79), we recognize that this equation is
equivalent to the prediction (123) of the 3-sphere model:

${\cal
E}_{\mathrm{L.R.}}({\mathbf{a}},{\mathbf{b}})\,=\int_{S^{2}}\\!\left\\{-\,\text{sign}(\mathbf{a}\cdot\mathbf{s}^{i}_{1})\;\text{sign}(\mathbf{b}\cdot\mathbf{s}^{i}_{2})\right\\}\,p({\mathbf{s}})\;d{\mathbf{s}}\approx\lim_{\,n\,\gg\,1}\\!\left[\frac{1}{n}\\!\sum_{k\,=\,1}^{n}\left\\{-\,\text{sign}({\bf
a}\cdot{\bf s}^{k}_{1})\;\text{sign}({\bf b}\cdot{\bf
s}^{k}_{2})\right\\}\right]=-\cos(\eta_{{\mathbf{a}}{\mathbf{b}}}).$ (127)

It is this prediction (127) that can be tested experimentally in a macroscopic
setting, as I have explained in detail in Christian 32. Note that the
summation in (127) involves sign functions of ordinary vectors, which can be
easily measured and calculated. In Christian 32 I have also provided a
simplified derivation of the correlation (127) that does not require using the
orientation of $S^{3}$.

## Conflict of Interest

The author declares no conflict of interests in this work.

## ORCID

Joy Christian https://orcid.org/0000-0002-8741-6943

##  Appendix A Proofs of Lemmas 2, 3, and 4, and of Equations (96) and (105)

###  A.1 Proof of Lemma 3.3

To verify the multiplicativity (50) of the norms $||{\mathbb{Q}}_{z}||$,
consider the left-hand side of (50) for two distinct elements of ${{\cal
K}^{\lambda}}$:

${\mathbb{Q}}_{z1}=\,{\bf q}_{r1}+{\bf
q}_{d1}\,\varepsilon\;\;\;\;\text{and}\;\;\;\;{\mathbb{Q}}_{z2}=\,{\bf
q}_{r2}+{\bf q}_{d2}\,\varepsilon\,.$ (A.1)

Since the pseudoscalar $\varepsilon$ commutes with every element of ${\cal
K}^{\lambda}$, the geometric product of the above elements gives

${\mathbb{Q}}_{z1}\,{\mathbb{Q}}_{z2}\,=\;\left({\bf q}_{r1}\,{\bf
q}_{r2}\,+\,{\bf q}_{d1}\,{\bf q}_{d2}\right)\,+\,\left({\bf q}_{r1}\,{\bf
q}_{d2}\,+\,{\bf q}_{d1}\,{\bf q}_{r2}\right)\,\varepsilon\,.$ (A.2)

where I have used ${\varepsilon^{2}=1}$. If we now use again the fact that
${\varepsilon}$, along with ${\varepsilon^{\dagger}=\varepsilon}$ and
${\varepsilon^{2}=1}$, commutes with every element of ${{\cal K}^{\lambda}}$
and consequently with all ${{\bf q}_{r}}$, ${{\bf q}^{\dagger}_{r}}$, ${{\bf
q}_{d}}$ and ${{\bf q}^{\dagger}_{d}}$, and work out
${{\mathbb{Q}}^{\dagger}_{z1}}$, ${{\mathbb{Q}}^{\dagger}_{z2}}$ and the
products ${{\mathbb{Q}}_{z1}{\mathbb{Q}}^{\dagger}_{z1}}$,
${\,{\mathbb{Q}}_{z2}{\mathbb{Q}}^{\dagger}_{z2}}$ and
${({\mathbb{Q}}_{z1}\,{\mathbb{Q}}_{z2})^{\dagger}}$ as

$\displaystyle{\mathbb{Q}}^{\dagger}_{z1}$ $\displaystyle=\,{\bf
q}^{\dagger}_{r1}+{\bf q}^{\dagger}_{d1}\,\varepsilon\,,$ (A.3)
$\displaystyle{\mathbb{Q}}^{\dagger}_{z2}$ $\displaystyle=\,{\bf
q}^{\dagger}_{r2}+{\bf q}^{\dagger}_{d2}\,\varepsilon\,,$ (A.4)
$\displaystyle{\mathbb{Q}}_{z1}\,{\mathbb{Q}}^{\dagger}_{z1}$
$\displaystyle=\,\left({\bf q}_{r1}\,{\bf q}^{\dagger}_{r1}\,+\,{\bf
q}_{d1}\,{\bf q}^{\dagger}_{d1}\right)\,+\,\left({\bf q}_{r1}\,{\bf
q}^{\dagger}_{d1}\,+\,{\bf q}_{d1}\,{\bf
q}^{\dagger}_{r1}\right)\,\varepsilon\,,$ (A.5)
$\displaystyle{\mathbb{Q}}_{z2}\,{\mathbb{Q}}^{\dagger}_{z2}$
$\displaystyle=\,\left({\bf q}_{r2}\,{\bf q}^{\dagger}_{r2}\,+\,{\bf
q}_{d2}\,{\bf q}^{\dagger}_{d2}\right)\,+\,\left({\bf q}_{r2}\,{\bf
q}^{\dagger}_{d2}\,+\,{\bf q}_{d2}\,{\bf
q}^{\dagger}_{r2}\right)\,\varepsilon\,,$ (A.6)
$\displaystyle\text{and}\;\;\;({\mathbb{Q}}_{z1}\,{\mathbb{Q}}_{z2})^{\dagger}\,$
$\displaystyle=\;\left({\bf q}^{\dagger}_{r2}\,{\bf q}^{\dagger}_{r1}\,+\,{\bf
q}^{\dagger}_{d2}\,{\bf q}^{\dagger}_{d1}\right)\,+\,\left({\bf
q}^{\dagger}_{d2}\,{\bf q}^{\dagger}_{r1}\,+\,{\bf q}^{\dagger}_{r2}\,{\bf
q}^{\dagger}_{d1}\right)\,\varepsilon\,,$ (A.7)

then, using (A.2) to (A.7), the norm relation (50) is not difficult to verify.
To that end, let us work out the geometric product
${({\mathbb{Q}}_{z1}\,{\mathbb{Q}}_{z2})({\mathbb{Q}}_{z1}\,{\mathbb{Q}}_{z2})^{\dagger}}$
using the expressions (A.2) and (A.7), which gives

$\displaystyle({\mathbb{Q}}_{z1}\,{\mathbb{Q}}_{z2})({\mathbb{Q}}_{z1}\,{\mathbb{Q}}_{z2})^{\dagger}$
$\displaystyle=\left\\{\left({\bf q}_{r1}\,{\bf q}_{r2}+{\bf q}_{d1}\,{\bf
q}_{d2}\right)\left({\bf q}^{\dagger}_{r2}\,{\bf q}^{\dagger}_{r1}+{\bf
q}^{\dagger}_{d2}\,{\bf q}^{\dagger}_{d1}\right)+\left({\bf q}_{r1}\,{\bf
q}_{d2}+{\bf q}_{d1}\,{\bf q}_{r2}\right)\left({\bf q}^{\dagger}_{d2}\,{\bf
q}^{\dagger}_{r1}+{\bf q}^{\dagger}_{r2}\,{\bf
q}^{\dagger}_{d1}\right)\right\\}$
$\displaystyle\;\;\;\;\;\;+\left\\{\left({\bf q}_{r1}\,{\bf q}_{d2}+{\bf
q}_{d1}\,{\bf q}_{r2}\right)\left({\bf q}^{\dagger}_{r2}\,{\bf
q}^{\dagger}_{r1}+{\bf q}^{\dagger}_{d2}\,{\bf
q}^{\dagger}_{d1}\right)+\left({\bf q}_{r1}\,{\bf q}_{r2}+{\bf q}_{d1}\,{\bf
q}_{d2}\right)\left({\bf q}^{\dagger}_{d2}\,{\bf q}^{\dagger}_{r1}+{\bf
q}^{\dagger}_{r2}\,{\bf q}^{\dagger}_{d1}\right)\right\\}\varepsilon\,.$ (A.8)

