> framework proposed in [1]. But that is not surprising, because, as
> acknowledged in [13], it has largely borrowed its claims from the claims
> made online by the author of [2]. More importantly, the claims made in [13]
> are equally incorrect, as I have demonstrated point by point in [3] and
> [14].

First of all, there is no statement in the paper by Lasenby [13] that he has
‘borrowed his claims from the claims made online by the author of [2]’.
Instead [13] says: ‘Note several of the points made here have been made
independently by Richard D. Gill and others in the discussion thread attached
to the Royal Society paper’.

Author response: Fair enough. It is of course possible to make the same
oversights independently.

Author action: I have revised item (3) of Section 2.2 in the manuscript and
replaced my statement with the following:

> 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
> framework proposed in [1]. However, in [13] it is acknowledged that “…
> several of the points made [in [13]] have been made independently by [the
> author of [2]] and others in the discussion thread attached to the Royal
> Society paper …” Thus it is not surprising that the claims in [13] are
> similar to those in [2]. More importantly, in [3] and [14] I have refuted
> the claims made in [13].

Secondly, since all JC’s alleged refutations are along the lines of what is in
his current response, i.e. based on demonstrably fallacious mathematics, it is
needless to say that all his assertions in the two paragraphs just referred
to, concerning lack of refutation of his work, are incorrect.

Author response: By now it is hopefully evident that there is no “fallacious
mathematics”, either in my original paper [1] or in the current manuscript. I
have demonstrated in my previous comments, in the manuscript, and in [3] and
[14] that the claims made in [2], [13], and the present review are not
correct. They are based on inconsistent use of product rules on the two sides
of (2.40), and misunderstandings of the 7-sphere framework I have proposed in
[1].

Author action: No action.

Reviewer # 7

Recommendation: Accept

Comments to the Author(s):

None.

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

Author action: No action.

Response to Reviewers — Round # 2:

Reviewer # 1

The response to my critique of Section 2.2 (1) is not adequate. Even according
to the 2015 revision of your 2011 preprint, equation 1 states that
$A(a,\lambda^{k})=+1$ if $\lambda^{k}=+1$, $A(a,\lambda^{k})=-1$ if
$\lambda^{k}=-1$, $B(b,\lambda^{k})=-1$ if $\lambda^{k}=+1$,
$B(b,\lambda^{k})=+1$ if $\lambda^{k}=-1$. From this it immediately follows
that $A(a,\lambda^{k})=-\,B(b,\lambda^{k})$ as Gill pointed out. You have
attempted to deny this conclusion in your various papers including reference
[6] but the arguments are not cogent and amount to denying the transitivity of
the equality relation which is not sensible at all.

Author response: The above argument by the reviewer is not correct. It is
based on a misreading the 3-sphere model inspired by Gill’s misinterpretation
of the model. The measurement functions ${\mathscr{A}}({\bf
a},\,{\lambda^{k}})$ and ${\mathscr{B}}({\bf b},\,{\lambda^{k}})$ in the model
are defined as results of interactions between spin bivectors ${{\bf L}({\bf
s},\,\lambda^{k})}$ and detector bivectors ${{\bf D}({\bf a})}$ and ${{\bf
D}({\bf b})}$:

$\displaystyle S^{3}\ni\,{\mathscr{A}}({\bf
a},\,{\lambda^{k}})\,:=\,\lim_{{\bf s}_{1}\,\rightarrow\,{\bf
a}}\left\\{-\,{\bf D}({\bf a})\,{\bf L}({\bf s}_{1},\,\lambda^{k})\right\\}$
$\displaystyle=\,\begin{cases}+\,1\;\;\;\;\;{\rm if}&\lambda^{k}\,=\,+\,1\\\
-\,1\;\;\;\;\;{\rm if}&\lambda^{k}\,=\,-\,1\end{cases}$
$\displaystyle\text{and}\;\;\;\;S^{3}\ni\,{\mathscr{B}}({\bf
b},\,{\lambda^{k}})\,:=\,\lim_{{\bf s}_{2}\,\rightarrow\,{\bf
b}}\left\\{+\,{\bf L}({\bf s}_{2},\,\lambda^{k})\,{\bf D}({\bf b})\right\\}$
$\displaystyle=\,\begin{cases}-\,1\;\;\;\;\;{\rm if}&\lambda^{k}\,=\,+\,1\\\
+\,1\;\;\;\;\;{\rm if}&\lambda^{k}\,=\,-\,1\,,\end{cases}$

where the orientation ${\lambda}$ of ${S^{3}}$ is assumed to be a random
variable with 50/50 chance of being ${+1}$ or ${-\,1}$ at the moment of the
creation of the spin pair that originates at the source situated in the
overlap of the backward light-cones of Alice and Bob. As such, the measurement
results are limiting scalar points of a quaternionic 3-sphere, taken to model
the geometry of the physical space in which the experiments are taking place.
The joint result ${\mathscr{A}}{\mathscr{B}}({\bf a},\,{\bf
b},\,{\lambda^{k}})$ is therefore a limiting scalar point of the product of
the quaternions $\left\\{-\,{\bf D}({\bf a})\,{\bf L}({\bf
s},\,\lambda^{k})\right\\}$ and $\left\\{+\,{\bf L}({\bf
s},\,\lambda^{k})\,{\bf D}({\bf b})\right\\}$. Since $S^{3}$ remains closed
under multiplication, the product of these quaternions is a third quaternion,
and therefore its limiting scalar point ${\mathscr{A}}{\mathscr{B}}({\bf
a},\,{\bf b},\,{\lambda^{k}})$ cannot be equal to $-1$ for all settings ${\bf
a}$ and ${\bf b}$ as claimed by Gill and the reviewer, unless the conservation
of the zero spin angular momentum that originates from the source is violated:

