# Response to ‘Comment on “Quantum correlations are weaved by the spinors of
the Euclidean primitives”’

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

###### Abstract

In this paper I respond to a critique of one of my papers previously published
in the Royal Society Open Science entitled “Quantum correlations are weaved by
the spinors of the Euclidean primitives.” Without engaging with the
geometrical framework presented in my paper, the critique incorrectly claims
that there are mathematical errors in it. I demonstrate that the critique is
based on a series of misunderstandings, and refute each of its claims of
error. I also bring out a number of logical, mathematical, and conceptual
errors from the critique and the critiques it relies on.

##  I Introduction

The geometric framework proposed in my paper RSOS  is based on a Clifford-
algebraic interplay between a quaternionic 3-sphere, or $S^{3}$, which I have
taken to model the geometry of three-dimensional physical space, and an
octonion-like 7-sphere, or $S^{7}$, which is an algebraic representation space
of this quaternionic 3-sphere. This framework overcomes Bell’s theorem by
reproducing quantum correlations local-realistically as geometrical effects,
without resorting to backward causation, superdeterminism, or any conspiracy
loophole. It is summed up in the following theorem proven in RSOS :

###### Theorem 3.1

Every quantum mechanical correlation can be understood as a classical, local,
realistic, and deterministic correlation among a set of points of an octonion-
like 7-sphere.

It is important to note that, contrary to the impression given in the critique
Gill-RSOS , it has not engaged with the actual contents of the framework
presented in RSOS . The readers are urged to read RSOS  for themselves to
appreciate this fact. For a comprehensive review and detailed mathematical
foundations of the 7-sphere framework for reproducing quantum correlations as
geometrical effects, see also Reply-to-Lasenby .

It is also important to note that I have already refuted many incorrect claims
made by the same author of the critique Gill-RSOS  in several of my previous
publications Aaronson ; Disproof ; IEEE-3 ; IEEE-4 . But since these claims
have been repeated in Gill-RSOS , some overlap in my point by point response
to them below is not avoidable.

##  II Point by point response to the critique Gill-RSOS

###  II.1 Incorrect claims in the first paragraph of the critique

Let me begin with a number of incorrect claims and mistakes in the first
paragraph of the critique.

  1. (1)

It is claimed in the first paragraph of Gill-RSOS  that in my papers published
between 2007 and 2021 I have proposed “a local hidden variable model (but not
always the same one)…” However, in all my publications on the subject Disproof
; IEEE-3 ; IEEE-4 ; IJTP ; IEEE-1 ; IEEE-2 , what I have proposed is one and
the same quaternionic 3-sphere model mentioned above. The critique’s claim is
thus an early indication that it has not quite understood what I have proposed
in my papers Disproof ; IEEE-3 ; IEEE-4 ; IJTP ; IEEE-1 ; IEEE-2 .

  2. (2)

This fact is confirmed as we read the paragraph further, which claims:
“Christian argues that Bell’s proof of his theorem is mathematically wrong.”
But nowhere have I made such a claim. The mathematical inequalities on which
Bell’s argument depends are trivially correct (cf. Eq. (4.9) of RSOS ). In
fact, they were discovered and proven by Boole some one hundred and eleven
years before the work of Bell Boole-1 ; Boole-2 . I have never questioned
Boole’s inequalities in any of my publications. Thus the claim made in the
critique is not correct.

  3. (3)

The next mistake in the Introduction is more serious. It is claimed that in my
paper RSOS  I connect the 3-sphere, or $S^{3}$, “to special relativity,
specifically to the solution of Einstein’s field equations known as Friedmann-
Robertson-Walker spacetime with a constant spatial curvature.” The same claim
appears in several preprint versions of the critique Gill-RSOS  that have been
posted on arXiv as well as in online discussions. The quoted sentence from
Gill-RSOS  thus exhibits a lack of understanding of the difference between the
special and general theories of relativity and how my proposed quaternionic
3-sphere model fits into the Friedmann-Robertson-Walker solution of Einstein’s
field equations of general relativity. In fact, much of the confusion in Gill-
RSOS  stems from its failure to understand the difference between strong
correlations within flat spacetime
${\mathrm{I\\!R}}\times{\mathrm{I\\!R}}^{3}$ and curved spacetime
${\mathrm{I\\!R}}\times S^{3}$.

  4. (4)

There is a further mathematical error in the first paragraph of Gill-RSOS .
Its last sentence reads: “He furthermore connects it to the 7-sphere $S^{7}$,
thought of as a quaternionic 3-sphere rather than a real 3-sphere.” The two
parts of the quoted sentence are even dimensionally incorrect. A 7-sphere
cannot be thought of as 3-sphere in any sense. Their dimensions are not the
same. Such misreadings of elementary facts indicate that critique’s reading of
RSOS  is mistaken and what it claims to be errors in RSOS  are, in fact,
errors in the reading of RSOS .

###  II.2 Less than correct narrative and claims in the rest of its
Introduction

In the rest of its Introduction, the critique Gill-RSOS  makes several claims
of equally questionable merit:

  1. (1)

It claims that in Gill-Entropy  a critique of my work was published. However,
what is published in Gill-Entropy  is not a critique of my work at all. In it,
a different model is presented based on a flat space, and using matrices and
vector algebra, instead of the quaternionic 3-sphere model using Geometric
Algebra I have proposed in my work Disproof ; IEEE-3 ; IEEE-4 ; IJTP ; IEEE-1
; IEEE-2 . This alternative model is then criticised, claiming the criticism
to be that of my model. The critique in Gill-Entropy  is thus oblivious to the
difference between its own flat space model and my 3-sphere model, and to the
fact that such a comparison amounts to committing a logical fallacy. Moreover,
in the abstract of Gill-Entropy  it is claimed that “Christian’s fundamental
idea is simple and quite original: he gives a probabilistic interpretation of
the fundamental GA equation $a\cdot b=(ab+ba)/2$.” However, I have not
proposed any such interpretation anywhere in my writings. Unfortunately, this
is not the only thing that is mistaken in Gill-Entropy . Several other
equations attributed to my work are also misconstrued in Gill-Entropy . For
example, nowhere in my writings there appear anything like Eq. (22) stipulated
in Gill-Entropy . It also does not follow mathematically from any other
equations I have written down anywhere without violating the conservation of
the initial zero spin angular momentum, as I have explained, for example, in
Subsection IV E of IEEE-3 , Subsection III E of IEEE-4 , and Section VIII of
IEEE-1 . There are also other oversights in Gill-Entropy , which I have
brought out in Aaronson ; Disproof ; IEEE-3 ; IEEE-4 . In particular, Gill-
Entropy  makes the same mathematical mistakes I bring out in detail below in
the Subsection II.7

  2. (2)

In the Introduction of Gill-RSOS  it is claimed that other papers have also
refuted my work. But to date, no one has refuted any part of my work, or
undermined it in any way. To be sure, there have been attempts of refutation,
but I have elucidated the errors in all such claims, for example in Aaronson ;
Disproof ; IEEE-3 ; IEEE-4  and references cited therein. See, especially,
Chapters 9 to 12 in Disproof .

  3. (3)

It is further claimed in the Introduction of Gill-RSOS  that Lasenby in
Lasenby-AACA  has independently made the same claims about the algebraic core
of the 7-sphere framework proposed in RSOS . However, in Lasenby-AACA  it is
acknowledged that “… several of the points made [in Lasenby-AACA ] have been
made independently by [the author of Gill-RSOS ] and others in the discussion
thread attached to the Royal Society paper …” Thus it is not surprising that
the claims in Lasenby-AACA  are similar to those in Gill-RSOS . More
importantly, in Reply-to-Lasenby  and Reply-Lasenby  I have refuted all of the
claims made in Lasenby-AACA .

  4. (4)

The Introduction in Gill-RSOS  is summed up by quoting a few disjoint
sentences out of context from Section II of my reply in IEEE-3 . This allows
Gill-RSOS  to misrepresent the argument I have presented in Section II of
IEEE-3  concerning Bell’s so-called “theorem”, to which I now turn.

###  II.3 Bell’s “theorem” is not a theorem in the mathematical sense

One of the most serious misconceptions exhibited in the critique Gill-RSOS  is
its presumption that Bell’s so-called “theorem” is a proven theorem in the
mathematical sense and therefore any critique of it must be wrong. But it is
important to appreciate the difference between the mathematical inequalities
used by Bell and his physical argument based on those inequalities. As noted
above, the mathematical inequalities discovered by Boole Boole-1 ; Boole-2  on
which Bell’s theorem depends are trivially correct. Moreover, while even
proven mathematical theorems may not be immune to refutations as so lucidly
explained by Lakatos Lakatos , Bell’s theorem is not a theorem in the
mathematical sense to begin with. It depends on a number of implicit and
explicit physical assumptions, which can be and have been questioned before,
not only by me Disproof ; IJTP ; RSOS ; IEEE-1 ; IEEE-2 ; IEEE-3 ; IEEE-4 ,
but also by many others (cf. footnote 1 in IEEE-1 ). If it were a theorem in
mathematical sense, then it would not require physical experiments for its
validity and any loophole (or “gap”) would render it invalid.

It is important to note that Bell’s own writings do not exhibit any such
misconception. Indeed, he actively sought strategies to overcome his
“theorem.” In the concluding sentence of Chapter 17 of his book Bell-2004 ,
Bell reminds us that “…what is proved by impossibility proofs is lack of
imagination.” More seriously, in Section 8 of Chapter 7 and Section 10 of
Chapter 24 of his book, Bell points out that his theorem depends on the
assumption of experimenters’ “free will”, which may turn out to be illusory.
Thus Bell was well aware of the other physical and metaphysical assumptions
that are necessary to support his theorem, in addition to those of locality
and realism.

In the Introduction and Section 4.2 of RSOS , and in Section II of IEEE-3  and
Section III of IEEE-4 , I have highlighted several other physical assumptions
that are necessary for Bell’s theorem to hold. Among these, there are two
assumptions that have been hitherto underappreciated, even though Bell himself
has discussed them at least indirectly. Let me bring them out in some detail
for clarity:

###  II.4 Assumption of the additivity of expectation values

The first among the two assumptions is the assumption of the additivity of
expectation values:

$\displaystyle\int_{\Lambda}{\mathscr{A}}({\bf a},\lambda){\mathscr{B}}({\bf
b},\lambda)\,p(\lambda)\,d\lambda+\\!\\!\int_{\Lambda}{\mathscr{A}}({\bf
a},\lambda){\mathscr{B}}({\bf
b^{\prime}},\lambda)\,p(\lambda)\,d\lambda+\\!\\!\\!\int_{\Lambda}\\!\\!{\mathscr{A}}({\bf
a^{\prime}},\lambda){\mathscr{B}}({\bf
b},\lambda)\,p(\lambda)\,d\lambda-\\!\\!\int_{\Lambda}\\!\\!{\mathscr{A}}({\bf
a^{\prime}},\lambda){\mathscr{B}}({\bf
b^{\prime}},\lambda)\,p(\lambda)\,d\lambda$
$\displaystyle\;\;\;\;=\\!\\!\int_{\Lambda}\\!\Big{\\{}\,{\mathscr{A}}({\bf
a},\lambda)\,{\mathscr{B}}({\bf b},\lambda)+{\mathscr{A}}({\bf
a},\lambda)\,{\mathscr{B}}({\bf b^{\prime}},\lambda)+{\mathscr{A}}({\bf
a^{\prime}},\lambda)\,{\mathscr{B}}({\bf b},\lambda)-{\mathscr{A}}({\bf
a^{\prime}},\lambda)\,{\mathscr{B}}({\bf
b^{\prime}},\lambda)\,\Big{\\}}\;p(\lambda)\,d\lambda,$ (1)

where ${\Lambda}$ is the space of hidden variables ${\lambda}$ and
${p(\lambda)}$ is the probability distribution of ${\lambda}$, so that

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

which is the usual expression of expectation value in the context of Bell’s
theorem Bell-2004 . Now it is universally accepted that without the validity
of Eq. (1), the Bell-CHSH inequalities cannot be derived. On the other hand,
given the assumption of experimenters’ ability to freely, randomly, or
spontaneously choose the detector directions ${\mathbf{a}}$ and
${\mathbf{b}}$, which amounts to assuming
$p(\lambda\,|\,{\mathbf{a}},{\mathbf{b}})=p(\lambda)$, mathematically Eq. (1)
follows at once. However, physically Eq. (1) harbours a non-trivial
assumption. This is obliquely recognized in the critique Gill-RSOS  in the
last paragraph of its Section 2(b):

> He also argues [in Oversight ] that Bell’s proof contains a fundamental
> error in reasoning: the Bell-CHSH inequality involves correlations obtained
> from different sub-experiments involving measurements of non-commuting
> observables, and (he says) therefore cannot be combined. However, in quantum
> mechanics, even if two observables do not commute, a real linear combination
> of those observables is another observable. By the linearity encapsulated in
> the basic rules of quantum mechanics, expectation values of linear
> combinations of non-commuting observables are the same linear combination of
> the expectation values of each observable separately. If a local hidden
> variables model reproduces the statistical predictions of quantum mechanics,
> then it must reproduce this linearity.

But the last sentence of the quoted paragraph is manifestly incorrect. It
reveals a profound lack of understanding of what is meant by a hidden variable
theory since the pioneering work of von Neumann vonNeumann . I have discussed
the problem with Eq. (1) in Section II of IEEE-3  and in Oversight . While
mathematically correct, Eq. (1) is physically meaningless within any hidden
variable theory. It is an assumption over and above those of locality and
realism. In fact, it is the same physical mistake that von Neumann’s former
theorem against general hidden variable theories harboured. For observables
that are not simultaneously measurable, such as those involved in Bell-test
experiments, the replacement of the sum of expectation values with the
expectation value of the sum, although respected in quantum mechanics, does
not hold for hidden variable theories. This was pointed out by Einstein and
Grete Hermann in the 1930s within the context of von Neumann’s theorem, and
thirty years later by Bell Bell-1966  and others, as I have explained in
Oversight .

The example Bell Bell-1966  gives to illustrate this problem is that of the
spin components of a fermion. A measurement of $\sigma_{x}$ can be made with a
suitably oriented Stern-Gerlach magnet. But the measurement of $\sigma_{y}$
would require a different orientation of the magnet. And the measurement of
the sum $(\sigma_{x}+\sigma_{y})$ would again require a third and quite a
different orientation of the magnet from the previous two. Consequently, the
result of the last measurement — i.e., an eigenvalue of
$(\sigma_{x}+\sigma_{y})$ — will not be the sum of an eigenvalue of
$\sigma_{x}$ plus that of $\sigma_{y}$. The additivity of the expectation
values, namely,
$\langle\,\psi\,|\,\sigma_{x}\,|\,\psi\,\rangle+\langle\,\psi\,|\,\sigma_{y}\,|\,\psi\,\rangle=\langle\,\psi\,|\,\sigma_{x}+\,\sigma_{y}\,|\,\psi\,\rangle$,
is a peculiar property of the quantum states $|\,\psi\,\rangle$. It would not
hold for individual eigenvalues of non-commuting observables in a dispersion-
free state of a hidden variable theory. In a dispersion-free state, every
observable would have a unique value equal to one of its eigenvalues. And
since there can be no linear relationship between the eigenvalues of non-
commuting observables, the additivity relation that holds for quantum
mechanical states would not hold for dispersion-free states Oversight .