Using (A.2), (A.7), and their product evaluated in (A.8), we can now evaluate
the left-hand side of (51) as follows:

$\displaystyle||{\mathbb{Q}}_{z1}\,{\mathbb{Q}}_{z2}||^{2}$
$\displaystyle=\left({\mathbb{Q}}_{z1}\,{\mathbb{Q}}_{z2}\right)\left({\mathbb{Q}}_{z1}\,{\mathbb{Q}}_{z2}\right)^{\dagger}$
(A.9) $\displaystyle=\Big{\\{}\left({\bf q}_{r1}\,{\bf q}_{r2}+{\bf
q}_{d1}\,{\bf q}_{d2}\right)+\left({\bf q}_{r1}\,{\bf q}_{d2}+{\bf
q}_{d1}\,{\bf q}_{r2}\right)\varepsilon\Big{\\}}\left\\{\left({\bf
q}^{\dagger}_{r2}\,{\bf q}^{\dagger}_{r1}+{\bf q}^{\dagger}_{d2}\,{\bf
q}^{\dagger}_{d1}\right)+\left({\bf q}^{\dagger}_{d2}\,{\bf
q}^{\dagger}_{r1}+{\bf q}^{\dagger}_{r2}\,{\bf
q}^{\dagger}_{d1}\right)\varepsilon\right\\}$ (A.10)
$\displaystyle=\left\\{\left({\bf q}_{r1}\,{\bf q}_{r2}+{\bf q}_{d1}\,{\bf
q}_{d2}\right)\left({\bf q}^{\dagger}_{r2}\,{\bf q}^{\dagger}_{r1}+{\bf
q}^{\dagger}_{d2}\,{\bf q}^{\dagger}_{d1}\right)+\left({\bf q}_{r1}\,{\bf
q}_{d2}+{\bf q}_{d1}\,{\bf q}_{r2}\right)\left({\bf q}^{\dagger}_{d2}\,{\bf
q}^{\dagger}_{r1}+{\bf q}^{\dagger}_{r2}\,{\bf
q}^{\dagger}_{d1}\right)\right\\}$
$\displaystyle\;\;\;\;\;\;\;\,+\left\\{\left({\bf q}_{r1}\,{\bf q}_{r2}+{\bf
q}_{d1}\,{\bf q}_{d2}\right)\left({\bf q}^{\dagger}_{d2}\,{\bf
q}^{\dagger}_{r1}+{\bf q}^{\dagger}_{r2}\,{\bf
q}^{\dagger}_{d1}\right)+\left({\bf q}_{r1}\,{\bf q}_{d2}+{\bf q}_{d1}\,{\bf
q}_{r2}\right)\left({\bf q}^{\dagger}_{r2}\,{\bf q}^{\dagger}_{r1}+{\bf
q}^{\dagger}_{d2}\,{\bf q}^{\dagger}_{d1}\right)\right\\}\varepsilon$ (A.11)
$\displaystyle=\Big{\\{}{\bf q}_{r1}\,{\bf q}_{r2}\,{\bf
q}^{\dagger}_{r2}\,{\bf q}^{\dagger}_{r1}\,+\,{\bf q}_{r1}\,{\bf q}_{r2}\,{\bf
q}^{\dagger}_{d2}\,{\bf q}^{\dagger}_{d1}\,+\,{\bf q}_{d1}\,{\bf q}_{d2}\,{\bf
q}^{\dagger}_{r2}\,{\bf q}^{\dagger}_{r1}\,+\,{\bf q}_{d1}\,{\bf q}_{d2}\,{\bf
q}^{\dagger}_{d2}\,{\bf q}^{\dagger}_{d1}$
$\displaystyle\,\;\;\;\;\;\;\;+\,{\bf q}_{r1}\,{\bf q}_{d2}\,{\bf
q}^{\dagger}_{d2}\,{\bf q}^{\dagger}_{r1}\,+\,{\bf q}_{r1}\,{\bf q}_{d2}\,{\bf
q}^{\dagger}_{r2}\,{\bf q}^{\dagger}_{d1}\,+\,{\bf q}_{d1}\,{\bf q}_{r2}\,{\bf
q}^{\dagger}_{d2}\,{\bf q}^{\dagger}_{r1}\,+\,{\bf q}_{d1}\,{\bf q}_{r2}\,{\bf
q}^{\dagger}_{r2}\,{\bf q}^{\dagger}_{d1}\Big{\\}}$
$\displaystyle\;\;\;\;\;\;\;\;\;\;\;+\Big{\\{}{\bf q}_{r1}\,{\bf q}_{r2}\,{\bf
q}^{\dagger}_{d2}\,{\bf q}^{\dagger}_{r1}\,+\,{\bf q}_{r1}\,{\bf q}_{r2}\,{\bf
q}^{\dagger}_{r2}\,{\bf q}^{\dagger}_{d1}\,+\,{\bf q}_{d1}\,{\bf q}_{d2}\,{\bf
q}^{\dagger}_{d2}\,{\bf q}^{\dagger}_{r1}\,+\,{\bf q}_{d1}\,{\bf q}_{d2}\,{\bf
q}^{\dagger}_{r2}\,{\bf q}^{\dagger}_{d1}$
$\displaystyle\,\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;+\,{\bf q}_{r1}\,{\bf
q}_{d2}\,{\bf q}^{\dagger}_{r2}\,{\bf q}^{\dagger}_{r1}\,+\,{\bf q}_{r1}\,{\bf
q}_{d2}\,{\bf q}^{\dagger}_{d2}\,{\bf q}^{\dagger}_{d1}\,+\,{\bf q}_{d1}\,{\bf
q}_{r2}\,{\bf q}^{\dagger}_{r2}\,{\bf q}^{\dagger}_{r1}\,+\,{\bf q}_{d1}\,{\bf