$\displaystyle{\mathscr{A}}{\mathscr{B}}({\bf a},\,{\bf
b},\,{\lambda^{k}})=\lim_{\begin{subarray}{c}{\mathbf{s}}_{1}\,\rightarrow\,{\mathbf{a}}\\\
{\mathbf{s}}_{2}\,\rightarrow\,{\mathbf{b}}\end{subarray}}\left\\{-\,{\bf
D}({\bf a})\,{\bf L}({\bf s}_{1},\,\lambda^{k})\right\\}\left\\{+\,{\bf
L}({\bf s}_{2},\,\lambda^{k})\,{\bf D}({\bf
b})\right\\}\longrightarrow\begin{cases}-1&\text{if}\;\,{\mathbf{s}}_{1}\not={\mathbf{s}}_{2}\\\
-\,{\mathbf{a}}\cdot{\mathbf{b}}&\text{if}\;\,{\mathbf{s}}_{1}={\mathbf{s}}_{2},\end{cases}$

because all bivectors are unit bivectors and ${\bf L}({\bf
s}_{1},\,\lambda^{k})\,{\bf L}({\bf s}_{2},\,\lambda^{k})=-1$ for
${\mathbf{s}}_{1}={\mathbf{s}}_{2}$ so that ${\bf D}({\bf a}){\bf D}({\bf
b})\longrightarrow-\,{\mathbf{a}}\cdot{\mathbf{b}}$ [7].

I have explained this many times before in detail; for example, in my
previously published responses [4, 6, 7]. The 3-sphere model is not a model of
after-the-event analysis followed by experimenters (i.e., of phenomenology)
but an ontological model of what the correlations will be if the geometry of
the physical space is that of $S^{3}$ instead of $\mathrm{I\\!R}^{3}$.

Author action: I have revised the relevant sentences in the manuscript at the
end of Section 2.2 (1) to: “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 [6], Subsection III E of [7], and Section
VIII of [9]. There are also other oversights in [13], which I have brought out
in [4–7]. In particular, [13] makes the same mathematical mistakes I bring out
in detail below in Subsection 2.7.”

The response to my and reviewer 5’s critiques of Section 2.4 is not entirely
adequate - if two expressions are mathematically equal and one of the
expressions represents a correct physical value, then so does the other
expression as it is the same value. What we have is that Eq. 1 is not even
mathematically valid for Bell type experiments as it is assuming that the
hidden variables $\Lambda$ form a single joint probability space for all spin
measurements and that one can integrate over this same $\Lambda$ in all cases.
That is indeed not the case if we consider eigenvalues of non-commuting
operators as statistical random variables, so I am not making a different
argument. However the wording of section 2.4 has been changed in the latest
version of the paper and appears more adequate now – I disagree with reviewer
2 that your argument rules out your own model.

Author response: Apart from the historical precedence of the argument I have
presented in Section 2.4 and the universal agreement among the experts in
foundations of quantum mechanics on the invalidity of the assumption of
additivity of expectation values for hidden variable theories, the advantage
of my argument is that it remains valid even if a single joint probability
space for all spin measurements is assumed as insisted on by the adherents of
Bell’s theorem. One has to adapt double standards – one standard for von
Neumann’s theorem and another for Bell’s theorem – to deny the invalidity of
the assumption of additivity of expectation values for hidden variable
theories.

I am pleased to know that the present reviewer disagrees with the claim by
Reviewer # 2 that my argument in Section 2.4 rules out my own model.

Author action: I have further improved the wording of Section 2.4 to make my
argument more comprehensible.

Reviewer 2 raises the question as to how the geometry of space-time is
relevant to Bell type experiments. This is something that you have indeed
repeatedly tried to explain in your papers and as noted in my own critique
whether your model is sensible and rigorous in this regard remains
questionable.

Author response: In my experience those who question the 3-sphere model of
quantum correlations do so because they do not realize that it is not a model
of after-the-event analysis followed by the experimenters but an ontological
model that predicts what the correlations would be if the geometry of the
physical space is that of a quaternionic 3-sphere instead of a flat
${\mathrm{I\\!R}^{3}}$. The present reviewer’s claim that the 3-sphere model
implies “$A(a,\lambda^{k})=-\,B(b,\lambda^{k})$” for all settings $a$ and $b$
“as Gill pointed out” is ultimately because of this category error in
understanding the model.

Author action: In Section 2.1 (3) of the revised manuscript, I have added the
following sentence: “In fact, much of the confusion in [2] 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
${\mathrm{I\\!R}}\times S^{3}$.”

The response to my critique of the calculation involving epsilon in Section
2.7, and similar concerns by reviewers 2 and 6 are far from adequate. In the
original paper on which Gill has commented the norm is defined as
$|\Omega|:=\Omega\cdot\Omega^{\dagger}\equiv\sqrt{(s)}$ – the square root of
the scalar part of $\Omega\cdot\Omega^{\dagger}$, as reviewer 6 also points
out.

Author response: This is not correct. In the original paper [1] the
orthogonality condition ${\bf q}_{r}\,{\bf q}^{\dagger}_{d}+{\bf q}_{d}\,{\bf
q}^{\dagger}_{r}=0$ is used as a normalization condition, defined in Eq.
(2.54) of Section 2.5, to consistently obtain scalar values for the norms on
both sides of the norm relation (2.59). The scalar values thus obtained happen
to be same as those obtained using the traditional definition
$||\Omega||=\sqrt{\Omega\cdot\Omega^{\dagger}}$ stated in Eq. (2.8) of Section
2.5. Reviewer # 2 and # 6 are also mistaken regarding this point, as I
explained in detail in the previous round of reviews. Moreover, Reviewer # 3
has now verified my proof of the norm relation in detail, whose reports I
reproduce below verbatim.