In summary, the problem with Eq. (1) is that, while the sum of expectation
values appearing on its left-hand side is mathematically equal to the
expectation value of the sum appearing on its right-hand side, and while this
equality holds in quantum mechanics, it does not hold for hidden variable
theories based on dispersion-free states Bell-1966 . That is because the
eigenvalue of a sum of operators is not the sum of eigenvalues when the
constituent operators are non-commuting, as in Bell-test experiments. In other
words, while the joint results
${\mathscr{A}}(\mathbf{a},\lambda){\mathscr{B}}(\mathbf{b},\lambda)$, etc., on
the left-hand side of Eq. (1) are possible eigenvalues of the spin operators
${\boldsymbol{\sigma}}_{1}\cdot{\bf
a}\,\otimes\,{\boldsymbol{\sigma}}_{2}\cdot{\bf b}$, etc., their summation

$\displaystyle{\mathscr{A}}({\bf a},\,\lambda)\,{\mathscr{B}}({\bf
b},\,\lambda)+{\mathscr{A}}({\bf a},\,\lambda)\,{\mathscr{B}}({\bf
b^{\prime}},\,\lambda)+{\mathscr{A}}({\bf
a^{\prime}},\,\lambda)\,{\mathscr{B}}({\bf b},\,\lambda)-{\mathscr{A}}({\bf
a^{\prime}},\,\lambda)\,{\mathscr{B}}({\bf b^{\prime}},\,\lambda)$ (3)

appearing as the integrand on the right-hand side of Eq. (1) is not an
eigenvalue of the operator

${\boldsymbol{\sigma}}_{1}\cdot{\bf
a}\,\otimes\,{\boldsymbol{\sigma}}_{2}\cdot{\bf
b}+{\boldsymbol{\sigma}}_{1}\cdot{\bf
a}\,\otimes\,{\boldsymbol{\sigma}}_{2}\cdot{\bf
b^{\prime}}+{\boldsymbol{\sigma}}_{1}\cdot{\bf
a^{\prime}}\,\otimes\,{\boldsymbol{\sigma}}_{2}\cdot{\bf
b}-{\boldsymbol{\sigma}}_{1}\cdot{\bf
a^{\prime}}\,\otimes\,{\boldsymbol{\sigma}}_{2}\cdot{\bf b^{\prime}},$ (4)

because the joint operators ${\boldsymbol{\sigma}}_{1}\cdot{\bf
a}\,\otimes\,{\boldsymbol{\sigma}}_{2}\cdot{\bf b}$, etc., do not commute with
each other. On the other hand, the very meaning of a hidden variable theory
dictates simultaneous assignment of definite eigenvalues to all observables of
the singlet system Oversight , including the one in (4), whether or not they
are actually measured (albeit this assignment has to be contextual in the
light of the Kochen-Specker theorem). But since the sum of results (3) is not
one of the eigenvalues of the summed operator in (4), its appearance on the
right-hand side of (1) is incorrect, making the replacement of the left-hand
side of Eq. (1) with its right-hand side at least physically invalid. But
without this replacement the absolute upper bound of 2 on the left-hand side
of Eq. (1) cannot be derived.

Once this oversight is removed from Bell’s “theorem” Bell-2004  and local
realism is implemented correctly by using the correct eigenvalue of (4) (which
I have worked out explicitly in Appendix A of Oversight ) instead of (3) on
the right-hand side of (1), the bounds on the left-hand side of (1) work out
to be $\pm 2\sqrt{2}$ instead of $\pm 2$ (as I have demonstrated, for example,
in Section V of Oversight ), thereby mitigating the conclusions of Bell’s
theorem Oversight . Consequently, what is ruled out by the Bell-test
experiments is not local realism as widely believed, but the assumption of the
additivity of expectation values, which does not hold in general for any
hidden variable theories to begin with.

###  II.5 Assumption of a flat and immutable spacetime

The second assumption necessary to support Bell’s theorem is that of immutable
spacetime. The formulation of the theorem thus neglects the mutable spacetime
geometries of Einstein’s theory of gravity. Note that it is not the strength
of gravity that is at stake here but the qualitative differences between
immutable flat spacetime and mutable curved spacetimes Newton-Cartan . This is
hinted at by Bell himself. In Section 8 of Chapter 7 of his book Bell-2004 ,
while exploring possible strategies that may be used to overcome his theorem,
he writes: “The space time structure has been taken as given here. How then
about gravitation?” Thus Bell seems to have anticipated using a solution of
Einstein’s field equations of general relativity to overcome his theorem, as I
have proposed in Disproof ; IJTP ; RSOS ; IEEE-1 ; IEEE-2 ; IEEE-3 ; IEEE-4 ;
Symmetric .

By contrast, in the critique Gill-RSOS  Bell’s theorem is stated using
“ordinary 3D Euclidean space”:

> Suppose that $X_{a}$ and $Y_{b}$ are a family of random variables on a
> single probability space, taking values in the set $\\{-1,+1\\}$, and where
> $a$ and $b$ denote directions in ordinary 3D Euclidean space, represented by
> unit vectors $a$, $b$. Then it is not possible that
> ${\mathbb{E}(X_{a}Y_{b})=-a\cdot b}$ for all $a$ and $b$.

Stated thus, the physical limitation of Bell’s theorem is conspicuous. It is
revealed in the following explicit assumption about the geometry of the three-
dimensional physical space: “… where $a$ and $b$ denote directions in ordinary
3D Euclidean space, represented by unit vectors $a$, $b$.” But why must we
allow such an outmoded view of physical space to make radical claims about the
fundamental nature of locality and reality after more than a century of
general-relativistic revolution in which spacetime geometry is rendered
dynamical and malleable? And why must we use vector “algebra” after more than
a century and a half of insights from Grassmann and Clifford on the correct
algebraic representation of the three-dimensional physical space? As I have
proposed in Disproof ; IJTP ; RSOS ; IEEE-1 ; IEEE-2 ; IEEE-3 ; IEEE-4 , there
are both theoretical and observational reasons that compel us to model
physical space as a closed and compact quaternionic 3-sphere, or $S^{3}$,
instead of a flat Euclidean space, or ${\mathrm{I\\!R}^{3}}$, both being
admissible spatial parts of one of the well-known cosmological solutions of
Einstein’s field equations of general relativity. Moreover, as explained in
Disproof ; IJTP ; RSOS ; IEEE-1 ; IEEE-2 ; IEEE-3 ; IEEE-4 , the correct
language to model $S^{3}$ as physical space is Geometric Algebra, not vector
“algebra.” But once the physical space is modelled as $S^{3}$ instead of
$\mathrm{I\\!R}^{3}$ using the powerful language of Geometric Algebra, the
correlation between the local results ${\mathscr{A}_{\bf a}}$ and
${\mathscr{B}_{\bf b}}$ observed by Alice and Bob inevitably turns out to be
${\cal E}({\mathscr{A}_{\bf a}}{\mathscr{B}_{\bf b}})=-{\bf a}\cdot{\bf b}$ as
I have proved in RSOS ; Disproof ; IJTP ; IEEE-1 ; IEEE-2 ; IEEE-3 ; IEEE-4 ;
Symmetric , contrary to the claims made in Gill-RSOS . For a comprehensive
proof of ${\cal E}({\mathscr{A}_{\bf a}}{\mathscr{B}_{\bf b}})=-{\bf
a}\cdot{\bf b}$ within the local-realistic geometry of $S^{3}$, I especially
recommend the derivations of the singlet correlations in Reply-to-Lasenby  and
Symmetric .

Given the many implicit assumptions required to prove Bell’s theorem in
addition to locality and realism I have brought out in the Introduction and
Section 4.2 of RSOS , in Section II of IEEE-3 , and in Section III of IEEE-4 ,
together with the two assumptions discussed above, the infallibilist view of
the theorem adhered to in the critique Gill-RSOS  (to borrow the term coined
by LakatosLakatos ) is not justified.

###  II.6 Orientation $\lambda$ of ${\cal K}^{\lambda}$ acts as a hidden
variable in the $S^{7}$ model

Another issue raised in the critique Gill-RSOS  concerns the orientation of
the vector space ${\cal K}^{\lambda}$ used in RSOS :

> A curious elementary mathematical error is that he defines two algebras,
> built from two 8-dimensional real vector spaces ${\cal K}^{+}$ and ${\cal
> K}^{-}$ by specifying a vector space basis for each algebra and
> multiplication tables for the 8 basis elements of each algebra. But they are
> the same algebra. The linear spans of those two bases are trivially the
> same. The multiplication operation is the same.

However, there is no such error in RSOS . I have not defined “two algebras”
built from “two” vector spaces ${\cal K}^{+}$ and ${\cal K}^{-}$ and
multiplication tables for “each algebra.” I have defined only one algebra,
namely the even subalgebra ${\cal K}^{\lambda}$ of the Clifford algebra
$\mathrm{Cl}_{4,0}$, with only one multiplication table for that algebra,
namely the Table 1 on page 8 of RSOS . The superscript ${\lambda=\pm}$ over
${\cal K}$ refers to the orientation (or handedness) of the corresponding
vector space. Moreover, there is no claim in RSOS  that the linear spans of
${\cal K}^{+}$ and ${\cal K}^{-}$ are different. ${\cal K}^{\lambda}$ is one
and the same algebra, or vector space, with two possible orientations. There
is nothing unusual or unorthodox about this concept. It is clearly explained
in Subsection 2.3 of RSOS  that ${\cal K}^{+}$ and ${\cal K}^{-}$ differ only
in their orientations: $\lambda=+$ or $\lambda=-$. If one of them (say ${\cal
K}^{+}$) is deemed right-handed, then the other one (${\cal K}^{-}$) is left-
handed, and vice versa, with the definition of orientation from Milnor Milnor
reproduced on page 7 of RSOS :

> Definition of Orientation: 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 ${{\rm
> det}(\omega_{ij})<0}$.

On the page 8 of RSOS , I have stated: “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 ${\cal K}^{\lambda}=\,{\rm span}\left\\{1,\,\lambda{\bf
e}_{x}{\bf e}_{y},\,\lambda{\bf e}_{z}{\bf e}_{x},\,\lambda{\bf e}_{y}{\bf
e}_{z},\,\lambda{\bf e}_{x}{\bf e}_{\infty},\,\lambda{\bf e}_{y}{\bf
e}_{\infty},\,\lambda{\bf e}_{z}{\bf e}_{\infty},\,\lambda I_{3}{\bf
e}_{\infty}\right\\},\;\lambda^{2}=1\Longleftrightarrow\lambda=\pm 1.$”

In the later sections in RSOS , ${\lambda}$ is taken as a Bell-type hidden
variable, with physical consequences. Therefore ${{\cal K}^{+}}$ and ${{\cal
K}^{-}}$ are physically not identical within the 7-sphere framework proposed
in RSOS .

###  II.7 Algebra ${{\cal K}^{\lambda}}$ used in RSOS  is not incompatible
with Hurwitz’s theorem

In its Section 2, the critique Gill-RSOS  claims that Hurwitz’s theorem
contradicts the mathematical claims in RSOS . However, the critique seems to
have missed Appendix A of RSOS  where I have explicitly discussed Hurwitz’s
theorem in detail, as well as its significance for the 7-sphere framework
presented in RSOS . Overlooking this discussion, the critique claims a
counterexample to Eq. (2.40) of RSOS , which says that for any multivectors
$X$ and $Y$ in ${\cal K}^{\lambda}$, the following composition law holds:

$||XY||=||X||\,||Y||,$ (5)

where the norms defined in RSOS  as $||X||:=\sqrt{XX^{\dagger}\,}$ using
geometric products are positive definite:

$||X||=0\iff X=0\,.$ (6)

This definition is consistent with how norms are defined for complex numbers
$c$, quaternions ${\bf q}$, and octonions ${\bf O}$. Namely, by
$||{c}||=\sqrt{{c}\,{c}^{\dagger}}$, $||{\bf q}||=\sqrt{{\bf q}{\bf
q}^{\dagger}}$, and $||{\bf O}||=\sqrt{{\bf O}{\bf O}^{\dagger}}$,
respectively RSOS ; Dray . Moreover, a geometric product between $X$ and
$X^{\dagger}$ in the definition $||X||:=\sqrt{XX^{\dagger}\,}$ is necessary
for maintaining consistency between the two sides of Eq. (5) because only a
geometric product between $X$ and $Y$ can produce a new multivector $Z=XY$ in
${\cal K}^{\lambda}$ appearing on its left-hand side. On the other hand,
because the algebra ${\cal K}^{\lambda}$ is a tensor product
${\mathbb{H}}\otimes\mathbb{C}^{\prime}$ of quaternions with split complex
numbers, its elements are of the form $X=\,{\bf q}_{r}+{\bf
q}_{d}\,\varepsilon$ with $\varepsilon^{2}=1$, and consequently the quadratic
form ${XX^{\dagger}}$ in general takes values in split complex numbers
$\mathbb{C}^{\prime}$ instead of real numbers.

Challenging the norm relation (5) (which I have proved in Section 2.5 of RSOS
, Appendix A.1 of Reply-to-Lasenby , and Appendix B of Eight ), the critique
in Gill-RSOS  alleges the following counterexample without providing a
definition of norm or specifying what it means by “taking norms”
mathematically:

> However, the author’s algebra has an element called the ‘pseudo-scalar’, I
> will denote it by $M$, such that $M^{2}=1$. It follows that
> $0=M^{2}-1=(M-1)(M+1)$. Taking norms, $0=||M-1||.||M+1||$. Hence $||M-1||=0$
> or $||M+1||=0$. Therefore $M-1=0$ or $M+1=0$, which implies that $M=1$ or
> $M=-1$. That is a contradiction.

The same claim appears in Gill-Entropy  and Gill-apology , after the
correction of a mistake I had pointed out in Aaronson .

But it is easy to see that this alleged counterexample continues to harbour
several mistakes. The first mistake in the quoted claim is immediately
obvious. It starts out with the equation $M^{2}=1$ and ends with the equations
$M=1$ or $M=-1$. And then it claims that “That is a contradiction.” However,
the argument $M^{2}=1\implies M=1$ or $M=-1$ by itself is not a contradiction.
To allege a contradiction one must assume that $M$ is a pseudoscalar. But then
it by no means follows from $||M-1||\,||M+1||=0$ that either $||M-1||=0$ or
$||M+1||=0$ as alleged in the critique, unless two different product rules are
employed on the two sides of equation (5) — a geometric product, namely
$(M-1)(M+1)=M^{2}-1$, to derive $0$ on the left-hand side of (5) and scalar
products such as in $\sqrt{(M-1)\cdot(M-1)^{\dagger}}$ to evaluate the norms
on its right-hand side. Needless to say, one can always derive a contradiction
from any mathematical equation by employing two different product rules on the
two sides of that equation111Some reviewers unjustifiably defended this
counterexample during the review process of this paper. In addition to Reply-
to-Lasenby , my detailed rebuttal to their defence is available online in the
Review History published along with this paper. All variants of the alleged
counterexample depend on inconsistent application of product rules on the two
sides of equation (5). For example, in the counterexample alleged in Lasenby-
AACA , a contradiction is achieved by employing geometric product between
$X=(1+\varepsilon)/\sqrt{2}$ and $Y=(1-\varepsilon)/\sqrt{2}$ on the left-hand
side of equation (5), whereas scalar products are used to evaluate its right-
hand side.. Barring that inconsistency, what the equation $||M-1||\,||M+1||=0$
says is that the geometric product between $||M-1||$ and $||M+1||$ must
vanish, precisely because $M$, and therefore $||M-1||$ and $||M+1||$, are no
longer scalar quantities, and a geometric product, namely $(M-1)(M+1)$, is
used to derive 0 on the left-hand side of (5).

Indeed, the notation used in the critique for its equation
“$0=||M-1||.||M+1||$” seems to tacitly acknowledge that $||M-1||$ and
$||M+1||$ are not scalars. For otherwise there would be no need to introduce a
“dot” between $||M-1||$ and $||M+1||$ indicating a scalar product between them
when no such dot appears in the relation (5). But even with the scalar product
introduced between $||M-1||$ and $||M+1||$ in an ad hoc manner, the critique’s
alleged conclusion that either $||M-1||=0$ or $||M+1||=0$ does not follow.
Therefore the critique’s claim of contradiction fails.