q}_{r2}\,{\bf q}^{\dagger}_{d2}\,{\bf q}^{\dagger}_{d1}\Big{\\}}\,\varepsilon$
(A.12) $\displaystyle=\Big{\\{}\varrho^{2}_{r1}\,\varrho^{2}_{r2}\,+\,{\bf
q}_{r1}\,{\bf q}_{r2}\,{\bf q}^{\dagger}_{d2}\,{\bf q}^{\dagger}_{d1}\,+\,{\bf
q}_{d1}\,{\bf q}_{d2}\,{\bf q}^{\dagger}_{r2}\,{\bf
q}^{\dagger}_{r1}\,+\,\varrho^{2}_{d1}\,\varrho^{2}_{d2}$
$\displaystyle\,\;\;\;\;\;\;\;+\,\varrho^{2}_{r1}\,\varrho^{2}_{d2}\,+\,{\bf
q}_{r1}\,{\bf q}_{d2}\,{\bf q}^{\dagger}_{r2}\,{\bf q}^{\dagger}_{d1}\,+\,{\bf
q}_{d1}\,{\bf q}_{r2}\,{\bf q}^{\dagger}_{d2}\,{\bf
q}^{\dagger}_{r1}\,+\,\varrho^{2}_{d1}\,\varrho^{2}_{r2}\Big{\\}}$
$\displaystyle\;\;\;\;\;\;\;\;\;\;\;+\Big{\\{}{\bf q}_{r1}\,{\bf q}_{r2}\,{\bf
q}^{\dagger}_{d2}\,{\bf q}^{\dagger}_{r1}\,+\,\varrho^{2}_{r2}\,{\bf
q}_{r1}\,{\bf q}^{\dagger}_{d1}\,+\,\varrho^{2}_{d2}\,{\bf q}_{d1}\,{\bf
q}^{\dagger}_{r1}\,+\,{\bf q}_{d1}\,{\bf q}_{d2}\,{\bf q}^{\dagger}_{r2}\,{\bf
q}^{\dagger}_{d1}$ $\displaystyle\,\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;+\,{\bf
q}_{r1}\,{\bf q}_{d2}\,{\bf q}^{\dagger}_{r2}\,{\bf
q}^{\dagger}_{r1}\,+\,\varrho^{2}_{d2}\,{\bf q}_{r1}\,{\bf
q}^{\dagger}_{d1}\,+\,\varrho^{2}_{r2}\,{\bf q}_{d1}\,{\bf
q}^{\dagger}_{r1}\,+\,{\bf q}_{d1}\,{\bf q}_{r2}\,{\bf q}^{\dagger}_{d2}\,{\bf
q}^{\dagger}_{d1}\Big{\\}}\,\varepsilon$ (A.13)
$\displaystyle=\Big{\\{}\\!\left(\varrho^{2}_{r1}+\varrho^{2}_{d1}\right)\\!\left(\varrho^{2}_{r2}+\varrho^{2}_{d2}\right)+{\bf
q}_{r1}\left({\bf q}_{r2}\,{\bf q}^{\dagger}_{d2}+{\bf q}_{d2}\,{\bf
q}^{\dagger}_{r2}\right){\bf q}^{\dagger}_{d1}+{\bf q}_{d1}\left({\bf
q}_{r2}\,{\bf q}^{\dagger}_{d2}+{\bf q}_{d2}\,{\bf
q}^{\dagger}_{r2}\right){\bf q}^{\dagger}_{r1}\Big{\\}}$
$\displaystyle\;\;\;\;\;\;\;\;\;\;\;+\Big{\\{}{\bf q}_{r1}\left({\bf
q}_{r2}\,{\bf q}^{\dagger}_{d2}+{\bf q}_{d2}\,{\bf
q}^{\dagger}_{r2}\right){\bf q}^{\dagger}_{r1}\,+\,\varrho^{2}_{r2}\left({\bf
q}_{r1}\,{\bf q}^{\dagger}_{d1}+{\bf q}_{d1}\,{\bf q}^{\dagger}_{r1}\right)$
$\displaystyle\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;+\varrho^{2}_{d2}\left({\bf
q}_{r1}\,{\bf q}^{\dagger}_{d1}+{\bf q}_{d1}\,{\bf
q}^{\dagger}_{r1}\right)+{\bf q}_{d1}\left({\bf q}_{r2}\,{\bf
q}^{\dagger}_{d2}+{\bf q}_{d2}\,{\bf q}^{\dagger}_{r2}\right){\bf
q}^{\dagger}_{d1}\Big{\\}}\,\varepsilon$ (A.14)
$\displaystyle=\left\\{\varrho^{2}_{r1}\,\varrho^{2}_{r2}+\varrho^{2}_{r1}\,\varrho^{2}_{d2}+\varrho^{2}_{d1}\,\varrho^{2}_{r2}+\varrho^{2}_{d1}\,\varrho^{2}_{d2}\,+\,\left({\bf
q}_{r1}\,{\bf q}^{\dagger}_{d1}+{\bf q}_{d1}\,{\bf
q}^{\dagger}_{r1}\right)\left({\bf q}_{r2}\,{\bf q}^{\dagger}_{d2}+{\bf
q}_{d2}\,{\bf q}^{\dagger}_{r2}\right)\right\\}$
$\displaystyle\;\;\;\;\;\;\;\;\;\;+\left\\{\left(\varrho^{2}_{r1}\,+\,\varrho^{2}_{d1}\right)\left({\bf
q}_{r2}\,{\bf q}^{\dagger}_{d2}+{\bf q}_{d2}\,{\bf
q}^{\dagger}_{r2}\right)+\left(\varrho^{2}_{r2}\,+\,\varrho^{2}_{d2}\right)\left({\bf
q}_{r1}\,{\bf q}^{\dagger}_{d1}+{\bf q}_{d1}\,{\bf
q}^{\dagger}_{r1}\right)\right\\}\,\varepsilon\,.$ (A.15)