Author action: In Section 2.8 of the revised manuscript I have summarized the
proof again for convenience.

This is scalar valued and positive definite and the standard rules for scalar
multiplication and positive definite norms apply and the square root being
used is the standard one of arithmetic. Thus placing an expression within a
square root so understood implies that the expression is a scalar.

Author response: All of the above properties remain true for the norms
obtained using the orthogonality condition ${\bf q}_{r}\,{\bf
q}^{\dagger}_{d}+{\bf q}_{d}\,{\bf q}^{\dagger}_{r}=0$ as a normalization
condition. I have shown this in detail in Refs. [3] and [25] cited in the
manuscript.

Author action: In Section 2.8 of the manuscript I have added a proof of the
positive definiteness of the norm.

But in your attempt to refute Gill’s counterexample you have changed the
definition of the norm to a non-scalar geometric magnitude as well as changed
to working with a geometric square root, which is not what appears in the
original paper.

Author response: This is not correct. I have not changed the definition of the
norm I have used in the original paper [1], as I explained above. The
definition I have used is stated in Eq. (2.54) of Section 2.5 in the paper
[1].

More importantly, the supposed counterexample by Gill and Lasenby fails as I
have demonstrated in the manuscript as well as in the previous round of
reviews. Reviewer # 3 has also verified in detail that the counterexample
fails.

Author action: No action.

Additionally the meaning of this square root that is being used needs further
elucidation, only a reference to an unpublished paper appears.

Author response: In addition to the elucidations in Refs. [3] and [25] cited
in the manuscript, Reviewer # 3 has also now elucidated the meaning of the
square root I have used for the hyperbolic or split complex numbers.

Author action: In Section 2.8 of the revised manuscript, I have added the
following footnote: “During the reviewprocess 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.”

Section 2.8 has been improved.

Author response: I thank the reviewer for this observation.

Author action: No action.

Reviewer # 2

Dear Editor,

I recommend that this paper will not be accepted for publication. Here is a
list of some reasons that support this opinion.

1\. “Bell’s theorem is not a theorem in mathematical sense”

I cannot even understand what a sentence like this means: in which other sense
can a theorem be a theorem?

Author response: Reviewer has quoted the heading of Section 2.3 of the
manuscript, while omitting the quotation marks on the word “theorem.” In the
first two paragraphs of Section 2.3 I have very clearly explained why Bell’s
so-called “theorem” is not a theorem in mathematical sense. Unlike it, a
mathematical theorem does not depend on physical and metaphysical assumptions
such as locality, realism, “free will” of experimenters, etc., and does not
require physical experiments for its validity. Any loophole (or “gap”) would
render a mathematical theorem invalid.

Author action: No action.

Bell’s theorem can be precisely formulated as a theorem, even if John Bell
didn’t do it by himself, and its conclusions can be trivially proved. Of
course, like any other theorem, Bell’s theorem does rely on certain
assumptions, and it could certainly be the case that these assumptions are not
fulfilled in the actual experiments that test the Bell inequalities. That
would make the Bell theorem irrelevant for understanding or interpreting the
experiments, but it does not make the theorem wrong. The theorem would be
wrong only if not all the assumptions needed to prove it were properly stated.

Author response: I have not claimed that the inequalities usually attributed
to Bell (which were in fact derived by George Boole some one hundred and
eleven years before Bell’s work) are wrong. The reviewer seems to have succumb
to the same confusion between Bell’s inequalities and Bell’s theorem as the
author of the critique [2] has.

Author action: No action.

Let me give a very simple example that could clarify the situation. Consider
the following theorem:

Theorem: Under the assumptions of Newton’s laws, the distance covered by a
body that falls freely with constant acceleration g is given by the expression

$\mathrm{d}=0.5*\mathrm{g}*\mathrm{t}^{2}+\mathrm{v}*\mathrm{t},$

where $\mathrm{t}$ is the time elapsed and $\mathrm{v}$ is the body’s initial
velocity.

This theorem is precisely formulated and can be trivially proven under the
assumptions that appear in the statement of the theorem. Nonetheless, if we
could perform an experiment in which we would let a body fall freely from an
altitude of 3,000 Km, we would find that the results of the experiment do not
fit the predictions of the above theorem. Of course, the reason why the
theorem fails to describe the said experiment is not that “the theorem is not
a theorem in the mathematical sense”, but the fact that the assumption of
constant acceleration under which the theorem was derived does not hold for a
body that falls freely from very high altitude.

The author fails to clearly make this simple observation and instead makes an
extremely confusing argumentation about Bell’s theorem not being a theorem
even though it can be trivially proven. The reader, obviously, gets absolutely
confused and learns nothing from the author’s paragraph.

Author response: The example presented by the reviewer does not depend on
metaphysical assumptions such as “realism” and “free will”, and therefore it
does not correspond to the problems with Bell’s so-called theorem. I have
explained the problem with Bell’s supposed theorem very clearly in the
Sections 2.3, 2.4, and 2.5 of the manuscript.

Author action: No action.

2\. “Assumption of the additivity of the expectation values”

A successful model for a physical system, e.g. Bell’s experiment, is a
precisely defined mathematical object that reproduces the characteristic
features of the collected experimental data about the system that we want to
describe or understand.

In a Bell experiment, four different physical observables are measured. Let me
call them E11, E12, E21, E22. The measured values for these magnitudes are, of
course, rational numbers. These four numbers are then added up (in the usual
way in which rational numbers are added up) and a fifth number
S=abs(E11+E12+E21-E22) is obtained.