Now it is easy to prove that geometric product between $||M-1||$ and $||M+1||$
does vanish. In RSOS , I have denoted the pseudoscalar in ${{\cal
K}^{\lambda}}$ by ${\varepsilon:={\bf e}_{1}{\bf e}_{2}{\bf e}_{3}{\bf
e}_{\infty}\not=\pm 1}$. It satisfies the properties
$\varepsilon^{\dagger}=\varepsilon$ and $\varepsilon^{2}=1$. Given this, what
the critique has considered are the following elements in ${{\cal
K}^{\lambda}}$:

$X=\varepsilon-1\;\;\;\;\text{and}\;\;\;\;Y=\varepsilon+1.$ (7)

But, to begin with, no such two-dimensional multivectors play any role
whatsoever in the 7-sphere framework presented in RSOS . Therefore, even if
such ad hoc two-dimensional objects lead to “contradiction” as alleged in the
critique Gill-RSOS , that would have no effect on or consequences for the
7-sphere framework. Nevertheless, it is instructive to play along with (7).
The question then is: Do such multivectors in ${{\cal K}^{\lambda}}$ satisfy
the norm relation (5)? If they do not, then the claim made in the critique
would be correct. But if they do, then the claim made in my paper would be
correct. To investigate the question, we begin with evaluating the left-hand
side of the norm relation (5):

$||XY||=||(\varepsilon-1)(\varepsilon+1)||=\left|\left|\varepsilon^{2}-1\right|\right|=\left|\left|1-1\right|\right|=||\,0\,||=0,$
(8)

where $\varepsilon^{2}=1$ is used. Next, using
$\varepsilon^{\dagger}=\varepsilon$, we evaluate the right-hand side of the
norm relation (5):

$||X||=||(\varepsilon-1)||=\sqrt{(\varepsilon-1)(\varepsilon-1)^{\dagger}}=\sqrt{(\varepsilon-1)(\varepsilon-1)}=\sqrt{2\,(1-\varepsilon)}=1-\varepsilon$
(9)

and

$||Y||=||(\varepsilon+1)||=\sqrt{(\varepsilon+1)(\varepsilon+1)^{\dagger}}=\sqrt{(\varepsilon+1)(\varepsilon+1)}=\sqrt{2\,(1+\varepsilon)}=1+\varepsilon,$
(10)

and therefore

$||X||\,||Y||=(1-\varepsilon)(1+\varepsilon)=1-\varepsilon^{2}=1-1=0.$ (11)

Comparing (8) and (11) we see that the norm relation (5) is satisfied for the
multivectors $X$ and $Y$ considered in (7). Thus, contrary to the claim in
Gill-RSOS , $X$ and $Y$ considered in (7) do not entail a counterexample to
(5). The contradiction to (5) is achieved in Gill-RSOS  by computing norms
incorrectly. Moreover, the non-scalar values of $||X||$ and $||Y||$ arrived at
in (9) and (10) reiterate the fact that $X$ and $Y$ considered in the critique
Gill-RSOS  are not parts of the 7-sphere framework proposed in RSOS .

###  II.8 Proof of the norm relation (5) for the eight-dimensional algebra
${\cal K}^{\lambda}$

Elsewhere Reply-to-Lasenby ; Eight , I have proved that the norm relation (5)
holds, without exception, for arbitrary $X$ and $Y$ in ${\cal K}^{\lambda}$,
and, as a special case, reduces to the one with scalar values for $||X||$,
$||Y||$, and $||XY||$. The proofs of (5) and (6) are straightforward222During
the review process of this paper, one of the reviewers independently verified
this proof of the norm relations (5) and (27) with detailed calculations and
comments, which are available online in the Review History published with this
paper.. As noted above, the algebra ${\cal K}^{\lambda}$ is a tensor product
$\mathbb{H}\otimes\mathbb{C}^{\prime}$ of quaternions with split complex
numbers Dray , but with conjugation (or “reverse”) affecting only the
quaternions RSOS . In other words, any multivectors $X$ and $Y$ in ${\cal
K}^{\lambda}$ are of the form $X=\,{\bf q}_{r1}+{\bf q}_{d1}\,\varepsilon$ and
$Y=\,{\bf q}_{r2}+{\bf q}_{d2}\,\varepsilon$ with $\varepsilon^{2}=+1$ and
$\varepsilon^{\dagger}=\varepsilon$, where ${\bf q}_{r1}$ and ${\bf q}_{d1}$
constituting $X$, for example, are two independent quaternions, which can be
written as ${\bf q}_{r1}=g_{1}+I_{3}{\bf u}_{1}$ and ${\bf
q}_{d1}=h_{1}+I_{3}{\bf v}_{1}$, where $g_{1}$ and $h_{1}$ are scalars,
${I_{3}={\bf e}_{1}{\bf e}_{2}{\bf e}_{3}}$ is the standard pseudoscalar in
three dimensions, and ${{\bf u}_{1}=u_{1x}{\bf e}_{x}+u_{1y}{\bf
e}_{y}+u_{1z}{\bf e}_{z}}$ and ${{\bf v}_{1}=v_{1x}{\bf e}_{x}+v_{1y}{\bf
e}_{y}+v_{1z}{\bf e}_{z}}$ are Cartesian vectors. As a result, the geometric
product $XX^{\dagger}$ between $X$ and $X^{\dagger}$ works out to be

$\displaystyle XX^{\dagger}\,$ $\displaystyle=\left({\bf q}_{r1}+{\bf
q}_{d1}\,\varepsilon\right)\left({\bf q}_{r1}+{\bf
q}_{d1}\,\varepsilon\right)^{\dagger}$ (12) $\displaystyle=\left({\bf
q}_{r1}\,{\bf q}^{\dagger}_{r1}\,+\,{\bf q}_{d1}\,{\bf
q}^{\dagger}_{d1}\right)+\left({\bf q}_{r1}\,{\bf q}^{\dagger}_{d1}\,+\,{\bf
q}_{d1}\,{\bf q}^{\dagger}_{r1}\right)\varepsilon$ (13)
$\displaystyle=(g_{1}^{2}+{\bf u}_{1}\cdot{\bf u}_{1}+h_{1}^{2}+{\bf
v}_{1}\cdot{\bf v}_{1})+(2\,g_{1}h_{1}+2\,{\bf u}_{1}\cdot{\bf
v}_{1})\,\varepsilon$ (14) $\displaystyle=\,\text{(a scalar)}+\text{(a
scalar)}\;\varepsilon\,.$ (15)

And likewise, using $Y=\,{\bf q}_{r2}+{\bf q}_{d2}\,\varepsilon$, together
with ${\bf q}_{r2}=g_{2}+I_{3}{\bf u}_{2}$ and ${\bf q}_{d2}=h_{2}+I_{3}{\bf
v}_{2}$, the geometric product $YY^{\dagger}$ between $Y$ and $Y^{\dagger}$
works out to be

$\displaystyle YY^{\dagger}\,$ $\displaystyle=\left({\bf q}_{r2}+{\bf
q}_{d2}\,\varepsilon\right)\left({\bf q}_{r2}+{\bf
q}_{d2}\,\varepsilon\right)^{\dagger}$ (16) $\displaystyle=\left({\bf
q}_{r2}\,{\bf q}^{\dagger}_{r2}\,+\,{\bf q}_{d2}\,{\bf
q}^{\dagger}_{d2}\right)+\left({\bf q}_{r2}\,{\bf q}^{\dagger}_{d2}\,+\,{\bf
q}_{d2}\,{\bf q}^{\dagger}_{r2}\right)\varepsilon$ (17)
$\displaystyle=(g_{2}^{2}+{\bf u}_{2}\cdot{\bf u}_{2}+h_{2}^{2}+{\bf
v}_{2}\cdot{\bf v}_{2})+(2\,g_{2}h_{2}+2\,{\bf u}_{2}\cdot{\bf
v}_{2})\,\varepsilon$ (18) $\displaystyle=\,\text{(a scalar)}+\text{(a
scalar)}\;\varepsilon\,.$ (19)

The positive definiteness (6) is now easy to prove. Since $X=\,{\bf
q}_{r1}+{\bf q}_{d1}\,\varepsilon$, it can be zero only if both ${\bf
q}_{r1}=0$ and ${\bf q}_{d1}=0$. But from (13) that implies $XX^{\dagger}=0$,
and therefore $\sqrt{XX^{\dagger}}=\|X\|=0$. Thus $X=0\Longrightarrow\|X\|=0$
is proved. Conversely, suppose $\|X\|=0$. Then again from (13) we have
$\|X\|=\sqrt{XX^{\dagger}}=\sqrt{\left({\bf q}_{r1}\,{\bf
q}^{\dagger}_{r1}+\,{\bf q}_{d1}\,{\bf q}^{\dagger}_{d1}\right)+\left({\bf
q}_{r1}\,{\bf q}^{\dagger}_{d1}+\,{\bf q}_{d1}\,{\bf
q}^{\dagger}_{r1}\right)\varepsilon\,}=0$. But that is possible only if the
quantities appearing in both parentheses under the square root are zero.
However, both ${\bf q}_{r1}\,{\bf q}^{\dagger}_{r1}$ and ${\bf q}_{d1}\,{\bf
q}^{\dagger}_{d1}$ in the first parenthesis $\left({\bf q}_{r1}\,{\bf
q}^{\dagger}_{r1}+\,{\bf q}_{d1}\,{\bf q}^{\dagger}_{d1}\right)$ are positive
definite scalars, and therefore they both must be zero for ${\bf q}_{r1}\,{\bf
q}^{\dagger}_{r1}+\,{\bf q}_{d1}\,{\bf q}^{\dagger}_{d1}$ to be zero. But
since ${\bf q}_{r1}\,{\bf q}^{\dagger}_{r1}$ and ${\bf q}_{d1}\,{\bf
q}^{\dagger}_{d1}$ are positive definite scalars, they can be zero only if
${\bf q}_{r1}$ and ${\bf q}_{d1}$ are zero. In other words, both ${\bf
q}_{r1}$ and ${\bf q}_{d1}$ must be zero for $\|X\|=0$ to hold. But ${\bf
q}_{r1}=0$ and ${\bf q}_{d1}=0$ implies that $X=\,{\bf q}_{r1}+{\bf
q}_{d1}\,\varepsilon=0$. Therefore, the converse $\|X\|=0\Longrightarrow X=0$
holds. The positive definiteness (6) is thus proved.

The proof of the norm relation (5) is also straightforward. Evidently, the
geometric products $XX^{\dagger}$ and $YY^{\dagger}$ written above resemble
split complex numbers because for any two quaternions (such as ${\bf q}_{r1}$
and ${\bf q}_{d1}$) the quantities appearing in the parentheses in (14) and
(18) are scalar quantities. Thus the products are of the form
$XX^{\dagger}=a+b\,\varepsilon$ and $YY^{\dagger}=c+d\,\varepsilon$, where
$a={\bf q}_{r1}\,{\bf q}^{\dagger}_{r1}+{\bf q}_{d1}\,{\bf q}^{\dagger}_{d1}$,
$b={\bf q}_{r1}\,{\bf q}^{\dagger}_{d1}+{\bf q}_{d1}\,{\bf q}^{\dagger}_{r1}$,
$c={\bf q}_{r2}\,{\bf q}^{\dagger}_{r2}+{\bf q}_{d2}\,{\bf q}^{\dagger}_{d2}$,
and $d={\bf q}_{r2}\,{\bf q}^{\dagger}_{d2}+{\bf q}_{d2}\,{\bf
q}^{\dagger}_{r2}$ are scalar quantities, with
$\varepsilon^{\dagger}=+\,\varepsilon$ instead of $-\,\varepsilon$.
Consequently, for the left-hand side of (5) we have

$\displaystyle||XY||=\sqrt{(XY)(XY)^{\dagger}}=\sqrt{(XY)\left(Y^{\dagger}X^{\dagger}\right)}=\sqrt{X\left(YY^{\dagger}\right)X^{\dagger}}$
$\displaystyle=\sqrt{X(c+d\,\varepsilon)X^{\dagger}}$ (20)
$\displaystyle=\sqrt{\left(XX^{\dagger}\right)(c+d\,\varepsilon)}$ (21)
$\displaystyle=\sqrt{(a+b\,\varepsilon)(c+d\,\varepsilon)}$ (22)
$\displaystyle=\sqrt{(ac+bd)+(ad+bc)\,\varepsilon\,},$ (23)

because, as noted, the pseudoscalar $\varepsilon$ satisfies
$\varepsilon^{2}=1$ and commutes with every element of ${\cal K}^{\lambda}$,
and consequently the identity
$(XY)^{\dagger}=\left(Y^{\dagger}X^{\dagger}\right)$ for $X$ and $Y$ in ${\cal
K}^{\lambda}$ is straightforward to verify. On the other hand, the right-hand
side of the equation (5) also works out to give the same quantity:

$||X||\,||Y||=\left(\sqrt{XX^{\dagger}}\;\right)\left(\sqrt{YY^{\dagger}}\;\right)=\sqrt{(a+b\,\varepsilon)(c+d\,\varepsilon)}=\sqrt{(ac+bd)+(ad+bc)\,\varepsilon\,}.$
(24)

Comparing (23) and (24) we see that when product rules are applied
consistently on the two sides of equation (5) using geometric products, both
of its sides work out to be identical to the square-root of the following
quantity, which also resembles a split complex or hyperbolic number,

$\displaystyle\left\\{\left(\varrho^{2}_{r1}+\varrho^{2}_{d1}\right)\left(\varrho^{2}_{r2}+\varrho^{2}_{d2}\right)\,+\,\left({\bf
q}_{r1}\,{\bf q}^{\dagger}_{d1}+{\bf q}_{d1}\,{\bf
q}^{\dagger}_{r1}\right)\left({\bf q}_{r2}\,{\bf q}^{\dagger}_{d2}+{\bf
q}_{d2}\,{\bf q}^{\dagger}_{r2}\right)\right\\}$
$\displaystyle\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;+\left\\{\left(\varrho^{2}_{r1}\,+\,\varrho^{2}_{d1}\right)\left({\bf
q}_{r2}\,{\bf q}^{\dagger}_{d2}+{\bf q}_{d2}\,{\bf
q}^{\dagger}_{r2}\right)+\left({\bf q}_{r1}\,{\bf q}^{\dagger}_{d1}+{\bf
q}_{d1}\,{\bf
q}^{\dagger}_{r1}\right)\left(\varrho^{2}_{r2}\,+\,\varrho^{2}_{d2}\right)\right\\}\,\varepsilon\,,$
(25)

thus proving the norm relation (5), where $\varrho_{r1}=\sqrt{{\bf
q}_{r1}\,{\bf q}^{\dagger}_{r1}}\,$, etc., and the quantities appearing in the
two curly brackets are scalar quantities. Comparing (13), (17), and (25), we
thus see that the coefficient algebra underlying ${\cal K}^{\lambda}$
resembles that of split complex numbers333Hurwitz’s theorem, as usually
stated, assumes that the coefficient algebra underlying the algebras such as
$\mathbb{R}$, $\mathbb{C}$, $\mathbb{H}$, and $\mathbb{O}$ is that of real
numbers. It is known, however, that if the norms are generalized to take
values in split complex numbers
$\mathbb{C}^{\prime}=\\{a+b\,\varepsilon\,|\,\varepsilon^{2}=1\\}$, then new
composition algebras can be formed. See the discussion, for example, in
Section 5.3 of Dray . instead of real numbers, and therefore what is just
proved does not contradict Hurwitz’s theorem RSOS ; Dray , contrary to the
claims made in the critiques Gill-RSOS ; Lasenby-AACA . In fact, the result
(25) holds for any multivectors $X$ and $Y$ in ${\cal K}^{\lambda}$, and
therefore the norm relation (5) holds for any multivectors $X$ and $Y$ in the
algebra ${\cal K}^{\lambda}$, because it is possible to work out square-root
of a hyperbolic number such as (25), as shown in Appendix A.1 of Reply-to-
Lasenby . The normalization or orthogonality condition specified in Eq. (2.54)
of RSOS  — which amounts to setting the coefficient ${\bf q}_{r1}\,{\bf
q}^{\dagger}_{d1}+{\bf q}_{d1}\,{\bf q}^{\dagger}_{r1}=0$ in (13) and ${\bf
q}_{r2}\,{\bf q}^{\dagger}_{d2}+{\bf q}_{d2}\,{\bf q}^{\dagger}_{r2}=0$ in
(17) so that the norms $||X||$ and $||Y||$ reduce, respectively, to scalar
values $\sqrt{\varrho_{r1}^{2}+\varrho_{d1}^{2}\,}$ and
$\sqrt{\varrho_{r2}^{2}+\varrho_{d2}^{2}\,}$ — then reduces the square-root of
the hyperbolic number (25) to the scalar quantity

$\sqrt{\left(\varrho^{2}_{r1}+\varrho^{2}_{d1}\right)\left(\varrho^{2}_{r2}+\varrho^{2}_{d2}\right)}\,.$
(26)

We have thus proved the norm relation (5) with scalar values, as originally
proved in RSOS  and Eight :

$||XY||=\sqrt{\left(\varrho^{2}_{r1}+\varrho^{2}_{d1}\right)\left(\varrho^{2}_{r2}+\varrho^{2}_{d2}\right)\,}=||X||\,||Y||.$
(27)

In particular, since the norm relation (5) holds for any multivectors $X$ and
$Y$ in ${\cal K}^{\lambda}$ and therefore also for those with scalar values
for their norms, if $X$ and $Y$ happen to be unit multivectors so that
$||X||=1$ and $||Y||=1$, then (27) necessitates that their product $Z=XY$ will
also be a unit multivector, $||Z||=||XY||=||X||\,||Y||=1\times 1=1$, contrary
to the claims made in Gill-RSOS  and Lasenby-AACA .