Again, since the quantities such as ${\bf q}_{r1}\,{\bf q}^{\dagger}_{d1}+{\bf
q}_{d1}\,{\bf q}^{\dagger}_{r1}$ are scalar quantities, we see that the above
product also resembles a split complex number similar to
${\mathbb{Q}}_{z}\,{\mathbb{Q}}^{\dagger}_{z}$ in (44). In other words, it is
a sum of a scalar and a pseudoscalar, as in (47).

Next, we can evaluate the right-hand side of the relation (51) using (A.5) and
(A.6) as follows:

$\displaystyle||{\mathbb{Q}}_{z1}||^{2}\,||{\mathbb{Q}}_{z2}||^{2}$
$\displaystyle=\left({\mathbb{Q}}_{z1}\,{\mathbb{Q}}^{\dagger}_{z1}\right)\left({\mathbb{Q}}_{z2}\,{\mathbb{Q}}^{\dagger}_{z2}\right)$
(A.16) $\displaystyle=\left\\{\left({\bf q}_{r1}\,{\bf q}^{\dagger}_{r1}+{\bf
q}_{d1}\,{\bf q}^{\dagger}_{d1}\right)+\left({\bf q}_{r1}\,{\bf
q}^{\dagger}_{d1}+{\bf q}_{d1}\,{\bf
q}^{\dagger}_{r1}\right)\varepsilon\right\\}\left\\{\left({\bf q}_{r2}\,{\bf
q}^{\dagger}_{r2}+{\bf q}_{d2}\,{\bf q}^{\dagger}_{d2}\right)+\left({\bf
q}_{r2}\,{\bf q}^{\dagger}_{d2}+{\bf q}_{d2}\,{\bf
q}^{\dagger}_{r2}\right)\varepsilon\right\\}$ (A.17)
$\displaystyle=\left\\{\left({\bf q}_{r1}\,{\bf q}^{\dagger}_{r1}+{\bf
q}_{d1}\,{\bf q}^{\dagger}_{d1}\right)\left({\bf q}_{r2}\,{\bf
q}^{\dagger}_{r2}+{\bf q}_{d2}\,{\bf q}^{\dagger}_{d2}\right)+\left({\bf
q}_{r1}\,{\bf q}^{\dagger}_{d1}+{\bf q}_{d1}\,{\bf
q}^{\dagger}_{r1}\right)\left({\bf q}_{r2}\,{\bf q}^{\dagger}_{d2}+{\bf
q}_{d2}\,{\bf q}^{\dagger}_{r2}\right)\right\\}$
$\displaystyle\;\;\;\;\;\;\;\;\;\;+\Big{\\{}\left({\bf q}_{r1}\,{\bf
q}^{\dagger}_{r1}+{\bf q}_{d1}\,{\bf q}^{\dagger}_{d1}\right)\left({\bf
q}_{r2}\,{\bf q}^{\dagger}_{d2}+{\bf q}_{d2}\,{\bf
q}^{\dagger}_{r2}\right)+\left({\bf q}_{r1}\,{\bf q}^{\dagger}_{d1}+{\bf
q}_{d1}\,{\bf q}^{\dagger}_{r1}\right)\left({\bf q}_{r2}\,{\bf
q}^{\dagger}_{r2}+{\bf q}_{d2}\,{\bf
q}^{\dagger}_{d2}\right)\Big{\\}}\,\varepsilon$ (A.18)
$\displaystyle=\left\\{\left(\varrho^{2}_{r1}+\varrho^{2}_{d1}\right)\left(\varrho^{2}_{r2}+\varrho^{2}_{d2}\right)\,+\,\left({\bf
q}_{r1}\,{\bf q}^{\dagger}_{d1}+{\bf q}_{d1}\,{\bf
q}^{\dagger}_{r1}\right)\left({\bf q}_{r2}\,{\bf q}^{\dagger}_{d2}+{\bf
q}_{d2}\,{\bf q}^{\dagger}_{r2}\right)\right\\}$
$\displaystyle\;\;\;\;\;\;\;\;\;\;+\left\\{\left(\varrho^{2}_{r1}\,+\,\varrho^{2}_{d1}\right)\left({\bf
q}_{r2}\,{\bf q}^{\dagger}_{d2}+{\bf q}_{d2}\,{\bf
q}^{\dagger}_{r2}\right)+\left({\bf q}_{r1}\,{\bf q}^{\dagger}_{d1}+{\bf
q}_{d1}\,{\bf
q}^{\dagger}_{r1}\right)\left(\varrho^{2}_{r2}\,+\,\varrho^{2}_{d2}\right)\right\\}\,\varepsilon$
(A.19)
$\displaystyle=\left\\{\varrho^{2}_{r1}\,\varrho^{2}_{r2}+\varrho^{2}_{r1}\,\varrho^{2}_{d2}+\varrho^{2}_{d1}\,\varrho^{2}_{r2}+\varrho^{2}_{d1}\,\varrho^{2}_{d2}\,+\,\left({\bf
q}_{r1}\,{\bf q}^{\dagger}_{d1}+{\bf q}_{d1}\,{\bf
q}^{\dagger}_{r1}\right)\left({\bf q}_{r2}\,{\bf q}^{\dagger}_{d2}+{\bf
q}_{d2}\,{\bf q}^{\dagger}_{r2}\right)\right\\}$
$\displaystyle\;\;\;\;\;\;\;\;\;\;+\left\\{\left(\varrho^{2}_{r1}\,+\,\varrho^{2}_{d1}\right)\left({\bf
q}_{r2}\,{\bf q}^{\dagger}_{d2}+{\bf q}_{d2}\,{\bf
q}^{\dagger}_{r2}\right)+\left(\varrho^{2}_{r2}\,+\,\varrho^{2}_{d2}\right)\left({\bf
q}_{r1}\,{\bf q}^{\dagger}_{d1}+{\bf q}_{d1}\,{\bf
q}^{\dagger}_{r1}\right)\right\\}\,\varepsilon\,,$ (A.20)

where I have used ${\bf q}_{r1}\,{\bf q}^{\dagger}_{r1}=\varrho^{2}_{r1}$,
${\bf q}_{d1}\,{\bf q}^{\dagger}_{d1}=\varrho^{2}_{d1}$, etc. Recalling from
(40) that the quantities such as ${\bf q}_{r1}\,{\bf q}^{\dagger}_{d1}+{\bf
q}_{d1}\,{\bf q}^{\dagger}_{r1}$ are scalar quantities, it is easy to see that
the above product (A.20) also resembles a split complex number similar to
${\mathbb{Q}}_{z}\,{\mathbb{Q}}^{\dagger}_{z}$ in (44).

More importantly, the right-hand sides of (A.20) and (A.15) are identical. We
have thus proved that, although norms in ${{\cal K}^{\lambda}}$ resemble split
complex numbers 20, 21 similarly to the product
${\mathbb{Q}}_{z}\,{\mathbb{Q}}^{\dagger}_{z}$ in (44) rather than scalar
numbers, the composition law (51) continues to hold for the algebra ${{\cal
K}^{\lambda}}$:

$||{\mathbb{Q}}_{z1}\,{\mathbb{Q}}_{z2}||^{2}=||{\mathbb{Q}}_{z1}||^{2}\,||{\mathbb{Q}}_{z2}||^{2}.$
(A.21)

Moreover, the square-root operation can be meaningfully applied to (51) by
treating the quantity appearing in (A.20) and (A.15) as a “pseudo-complex” or
hyperbolic number $a+\varepsilon\,b$, with $a$ and $b$ being scalars:

$\displaystyle\sqrt{a+\varepsilon\,b\,}$
$\displaystyle=\sqrt{\rho\,\\{\cosh{(\phi)}+\varepsilon\,\sinh{(\phi)}\\}}$
(A.22) $\displaystyle=\sqrt{\rho\,\exp{(\varepsilon\,\phi)}}$ (A.23)
$\displaystyle=\sqrt{\rho\,}\,\exp{\left(\varepsilon\,\frac{\phi}{2}\right)}$
(A.24)
$\displaystyle=\sqrt{\rho\,}\left\\{\cosh{\left(\frac{\phi}{2}\right)}+\varepsilon\,\sinh{\left(\frac{\phi}{2}\right)}\right\\}.$
(A.25)

For details see Sobczyk 21. This allows us to write the composition law (A.21)
in the square-root form as the norm relation (50):

$||{\mathbb{Q}}_{z1}\,{\mathbb{Q}}_{z2}||=||{\mathbb{Q}}_{z1}||\,||{\mathbb{Q}}_{z2}||.$
(A.26)

###  A.2 Proof of Lemma 3.4

To prove this lemma, we begin by recognizing that the algebra ${{\cal
K}^{\lambda}}$ remains closed under multiplication. Suppose ${X}$ and ${Y}$
are two multivectors in ${{\cal K}^{\lambda}}$. Then they can be expanded in
the graded basis of ${{\cal K}^{\lambda}}$ as:

${X}=\,x_{0}+x_{1}\,\lambda{\bf e}_{x}{\bf e}_{y}+x_{2}\,\lambda{\bf
e}_{z}{\bf e}_{x}+x_{3}\,\lambda{\bf e}_{y}{\bf e}_{z}+x_{4}\,\lambda{\bf
e}_{x}{\bf e}_{\infty}+x_{5}\,\lambda{\bf e}_{y}{\bf
e}_{\infty}+x_{6}\,\lambda{\bf e}_{z}{\bf e}_{\infty}+x_{7}\,\lambda I_{3}{\bf
e}_{\infty}$ (A.27)

and

${Y}=\,y_{0}+y_{1}\,\lambda{\bf e}_{x}{\bf e}_{y}+y_{2}\,\lambda{\bf
e}_{z}{\bf e}_{x}+y_{3}\,\lambda{\bf e}_{y}{\bf e}_{z}+y_{4}\,\lambda{\bf
e}_{x}{\bf e}_{\infty}+y_{5}\,\lambda{\bf e}_{y}{\bf
e}_{\infty}+y_{6}\,\lambda{\bf e}_{z}{\bf e}_{\infty}+y_{7}\,\lambda I_{3}{\bf
e}_{\infty}\,.$ (A.28)

Using the multiplication table for ${{\cal K}^{\lambda}}$ (Table 1), it is
then easy to verify that their product ${{Z}={X}{Y}}$ also belongs to ${{\cal
K}^{\lambda}}$:

$\displaystyle{Z}$ $\displaystyle=\,z_{0}+z_{1}\,\lambda{\bf e}_{x}{\bf
e}_{y}+z_{2}\,\lambda{\bf e}_{z}{\bf e}_{x}+z_{3}\,\lambda{\bf e}_{y}{\bf
e}_{z}+z_{4}\,\lambda{\bf e}_{x}{\bf e}_{\infty}+z_{5}\,\lambda{\bf e}_{y}{\bf
e}_{\infty}+z_{6}\,\lambda{\bf e}_{z}{\bf e}_{\infty}+z_{7}\,\lambda I_{3}{\bf
e}_{\infty}$ $\displaystyle={X}{Y}\in{\cal K}^{\lambda}.$ (A.29)

This property can be verified also by writing the two distinct elements of
${{\cal K}^{\lambda}}$ in the form

${\mathbb{Q}}_{z1}=\,{\bf q}_{r1}+{\bf
q}_{d1}\,\varepsilon\;\;\;\text{and}\;\;\;{\mathbb{Q}}_{z2}=\,{\bf
q}_{r2}+{\bf q}_{d2}\,\varepsilon\,.$ (A.30)

Then, using $\varepsilon^{2}=1$ and the fact that $\varepsilon$ commutes with
every element of ${\cal K}^{\lambda}$, their product works out to be

${\mathbb{Q}}_{z3}:={\mathbb{Q}}_{z1}\,{\mathbb{Q}}_{z2}\,=\,\left({\bf
q}_{r1}\,{\bf q}_{r2}\,+\,{\bf q}_{d1}\,{\bf q}_{d2}\right)\,+\,\left({\bf
q}_{r1}\,{\bf q}_{d2}\,+\,{\bf q}_{d1}\,{\bf q}_{r2}\right)\,\varepsilon\,.$
(A.31)

But since $\left({\bf q}_{r1}\,{\bf q}_{r2}+{\bf q}_{d1}\,{\bf q}_{d2}\right)$
and $\left({\bf q}_{r1}\,{\bf q}_{d2}+{\bf q}_{d1}\,{\bf q}_{r2}\right)$ are
both quaternions,
${\mathbb{Q}}_{z3}={\mathbb{Q}}_{z1}\,{\mathbb{Q}}_{z2}\in{\cal K}^{\lambda}$.

The next question is whether the above property is satisfied by those
multivectors of ${{\cal K}^{\lambda}}$ that satisfy the orthogonality
condition $\sigma_{c}^{2}={\bf q}_{r}\,{\bf q}^{\dagger}_{d}+{\bf q}_{d}\,{\bf
q}^{\dagger}_{r}=0$ imposed on them for the purpose of normalization. It turns
out that the orthogonality of the quaternions ${{\bf q}_{r}}$ and ${{\bf
q}_{d}}$ in ${\mathbb{Q}}_{z}$ is preserved under multiplication of the
elements of ${\cal K}^{\lambda}$. To prove this, suppose the orthogonality
conditions

${\bf q}_{r1}\,{\bf q}^{\dagger}_{d1}+{\bf q}_{d1}\,{\bf
q}^{\dagger}_{r1}=0\;\;\;\text{and}\;\;\;{\bf q}_{r2}\,{\bf
q}^{\dagger}_{d2}+{\bf q}_{d2}\,{\bf q}^{\dagger}_{r2}=0$ (A.32)

are satisfied for the two distinct elements considered in (A.30). If we now
define the following two quaternions,

${\bf q}_{r3}\,:=\,\left({\bf q}_{r1}\,{\bf q}_{r2}\,+\,{\bf q}_{d1}\,{\bf
q}_{d2}\right)\;\;\;\text{and}\;\;\;{\bf q}_{d3}\,:=\,\left({\bf q}_{r1}\,{\bf
q}_{d2}\,+\,{\bf q}_{d1}\,{\bf q}_{r2}\right),$ (A.33)

then the product (A.31) of the multivectors considered in (A.30) can be
expressed as the following third multivector within ${\cal K}^{\lambda}$:

${\mathbb{Q}}_{z3}=\,{\bf q}_{r3}+{\bf q}_{d3}\,\varepsilon\,.$ (A.34)

It is now easy to prove that the quaternions ${{\bf q}_{r3}}$ and ${{\bf
q}_{d3}}$ are also orthogonal, or equivalently, that ${{\bf q}_{r3}\,{\bf
q}^{\dagger}_{d3}+{\bf q}_{d3}\,{\bf q}^{\dagger}_{r3}=0}$:

$\displaystyle{\bf q}_{r3}\,{\bf q}^{\dagger}_{d3}+{\bf q}_{d3}\,{\bf
q}^{\dagger}_{r3}\,$ $\displaystyle=\,\left({\bf q}_{r1}\,{\bf
q}_{r2}\,+\,{\bf q}_{d1}\,{\bf q}_{d2}\right)\left({\bf q}_{r1}\,{\bf
q}_{d2}\,+\,{\bf q}_{d1}\,{\bf q}_{r2}\right)^{\dagger}+\,\left({\bf
q}_{r1}\,{\bf q}_{d2}\,+\,{\bf q}_{d1}\,{\bf q}_{r2}\right)\left({\bf
q}_{r1}\,{\bf q}_{r2}\,+\,{\bf q}_{d1}\,{\bf q}_{d2}\right)^{\dagger}$ (A.35)
$\displaystyle=\,\left({\bf q}_{r1}\,{\bf q}_{r2}\,+\,{\bf q}_{d1}\,{\bf
q}_{d2}\right)\left({\bf q}_{d2}^{\dagger}\,{\bf q}_{r1}^{\dagger}\,+\,{\bf
q}_{r2}^{\dagger}\,{\bf q}_{d1}^{\dagger}\right)+\,\left({\bf q}_{r1}\,{\bf
q}_{d2}\,+\,{\bf q}_{d1}\,{\bf q}_{r2}\right)\left({\bf
q}_{r2}^{\dagger}\,{\bf q}_{r1}^{\dagger}\,+\,{\bf q}_{d2}^{\dagger}\,{\bf
q}_{d1}^{\dagger}\right)$ (A.36) $\displaystyle=\,{\bf q}_{r1}\,{\bf
q}_{r2}\,{\bf q}_{d2}^{\dagger}\,{\bf q}_{r1}^{\dagger}\,+\,{\bf q}_{r1}\,{\bf
q}_{r2}\,{\bf q}_{r2}^{\dagger}\,{\bf q}_{d1}^{\dagger}\,+\,{\bf q}_{d1}\,{\bf
q}_{d2}\,{\bf q}_{d2}^{\dagger}\,{\bf q}_{r1}^{\dagger}\,+\,{\bf q}_{d1}\,{\bf
q}_{d2}\,{\bf q}_{r2}^{\dagger}\,{\bf q}_{d1}^{\dagger}$
$\displaystyle\;\;\;\;\;\;\;\;+\,{\bf q}_{r1}\,{\bf q}_{d2}\,{\bf
q}_{r2}^{\dagger}\,{\bf q}_{r1}^{\dagger}\,+\,{\bf q}_{r1}\,{\bf q}_{d2}\,{\bf
q}_{d2}^{\dagger}\,{\bf q}_{d1}^{\dagger}\,+\,{\bf q}_{d1}\,{\bf q}_{r2}\,{\bf
q}_{r2}^{\dagger}\,{\bf q}_{r1}^{\dagger}\,+\,{\bf q}_{d1}\,{\bf q}_{r2}\,{\bf
q}_{d2}^{\dagger}\,{\bf q}_{d1}^{\dagger}$ (A.37) $\displaystyle=\,{\bf
q}_{r1}\left\\{{\bf q}_{r2}\,{\bf q}_{d2}^{\dagger}+{\bf q}_{d2}\,{\bf
q}_{r2}^{\dagger}\right\\}{\bf
q}_{r1}^{\dagger}\,+\,\varrho_{r2}^{2}\left\\{{\bf q}_{r1}\,{\bf
q}_{d1}^{\dagger}+{\bf q}_{d1}\,{\bf q}_{r1}^{\dagger}\right\\}$
$\displaystyle\;\;\;\;\;\;\;\;+\,{\bf q}_{d1}\left\\{{\bf q}_{r2}\,{\bf
q}_{d2}^{\dagger}+{\bf q}_{d2}\,{\bf q}_{r2}^{\dagger}\right\\}{\bf
q}_{d1}^{\dagger}\,+\,\varrho_{d2}^{2}\left\\{{\bf q}_{r1}\,{\bf
q}_{d1}^{\dagger}+{\bf q}_{d1}\,{\bf q}_{r1}^{\dagger}\right\\}$ (A.38)
$\displaystyle=\,\left(\varrho_{r1}^{2}+\varrho_{d1}^{2}\right)\left\\{{\bf
q}_{r2}\,{\bf q}_{d2}^{\dagger}+{\bf q}_{d2}\,{\bf
q}_{r2}^{\dagger}\right\\}\,+\,\left(\varrho_{r2}^{2}+\varrho_{d2}^{2}\right)\left\\{{\bf
q}_{r1}\,{\bf q}_{d1}^{\dagger}+{\bf q}_{d1}\,{\bf q}_{r1}^{\dagger}\right\\}$
(A.39) $\displaystyle=\,0.$ (A.40)

Here Eq. (A.39) follows from Eq. (A.38) because, as shown in Eq. (48), the
quantities such as ${\bf q}_{r1}\,{\bf q}_{d1}^{\dagger}+{\bf q}_{d1}\,{\bf
q}_{r1}^{\dagger}$ are purely scalar quantities, and therefore we can use
${\bf q}_{r1}\,{\bf q}^{\dagger}_{r1}=\varrho^{2}_{r1}$, ${\bf q}_{d1}\,{\bf
q}^{\dagger}_{d1}=\varrho^{2}_{d1}$, etc., as before, to reduce Eq. (A.38) to
Eq. (A.39). Then, the orthogonality conditions ${\bf q}_{r1}\,{\bf
q}^{\dagger}_{d1}+{\bf q}_{d1}\,{\bf q}^{\dagger}_{r1}=0$ and ${\bf
q}_{r2}\,{\bf q}^{\dagger}_{d2}+{\bf q}_{d2}\,{\bf q}^{\dagger}_{r2}=0$ from
(A.32) reduces the RHS of Eq. (A.39) to zero.Consequently, the 7-sphere
defined in (54) remains closed under multiplication, analogously to the
octonionic 7-sphere.

###  A.3 Proof of Lemma 3.5

Let ${U}$ and ${V}$ be any two multivectors in ${\cal K}^{\lambda}$ defined by
(34), subject to the normalization condition $\sigma_{c}^{2}={\bf q}_{r}{\bf
q}^{\dagger}_{d}+{\bf q}_{d}{\bf q}^{\dagger}_{r}=0$. Then their norms
$||{U}||$ and $||{V}||$ are scalar numbers. Therefore, the product of their
norms $||{U}||\,||{V}||$ can vanish if and only if either $||{U}||=0$ or
$||{V}||=0$. And the positive definiteness (49) of the norms implies that the
latter is possible if and only if either ${U}=0$ or ${V}=0$. In other words,
if both ${U}\not=0$ and ${V}\not=0$, then $||{U}||\,||{V}||\not=0$, unlike
that in (68). The latter result, namely ${||{X}||\,||{Y}||=0}$ for ${X}\not=0$
and ${Y}\not=0$, occurred because of the hyperbolic norm values
$||{X}||=\sqrt{2\,(1-\varepsilon)}\,$ and $||{Y}||=\sqrt{2\,(1+\varepsilon)}$,
which is not possible for the scalar norm values of $||{U}||$ and $||{V}||$.
But that implies ${||{U}{V}||\not=0}$ because, according to (53),
$||{U}||\,||{V}||=||{U}{V}||$. The positive definiteness (49) of the norms
then implies ${{U}{V}\not=0}$. Thus for arbitrary ${U}\not=0$ and ${V}\not=0$
we have proved that ${{U}{V}\not=0}$. Therefore the normed algebra ${\cal
K}^{\lambda}$ in (34) subject to (48) is a division algebra.

###  A.4 Proof of Eq. (97)

To prove the equality displayed in Eq. (97), we begin with the equalities (84)
= (80) and (90) = (86), which are as follows:

$\lim_{{\mathbf{s}}_{1}\,\rightarrow\,\mu_{1}{\mathbf{a}}}\left\\{\,+\,{\mathbf{q}}(\eta_{{\mathbf{a}}{\mathbf{s}}_{1}},\,{\mathbf{r}}_{1})\right\\}=\lim_{{\mathbf{s}}_{1}\,\rightarrow\,\mu_{1}{\mathbf{a}}}\left\\{-\,{\mathbf{D}}({\mathbf{a}})\,{\mathbf{L}}({\mathbf{s}}_{1})\right\\}$
(A.41)

and

$\lim_{{\mathbf{s}}_{2}\,\rightarrow\,\mu_{2}{\mathbf{b}}}\left\\{\,-\,{\mathbf{q}}(\eta_{{\mathbf{s}}_{2}{\mathbf{b}}},\,{\mathbf{r}}_{2})\right\\}=\lim_{{\mathbf{s}}_{2}\,\rightarrow\,\mu_{2}{\mathbf{b}}}\left\\{+\,{\mathbf{L}}({\mathbf{s}}_{2})\,{\mathbf{D}}({\mathbf{b}})\right\\}.$
(A.42)