A successful model for (any settings of) this experiment must reproduce all
these five numbers, and not only the last one. The author, nonetheless, argues
that his model’s prediction for S does not equal the usual addition of its
predictions for E11, E12, E21 E22. If his claim is correct, then his model is
automatically ruled out because it does not reproduce the experimental data.

Author response: The last claim made by the reviewer is not correct. This is
also observed by Reviewer # 1 who has noted that “I disagree with reviewer 2
that your argument rules out your own model.” The problem with Bell argument
is that what the reviewer calls “a fifth number S=abs(E11+E12+E21-E22)” is not
a correct eigenvalue of the corresponding quantum mechanical operator, and
therefore it does not belong to the right-hand side of Eq. (1) in any hidden
variable theory based on dispersion-free states. It is thus the same mistake
that invalidated von Neumann’s theorem against hidden variable theories. I
have explained this very clearly with necessary details in Section 2.4 of the
manuscript. When the sum E11+E12+E21-E22 is replaced with the correct
eigenvalue of the corresponding quantum mechanical operator, the bounds on the
CHSH correlator exceeds from $\pm 2$ to $\pm 2\sqrt{2}$, thereby permitting
models like the one I have proposed to reproduce strong correlations instead
of ruling them out.

Author action: I have improved some wording in Section 2.4 to make my argument
clearer. To explain the above points more clearly, I have also added the
example of Stern-Gerlach magnet Bell has used in Chapter 1 of his book.

3\. “Assumption of a flat and immutable spacetime”

The author claims that his model succeeds to reproduce the statistical
features of the experimental data collected in actual Bell’s experiments by
modeling the spatial geometry of the Universe as a closed sphere. He further
argues that collected cosmological data even supports this geometry.

The question then is: how is it possible that the geometry of the whole
Universe is relevant in order to describe a ‘tabletop’ experiment? Violation
of the Bell inequality has been observed in experiments performed between two
stations separated at most a few hundred kilometers. The visible patch of our
Universe is more than twenty orders of magnitude larger than this distance!!

Since a sphere is different from a homogeneous and isotropic flat manifold
only in their local curvature and their topology, the author’s model for the
Bell experiment must be sensible to either the curvature of the sphere, its
topology, or both. According to the cosmological evidence cited by the author,
the curvature of the Universe is at most only very slightly different from
zero. Hence, the author’s model must be sensible to the topology of the sphere
(in fact, this is already obvious from the author’s example with a Moebius
strip, which is different from a usual closed strip only in their topologies).

That means, that the author’s model for the tabletop Bell’s experiment do not
comply with the usual causality constraints, since the correlations generated
in the short time interval elapsed between the emission and detection of the
pair of entangled particles are sensible to the topology of the whole
Universe.

Author response:

I have addressed the issue raised by the reviewer in Answer 4 of Appendix B of
Ref. [10] cited in the manuscript. The 3-sphere model presented in [1] does
not require Bell-test experiments to be performed over cosmological distances.
They could be performed on a table top and the 3-sphere model would then still
provide a viable explanation of the observed results. I have proposed just
such a table top experiment in Refs [8] and [22] cited in the manuscript. The
curvature of the Universe is not of direct relevance for the 3-sphere model or
the proposed experiment. The geometry of the Universe is indeed irrelevant for
the Bell-test experiments performed on Earth. The 3-sphere model concerns the
intrinsic geometry of the physical space, since it is a part of a solution of
the field equations of general relativity. In the toy model of the Möbius
world I have discussed in Ref. [10], the twist in the strip responsible for
the strong correlations is an extrinsic twist in the geometry of the strip,
while the twists in the Hopf bundle of $S^{3}$ are intrinsic to $S^{3}$.
Consequently, unlike in the Möbius world where relative handedness of spins
depend on the revolutionary distance between them, in the real world the
relative handedness of quaternions that constitute the 3-sphere reflects their
intrinsic spinorial characteristics, independently of any distance between
them. Therefore the strong correlations predicted by the 3-sphere model and
observed in the Bell-test experiments performed in terrestriallaboratories can
be taken as evidence that the intrinsic geometry of the physical space is that
of $S^{3}$ and not of ${{\mathrm{I\\!R}}^{3}}$.

Author action: No action.

4\. “Algebra ${\cal K}^{\lambda}$ used in [1] is not incompatible with
Hurwitz’s theorem”

The author argues that he has proven this point somewhere else, but no clear
explanation can be found neither in the document that he cites, so I will
again raise the question:

Author response: This claim by the reviewer is not credible. A very clear and
explicit proof of the norm relation in question is already provided in Section
2.5 of the original paper [1], and again with exhaustive detail in [3] and
[25]. Moreover, now Reviewer # 3 has independently verified my proof
explicitly, which I include and discuss below.

Author action: In Section 2.8 of the revised manuscript I have summarized the
proof again for convenience.

Let me assume, as the author claims, that in the said algebra the following
identity

$||\mathrm{X}\mathrm{Y}||=||\mathrm{X}||\;||\mathrm{Y}||,$

holds for any pair of objects X,Y that belong to the said algebra, where by
definition $||X||$ and $||Y||$ are non-negative real numbers. As I said, I am
willing to accept that this identity indeed holds for the said algebra.

The author further claims that in his algebra ${\cal K}^{\lambda}$ there exist
an element epsilon different from I (the identity of the algebra) and
different from -I, that fulfills the identity

$\mathrm{epsilon}^{2}=\mathrm{I}.$

Now, using the fact that epsilon is an element of the algebra we can write the
last identity as

(epsilon + I) (epsilon - I) = 0

and taking X=epsilon + I and Y=epsilon - I and using the above identity, which
as I said I did not dispute, we find

$0=||0||=||\text{(epsilon + I) (epsilon-I)}||=||\text{epsilon +
I}||\,||\text{epsilon - I}||,$

which implies that either epsilon+I = 0 or epsilon-I = 0, since for a normed
algebra $||X||$ = 0 if and only if X=0.