Eq. (27) also leads us to define a scalar-valued norm for any multivector
${\mathbb{Q}}_{z}=\,{\bf q}_{r}+{\bf q}_{d}\,\varepsilon$ in ${\cal
K}^{\lambda}$,

$||{\mathbb{Q}}_{z}||:=\sqrt{\left({\mathbb{Q}}_{z}{\mathbb{Q}}^{\dagger}_{z}\right)\\!\Big{|}_{\,\varepsilon\cdot\left({\mathbb{Q}}_{z}{\mathbb{Q}}^{\dagger}_{z}\right)\,=\,0}\,}\;,$
(28)

which gives the same value for the norm as that obtained using the traditional
ad hoc definition in Geometric Algebra (cf. Eq. (2.8) in RSOS ). This is
explained also with more detail in Eight . As explained in Eqs. (2.54) to
(2.56) in RSOS , the quadratic form in the definition (28) with split-complex
image is

${\mathbb{Q}}_{z}{\mathbb{Q}}^{\dagger}_{z}=\left({\bf q}_{r}\,{\bf
q}^{\dagger}_{r}+{\bf q}_{d}\,{\bf q}^{\dagger}_{d}\right)+\left({\bf
q}_{r}\,{\bf q}^{\dagger}_{d}+{\bf q}_{d}\,{\bf
q}^{\dagger}_{r}\right)\varepsilon=\left(\varrho_{r}^{2}+\varrho_{d}^{2}\right)+\left({\bf
q}_{r}\,{\bf q}^{\dagger}_{d}+{\bf q}_{d}\,{\bf
q}^{\dagger}_{r}\right)\varepsilon$ (29)

and
$\varepsilon\cdot\left({\mathbb{Q}}_{z}{\mathbb{Q}}^{\dagger}_{z}\right)={\bf
q}_{r}\,{\bf q}^{\dagger}_{d}+{\bf q}_{d}\,{\bf q}^{\dagger}_{r}$ is the
scalar coefficient of $\varepsilon$. It is easy to see that definition (28)
gives a scalar value for the norm as the Pythagorean distance from the origin
in eight dimensions,

$||{\mathbb{Q}}_{z}||=\sqrt{\varrho_{r}^{2}+\varrho_{d}^{2}\,}=\sqrt{g^{2}+u_{x}^{2}+u_{y}^{2}+u_{z}^{2}+h^{2}+v_{x}^{2}+v_{y}^{2}+v_{z}^{2}\,},$
(30)

with the notation ${\bf q}_{r}=g+I_{3}{\bf u}$ and ${\bf q}_{d}=h+I_{3}{\bf
v}$, and therefore the norms defined by (28) are also positive definite:
$||{\mathbb{Q}}_{z}||=0\iff{\mathbb{Q}}_{z}=0$. More importantly, definition
(28) immediately leads to the composition law (27) with scalar values.
Consequently, the algebra ${\cal K}^{\lambda}$ considered in RSOS  is a normed
division algebra with respect to the norm defined in (28), as I have proved in
Reply-to-Lasenby  and Eight .

Moreover, the definition (28) allows us to construct the following 7-sphere
embedded in ${\cal K}^{\lambda}$:

${\cal K}^{\lambda}\hookleftarrow S^{7}=\,\left\\{\,{\mathbb{Q}}_{z}=\,{\bf
q}_{r}+{\bf
q}_{d}\,\varepsilon\;\Big{|}\;||{\mathbb{Q}}_{z}||=\sqrt{\left({\mathbb{Q}}_{z}{\mathbb{Q}}^{\dagger}_{z}\right)\\!\Big{|}_{\,{\bf
q}_{r}\,{\bf q}^{\dagger}_{d}+{\bf q}_{d}\,{\bf
q}^{\dagger}_{r}\,=\,0}\,}\,=\sqrt{\varrho_{r}^{2}+\varrho_{d}^{2}\,}\right\\}.$
(31)

Theorem 3.1 stated in the Introduction and proved in RSOS  refers to this
7-sphere. It is also worth noting that definition (28) of the norm is valid
also for the real and complex numbers, quaternions, and octonions, because for
them the coefficient of $\varepsilon$ is naturally zero, where the complex
numbers and quaternions are isomorphic, respectively, to the even sub-algebras
of the Clifford algebras ${\mathrm{Cl}_{2,0}}$ and ${\mathrm{Cl}_{3,0}}$,
whereas the algebra ${\cal K}^{\lambda}$ is isomorphic to the even subalgebra
of the sixteen-dimensional algebra ${\mathrm{Cl}_{4,0}}$. Thus, the even sub-
algebras of the Clifford algebras ${\mathrm{Cl}_{1,0}}$,
${\mathrm{Cl}_{2,0}}$, ${\mathrm{Cl}_{3,0}}$, and ${\mathrm{Cl}_{4,0}}$ form
associative norm division algebras with respect to the norm defined in (28),
isomorphic, respectively, to ${\mathbb{R}}$, ${\mathbb{C}}$, ${\mathbb{H}}$,
and ${\cal K}^{\lambda}$, in the only possible dimensions 1, 2, 4, and 8 RSOS
; Reply-to-Lasenby ; Eight .

###  II.9 The $S^{7}$ model concerns the workings of Nature, not computers

The critique Gill-RSOS  puts forward two arguments that it claims are computer
science versions of “Bell’s core result.” But these arguments are as devoid of
physical content as “Bell’s core result”, which is simply Boole’s inequalities
discussed in Subsection II.3 above. They ignore the obvious fact that Alice
and Bob are confined to perform their experiments within a three-dimensional
physical space that is common to both of them. If this space is implicitly
assumed to be a flat Euclidean space in the arguments put forward in Gill-RSOS
, then they are subject to the same criticism I have presented in Subsection
II.5. In particular, there is no indication that the geometry and topology of
the quaternionic 3-sphere shared between Alice and Bob have been taken into
account in either of the arguments. Therefore, they are not relevant for the
physical experiments performed by Alice and Bob even within
$\mathrm{I\\!R}^{3}$. To put this more charitably, at best the arguments in
Gill-RSOS  prove the poverty of computer programs that fail to capture the
correct geometrical properties of $S^{3}$ and $S^{7}$, and therefore it is not
surprising that they fail to reproduce the observed strong correlations.

By contrast, it is evident from the detailed, line-by-line annotations
included in the codes of the two numerical simulations presented in RSOS  that
they are faithful implementations of the physically motivated $S^{7}$ model in
which $S^{7}$ is the algebraic representation space of a quaternionic
3-sphere. In particular, contrary to the claim made in the critique Gill-RSOS
, the following line in the code,

$\texttt{if(lambda==1) \\{q=NA NB;\\} else \\{q=NB NA;\\}},$

is a correct implementation of the orientation ${\lambda}$ in the 7-sphere
model presented in RSOS . It switches the order of the spin bivectors with
respect to that of the detector bivectors, thereby shuffling the alternative
orientations of the 7-sphere, precisely as necessitated in the model. In other
words, the line in question is necessitated by the model itself. In fact, the
line in question is the very essence of the 7-sphere model. Moreover, by now
the codes have been independently translated by several programmers into
different computer languages, such as Python, Maple, R, and Mathematica IEEE-3
; IEEE-4 .

It is, however, important to note that the computer codes included in RSOS
are, by themselves, not the model, or even proofs of the model. Their purpose
is to demonstrate how the model works. They are pedagogical tools that
demonstrate the analytical derivations presented in RSOS . Needless to say,
the analytical derivations stand on their own and do not require numerical
simulations for their validity. On the other hand, the numerical simulations
in RSOS  provide additional support to the analytical derivations, because
they are both pedagogically and statistically illuminating.

Ultimately, the merits of the geometrical framework presented in RSOS  can
only be judged through Nature. Therefore, in IJTP ; Symmetric  I have proposed
an experiment, set in a macroscopic domain, that may be able to falsify the
3-sphere hypothesis underlying the framework presented in RSOS .

##  III Conclusion

Without engaging with the actual local-realistic framework for reproducing
quantum correlations I have proposed in RSOS , the critique in Gill-RSOS  has
claimed that there are errors in my papers published in RSOS ; Disproof ;
IEEE-3 ; IEEE-4 ; IJTP ; IEEE-1 ; IEEE-2 . In this response, I have
demonstrated that there are no such errors in RSOS ; Disproof ; IEEE-3 ;
IEEE-4 ; IJTP ; IEEE-1 ; IEEE-2 . In fact, all of the claims made in Gill-RSOS
are either mistaken or irrelevant to the 7-sphere framework I have presented
in RSOS . Moreover, I have brought out a number of logical, mathematical,
physical, and conceptual mistakes from the critique Gill-RSOS  and other
arguments it has relied on (such as those in Lasenby-AACA ). As a result, the
conclusions drawn in Gill-RSOS  are neither valid nor justified. I urge the
readers to also consult my refutations in IEEE-3 ; IEEE-4  of the similar
claims made by the author of Gill-RSOS . Unfortunately, in the critiques Gill-
RSOS  and Lasenby-AACA  the main message of RSOS  has been misunderstood and
sidelined. Therefore, in Reply-to-Lasenby  and Symmetric  I have once again
reviewed the mathematical and conceptual bases of the local-realistic
framework presented in RSOS ; Disproof ; IEEE-3 ; IEEE-4 ; IJTP ; IEEE-1 ;
IEEE-2  to make its message more transparent.

## Acknowledgements

The author thanks Prof. Tevian Dray for his insightful comments on Ref. Eight

## References

  * (1) Christian J. 2018 Quantum correlations are weaved by the spinors of the Euclidean primitives. R. Soc. Open Sci. 5, 180526. (doi:10.1098/rsos.180526)
  * (2) Gill RD. 2022 Comment on ‘Quantum correlations are weaved by the spinors of the Euclidean primitives.’ R. Soc. Open Sci. 9, 201909. (doi.org/10.1098/rsos.201909)
  * (3) Christian J. 2022 Local origins of quantum correlations rooted in geometric algebra. (doi:10.48550/arXiv.2205.11372)
  * (4) Christian J. 2019 Refutation of Scott Aaronson’s critique of my disproof of Bell’s theorem (doi:10.5281/zenodo.6970787)
  * (5) Christian J. 2014 Disproof of Bell’s Theorem: Illuminating the Illusion of Entanglement. Second Edition. Boca Raton, Florida: Brwonwalker Press.
  * (6) Christian J. 2021 Reply to “Comment on ‘Dr. Bertlmann’s socks in the quaternionic world of ambidextral reality.” IEEE Access. 9, 72161. (doi:10.1109/ACCESS.2021.3076449)
  * (7) Christian J. 2022 Reply to “Comment on ‘Bell’s theorem versus local realism in a quaternionic model of physical space.” IEEE Access. 10, 14429-14439. (doi:10.1109/ACCESS.2022.3146421)
  * (8) Christian J. 2015 Macroscopic observability of spinorial sign changes under ${2\pi}$ rotations. Int. J. Theor. Phys. 54, 20-46. (doi:10.1007/s10773-014-2412-2)
  * (9) Christian J. 2019 Bell’s theorem versus local realism in a quaternionic model of physical space. IEEE Access. ${\bf 7}$, 133388. (doi:10.1109/ACCESS.2019.2941275)
  * (10) Christian J. 2020 Dr. Bertlmann’s socks in the quaternionic world of ambidextral reality. IEEE Access. ${\bf 8}$, 191028. (doi:10.1109/ACCESS.2020.3031734)
  * (11) Boole G. 1853 An Investigation of the Laws of Thought, on which are founded the Mathematical Theories of Logic and Probabilities. London: Macmillan.
  * (12) Boole G. 1862 On the theory of probabilities. Philos. Trans. Roy. Soc. London. 152, 225–252. (doi:10. 1098/rstl.1862.0015)
  * (13) Gill RD. 2020 Does geometric algebra provide a loophole to Bell’s theorem? Entropy. 22, no. 1, Art. no. 61. (doi:10.3390/e22010061)
  * (14) Lasenby AN. 2020 A 1d up approach to conformal geometric algebra: applications in line-fitting and quantum mechanics. Adv. Appl. Clifford Algebras. 30, no. 22. (doi:10.1007/s00006-020-1046-0)
  * (15) Christian J. 2020 Reply to “A 1d up approach to conformal geometric algebra: applications in line fitting and quantum mechanics.” (doi:10.5281/zenodo.6752773)
  * (16) Lakatos I. 2015 Proofs and Refutations: The Logic of Mathematical Discovery. Cambridge: Cambridge University Press.
  * (17) Bell JS. 2004 Speakable and Unspeakable in Quantum Mechanics. Revised edition. Cambridge: Cambridge University Press.
  * (18) Christian J. 2017 Oversights in the respective theorems of von Neumann and Bell are homologous. (doi:10.48550/arXiv.1704.02876)
  * (19) Von Neumann J. 1955 Mathematical Foundations of Quantum Mechanics. Princeton, NJ: Princeton University Press.
  * (20) Bell JS. 1966 On the problem of hidden variables in quantum mechanics. Rev. Mod. Phys. 38, 447–452 (doi:10.1103/RevModPhys.38.447)
  * (21) Christian J. 1997 Exactly soluble sector of quantum gravity. Physical Review D. 56, 4844–4877. (doi:10.1103/PhysRevD.56.4844)
  * (22) Christian J. 2022 Symmetric derivation of the singlet correlations within a quaternionic 3-sphere. (doi:10.48550/arXiv.2204.10288)
  * (23) Milnor JW. 1997 Topology from the Differentiable Viewpoint. Princeton, NJ: Princeton University Press.
  * (24) Dray T. and Manogue CA. 2015 The Geometry of the Octonions. Singapore: World Scientific.
  * (25) Christian J. 2019 Eight-dimensional octonion-like but associative normed division algebra. (doi:10.48550/arXiv.1908.06172)
  * (26) Gill RD. 2021 Correction to “Does geometric algebra provide a loophole to Bell’s theorem?” Entropy. 23, no. 5, Art. no. 631. (doi:10.3390/e23050631)

Peer Review History of the Paper:

Response to Reviewers — Round # 1:

Reviewer # 1

Sections 1 and 3 are introduction and conclusion, comments will focus on
section 2:

Section 2.1: All points made here are valid and show that Gill has indeed
misunderstood a lot.

Author response: I thank the reviewer for this comment.

Author action: No action.

Section 2.2 (1): The response here regarding Gill’s critique in reference 12
is largely valid except for the comments regarding Eq. (22) “It also does not
follow mathematically from any other equations I have written down anywhere.”
However Eq. (22) follows from equations (1) and (2) of your 2011 preprint for
example …

Author response: Eq. (22) of [12] does not follow from equations (1) and (2)
of my 2011 preprint, which is a one-page summary I produced to meet frequent
request by some that they do not have time to read my longer papers.