Using these equalities, the product of limits on the left-hand side of (97)
can be simplified as follows:

$\displaystyle\left[\lim_{{\mathbf{s}}_{1}\,\rightarrow\,\mu_{1}{\mathbf{a}}}\left\\{+\,{\mathbf{q}}(\eta_{{\mathbf{a}}{\mathbf{s}}_{1}},\,{\mathbf{r}}_{1})\right\\}\right]\left[\lim_{{\mathbf{s}}_{2}\,\rightarrow\,\mu_{2}{\mathbf{b}}}\left\\{-\,{\mathbf{q}}(\eta_{{\mathbf{s}}_{2}{\mathbf{b}}},\,{\mathbf{r}}_{2})\right\\}\right]$
$\displaystyle=\,\bigg{[}\lim_{{\mathbf{s}}_{1}\,\rightarrow\,\mu_{1}{\mathbf{a}}}\left\\{-\,{\mathbf{D}}({\mathbf{a}})\,{\mathbf{L}}({\mathbf{s}}_{1})\right\\}\bigg{]}\left[\lim_{{\mathbf{s}}_{2}\,\rightarrow\,\mu_{2}{\mathbf{b}}}\left\\{+\,{\mathbf{L}}({\mathbf{s}}_{2})\,{\mathbf{D}}({\mathbf{b}})\right\\}\right]$
(A.43)
$\displaystyle=\,\bigg{[}\lim_{{\mathbf{s}}_{1}\,\rightarrow\,\mu_{1}{\mathbf{a}}}\left\\{-I_{3}{\mathbf{a}}\,I_{3}{\mathbf{s}}_{1}\right\\}\bigg{]}\left[\lim_{{\mathbf{s}}_{2}\,\rightarrow\,\mu_{2}{\mathbf{b}}}\left\\{+I_{3}{\mathbf{s}}_{2}\,I_{3}{\mathbf{b}}\right\\}\right]$
(A.44)
$\displaystyle=\,\bigg{[}\lim_{{\mathbf{s}}_{1}\,\rightarrow\,\mu_{1}{\mathbf{a}}}\left\\{{\mathbf{a}}\,{\mathbf{s}}_{1}\right\\}\bigg{]}\left[\lim_{{\mathbf{s}}_{2}\,\rightarrow\,\mu_{2}{\mathbf{b}}}\left\\{-\,{\mathbf{s}}_{2}\,{\mathbf{b}}\right\\}\right]$
(A.45)
$\displaystyle=\,\left[\mu_{1}{\mathbf{a}}\,{\mathbf{a}}\right]\left[-\mu_{2}{\mathbf{b}}\,{\mathbf{b}}\right]$
(A.46) $\displaystyle=-\mu_{1}\mu_{2}\,,$ (A.47)

because all vectors involved in the above equations are unit vectors and I
have used $(I_{3})^{2}=-1$ and the fact that the pseudoscalar $I_{3}$ commutes
with all other elements of $\mathrm{Cl}_{3,0}$. Similarly, using (A.41) and
(A.42) the right-hand side of (97) can be simplified to

$\displaystyle\lim_{\begin{subarray}{c}{\mathbf{s}}_{1}\,\rightarrow\,\mu_{1}{\mathbf{a}}\\\
{\mathbf{s}}_{2}\,\rightarrow\,\mu_{2}{\mathbf{b}}\end{subarray}}\Big{\\{}-{\mathbf{q}}(\eta_{{\mathbf{a}}{\mathbf{s}}_{1}},\,{\mathbf{r}}_{1})\,{\mathbf{q}}(\eta_{{\mathbf{s}}_{2}{\mathbf{b}}},\,{\mathbf{r}}_{2})\Big{\\}}$
$\displaystyle=\lim_{\begin{subarray}{c}{\mathbf{s}}_{1}\,\rightarrow\,\mu_{1}{\mathbf{a}}\\\
{\mathbf{s}}_{2}\,\rightarrow\,\mu_{2}{\mathbf{b}}\end{subarray}}\left[\left\\{-\,{\mathbf{D}}({\mathbf{a}})\,{\mathbf{L}}({\mathbf{s}}_{1})\right\\}\left\\{+\,{\mathbf{L}}({\mathbf{s}}_{2})\,{\mathbf{D}}({\mathbf{b}})\right\\}\right]$
(A.48)
$\displaystyle=\lim_{\begin{subarray}{c}{\mathbf{s}}_{1}\,\rightarrow\,\mu_{1}{\mathbf{a}}\\\
{\mathbf{s}}_{2}\,\rightarrow\,\mu_{2}{\mathbf{b}}\end{subarray}}\left[\left\\{-I_{3}{\mathbf{a}}\,I_{3}{\mathbf{s}}_{1}\right\\}\left\\{+I_{3}{\mathbf{s}}_{2}\,I_{3}{\mathbf{b}})\right\\}\right]$
(A.49)
$\displaystyle=\lim_{\begin{subarray}{c}{\mathbf{s}}_{1}\,\rightarrow\,\mu_{1}{\mathbf{a}}\\\
{\mathbf{s}}_{2}\,\rightarrow\,\mu_{2}{\mathbf{b}}\end{subarray}}\left[\left\\{{\mathbf{a}}\,{\mathbf{s}}_{1}\right\\}\left\\{-\,{\mathbf{s}}_{2}{\mathbf{b}})\right\\}\right]$
(A.50)
$\displaystyle=\lim_{\begin{subarray}{c}{\mathbf{s}}_{1}\,\rightarrow\,\mu_{1}{\mathbf{a}}\\\
{\mathbf{s}}_{2}\,\rightarrow\,\mu_{2}{\mathbf{b}}\end{subarray}}\left[-\,{\mathbf{a}}\,{\mathbf{s}}_{1}\,{\mathbf{s}}_{2}{\mathbf{b}}\right]$
(A.51)
$\displaystyle=\left[-\mu_{1}{\mathbf{a}}\,{\mathbf{a}}\,\mu_{2}{\mathbf{b}}\,{\mathbf{b}}\right]$
(A.52) $\displaystyle=-\mu_{1}\mu_{2}.$ (A.53)

Since the right-hand sides of (A.47) and (A.53) are identical, “the product of
limits = limits of product” holds, as claimed in (97).