That is, either epsilon=I or epsilon= -I, in contradiction with the author’s
statement in eq.(2.51) that there exists in the algebra ${\cal K}^{\lambda}$
and element epsilon different from I and -I for which holds the identity
epsilon2=I.

This derivation shows that the only way to escape this contradiction is that
either eq.(2.40) does not hold, or the statement $||X||$ = 0 iif X = 0. Since
the author claims that eq.(2.40) holds (and I did not dispute his claim), the
only option left is that [there] exists X.neq.0 for which $||X||$ = 0, but in
such a case ${\cal K}^{\lambda}$ is not a normed algebra.

Author response: In the last round of reviews I have already pointed out the
mathematical oversights in the above argument in considerable detail.
Unfortunately, the reviewer has repeated the same oversights again without
recognizing them. Since I have also discussed the issue raised by the reviewer
in the Sections 2.7 and 2.8 of the manuscript, and since Reviewer # 3 has now
verified my proof of the norm relation
$||\mathrm{X}\mathrm{Y}||=||\mathrm{X}||\;||\mathrm{Y}||$ in detail, I will
only point out the main oversight in the reviewer’s argument. The oversight is
evident in the following equation:

$0=||0||=||\text{(epsilon + I) (epsilon-I)}||=||\text{epsilon +
I}||\,||\text{epsilon - I}||.$

Notice that a geometric product between X = (epsilon + I) and Y = (epsilon -
I) is used by the reviewer to obtain XY = 0 on the left-hand side of this
equation, whereas scalar products are tacitly used to guess scalar-valued
norms on its right-hand side (without defining what norm $||\cdot||$ is).
Indeed, it is inevitable to use a geometric product on the left-hand side
because otherwise the resulting multivector Z = XY would not be a member of
the algebra ${\cal K}^{\lambda}$. Moreover, if the scalar products are not
used to evaluate norms on the right-hand side but geometric products are used
instead to match the use of geometric product on the left-hand side, then the
norms would not be scalar-valued. They would be equal to $||\text{epsilon +
I}||=\text{I + epsilon}$ and $||\text{epsilon - I}||=\text{I - epsilon}$,
giving zero right-hand side:

$||\text{epsilon + I}||\,||\text{epsilon - I}||=\text{(I + epsilon) (I -
epsilon)}=\text{I - epsilon}^{2}=\text{I - I = 0}.$

Therefore the conclusion drawn by the reviewer that either (I + epsilon) = 0
or (I - epsilon) = 0 does not follow. Consequently, the alleged contradiction
drawn by the reviewer that either epsilon = I or epsilon = -I does not follow.

One can always draw a contradiction from any mathematical equation by
employing two different rules on the two sides of the equation (in this case
product rules). The oversight in the reviewer’s argument is thus quite
elementary.

Author action: To be fair to the reviewer’s point of view, in a footnote in
Section 2.8 I have added the following footnote: “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 published along with this paper.”

I think that I have enumerated a long enough list of serious criticisms that
justifies my recommendation of rejection.

Author response: I have demonstrated above that none of the criticisms by the
reviewer is correct. They are basedon technical oversights. Therefore the
reviewer’s recommendation for the rejection of my manuscript is not justified.

Author action: No action.

Reviewer # 3

I have confirmed your norm for the dual quaternions, and it is interesting how
it works, and that the unit dual, epsilon, squares to +1 and commutes with
quaternions. The norm of a dual quaternion always giving a dual number a +
epsilon b, a,b Real.

Author response: I am grateful to the reviewer for verifying the norm relation
$\|XY\|=\|X||\,\|Y\|$ in full detail.

Below I reproduce the reviewer’s calculations verbatim, with some comments. It
is surprising that while the validity of the norm relation has been challenged
by Gill, Lasenby, and others despite its proof I have presented in [1], [3],
and [25], the present reviewer is the only person I am aware of who has
bothered to verify the proof in detail to appreciate how it actually works.
Others have merely alleged a counterexample, or that the norm relation
contradicts Hurwitz’stheorem, without digging into details or pointing out an
error in the proof I have presented in [1], [3], and [25].

Author action: In Section 2.8 of the revised manuscript I have reproduced a
succinct version of the proof.

1\. Not sure therefore why you are requiring the two quaternions (inside a
dual quaternion) to be orthogonal? The subscript ‘s’ on your norm actually
confused me, seems unnecessary.

Author response: The orthogonality condition ${\bf q}_{r}\,{\bf
q}^{\dagger}_{d}+{\bf q}_{d}\,{\bf q}^{\dagger}_{r}=0$ is the normalization
condition I have used in Eq. (2.54) of the Section 2.5 of the original paper
[1] to consistently obtain scalar values for the norms on both sides of the
norm relation (2.59). This then allows us to construct a 7-sphere with a
scalar radius, as in Eq. (2.60) of [1].

By contrast the standard definition (2.8) of the norm (stated in the preamble
of the introductory Section 2 of [1] that summarizes the basic concepts of
Geometric Algebra) picks out the scalar part of the geometric product
$\Omega\Omega^{\dagger}$, which is indicated by the subscript ‘s’. This
definition is not used in calculating the norms in [1] because it is not
consistently applicable in an equation like $\|XY\|=\|X||\,\|Y\|$ that
involves geometric product on at least one side.

Author action: No action.

2\. Can your S3 space be viewed as describing the wave function of the two
particles?