The 2011 preprint cites six other much longer and detailed preprints I had
written prior to 2011, explaining all aspects of the quaternionic 3-sphere
model. By 2020 when Gill’s paper [12] was published, I had posted ten other
papers on arXiv, again explaining all aspects of the 3-sphere model in more
detail, including the RSOS paper [1] and two papers published in IEEE Access.
What is more, my 2011 preprint itself was updated in 2015. That Gill has been
aware of all of these developments is evident from the first line in the
abstract of [12] and numerous online discussions. And yet, Gill’s paper [12]
cites the unpublished first version of my 2011 preprint, disregarding sixteen
other papers I had written prior to 2020, to support its claim that Eq. (22)
of [12] is what the quaternionic 3-sphere model predicts.

Author action: No action.

… and if this is being misunderstood it indicates that you are not using
mathematical notation clearly and correctly. I suggest that the discussion of
Eq. (22) be removed from the paper.

Author response: Equations (1) and (2) in my 2011 preprint are indeed
misconstrued by Gill in [12]. Eq. (22) of [12] does not follow from them since
they are defined as products of two bivectors, representing the physical
interaction during the detection processes between detector bivectors chosen
by Alice and Bob and the spin bivectors originating at the source. This is
evident from the definitions (3) and (4) of the bivectors and the notation
$\lambda$, which, in the literature on Bell’s theorem, represents the initial
state of the spin system originating at the source. As a result, using (1) and
(2), the product $AB$ of the measurement outcomes is a product of two non-pure
quaternions in general, and therefore cannot be equal to $-1$ for all settings
${\bf a}$ and ${\bf b}$ as claimed in [12]. This is evident from the
correlations (7), which are computed in the preprint using the Pearson
correlation coefficient. All this is misunderstood in [12].

Thus the notations in my 2011 preprint, or in any other paper, are just fine,
and I have also used them correctly. The real reason for misunderstanding is
elsewhere and it is conceptual. What is not understood is that the
quaternionic 3-sphere model is not a model of after-the-event analysis
followed by experimenters to compute correlations, but a theoretical model
that predicts that the correlations will be strong if produced within a
quaternionic 3-sphere (rather than a flat Euclidean space), taken as the 3D
physical space. I have explained this conceptual difference many times before,
especially in my replies [5] and [6] to Gill, and references therein. Once
this conceptual difference is understood, it becomes evident that Eq. (22) of
[12] does not follow from the definitions (1) and (2) stated in my 2011
preprint.

Author action: I have revised the last but one sentence of Section 2.2 (1) to:
“It also does not follow mathematically from any other equations I have
written down anywhere, as I have explained in [6].”

Section 2.2 (2) - (4) are subjective - readers can judge for themselves.

Section 2.3 makes valid points, “Bell’s Theorem” and indeed other similar “no-
go theorems” are indeed not mathematical theorems and are typically flim-flam
based on erroneously and irrationally claiming that realist models must have a
structure of a single joint probability space despite such a model neither
being implied by nor implying realism. Indeed “Bell’s Theorem” and other “no-
go theorems” have been thoroughly and repeatedly debunked in the physics
literature and are considered to have no relevance to modern approaches to QM
such as the Consistent Histories formalism which explicitly rejects the idea
that QM is non-realist or non-local while recognizing the need for different
probability spaces for different frameworks.

Author response: In my approach, I have not followed the Consistent Histories
formalism or any other version or interpretation of quantum mechanics. Not a
single concept from quantum mechanics is used in the 3-sphere model. As such,
it is independent of any interpretation of quantum mechanics. The Consistent
History formalism is not a universally accepted formalism among the experts in
foundations of quantum mechanics.

Author action: No action.

Section 2.4 starts off making correct observations but becomes confused
towards the end. The notion of Eq. (1) being mathematically valid but not
physically valid makes no sense.

Author response: There is no confusion of any kind in Section 2.4. The notion
of Eq. (1) being mathematically valid but physically invalid makes perfect
sense. It is not something I have invented myself. It is an argument well
known and near universally accepted in the foundations of quantum mechanics
within the context of von Neumann’s theorem against hidden variable theories.
The problem with Eq. (1) was recognized by Einstein in the mid 1930s (see Ref.
[1] and [2] below). It was also recognized by Grete Hermann around the same
time (see Ref. [3] below). Thus the problem has been well known in foundations
of quantum mechanics for nearly nine decades. One of the most lucid
explanations of the problem can be found in Section 3 of Chapter 1 in Bell’s
book. The chapter has been reprinted from one of his papers (see Ref. [4]
below). The same problem with Eq. (1) in the context of von Neumann’s theorem
is also discussed by several other well known experts in foundations of
quantum mechanics (see Refs. [5] to [10] below).

Briefly, the problem is that, while the sum of expectation values is
mathematically the same as the expectation value of the sum, as in the
assumption that allows the replacement of the left-hand side of Eq. (1) in the
manuscript by its right-hand side, and while this assumption is valid in
quantum mechanics, it is not valid for any hidden variable theory involving
dispersion-free states, because the eigenvalue of a sum of operators is not
the sum of individual eigenvalues when the constituent operators do not
commute, as in the case of Bell-test experiments. This makes Eq. (1)
physically invalid. But without it the absolute upper bound of 2 on the CHSH
correlator cannot be derived. Therefore the problem I have pointed out in
Section 2.4 is very serious. It invalidates all derivations of the bound of 2.

[1] J. von Neumann, Mathematical Foundations of Quantum Mechanics (Princeton
University Press, Princeton, 1955).

[2] A. Shimony, Search for a Naturalistic World View, vol. II (Cambridge
University Press, Cambridge, 1993), p. 89.

[3] G. Hermann, Die naturphilosophischen Grundlagen der Quantenmechanik,
Abhandlungen der Fries’schen Schule 6, 75 (1935).

[4] J. S. Bell, On the problem of hidden variables in quantum mechanics, Rev.
Mod. Phys. 38, 447 (1966).

[5] J. M. Jauch and C. Piron, Can hidden variables be excluded in quantum
mechanics?, Helv. Phys. Acta 36, 827 (1963).

[6] A. Siegel, in Differential Space, Quantum Systems, and Prediction, eds: N.
Wiener, A. Siegel, B. Rankin, and W. T. Martin (MIT Press, Cambridge, MA,
1966).

[7] S. Kochen and E. P. Specker, The problem of hidden variables in quantum
mechanics, J. Math. Mech. 17, 59 (1967).

[8] M. Jammer, The Philosophy of Quantum Mechanics (John Wiley and Sons, New
York, NY, 1974).

[9] N. D. Mermin, Hidden variables and the two theorems of John Bell, Rev.
Mod. Phys. 65, 803 (1993).

[10] N. D. Mermin and R. Schack, Homer nodded: von Neumann’s surprising
oversight, Found. Phys. 48, 1007 (2018).

Author action: I have revised Section 2.4 to explain the problem and the above
points more clearly.

However it is indeed the case that Eq. (1) cannot be derived for sensible
realist models of Bell experiments but this is because different probability
spaces are required for incompatible measurement directions so that neither of
the (mathematically equal) sides of Eq. (1) would be valid. And indeed that
such different probability spaces are required is precisely what the standard
QM formalism tells us and this is indeed related to the fact that spin is
described by spin operators and their eigenvalues with non-commuting operators
producing different probability spaces even for a fixed wavefunction - the
eigenvalues are not simple random variables defined on a single joint
probability space and cannot be treated as such.

Author response: The argument mentioned by the reviewer is a different
argument. Elsewhere I have presented similar arguments, but without appealing
to any quantum mechanical concepts (see, for example, references [1], [5],
[6], [8], and [9] cited in the manuscript). Since Bell’s theorem is not a
theorem within quantum mechanics, it cannot be undermined by arguments that
use quantum mechanical concepts. That would only reinforce the view that
overcomingBell’s argument necessitates quantum concepts. In any event,
discussing such an argument again is beyond the scope of the current
manuscript, the purpose of which is to respond to the specific issues brought
up in the critique by Gill.

Author action: No action.

I would recommend that the discussion of mathematically valid vs physically
valid be removed from this section and replaced with comments emphasizing the
need for different distributions for different measurement directions. Rather
than indicating that realism is wrong, this is a simple consequence of the
fact that when dealing with complete states described by a function on a
continuum instead of a finite sequence of numbers (as in classical mechanics)
there is no single consistent way to measure the size of subsets of states
corresponding to various measurement outcomes - a purely mathematical fact.

Author response: That is not the argument I have put forward in the
manuscript. Although, as noted above, elsewhere I have presented similar
arguments, but without appealing to any quantum mechanical concepts (see, for
example, references [1], [5], [6], [8], and [9] cited in the manuscript).
Moreover, I have not claimed anywhere that realism is wrong. Physical
invalidity of Eq. (1) does not imply denial of realism. On the contrary, what
I have argued is that Eq. (1), and consequently Bell’s argument, does not
implement realism correctly by assigning definite values to all possible
observables of the spin system, regardless of whether they are actually
observed. In particular, it assigns incorrect definite value to the observable
that is a sum of those observables appearing on the left-hand side of Eq. (1).

There is no reason to remove the argument I have presented in Section 2.4. It
is one of the best and ironic arguments against Bell’s theorem. The argument
is ironic because Bell himself has made the problem with Eq. (1) well known
among the experts in the foundations of quantum mechanics within the context
of von Neumann’s theorem (see the references I have listed above). What I have
shown in Section 2.4 is that this problem also undermines Bell’s theorem
because Eq. (1) — encoding the assumption of the additivity of expectation
values — is essential for both theorems.

Author action: I have revised Section 2.4 to explain the problem and the above
points more clearly.

Section 2.5 makes a valid point, Gill ignores your space-time model and argues
using a different model.

Author response: Indeed. That is the main defect of the critiques of my
3-sphere model by Gill and others.

Author action: No action.

However whether your proofs of the correlations using your model are
mathematically sensible and rigorous is questionable, but this remains for the
reader to decide themselves.

Author response: Fair enough. While everything in science is questionable, in
my view the 3-sphere model is not only mathematically sensible and rigorous
but also adopted by Nature, which is the ultimate judge of its validity via
experiment.

Author action: I have added reference [21] in the revised version of the
manuscript. It discusses anew a macroscopic experiment I have proposed in [7]
to test the quaternionic 3-sphere hypothesis. Thus it is a falsifiable model.
It is my latest preprint on the arXiv that summarizes my approach to quantum
correlations, and presents a self-contained and simpler derivation of the
singlet correlations within $S^{3}$ that, I hope, will remove any remaining
doubts in the validity of the 3-sphere model.

Section 2.6 makes a valid point, both Lasenby and Gill have misunderstood that
you are referring to different oriented vector spaces not to different vector
spaces regardless of orientation.

Author response: I thank the reviewer for this observation.

Author action: No action.

Section 2.7 starts with a valid point, your original paper shows that you are
well aware of Hurwitz’s theorem.

Author response: I thank the reviewer for this observation. I have discuss
Hurwitz’s theorem in Appendix A of [1].

Author action: No action.

However your reply to Gill’s counterexample is not mathematically correct

Author response: My reply to Gill’s counterexample is mathematically correct.
It is Gill’s counterexample that is based on a number of elementary mistakes,
as I have demonstrated in both versions of the manuscript.

I have addressed this issue with considerably more detail in my responses
below to Review # 2 and Review # 6.

Author action: In the revised manuscript I have improved my refutation of
Gill’s alleged counterexample.

and shows that there is a problem in your definition of the algebra
$K^{\lambda}$ -

Author response: There is no problem with the definition of the algebra ${\cal
K}^{\lambda}$. It is simply the eight-dimensional even subalgebra of the
algebra $\mathrm{Cl}_{4,0}$. Thus the algebra ${\cal K}^{\lambda}$ is a
perfectly well defined, associative Clifford algebra.

Author action: No action.

this is the issue that Lasenby raised in his critique and although he has
misunderstood parts of your paper such as the fact that $K^{+}$ and $K^{-}$
are different oriented spaces, he is correct on his point about the properties
of your pseudoscalar.

Author response: Lasenby’s critique based on a variant of the alleged
counterexample is also incorrect. In it, the contradiction is achieved by
applying two different product rules on the two sides of the relation
$||XY||=||X||\,||Y||$.

I have addressed this issue with considerable more detail in my responses
below to Review # 2 and Review # 6.

Author action: No action.

The last expressions in (6) and (7) of your reply show that the epsilon must
in fact be a scalar. This attempt to refute Gill’s counterexample should be
removed.

Author response: The last expressions in (6) and (7) by no means show that
$\varepsilon$ must be a scalar. They are results of elementary computation of
norms and work out to be split complex numbers, with a scalar part and a
pseudoscalar part. The fact that they are not scalars only indicates that they
are not parts of the 7-sphere framework proposed in my paper [1]. The ad hoc
two-dimensional objects considered by Gill and Lasenby play no role whatsoever
in the 7-sphere framework proposed in [1]. For a fuller explanation see my
responses below to Review # 2 and Review # 6.

Author action: I have revised Section 2.7 to make the above points more clear.

Section 2.8 Gill’s so called computer science no-go theorems are indeed not
relevant as they apply to computer simulations that are obviously too limited
to model the differing probability spaces for different measurement directions
implied by QM. However the code routines you provide in the original paper are
not actually simulations of Bell experiments they merely calculate the
individual correlations. Here mention should be made to your RPubs code which
is a simulation.

Author response: That is correct. The codes presented in the original 2018
paper are not meant to be simulations of Bell-test experiments. They are
included in the paper as validations of the theoretical calculations presented
therein.

Author action: I have revised Section 2.8 considerably to make this point more
clear.

Reviewer # 2

Dear Editor,

I must recommend that this paper will not be accepted for publication. The
paper aims to provide a causal explanation for the violation of the Bell
inequality observed in experiments. However, the arguments presented by the
author are either unconvincing or plainly wrong.

For example, the arguments presented by the author in section 2.4 of the paper
seem to rule out his own model.

Author response: On the contrary, what is argued in Section 2.4 of the
manuscript is that the bounds of $\pm 2$ on the CHSH correlator are not
correct because they are based on the assumption of the additivity of
expectation values, which is well known to be invalid for any hidden variable
theory involving dispersion-free states. But since the bounds of $\pm 2$ are
not valid, Bell’s theorem does not rule out models like the one I have
presented in my original Royal Society paper published in 2018. The correct
bounds on the CHSH correlator are $\pm 2\sqrt{2}$, which are respected by my
model.

Author action: I have revised Section 2.4 of the manuscript to make this point
more clear.

Actual experiments measure the correlation between the outcomes of two
detectors for certain settings of the latter. The CHSH correlator is obtained
after all these four needed individual correlations have been measured. The
author argues that in his model the CHSH correlator is not an additive
function of the four involved correlations, hence implying that his model does
not reproduce the individual correlations measured in experiments.

Author response: What is claimed by the reviewer above is not what is
presented in Section 2.4 of the manuscript. Please refer to my previous answer
for a brief summary of what is actually presented in Section 2.4 of the
manuscript.

Author action: No action.

Neither the hand-waving arguments provided by the author in section 2.5 are
more convincing. I agree with the author that any implicit or explicit
assumption needed in the derivation of Bell’s theorem must be carefully
examined before radical claims are made about the nature of physical reality.
But how is the geometry of space-time relevant for explaining Bell’s
experiments? How are the cosmological solutions of GR relevant to tabletop
Bell’s experiments? Why are high-energy particle physics experiments precisely
reproduced by quantum field theories formulated in flat Minkowski space-time
while general relativity would be needed in order to reproduce the Bell
experiments?

Author response: The arguments presented in Section 2.5 of the manuscript are
hardly “hand-waving.” They are meant to address a specific statement made in
the critique [2]. On the other hand, I do appreciate that more detailed
answers to the questions raised above by the reviewer are desirable.
Unfortunately, these questions cannot be addressed adequately in a Reply to
Comment paper, the length of which is editorially restricted to the length of
the Comment paper [2]. But I have indeed addressed all of the questions raised
by the reviewer in extensive details for the past fifteenyears, in the
references [1], [3], [4], [5], [6], [7], [8], [9], [14], [17], [19], and [21]
of the revised version of the manuscript.