###  A.5 Proof of Eq. (118)

Let us begin by expanding the two quaternions appearing on the left-hand side
of Eq. (98) in the standard form as follows:

${\mathbf{q}}(\eta_{{\mathbf{a}}{\mathbf{s}}_{1}},\,{\mathbf{r}}_{1})=\cos(\eta_{{\mathbf{a}}{\mathbf{s}}_{1}})+(I_{3}{\mathbf{r}}_{1})\,\sin(\eta_{{\mathbf{a}}{\mathbf{s}}_{1}})$
(A.54)

and

${\mathbf{q}}(\eta_{{\mathbf{s}}_{2}{\mathbf{b}}},\,{\mathbf{r}}_{2})=\cos(\eta_{{\mathbf{s}}_{2}{\mathbf{b}}})+(I_{3}{\mathbf{r}}_{2})\,\sin(\eta_{{\mathbf{s}}_{2}{\mathbf{b}}}).$
(A.55)

The quaternion
${\mathbf{q}}(\eta_{{\mathbf{u}}{\mathbf{v}}},\,{\mathbf{r}}_{0})$ on the
right-hand side of (98) is then the product of these quaternions and can be
evaluated as follows:

$\displaystyle{\mathbf{q}}(\eta_{{\mathbf{u}}{\mathbf{v}}},\,{\mathbf{r}}_{0})$
$\displaystyle=\\{{\mathbf{q}}(\eta_{{\mathbf{a}}{\mathbf{s}}_{1}},\,{\mathbf{r}}_{1})\\}\\{{\mathbf{q}}(\eta_{{\mathbf{s}}_{2}{\mathbf{b}}},\,{\mathbf{r}}_{2})\\}$
(A.56)
$\displaystyle=\\{\cos(\eta_{{\mathbf{a}}{\mathbf{s}}_{1}})+(I_{3}{\mathbf{r}}_{1})\,\sin(\eta_{{\mathbf{a}}{\mathbf{s}}_{1}})\\}\\{\cos(\eta_{{\mathbf{s}}_{2}{\mathbf{b}}})+(I_{3}{\mathbf{r}}_{2})\,\sin(\eta_{{\mathbf{s}}_{2}{\mathbf{b}}})\\}$
(A.57)
$\displaystyle=\cos(\eta_{{\mathbf{a}}{\mathbf{s}}_{1}})\cos(\eta_{{\mathbf{s}}_{2}{\mathbf{b}}})+(I_{3}{\mathbf{r}}_{2})\cos(\eta_{{\mathbf{a}}{\mathbf{s}}_{1}})\sin(\eta_{{\mathbf{s}}_{2}{\mathbf{b}}})+(I_{3}{\mathbf{r}}_{1})\cos(\eta_{{\mathbf{s}}_{2}{\mathbf{b}}})\sin(\eta_{{\mathbf{a}}{\mathbf{s}}_{1}})+(I_{3}{\mathbf{r}}_{1})(I_{3}{\mathbf{r}}_{2})\sin(\eta_{{\mathbf{a}}{\mathbf{s}}_{1}})\sin(\eta_{{\mathbf{s}}_{2}{\mathbf{b}}}).$
(A.58)

Here $(I_{3}{\mathbf{r}}_{1})(I_{3}{\mathbf{r}}_{2})$ appearing in the last
term is a geometric product of two bivectors, which can be expanded as
follows:

$\displaystyle(I_{3}{\mathbf{r}}_{1})(I_{3}{\mathbf{r}}_{2})$
$\displaystyle=-{\mathbf{r}}_{1}\,{\mathbf{r}}_{2}$ (A.59)
$\displaystyle=-{\mathbf{r}}_{1}\cdot{\mathbf{r}}_{2}-{\mathbf{r}}_{1}\wedge{\mathbf{r}}_{2}$
(A.60)
$\displaystyle=-{\mathbf{r}}_{1}\cdot{\mathbf{r}}_{2}-I_{3}({\mathbf{r}}_{1}\times{\mathbf{r}}_{2}),$
(A.61)

where $I_{3}^{2}=({\bf e}_{x}{\bf e}_{y}{\bf e}_{z})({\bf e}_{x}{\bf
e}_{y}{\bf e}_{z})=-1$ is used. Substituting the right-hand side of (A.61)
into (A.58), the latter simplifies to

$\displaystyle{\mathbf{q}}(\eta_{{\mathbf{u}}{\mathbf{v}}},\,{\mathbf{r}}_{0})=\cos(\eta_{{\mathbf{a}}{\mathbf{s}}_{1}})\cos(\eta_{{\mathbf{s}}_{2}{\mathbf{b}}})+(I_{3}{\mathbf{r}}_{2})\cos(\eta_{{\mathbf{a}}{\mathbf{s}}_{1}})$
$\displaystyle\sin(\eta_{{\mathbf{s}}_{2}{\mathbf{b}}})+(I_{3}{\mathbf{r}}_{1})\cos(\eta_{{\mathbf{s}}_{2}{\mathbf{b}}})\sin(\eta_{{\mathbf{a}}{\mathbf{s}}_{1}})$
$\displaystyle-({\mathbf{r}}_{1}\cdot{\mathbf{r}}_{2})\sin(\eta_{{\mathbf{a}}{\mathbf{s}}_{1}})\sin(\eta_{{\mathbf{s}}_{2}{\mathbf{b}}})-\\{I_{3}({\mathbf{r}}_{1}\times{\mathbf{r}}_{2})\\}\sin(\eta_{{\mathbf{a}}{\mathbf{s}}_{1}})\sin(\eta_{{\mathbf{s}}_{2}{\mathbf{b}}}).$
(A.62)

Next, we wish to express the quaternion
${\mathbf{q}}(\eta_{{\mathbf{u}}{\mathbf{v}}},\,{\mathbf{r}}_{0})$ in the
standard form, with $\eta_{{\mathbf{u}}{\mathbf{v}}}$ being half the rotation
angle of the quaternion:

${\mathbf{q}}(\eta_{{\mathbf{u}}{\mathbf{v}}},\,{\mathbf{r}}_{0})=\cos(\eta_{{\mathbf{u}}{\mathbf{v}}})+(I_{3}{\mathbf{r}}_{0})\,\sin(\eta_{{\mathbf{u}}{\mathbf{v}}}).$
(A.63)

This can be done by collecting the sum of all scalar parts from (A.62) and
identifying it with $\cos(\eta_{{\mathbf{u}}{\mathbf{v}}})$, and collecting
the sum of all bivector parts from (A.62) and identifying it with
$I_{3}\\{{\mathbf{r}}_{0}\,\sin(\eta_{{\mathbf{u}}{\mathbf{v}}})\\}$:

$\cos(\eta_{{\mathbf{u}}{\mathbf{v}}})=\\{\cos(\eta_{{\mathbf{a}}{\mathbf{s}}_{1}})\cos(\eta_{{\mathbf{s}}_{2}{\mathbf{b}}})-({\mathbf{r}}_{1}\cdot{\mathbf{r}}_{2})\sin(\eta_{{\mathbf{a}}{\mathbf{s}}_{1}})\sin(\eta_{{\mathbf{s}}_{2}{\mathbf{b}}})\\}$
(A.64)

and

$I_{3}\\{{\mathbf{r}}_{0}\,\sin(\eta_{{\mathbf{u}}{\mathbf{v}}})\\}=I_{3}\\{{\mathbf{r}}_{1}\cos(\eta_{{\mathbf{s}}_{2}{\mathbf{b}}})\sin(\eta_{{\mathbf{a}}{\mathbf{s}}_{1}})+{\mathbf{r}}_{2}\cos(\eta_{{\mathbf{a}}{\mathbf{s}}_{1}})\sin(\eta_{{\mathbf{s}}_{2}{\mathbf{b}}})-({\mathbf{r}}_{1}\times{\mathbf{r}}_{2})\,\sin(\eta_{{\mathbf{a}}{\mathbf{s}}_{1}})\sin(\eta_{{\mathbf{s}}_{2}{\mathbf{b}}})\\}.$
(A.65)

Using the definitions in (9) and (13) of the axis vectors ${\mathbf{r}}_{1}$