Author response: In my view wave functions and entanglement are merely
placeholders for the observed quantum correlations. They are not fundamental
to Nature. The $S^{3}$ space facilitates a local-realistic model for the
observed correlations between two or many particles without needing a wave
function. Thus it does the job that is traditionally accomplished by quantum
mechanical wave functions via the usual Hilbert space formulation of quantum
mechanics.

Author action: No action.

3\. Your S3 space could also be viewed as arising from the space of unit
quaternions used to rotate vectors in 3D space?

Author response: Yes, $S^{3}$ is a set of all unit quaternions. It is a
quaternionic 3-sphere that can be defined as

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

where ${{\mathbf{J}}({\mathbf{r}})=I_{3}{\mathbf{r}}}$ is a unit bivector (or
a pure quaternion) rotating about an axis vector ${{\bf
r}\in{\mathrm{I\\!R}}^{3}}$ with rotation angle ranging from
${0\leq\theta<4\pi}$, $\varrho_{r}$ is the radius of the corresponding
3-sphere, and $I_{3}={\mathbf{e}}_{x}{\mathbf{e}}_{y}{\mathbf{e}}_{z}$ is the
standard trivector.

Author action: No action.

I decided to check the criticisms of the other reviewers regarding JC’s norm
that he defined for the dual quaternion, pdf document attached.

My calculations show that the norm is correct as defined by JC, coupled with
his condition of orthogonal quaternions, which then creates a norm within the
Real numbers. Hence, the counter-example proposed with
$\mathrm{e}^{2}-1=(\mathrm{e}-1)(\mathrm{e}+1)=0$, does not invalidate this
norm, as both $\mathrm{e}-1$ and $\mathrm{e}+1$ are not orthogonal
quaternions. Nevertheless, for this counter-example, they actually do satisfy
the norm anyway, provided the square root is taken properly, see attached. I
also explicitly calculate the square root of $\mathrm{e}-1$ and
$\mathrm{e}+1$, to help show this, which JC does not do.

Author response: I am grateful to the reviewer for verifying my proof of the
norm relation and recognizing that the alleged counterexample does not
invalidate it. I agree with the reviewer’s calculations reproduced below. It
is easy to see that they mostly parallel the proofs I have presented in [1],
[3], and [25], apart for the useful addition of explicit square roots of
$1-\varepsilon$ and $1+\varepsilon$.

I do calculate explicit square roots for the general elements of ${\cal
K}^{\lambda}$ in Appendix A.1 of Ref. [3] using the exponential method for
taking square roots of hyperbolic numbers. See equations (A.22) to (A.25) in
Appendix A.1 of [3].

Author action: I have adapted the reviewer’s explicit expressions for the
square roots of $1-\varepsilon$ and $1+\varepsilon$ in Section 2.8 of the
revised manuscript.

Your have created an impressive theory, and I hope the ultimate test of an
experiment can be performed soon, as you suggest.

Author response: I thank the reviewer for these kind words.

Author action: No action.

Calculation by Reviewer # 3 — Dual quaternions and their norm:

##  IV Introduction

It is shown here that JC’s generalized norm is satisfied, both in the specific
case, using the expansion $\varepsilon^{2}-1=(\varepsilon-1)(\varepsilon+1)=0$
as well as in the general case, provided that the square root operation uses
the principal square root.

Extra clarity is added to this issue through explicitly calculating each
individual square root.

##  V Analysis

###  V.1 The special case with
$\varepsilon^{2}-1=(\varepsilon-1)(\varepsilon+1)=0$

We have the algebraic expansion
$\varepsilon^{2}-1=(\varepsilon-1)(\varepsilon+1)=0$, and selecting
$Z=\varepsilon^{2}-1$, and $X=\varepsilon-1,Y=\varepsilon+1$ we wish to show
that the norm relation

$\|Z\|=\|XY\|=\|X||\|Y\|$ (51)

is satisfied where $\|Z\|=\sqrt{ZZ^{\dagger}}$. We also have defined
$\varepsilon=e_{x}e_{y}e_{z}e_{\infty}$, where
$\varepsilon^{\dagger}=\varepsilon$ and $\varepsilon^{2}=1$

Now

$\displaystyle\|X\|=\sqrt{(\varepsilon-1)(\varepsilon-1)^{\dagger}}=\sqrt{(\varepsilon-1)(\varepsilon-1)}=\sqrt{2(1-\varepsilon)},$
(52)
$\displaystyle\|Y\|=\sqrt{(\varepsilon+1)(\varepsilon+1)^{\dagger}}=\sqrt{(\varepsilon+1)(\varepsilon+1)}=\sqrt{2(1+\varepsilon)}.$

We can now explicitly calculate the square root of each individual term. That
is

$\displaystyle\sqrt{1-\varepsilon}=\left(\frac{1}{\sqrt{2}}-\frac{\varepsilon}{\sqrt{2}}\right),$
(53)
$\displaystyle\sqrt{1+\varepsilon}=\left(\frac{1}{\sqrt{2}}+\frac{\varepsilon}{\sqrt{2}}\right).$

Its obviously important for consistency to take the principal, positive square
root in both cases. These square roots can be confirmed through squaring, that
is

$\left(\frac{1}{\sqrt{2}}\pm\frac{\varepsilon}{\sqrt{2}}\right)^{2}=1\pm\varepsilon,$
(54)

as required.