Author action: I have added reference [21] in the paper. It addresses some of
the questions raised by the reviewer.

Finally, I wish to add that the author’s argument in section 2.7 seems to be
intentionally misleading. The author claims that his eq.(3) holds for any
elements X, Y belonging to a certain algebra. Furthermore, the author claims
that the said algebra contains an element epsilon, other than the identity I
and its opposite -I, which satisfies the constraint epsilon2=I. Nonetheless,
in the comment to his paper by R.Gill, it has been noticed to the author that
these two claims do contradict each other, as it can be readily shown as
follows. First, epsilon2 = I implies that (epsilon-I)(epsilon+I)=0, while
eq.(3) then implies that
$||(\text{epsilon-I})(\text{epsilon+I})||=||\text{epsilon-I}||\,||\text{epsilon+I}||=0$,
from which it follows that either epsilon = I or epsilon = -I, in
contradiction with the starting assumptions.

In section 2.7 the author claims to refute this trivial proof. Instead, the
author waves his hands and misleads the reader to think that the criticism is
about if eq.(3) either holds or not in his algebra. The author then shows that
eq.(3) holds in a particular case and he claims to have refuted the criticism.
I find this author’s strategy rather troublesome.

Author response: I am afraid the above argument by the reviewer has missed the
point I have made entirely. To begin with, if (as the reviewer claims) the
argument in the critique [2] is not “about if eq.(3) either holds or not in”
the algebra ${\cal K}^{\lambda}$ considered in my original 2018 paper, then
the critique’s point is irrelevant and there is no reason for me address it.
But of course the critique is indeed about whether eq.(3), namely the norm
relation $||XY||=||X||\,||Y||$, holds or not in the algebra ${\cal
K}^{\lambda}$ considered in my original 2018 paper.

Secondly, the innocuous looking starting equation

$\varepsilon^{2}=1$ (32)

in the critique [2] is not as innocent as the reviewer seems to think. It
displays a scalar value, $1$, of a geometric product of the pseudoscalar
$\varepsilon$ with itself: $\varepsilon\varepsilon=1$, which is the
fundamental product in geometric algebra, where, by definition,

$\varepsilon:={\bf e}_{1}{\bf e}_{2}{\bf e}_{3}{\bf e}_{4}\not=\pm 1.$ (33)

Therefore, consistency requires that any equation that is claimed to follow
from $\varepsilon^{2}=1$ must also use only geometric products on both sides
of the equation $||XY||=||X||\,||Y||$, or on both sides of any equation for
that matter. It would be illogical to use one product rule on the left-hand
side of eq.(3) and a different product rule on its right-hand side.

With this in mind, it is easy to reveal in which steps the critique’s and the
reviewer’s argument involves inconsistency in deriving the alleged
contradiction. To that end, the next step in the critique’s and the reviewer’s
argument, namely

$(\varepsilon-1)(\varepsilon+1)=0,$ (34)

is correct because it only involves geometric product, and consequently the
following equation would indeed hold,

$||(\varepsilon-1)(\varepsilon+1)||=0,$ (35)

provided we only use a single product rule consistently to arrive at it,
namely, the fundamental geometric product rule. But if we end up using a
scalar product to evaluate the norm in (35) at any stage, then that would
amount to introducing inconsistency by employing two entirely different
product rules to evaluate the norm $||(\varepsilon-1)(\varepsilon+1)||$.

Next, the critique and the reviewer correctly infer from (35) and norm
relation $||XY||=||X||\,||Y||$, or eq.(3), that

$0=||(\varepsilon-1)(\varepsilon+1)||=||(\varepsilon-1)||\;||(\varepsilon+1)||,$
(36)

and therefore

$||(\varepsilon-1)||\;||(\varepsilon+1)||=0.$ (37)

But contrary to the claim made by the reviewer and the claim in the critique
[2], the above equation by no means necessitates that either
$||(\varepsilon-1)||=0$ or $||(\varepsilon+1)||=0$ must hold, unless one
incorrectly assumes, a priori, that $\varepsilon$ is a scalar quantity. All it
says is that the geometric product between $||(\varepsilon-1)||$ and
$||(\varepsilon+1)||$ must vanish. That is because $\varepsilon$ is not a
scalar quantity but a geometric quantity that satisfies $\varepsilon^{2}=1$
via the geometric product of $\varepsilon$ with itself. Consequently, it by no
means follows from (37) that either $(\varepsilon-1)$ or $(\varepsilon+1)$
must vanish without circularly assuming, a priori, that $\varepsilon$ is a
scalar quantity. And we already know from definition (33) that
$\varepsilon\not=\pm 1$. Consequently, neither $(\varepsilon-1)$ nor
$(\varepsilon+1)$ can vanish. Moreover, as explained above, no product rule
other than geometric product rule must be used to evaluate the norms
$||(\varepsilon-1)||$ and $||(\varepsilon+1)||$ in (37), because what is used
to evaluate the left-hand side of (36), namely, to work out the result (35),
is geometric product. Now, using $\varepsilon^{\dagger}=\varepsilon$, we can
evaluate the norms:

$||(\varepsilon-1)||=\sqrt{(\varepsilon-1)(\varepsilon-1)^{\dagger}}=\sqrt{(\varepsilon-1)(\varepsilon-1)}=\sqrt{2\,(1-\varepsilon)}$
(38)

and

$||(\varepsilon+1)||=\sqrt{(\varepsilon+1)(\varepsilon+1)^{\dagger}}=\sqrt{(\varepsilon+1)(\varepsilon+1)}=\sqrt{2\,(1+\varepsilon)}\,,$
(39)

which are indeed non-scalar geometric quantities. Consequently, and
inevitably, the right-hand side of (37) vanishes:

$||(\varepsilon-1)||\;||(\varepsilon+1)||=\left(\sqrt{2\,(1-\varepsilon)}\right)\left(\sqrt{2\,(1+\varepsilon)}\right)=2\sqrt{(1-\varepsilon)(1+\varepsilon)}=2\sqrt{\left(1-\varepsilon^{2}\right)\,}=2\sqrt{0\,}=0.$
(40)

We thus see that, contrary to the claim made by the reviewer and in the
critique [2], no contradiction arises unless mathematical manipulations end up
involving two different product rules on the two sides of the norm relation
(3).

Moreover, given the fact that I have explicitly proved the norm relation
$||XY||=||X||\,||Y||$ in [1] and [3] for any arbitrary multivectors $X$ and
$Y$ in ${\cal K}^{\lambda}$, it is not surprising that any attempt to claim
contradiction via counterexample would necessarily involve inconsistency of
one kind or another, however cleverly hidden in the alleged counterexample.

Author action: I have revised Section 2.7 of the manuscript to briefly explain
the above points more clearly.

Reviewer # 3

A well considered, convincing and concise response. You have taken a nice
approach, referring critics back to original papers.

I note that the 3-sphere S3 has been identified by various other authors as a
key property of various physical theories, and so you do have valid grounds
for your arguments.

Author response: I thank the reviewer for these kind remarks and for
recommending my manuscript for publication.

Author action: No action.

I presume you are looking at extending your ideas to produce an alternate
formulation of QM?

Author response: Yes, that is correct. In the spirit of Einstein’s views, I
have argued that a completely local-realistic formulation of quantum mechanics
is possible. In my 2018 paper [1] I have taken the first steps in this
direction.

Author action: No action.

Reviewer # 4

The author responds to a critique of one of his papers entitled “Quantum
correlations are weaved by the spinors of the Euclidean primitives,” which was
published in the Royal Society Open Science. The paper “Quantum correlations
are weaved by the spinors of the Euclidean primitives” describes a geometric
framework based on a Clifford-algebraic interaction between the quaternionic
3-sphere, or S3, which describes the geometry of physical space, and an
octonion-like 7-sphere, or S7, which is the algebraic representation space of
the quaternionic 3-sphere. Without resorting to retrocausality or
superdeterminism, the framework described in the paper circumvents Bell’s
theorem by local-realistically recreating all quantum correlations.

According to the author, the critique wrongly asserts that it has several
flaws and does not interact with the model described in his paper. The author
argues that the criticism is founded on misconceptions and disproves several
assertions made in the critique.

The author begins by stating some incorrect claims and points out some of the
mistakes in the critique. Some of the claims made by the author are that he
has not proposed any locally hidden variable model but a quaternionic 3-sphere
model, and neither argued that Bell’s proof of his theorem is mathematically
wrong. The author notes that one of the gravest errors in the critique is the
assumption that Bell’s so-called ”theorem” is a mathematically proven theorem,
and hence any criticism of it must be incorrect. The author explains that
Bell’s theorem is not a theorem in the mathematical sense and is based on a
variety of implicit and explicit physical assumptions that have been
questioned from the past. He further explained that if this theorem is
considered a mathematical theorem, actual investigations would not be required
to prove its validity, and any loophole would render it invalid.

Further, the author discusses the assumption of the additivity of expectation
values and notes that the substitution of the sum of expectation values with
the expectation value of the sum, while accepted in quantum physics, does not
hold for hidden variable theories for observables that are not simultaneously
measurable, such as those used in Bell-test experiments. This observation has
been noted earlier in the 1930s, Einstein and Grete Hermann in the context of
von Neumann’s theorem, then Bell, and by several others.

The response delves into the foundational aspects of quantum mechanics
concerning Bell’s inequality, quantum correlations, geometric algebra,
assumption of flat and immutable spacetime, and the S7 model. In my view, this
response is worth publishing and will be helpful to the scientific community
as it sheds light on several topics and the interplay between them.

Author response: I thank the reviewer for the positive review and for
recommending my manuscript for publication.

Author action: No action.

Reviewer # 5

I review the ‘Response to ‘Comment on “Quantum correlations are weaved by the
spinors of the Euclidean primitives””. I neither comment on the qualities of
Joy Christian’s article published in RSOS in 2018, nor on the ones of Richard
Gill’s comment on it. Concerning Christian’s response, I try to judge its
scientific merits, whether it adds new material or arguments to the discussion
in the preceding papers and whether it engages with Gill’s comment. It will be
enough to consider Christian’s sec.s 2.4 and 2.5., where he discusses two
“hitherto underappreciated” assumptions “that are necessary to derive Bell’s
theorem”.

Author response: The purpose of my current manuscript is to respond to the
Comment paper by Gill on my original 2018 paper. Because Review # 5 is focused
only on Sections 2.4 and 2.5 of the manuscript, it is not surprising thatsome
of the comments below by the reviewer lack the benefit of context. My original
paper is an indispensable context.

Author action: In the abstract and Introduction of the revised version of the
manuscript I have made the context of the paper more transparent.

In sec. 2.4, Christian discusses a “hitherto underappreciated” assumption that
is “necessary to derive Bell’s theorem”: the “assumption of the additivity of
expectation values”. Gill, in his comment, does not say anything about
additivity and so Christian’s discussion introduces a new topic.

Author response: This claim by the reviewer is incorrect. What I have
discussed in Section 2.4 of the manuscript is a direct response to the claim
made in the critique [2] concerning the assumption of the additivity of
expectation values. I have quoted this claim verbatim from the critique [2] in
my manuscript. It appears as follows:

> He also argues [in [17]] that Bell’s proof contains a fundamental error in
> reasoning: the Bell-CHSH inequality involves correlations obtained from
> different sub-experiments involving measurements of non-commuting
> observables, and (he says) therefore cannot be combined. However, in quantum
> mechanics, even if two observables do not commute, a real linear combination
> of those observables is another observable. By the linearity encapsulated in
> the basic rules of quantum mechanics, expectation values of linear
> combinations of non-commuting observables are the same linear combination of
> the expectation values of each observable separately. If a local hidden
> variables model reproduces the statistical predictions of quantum mechanics,
> then it must reproduce this linearity.

Evidently, the reviewer seems to have missed the yellow highlighted sentence.
It states nothing but the assumption of the additivity of expectation values
required in the proofs of both von Neumann’s and Bell’s theorems. Thus,
contrary to the reviewer’s claim, I have not introduced “a new topic” but
responded only to the issue raised in the critique [2].

Author action: No action.

Christian claims that one of Gill’s claims (the last sentence of Gill’s sec.
2b and the last sentence of Christian’s quotation of Gill) is “manifestly
incorrect”. Christian elaborates:

> “While mathematically correct, Eq. (1) is physically meaningless within any
> hidden variable theory. […] The problem with Eq. (1) is that, while the sum
> of expectation values is mathematically the same as the expectation value of
> the sum, as in the assumption that allows us to mathematically replace the
> left-hand side of Eq. (1) above with its right-hand side, and while this
> assumption is valid in quantum mechanics […], it is not valid for any hidden
> variable theory involving dispersion-free states, […]. This makes the
> replacement of the left-hand side of Eq. (1) with its right-hand side
> physically invalid. [… Once the fallacious Eq. (1)] is removed from Bell’s
> argument and local realism is implemented correctly, the bounds on the
> right-hand side of Eq. (1) work out to be $\pm 2\sqrt{2}$ instead of $\pm
> 2$, thereby mitigating the conclusions of Bells theorem.”

Looking at the underlined phrases, we find Christian saying that “Eq. (1)” is
“mathematically correct” and “valid in quantum mechanics” but also “physically
meaningless”, even “physically invalid” in certain circumstances and finally
“fallacious”. So Christian claims that “Eq. (1)” is “mathematically correct”
but physically false. So, he asserts that some claim is both true and false,
which in the present context is unacceptable. Alternatively, we can take
Christian as meaning that the equation stands for an argument and that this
argument is “mathematically correct” and “valid in quantum mechanics” but also
“physically invalid”. It can indeed happen that an argument is valid, given a
fixed set of premises, and “invalid”, given another. But for clarity, the
different premises must be made explicit. As the discussion stands, it
suggests incoherent argumentation and should not be published in a scientific
context.

Author response: The argument I have presented in Section 2.4 of the
manuscript is not one I have invented myself. It cannot be brushed aside as
the above comments by the reviewer purports to do. The problem was recognized
by Einstein in the context of von Neumann’s theorem against hidden variable
theories in the mid 1930s (see Ref. [1] and [2] below). It was also recognized
by Grete Hermann around the same time (see Ref. [3] below). Thus the problem
has been well known in foundations of quantum mechanics for nearly nine
decades. One of the most lucid explanations of the problem can be found in
Section 3 of Chapter 1 in Bell’s book. The chapter has been reprinted from one
of his papers (see Ref. [4] below). The same problem with von Neumann’s
theorem is also discussed by several other well known experts in foundations
of quantum mechanics (see Refs. [5] to [10] below).

Briefly, the problem is that, while the sum of expectation values is
mathematically the same as the expectation value of the sum, as in the
assumption that allows the replacement of the left-hand side of Eq. (1) in the
manuscript by its right-hand side, and while this assumption is valid in
quantum mechanics, it is not valid for any hidden variable theory involving
dispersion-free states, because the eigenvalue of a sum of operators is not
the sum of individual eigenvalues when the constituent operators do not
commute, as in the case of Bell-test experiments. This makes Eq. (1)
physically invalid. But without it the absolute upper bound of 2 on the CHSH
correlator cannot be derived. Therefore the problem I have pointed out in
Section 2.4 is very serious. It invalidates all derivations of the bound of 2.

[1] J. von Neumann, Mathematical Foundations of Quantum Mechanics (Princeton
University Press, Princeton, 1955).

[2] A. Shimony, Search for a Naturalistic World View, vol. II (Cambridge
University Press, Cambridge, 1993), p. 89.

[3] G. Hermann, Die naturphilosophischen Grundlagen der Quantenmechanik,
Abhandlungen der Fries’schen Schule 6, 75 (1935).

[4] J. S. Bell, On the problem of hidden variables in quantum mechanics, Rev.
Mod. Phys. 38, 447 (1966).

[5] J. M. Jauch and C. Piron, Can hidden variables be excluded in quantum
mechanics?, Helv. Phys. Acta 36, 827 (1963).

[6] A. Siegel, in Differential Space, Quantum Systems, and Prediction, eds: N.
Wiener, A. Siegel, B. Rankin, and W. T. Martin (MIT Press, Cambridge, MA,
1966).

[7] S. Kochen and E. P. Specker, The problem of hidden variables in quantum
mechanics, J. Math. Mech. 17, 59 (1967).

[8] M. Jammer, The Philosophy of Quantum Mechanics (John Wiley and Sons, New
York, NY, 1974).

[9] N. D. Mermin, Hidden variables and the two theorems of John Bell, Rev.
Mod. Phys. 65, 803 (1993).

[10] N. D. Mermin and R. Schack, Homer nodded: von Neumann’s surprising
oversight, Found. Phys. 48, 1007 (2018).

Author action: I have revised Section 2.4 to explain the problem and the above
points more clearly.

I turn to sec. 2.5. It can be summed up by the following quotation:

> “The second assumption necessary to support Bell’s theorem is that of
> immutable spacetime. [… If we abandon this assumption and regard spacetime
> as mutable, then we can] model physical space as a closed and compact
> quaternionic 3-sphere, or $S^{3}$, instead of a flat Euclidean space, or
> ${\mathrm{I\\!R}^{3}}$ […]. But once the physical space is modelled as
> $S^{3}$ instead of ${\mathrm{I\\!R}^{3}}$ […], the correlation between the
> results Aa and Bb […] inevitably turns out to be E(AaBb) = -a$\cdot$b […]”.

(The sphere $S^{3}$ is a topic briefly mentioned by Gill without any
discussion.)

Author response: The main defect of the critique [2] is this fact noted by the
reviewer. Namely, that “The sphere $S^{3}$ is a topic briefly mentioned by
Gill without any discussion.” In other words, the critique [2] is published
with disregard for the actual model proposed in my original 2018 paper [1].
Unfortunately, this review comment has the same issue.

Author action: No action.

Here, the last sentence claims that, given the assumption of S3, “the
correlation between the results Aa and Bb […] inevitably turns out to be
E(AaBb) = -a$\cdot$b”. Now, given a suitable choice of vectors A, A’, B, B’,
this equation “E(AaBb) = -a · b” implies the right side of “Eq. (1)” above
with the values “$\pm 2\sqrt{2}$ instead of $\pm 2$” as bounds. Given such a
choice of vectors, both results are equivalent. But this raises the question
why we need two arguments – the argument from S3 and the one from ‘false Eq.
(1)’ – to disable Bell’s theorem. Introducing S3 is claimed to be sufficient
(“inevitably”) for this, so the preceding discussion of additivity seems
superfluous. On the other hand, the introduction of S3 is contentious and
apparently unnecessary for the argument in sec. 2.4, which deals with Eq. (1)
and an undisclosed alternative. The argument in sec. 2.4 suggests that just by
removing the ‘false Eq. (1)’, we can derive the bounds on the right-hand side
of Eq. (1) […] to be “$\pm 2\sqrt{2}$ instead of $\pm 2$”. Accordingly,
Christian cannot make plausible that 2.4. and 2.5 are two “hitherto
underappreciated” assumptions “that are [both] necessary to derive Bell’s
theorem”. Instead, one of these assumptions is claimed to be itself sufficient
to disable the theorem, for the other this is at least suggested.

Author response: The manuscript is a response to a comment paper on my 2018
paper. It is not a comprehensive review paper on my quaternionic 3-sphere
model for understanding quantum correlations. That model, as a constructive
counterexample to Bell’s theorem, is presented in the references [1], [3],
[4], [5], [6], [7], [8], [9], [14], [17], and [19] cited in the manuscript.
Moreover, the length of my response paper is editorially restricted to no more
than the length of the published comment paper [2]. My responses in Sections
2.4 and 2.5 of the manuscript are a direct response totwo of the issues raised
in the comment paper [2]. Among these, Section 2.4 concerns a formal aspect of
the proof of Bell’s theorem that has been discussed in the comment paper [2].
That formal aspect does not concern the quaternionic 3-sphere model of the
quantum correlations, viewed as a counterexample to Bell’s theorem. In the
manuscript I have not mixed the issues concerning the formal aspect of Bell’s
theorem and those concerning my specific counterexample. It is unclear to me
why the reviewer has tried to mix up these two aspects and tried to read
between the lines. I suspect that is because of the reviewer’s acknowledged
neglect of my original 2018 paper and the comment paper [2].

Author action: I have added reference [21] in the paper. It addresses some of
the questions raised by the reviewer.

All in all, the author’s response contains two key sections that are
scientifically insufficient and do not address the comment to which it
responds. The response seems fruitless and should not be published. I doubt
whether a revised version could improve the argument.

Author response: It should be evident from my response above that neither of
the two claims made by the reviewer hold in the context of my paper.
Consequently, the reviewer’s conclusion regarding my response paper does not
follow. It is important that my response paper is published because it brings
out the oversights contained in the critique [2] of my 2018 paper [1].

Author action: No action.

Reviewer # 6

This is not a comprehensive report on the response by Joy Christian, since
there are aspects of both the original paper, and the response, that I have
not gone through in detail.

However, the aspects that I have gone through in detail, and that I’m familiar
with, are sufficient to make it quite clear that the original paper should not
have been published in the form it was, and that many of the same grave errors
are present in the Christian response under present consideration. No further
useful clarifications or statements that could change the conclusions about
these errors are made in the response, hence what I will do here is list these
same errors again, and point out where in the response they are repeated. They
are easily serious enough to say that the response should not be published.

Author response: I do not agree with the reviewer’s assessment of either my
original 2018 paper or of the current manuscript under consideration. In what
follows I demonstrate that each of the claim of error made by the reviewer
does not hold, just as the claims made about my 2018 paper in [2] and in [13]
do not hold. As noted in my manuscript, in [3] and [14] I have already brought
out all of the oversights in [13] in considerable detail. Since these
oversights are repeated in this review, in what follows I bring out the
oversights in [2] and in [13] again.

Author action: No action

The foremost and easiest to discuss problem, is that JC still believes that in
the algebra he calls ${\cal K}^{\lambda}$, then for any multivectors $X$ and
$Y$ within it, the following composition law for the norms holds (his equation
(3) in the current response and (2.40) in the original paper):

$||XY||=||X||\,||Y||$ (1.1)

This seems to be important to JC, and a good part of the first part of the
original paper is devoted to ‘establishing’ it, and in the response, it
occupies about a page.

Author response: The above equation, which is equation (2.40) in the original
2018 paper, is not based on a “belief.” There are no “beliefs” in my paper.
Every mathematical equation in it is proved rigorously. The above equation is
explicitly proved in Section 2.5 of the 2018 paper, with more details of the
proof provided in Appendix A.1 of [3] and in Appendix B of
https://arxiv.org/pdf/1908.06172.pdf. To date, no one has identified a mistake
in my proof of (2.40). In particular, Gill in [2], Lasenby in [13], and the
present reviewer have not identified any mistake in my proof of (2.40)
presented in [1] and [3]. All they have managed to do is allege a
counterexample, which I have demonstrated in [3] and [14] to be inapplicable.
It is based on internal inconsistency. The contradiction is arrived in it
byinconsistent use of the product rules on the two sides of equation (2.40)
[which, as noted, is the same as (1.1) above].

The outline of the proof of (2.40) is straightforward to follow. The algebra
${\cal K}^{\lambda}$ is the eight-dimensional even subalgebra of the sixteen-
dimensional Clifford algebra $\mathrm{Cl}_{4,0}$. As such, any general
multivectors $X$ and $Y$ in ${\cal K}^{\lambda}$ are of the form

$X=\,{\bf q}_{r1}+{\bf q}_{d1}\,\varepsilon$ (41)

and

$Y=\,{\bf q}_{r2}+{\bf q}_{d2}\,\varepsilon\,,$ (42)

where ${\bf q}_{r1}$ and ${\bf q}_{d1}$ constituting $X$ are two independent
quaternions and $\varepsilon:={\bf e}_{1}{\bf e}_{2}{\bf e}_{3}{\bf
e}_{4}\not=\pm 1$ that satisfies $\varepsilon^{2}=1$, with its reverse
$\varepsilon^{\dagger}$ satisfying $\varepsilon^{\dagger}=\varepsilon$. As a
result, the geometric product $XX^{\dagger}$ is of the form

$\displaystyle XX^{\dagger}\,$ $\displaystyle=\left({\bf q}_{r1}+{\bf
q}_{d1}\,\varepsilon\right)\left({\bf q}_{r1}+{\bf
q}_{d1}\,\varepsilon\right)^{\dagger}$ (43) $\displaystyle=\left({\bf
q}_{r1}\,{\bf q}^{\dagger}_{r1}\,+\,{\bf q}_{d1}\,{\bf
q}^{\dagger}_{d1}\right)+\left({\bf q}_{r1}\,{\bf q}^{\dagger}_{d1}\,+\,{\bf
q}_{d1}\,{\bf q}^{\dagger}_{r1}\right)\varepsilon$ (44)
$\displaystyle=\,\text{(a scalar)}\,+\,\text{(a scalar)}\,\varepsilon\,,$ (45)

and likewise for $YY^{\dagger}$. Note that this quantity resembles a split
complex or hyperbolic number. Consequently, when product rules are applied
consistently on the two sides of equation (2.40) using only geometric
products, both sides turn out to be identical to the square-root of the
following quantity, which also resembles a split complex number:

$\displaystyle\left\\{\left(\varrho^{2}_{r1}+\varrho^{2}_{d1}\right)\left(\varrho^{2}_{r2}+\varrho^{2}_{d2}\right)\,+\,\left({\bf
q}_{r1}\,{\bf q}^{\dagger}_{d1}+{\bf q}_{d1}\,{\bf
q}^{\dagger}_{r1}\right)\left({\bf q}_{r2}\,{\bf q}^{\dagger}_{d2}+{\bf
q}_{d2}\,{\bf q}^{\dagger}_{r2}\right)\right\\}$
$\displaystyle\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;+\left\\{\left(\varrho^{2}_{r1}\,+\,\varrho^{2}_{d1}\right)\left({\bf
q}_{r2}\,{\bf q}^{\dagger}_{d2}+{\bf q}_{d2}\,{\bf
q}^{\dagger}_{r2}\right)+\left({\bf q}_{r1}\,{\bf q}^{\dagger}_{d1}+{\bf
q}_{d1}\,{\bf
q}^{\dagger}_{r1}\right)\left(\varrho^{2}_{r2}\,+\,\varrho^{2}_{d2}\right)\right\\}\,\varepsilon\,,$
(46)

where $\varrho_{r1}=\sqrt{{\bf q}_{r1}\,{\bf q}^{\dagger}_{r1}}\,$, etc., are
scalars, and therefore the quantities appearing in the two curly brackets are
also scalars. This is a completely general result. It holds for any
multivectors $X$ and $Y$ in the algebra ${\cal K}^{\lambda}$. And therefore
the norm relation $||XY||=||X||\,||Y||$ holds for any multivectors $X$ and $Y$
in the algebra ${\cal K}^{\lambda}$. I stress again that, to date, no one has
found a mistake in the above proof. It takes less than half an hour, with some
familiarity in geometric algebra and quaternions, to verify the proof provided
in [1] and [3]. The normalization or orthogonality condition (2.54) in [1],
namely ${\bf q}_{r}\,{\bf q}^{\dagger}_{d}+{\bf q}_{d}\,{\bf
q}^{\dagger}_{r}=0$, then reduces the hyperbolic number (46) to the scalar
quantity

$\sqrt{\left(\varrho^{2}_{r1}+\varrho^{2}_{d1}\right)\left(\varrho^{2}_{r2}+\varrho^{2}_{d2}\right)}\,,$
(47)

giving the result

$||XY||=\sqrt{\left(\varrho^{2}_{r1}+\varrho^{2}_{d1}\right)\left(\varrho^{2}_{r2}+\varrho^{2}_{d2}\right)\,}=||X||\,||Y||$
(48)

as a special case. Consequently, since the norm relation (2.40) holds for any
multivectors $X$ and $Y$ in ${\cal K}^{\lambda}$ and thus even for those with
scalar values for their norms, if $X$ and $Y$ happen to be unit multivectors
so that $||X||=1$ and $||Y||=1$, then (27) necessitates that their product
$Z=XY$ will also be a unit multivector: $||Z||=||XY||=||X||\,||Y||=1$.

Author action: In the revised Section 2.7 of the manuscript I have explained
the above facts more clearly.

If it were true, then JC would have found an associative version of the 8
dimensional normed division algebra that is normally represented by the non-
associative octonions. Indeed this part of the original paper was also written
up separately in the paper: ‘Eight-dimensional octonion-like but associative
normed division algebra’ (arXiv:1908.06172) which was published in
‘Communications in Algebra’ in 2020. The editors of Communications in Algebra
have since retracted the article, having realised that what it describes in
the title is specifically ruled out by Hurwitz’s theorem, and the reference is
now vol. 49, no. 2, 905–914 (2021), which points to the retracted version.

Author response: The reviewer does not seem to have all the facts behind the
retraction from Communications in Algebra of my paper
https://doi.org/10.48550/arXiv.1908.06172. The paper underwent more than six
months and two rounds of peer-review process, and remained published online
for about two months, until Richard D. Gill, the author of critique [2],
complained to the journal with the goal of having it retracted. A retraction
of any paper, however, is not a proof of a flaw in the paper, unless such a
proof is provided by the journal. Indeed, to date no one, including any of the
reviewers of the current manuscript, has identified a flaw in the actual proof
of the norm relation $||XY||=||X||\,||Y||$ I have presented in my papers [1]
and [3]. The only objection to it is raised by means of a counterexample.
However, in what follows I demonstrate that the alleged counterexample in [13]
is misplaced.

For verification, I had also sent my retracted paper to one of the well known
experts on the subject of normed division algebras; namely, to Prof. Tevian
Dray of Oregon State University in the US, who is a coauthor of the well known
book on the subject: The Geometry of the Octonions. I asked him (1) whether he
was able to confirm that my proof was correct, and if so, (2) whether he was
surprised by it. His answers were the following:

> (1) As you are likely aware, this algebra is H$\times$C’, that is, the
> tensor product of the quaternions with the split complex numbers, but with
> an interesting twist. Specifically, just as with the complexified
> quaternions, you assume that conjugation affects H, but not C – or, in this
> case, C’. For the complexified quaternions, the resulting norm is complex,
> rather than real, but the composition law still holds; here, the norm is
> split complex, but again, the composition law remains intact.
>
> (2) No, I wasn’t particularly surprised – once I had figured out what you
> were doing. The Hurwitz theorem as usually stated assumes that the
> underlying coefficient algebra is real; for you, it is the split complex
> numbers.

By “composition law” Prof. Dray meant the norm relation (1.1) under
discussion, or Eq. (2.40) of my paper [1].

Author action: No action.

Specific counterexamples to equation (1.1) (equation (3) in JC’s response),
have been given by Gill and Lasenby, and are discussed and allegedly refuted
in the response.

So how is it that JC thinks his claims are still defensible, and that he has
refuted the concrete counterexamples? Clearly his own analysis in the response
must contain mathematical errors, so we need to find these.

Author response: On the contrary, as I demonstrate below, the mathematical
errors are in fact contained in the alleged counterexample and all of its
variants. This should not be surprising, because, as I have noted before,
equation (2.40) in [1] is rigorously proven in [1] and [3]. Therefore any
alleged counterexample must necessarily contain errors or inconsistencies,
however hidden they may be. The only remaining task is to bring out these
inconsistencies explicitly.

Author action: No action.

Shortly after (his) equation (3), he quotes from Gill’s comments the following
passage:

> However, the author’s algebra has an element called the “pseudo-scalar”, I
> will denote it by $M$, such that $M^{2}=1$. It follows that
> $0=M^{2}-1=(M-1)(M+1)$. Taking norms, $0=||M-1||.||M+1||$. Hence $||M-1||=0$
> or $||M+1||=0$. Therefore $M-1=0$ or $M+1=0$, which implies that $M=1$ or
> $M=-1$. That is a contradiction.