Hence

$\displaystyle\|X\|=\sqrt{2}\sqrt{(1-\varepsilon)}=\sqrt{2}\left(\frac{1}{\sqrt{2}}-\frac{\varepsilon}{\sqrt{2}}\right)=1-\varepsilon$
(55)
$\displaystyle\|Y\|=\sqrt{2}\sqrt{(1+\varepsilon)}=\sqrt{2}\left(\frac{1}{\sqrt{2}}+\frac{\varepsilon}{\sqrt{2}}\right)=1+\varepsilon.$

Author comment: I have adapted the above expressions for the square roots in
Section 2.8 of the manuscript.
Therefore

$\|X\|\|Y\|=(1-\varepsilon)(1+\varepsilon)=1-\varepsilon^{2}=0,$ (56)

using $\varepsilon^{2}=1$. We also have
$\|Z\|=\left\|\varepsilon^{2}-1\right\|=\|0\|=0$.

Hence, we have shown explicitly that
$\varepsilon^{2}-1=(\varepsilon-1)(\varepsilon+1)=0$ satisfies the norm
relation.

###  V.2 Confirming the norm in the general case

We firstly confirm that $\varepsilon$ commutes with all quaternions, such as

$\displaystyle
q_{r}=a+v_{1}e_{y}e_{z}+v_{2}e_{z}e_{x}+v_{3}e_{x}e_{y}=a+I_{3}\boldsymbol{v},$
(57) $\displaystyle
q_{d}=b+w_{1}e_{y}e_{z}+w_{2}e_{z}e_{x}+w_{3}e_{x}e_{y}=b+I_{3}\boldsymbol{w},$
(58)

where $I_{3}=e_{x}e_{y}e_{z}$. For clarity we have used regular Cartesian
vectors $\boldsymbol{v}=$ $v_{1}e_{x}+v_{2}e_{y}+v_{3}e_{z}$ and
$\boldsymbol{w}=w_{1}e_{x}+w_{2}e_{y}+w_{3}e_{z}$. We can confirm that

$I_{3}\boldsymbol{v}=e_{x}e_{y}e_{z}\left(v_{1}e_{x}+v_{2}e_{y}+v_{3}e_{z}\right)=v_{1}e_{y}e_{z}+v_{2}e_{z}e_{x}+v_{3}e_{x}e_{y},$
(59)

as required. Then we can write a dual quaternion

$Q=q_{r}+\varepsilon q_{d}$ (60)

and the norm squared of $Q$ is

$\|Q\|^{2}=QQ^{\dagger}=\left(q_{r}+\varepsilon
q_{d}\right)\left(q_{r}^{\dagger}+\varepsilon
q_{d}^{\dagger}\right)=q_{r}q_{r}^{\dagger}+q_{d}q_{d}^{\dagger}+\varepsilon\left(q_{r}q_{d}^{\dagger}+q_{d}q_{r}^{\dagger}\right).$
(61)

Now

$q_{r}q_{r}^{\dagger}+q_{d}q_{d}^{\dagger}=a^{2}+\boldsymbol{v}^{2}+b^{2}+\boldsymbol{w}^{2}=a^{2}+v_{1}^{2}+v_{2}^{2}+v_{3}^{2}+b^{2}+w_{1}^{2}+w_{2}^{2}+w_{3}^{2},$
(62)

a non-negative scalar, giving the Pythagorean distance in eight dimensions.
Also

$q_{r}q_{d}^{\dagger}+q_{d}q_{r}^{\dagger}=\left(a+I_{3}\boldsymbol{v}\right)\left(b-I_{3}\boldsymbol{w}\right)+\left(b+I_{3}\boldsymbol{w}\right)\left(a-I_{3}\boldsymbol{v}\right)=2(ab+\boldsymbol{v}\cdot\boldsymbol{w}),$
(63)

giving a pure scalar result, as the dot product of two vectors is a scalar.
Hence, the norm squared of the dual quaternion is a commuting dual number
(also called a hyperbolic or split complex number) $c+\varepsilon d$, where
$c,d\in\Re$.
Author comment: I have adapted some of this language in my proof presented in
Section 2.8 of the manuscript.

Now for two dual quaternions $X,Y$, we have the norm squared

$\|XY\|^{2}=(XY)(XY)^{\dagger}=XYY^{\dagger}X^{\dagger}=X\|Y\|^{2}X^{\dagger}=XX^{\dagger}\|Y\|^{2}=\|X\|^{2}\|Y\|^{2}$
(64)

a completely general result, relying on the norm being a commuting dual
number. The square root of a dual number is

$\sqrt{c+\varepsilon
d}=\frac{b}{\sqrt{2}\sqrt{c\pm\sqrt{c^{2}-d^{2}}}}+\frac{\varepsilon}{\sqrt{2}}\sqrt{c\pm\sqrt{c^{2}-d^{2}}}$
(65)

However, we can see from the norm, that the real part is always greater or
equal to the pseudoscalar part, and hence $c>=d$. Hence, the roots of these
dual numbers will also be dual (and not complex), although they will be two-
valued, in general.

We thus need care when distributing the square root over two dual numbers
$\sqrt{\|X\|^{2}\|Y\|^{2}}$, through selecting the principal root in each
case. The double valued nature of the square root implies the equation will be
satisfied at two different values.

###  V.3 The orthogonality condition

If we specify an orthogonality condition of
$q_{r}q_{d}^{\dagger}+q_{d}q_{r}^{\dagger}=0$ then the norm will always be a
non-negative real number and so we can immediately write

$\|XY\|=\|X\|\|Y\|,$ (66)

as we will only be taking roots over non-negative Real numbers.

If we consider again the special test case of
$\varepsilon^{2}-1=(\varepsilon-1)(\varepsilon+1)=0$, we can see immediately
that with multivectors $X=\varepsilon-1$ and $Y=\varepsilon+1$ that they do
not satisfy the orthogonality condition and so are not in the restricted class
of multivectors with a non-negative real valued norm. Nevertheless, as shown,
they still satisfy the norm relation.