JC then says

> But it is not difficult to see that this alleged counterexample harbours
> several elementary errors. The first error in the quoted claim is
> immediately obvious. It starts out with the equation $M^{2}=1$ and ends with
> the equations $M=1$ or $M=-1$. And then it claims that “That is a
> contradiction.”

This is a completely fallacious point of course. The original object $M$ is
meant to be the pseudoscalar of the space, hence finding out that $M$ has to
be the scalar 1 or -1, is indeed a contradiction, amounting to saying that the
pseudoscalar $=\pm 1$, which is impossible in any dimension above one.

Author response: Far from being “fallacious”, what is quoted above by the
reviewer from the manuscript is merely my first observation of an elementary
logical mistake in the counterexample alleged in the critique [2], and it is a
perfectly valid observation. The argument in the critique that
$M^{2}=1\implies M=1$ or $M=-1$ is not a contradiction by itself. To allege a
contradiction one must assume that $M$ is a pseudoscalar. But then it by no
means follows from

$||M-1||\,||M+1||=0$ (49)

that either $||M-1||=0$ or $||M+1||=0$ must hold, as alleged in the critique
[2]. On the contrary, all the equation (49) says is that the geometric product
between $||M-1||$ and $||M+1||$ must vanish, precisely because $||M-1||$ and
$||M+1||$ are no longer scalars. And the geometric product does vanish, as I
have demonstrated in the manuscript. Ihave explained this point in greater
detail in my response to Reviewer # 2 [see discussion around Eqs. (32) to
(11)].

Author action: In the revised version of Section 2.7, I have made the above
points more clear.

Next JC considers the assignments $X=\epsilon-1$, $Y=\epsilon+1$, where
$\epsilon$ is his notation for the pseudoscalar.

Author response: Just to be clear, I have not considered such two-dimensional
multivectors in my original paper [1]. They do not play any role whatsoever in
the 7-sphere framework I have proposed in [1]. They have been mistakenly
assumed in [13] and [2] to construct a counterexample. Therefore, even if such
ad hoc two-dimensional objects lead to “contradiction” as alleged in [2] and
[13], that would have no effect on or consequences for the 7-sphere framework.

Author action: No action.

He purports to show that the norm relation (1.1) is satisfied in this case,
since each side is zero. This would then disprove Gill’s claims for this case,
which would assign non-zero norms to both $X$ and $Y$, but have the norm of
their product as zero.

Author response: That is correct.

Author action: No action.

Now, in equation (2.8) of the original paper, JC correctly defines the norm of
an object $\Omega$ as

$||\Omega||=\sqrt{\langle\Omega\,\tilde{\Omega}\rangle_{s}}$ (1.2)

where $\tilde{\Omega}$ is the reverse of $\Omega$ (JC writes
$\Omega^{\dagger}$ instead, but it is definitely defined as the reverse) and
$\langle\dots\rangle_{s}$ denotes taking the scalar part of the object in the
angle brackets.

Author response: Here the adjective “correctly” is not what I have used
anywhere to describe the ad hoc definition (1.2) of a norm of an isolated
multivector in geometric algebra. The definition (2.8) of a norm in my paper
[1] is from the introductory section in which I summarize the standard
geometric algebra going back to Clifford and Grassmann.

What the reviewer does not mention is that (1.2) above, or (2.8) in [1], is
not the definition of the norm that is used in section 2.5 of my paper [1],
where the norm relation $||XY||=||X||\,||Y||$ is proved that evidently
involves a geometric product of two multivector X and Y on its left-hand side.
The equivalent definition used to prove the norm relation (2.40) is explicitly
stated in (2.54) of [1] in which the non-scalar part of the geometric product
$XX^{\dagger}$ is set to zero, with explanation of why that definition is used
giving the same value for the norm. It is unclear to me why the reviewer
insists on using definition (2.8) from the introductory section and ignores
definition (2.54) discussed in Section 2.5.

To begin with, the definition (2.8) is ad hoc because it goes against the very
essence of geometric algebra in which the fundamental product is geometric
product, not the scalar product being picked out in $\langle\dots\rangle_{s}$
by disregarding the rich algebra and geometry contained in the non-scalar part
prematurely. In addition, definition (2.8) leads to serious inconsistency when
products of two or more multivectors are involved in an equation such as
$||XY||=||X||\,||Y||$. Notice that, if we were to use definition (2.8) to
evaluate the norms on the two sides of this equation, then on the left-hand
side of it we would end up using two different product rules, the geometric
product rule between $X$ and $Y$ and the scalar product rule for evaluating
the norm $||XY||$. Whereas on its right-hand side we would end up using only
the scalar product rule for evaluating $||X||$ and $||Y||$. This inconsistent
application of two different product rules on the two sides of the norm
relation is the root cause of the contradiction in the alleged counterexample
in [13].

It is also worth noting that no such inconsistency arises in verifying the
same norm relation for the complex numbers $c$, quaternions ${\bf q}$, and
octonions ${\bf O}$, for which the norms are defined by
$||{c}||=\sqrt{{c}\,{c}^{\dagger}}$, $||{\bf q}||=\sqrt{{\bf q}{\bf
q}^{\dagger}}$, and $||{\bf O}||=\sqrt{{\bf O}{\bf O}^{\dagger}}$,
respectively. In all three cases, only geometric products are used on both
sides of the relation $||XY||=||X||\,||Y||$. In particular, nowhere in the
evaluation of these norms scalar products are used in an ad hoc manner as done
in [13].

Author action: No action.

By taking this scalar part, we obtain something that, provided it is non-
negative, we can validly take a square root of, and in fact for all objects in
the algebra JC is considering, $\langle\Omega\,\tilde{\Omega}\rangle_{s}$ will
be greater or equal to zero, and the norm is well-defined.

Author response: The norm defined by setting the non-scalar part of $X$ equal
to zero is equally well-defined. For details see Section 2.5 of [1] and
Sections 3.3 and 3.4 of [3]. As noted, both definitions give the same value
for norm.

Author action: No action.

However, in the current response, JC says that norms are defined as (see line
after his equation (3))

$||X||=\sqrt{X\tilde{X}}$ (1.3)

i.e. he has missed out taking the scalar part of the geometric product
$X\tilde{X}$ before the square root is taken. This is an incorrect version of
the norm, and it is because of using this new (and incorrect) definition that
JC gets the results in relation to (1.1) that are claimed.

Author response: By now it should be evident that these claims are misplaced.
As noted above, (1.3) is not a “new” definition of the norm I have invented
just for my response to the critique [2]. And I have not “missed out taking
the scalar part of the geometric product $X\tilde{X}$ before the square root
is taken.” The definition (1.3) employing the fundamental geometric product is
the correct definition of the norm that does not discard non-scalar parts of
geometric quantities prematurely, and is consistently applicable on both sides
of the norm relation $||XY||=||X||\,||Y||$.

Author action: No action.

Specifically, in his equation (6), the chain of equalities should be

$\displaystyle||X||=||(\epsilon-1)||$
$\displaystyle=\sqrt{\langle(\epsilon-1)\widetilde{(\epsilon-1)}\rangle_{s}}=\sqrt{\langle\epsilon^{2}-2\epsilon+1\rangle_{s}}$
$\displaystyle=\sqrt{\langle 2(1-\epsilon)\rangle_{s}}=\sqrt{2}$ (1.4)

and similarly we obtain $||Y||=\sqrt{2}$ also, meaning $||X||\,||Y||=2$, in
contradiction to (1.1) since $||XY||=||\epsilon^{2}-1||=0$.

Author response: The inconsistency in this argument is contained in the last,
unnumbered equation, namely, in

$||XY||=||\varepsilon^{2}-1||=0.$ (50)

Notice that, while in (1.4) the norms $||X||=\sqrt{2}$ and $||Y||=\sqrt{2}$
belonging to the right-hand side of the norm relation $||XY||=||X||\,||Y||$
are computed using scalar products, the above equation belonging to the left-
hand side of the norm relation is computed using both geometric product
between $X$ and $Y$ and a scalar product for the norm $||XY||$. This mixing of
different product rules on the two sides of norm relation is inevitable for
the reviewer, because a new multivector $Z=XY$ belonging to the algebra ${\cal
K}^{\lambda}$ can only be obtained by a geometric product between $X$ and $Y$.
Thus the alleged contradiction in the reviewer’s argument has been achieved by
inconsistently applying of the product rules on the two sides of the norm
relation (1.1). The only way to avoid that inconsistency is by using geometric
products on both sides of (1.1), as I have done in the manuscript. But then no
contradiction arises and the alleged counterexample in the critiques [2] and
[13] fails, as it must in the light of the general result (2.40) proved in [1,
3].

In summary, the only consistent way to derive the norm relation
$||XY||=||X||\,||Y||$ with all scalar values is via the procedure I have
described in the paragraph containing equations (12) to (27) and proved
rigorously in [1] and [3].

Author action: No action.

Thus the claims by JC in his response are demonstrably false, and rest on an
incorrect definition of the norm. It is worth emphasising that it is incorrect
even according to JC, since the correct definition is given [in] his original
paper (equation (2.8) there, as already referred to).

Author response: It should be evident by now that the demonstrations I have
provided refuting the counterexamples in [2] and [13] are correct. Moreover,
as already noted, the definition (2.8) of a norm in my 2018 paper [1] is from
the introductory section in which I summarize the standard geometric algebra
going back to Clifford and Grassmann. That definition is not used in Section
2.5 of [1] where the norm relation $||XY||=||X||\,||Y||$ is proved in full
generality.

Author action: No action.

Note the expressions that JC gives for the norms of $X$ and $Y$ individually,
in equations (6) and (7) of the response, namely $\sqrt{2(1-\epsilon)}$ and
$\sqrt{2(1+\epsilon)}$, are clearly incorrect, since they involve square roots
of scalar and pseudoscalar combinations, which we could work out if desired,
but cannot be norms since they are not non-negative scalars.

Author response: The fact that the norms $||X||=\sqrt{2(1-\varepsilon)}$ and
$||Y||=\sqrt{2(1+\varepsilon)}$ computed in my manuscript are not scalars does
not prove anything. It only demonstrates the trivial fact that the ad hoc
invention of two-dimensional multivectors $X=\varepsilon-1$ and
$Y=\varepsilon+1$ leads to non-scalar norms, and therefore those $X$ and $Y$
are not a part of the set of multivectors that are normalizable to scalars and
constitute the 7-sphere constructed in my original paper [1].

To put this differently, the two-dimensional objects $\varepsilon-1$ and
$\varepsilon+1$ used to allege a counterexample form merely a side show. They
do not play any role whatsoever in the 7-sphere framework proposed in my
original 2018 paper [1]. As aresult, even if we accept the contradiction
alleged in [2] and [13], it would not affect the framework proposed in [1].

Author action: To be fair to the reviewer’s point of view, in the revised
version of the manuscript I have added the following footnote on page 7 for
the benefit of the readers: “Some reviewers unjustifiably defended this
counterexample during the review process of this paper. In addition to [3], my
detailed rebuttal to their defence is available online in the Review History
that is published along with this paper. All variants of the alleged
counterexample depend on inconsistent application of product rules on the two
sides of the norm relation (5).”

Another area it is worth commenting on is Section 2.6 of JC’s response. Here
he is responding to Gill’s criticism that:

> A curious elementary mathematical error is that he defines two algebras,
> built from two 8-dimensional real vector spaces ${\cal K}^{+}$ and ${\cal
> K}^{-}$ by specifying a vector space basis for each algebra and
> multiplication tables for the 8 basis elements of each algebra. But they are
> the same algebra. The linear spans of those two bases are trivially the
> same. The multiplication operation is the same.

JC’s reply in Section 2.6 of the current response is:

> However, there is no such error in [1] [the original RSOS paper]. I have
> neither defined two different algebras, nor claimed that ${\cal K}^{+}$ and
> ${\cal K}^{-}$ span different spaces.

However, in equations (2.33) and (2.34) of the original paper, JC writes
${\cal K}^{+}$ as the span of a certain set of elements, and then ${\cal
K}^{-}$ as the span of the same set but with some minus signs introduced. He
thus seems to believe that ${\cal K}^{+}$ and ${\cal K}^{-}$ are different,
since if not, why have a different symbol for them, but of course Gill’s point
is correct in that as defined they are identical. This is a trivial matter,
but it is telling that JC is unable to accept that what he has written here in
the original paper doesn’t make sense mathematically. Of course we are free to
define two different orientations for sets of basis elements, but what is
being pointed out is that his definitions (2.33) and (2.34) fail to do this,
and it is no good his reply saying that ‘there is no such error in [1]’ —
there is.

Author response: What is raised in the above comments by the reviewer is a
non-issue. As I have explained in the manuscript, there is no error in [1] in
this regard. The only difficulty here is that [2], [13], and the present
reviewer have overlooked a straightforward concept in linear algebra and
differential geometry. Namely, regarding an orientable vector space such as
${\cal K}$ (or a manifold such as $S^{7}$) with unspecified orientation
$\lambda$. I have already explained this concept adequately in Section 2.3 of
my original paper [1] and in Section 2.6 of the current manuscript. No further
comment is necessary. However, it may be helpful to note a comment from
Reviewer # 1 in support of my view on this matter:

> Reviewer # 1: “Section 2.6 makes a valid point, both Lasenby and Gill have
> misunderstood that you are referring to differently oriented vector spaces
> not to different vector spaces regardless of orientation.”

As noted in [1] and the manuscript, since $\lambda$ plays the role of a hidden
variable as understood within Bell’s local-realistic framework, ${\cal K}^{+}$
and ${\cal K}^{-}$ specified in the equations (2.33) and (2.34) of my 2018
paper [1] are physically not identical.

Author action: No action.

Finally, it is maybe worth commenting on some points that JC makes in Section
2 of his response.

In 2.1(3) he says:

> The next mistake in the Introduction is more serious. It is claimed that in
> my paper [1] I connect the 3-sphere, or $S^{3}$, “to special relativity,
> specifically to the solution of Einstein’s field equations known as
> Friedmann-Robertson-Walker spacetime with a constant spatial curvature.”
> This is not a misprint or oversight. The same claim appears in several
> preprint versions of the critique [2] that have been posted on arXiv. The
> quoted sentence from [2] thus exhibits a lack of understanding of the
> difference between the special and general theories of relativity and how my
> proposed quaternionic 3-sphere model fits into the Friedmann-Robertson-
> Walker solution of Einstein’s field equations of general relativity.

While I agree that ‘general relativity’ should have been said instead of
‘special relativity’, there is no way in which this can be brought forward as
a serious error, and thought worth highlighting in this way. It is a trivial
error, and has no consequences for any of Gill’s criticisms.

Author response: I do not agree with the reviewer’s opinion. To confuse
general relativity with special relativity is a serious mistake. And it is not
the first time the author of the critique [2] has made such a mistake. It is
unfortunate that it is now published in a Royal Society journal. It is even
more serious considering that it appears in the context of my work that is
fundamentally based on a non-flat geometry of the physical space. Much of the
confusion in the critique [2] stems from its failure to understand the
difference between the strong correlations within a flat spacetime such as
${\mathrm{I\\!R}}\times{\mathrm{I\\!R}}^{3}$ and a curved spacetime such as
${\mathrm{I\\!R}}\times S^{3}$. It is therefore entirely appropriate for me to
draw attention tothis mistake. Moreover, I am not willing to let the readers
go away with thinking that the mistake is contained in my original paper. It
is my scientific duty to point out that the mistake is in the critique and not
in my original paper.

Author action: No action.

In 2.2(2) and (3) JC says:

> In the Introduction of [2] [the Gill comment] it is claimed that other
> papers have also refuted my work. But to date, no one has refuted any part
> of my work, or undermined it in any way. To be sure, there have been
> attempts of refutation, but I have elucidated the errors in all such claims,
> for example in [4-6] and references cited therein. See, especially, Chapters
> 9 to 12 in [4]

> It is further claimed in the Introduction of [2] that Lasenby in [13] has
> independently made the same claims about the algebraic core of the 7-sphere