##  VI Discussion

One of the points of contention with $\mathrm{JC}$ ’s ’norm’ is that it is not
conventional to call it a norm, which should be a non-negative scalar
quantity. However, accepting JC’s definition of a generalized norm over the
dual numbers, it does appear that the norm relation $\|XY\|=\|X\|\|Y\|$ is
valid in general, provided that the principal root is selected (similar to
complex numbers). Specifically, the roots of dual numbers are two-valued, and
so they need to be matched up on both sides of the equation.

We show explicitly that the alleged counter example using
$\varepsilon^{2}-1=(\varepsilon-$ 1) $(\varepsilon+1)=0$ does not disprove
this norm relation.
Author comment: Moreover, using the orthogonality condition ${\bf q}_{r}\,{\bf
q}^{\dagger}_{d}+{\bf q}_{d}\,{\bf q}^{\dagger}_{r}=0$ as a normalization
condition, the resulting norms reduced to scalar-valued norms, as explained in
more detail in Section 2.8 of the manuscript.
Author action: In Section 2.8 of the revised manuscript, I have added the
following footnote: “During the reviewprocess 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.”

Another calculation by Reviewer # 3 — Dual quaternions:

Author comment: The following calculation by Reviewer # 3 appears to be an
earlier version of the one above. It is less detailed than the previous
version, but I am including it here for the sake of completeness of the
records.

## 1 Introduction

We have $\varepsilon=e_{x}e_{y}e_{z}e_{\infty}$, where
$\varepsilon^{\dagger}=\varepsilon$ and $\varepsilon^{2}=1$ and $\varepsilon$
commutes with all quaternions, such as

$\displaystyle
q_{r}=a+v_{1}e_{y}e_{z}+v_{2}e_{z}e_{x}+v_{3}e_{x}e_{y}=a+I_{3}\boldsymbol{v},$
(1) $\displaystyle
q_{d}=b+w_{1}e_{y}e_{z}+w_{2}e_{z}e_{x}+w_{3}e_{x}e_{y}=b+I_{3}\boldsymbol{w},$

where $I_{3}=e_{x}e_{y}e_{z}$. For clarity we can use regular Cartesian
vectors $\boldsymbol{v}=v_{1}e_{x}+$ $v_{2}e_{y}+v_{3}e_{z}$ and
$\boldsymbol{w}=w_{1}e_{x}+w_{2}e_{y}+w_{3}e_{z}$. We can then confirm that

$I_{3}\boldsymbol{v}=e_{x}e_{y}e_{z}\left(v_{1}e_{x}+v_{2}e_{y}+v_{3}e_{z}\right)=v_{1}e_{y}e_{z}+v_{2}e_{z}e_{x}+v_{3}e_{x}e_{y},$
(2)

as required. Then we can write a dual quaternion

$Q=q_{r}+\varepsilon q_{d}$ (3)

and the norm squared of $Q$ is

$\|Q\|^{2}=QQ^{\dagger}=\left(q_{r}+\varepsilon
q_{d}\right)\left(q_{r}^{\dagger}+\varepsilon
q_{d}^{\dagger}\right)=q_{r}q_{r}^{\dagger}+q_{d}q_{d}^{\dagger}+\varepsilon\left(q_{r}q_{d}^{\dagger}+q_{d}q_{r}^{\dagger}\right).$
(4)

Now

$q_{r}q_{d}^{\dagger}+q_{d}q_{r}^{\dagger}=\left(a+I_{3}\boldsymbol{v}\right)\left(b-I_{3}\boldsymbol{w}\right)+\left(b+I_{3}\boldsymbol{w}\right)\left(a-I_{3}\boldsymbol{v}\right)=2(ab+\boldsymbol{v}\cdot\boldsymbol{w}),$
(5)

giving a pure scalar result, as the dot product of two vectors is a scalar.
Hence, the norm squared of the dual quaternion is a commuting dual number
$c+\varepsilon d$, where $c,d\in\Re$.

Now for two dual quaternions $X,Y$, we have the norm squared

$\|XY\|^{2}=(XY)(XY)^{\dagger}=XYY^{\dagger}X^{\dagger}=X\|Y\|^{2}X^{\dagger}=XX^{\dagger}\|Y\|^{2}=\|X\|^{2}\|Y\|^{2}$
(6)

a completely general result, relying on the norm being a commuting dual
number.

Taking square roots, in order to find the norm could be problematic over the
dual numbers, as the dual numbers may not contain the square root (if they
require a square root of minus one) and care will be needed to distribute the
square root over two dual numbers when finding $\sqrt{\|X\|^{2}\|Y\|^{2}}$.

However, if we specify an orthogonality condition, of
$q_{r}q_{d}^{\dagger}+q_{d}q_{r}^{\dagger}=0$ then the norm will always be a
non-negative real number and so we can write

$\|XY\|=\|X\|\|Y\|.$ (7)

If we consider the special test case of
$\varepsilon^{2}-1=(\varepsilon-1)(\varepsilon+1)=0$, we can see immediately
that with multivectors $X=(\varepsilon-1)$ and $Y=(\varepsilon+1)$ they do not
satisfy the orthogonality condition, and so are not valid dual quaternions to
use in the norm. However, we can proceed regardless to see if the norm is
satisfied in this case. We find the norm

$\|X\|=\sqrt{(\varepsilon-1)(\varepsilon-1)}=\sqrt{2(1-\varepsilon)}.$ (8)

Now, in this case the square root does exist within the dual numbers, with
$\sqrt{1-\varepsilon}=\frac{1}{\sqrt{2}}-\frac{\varepsilon}{\sqrt{2}}$ and so
$\|X\|=1-\varepsilon$, and we can also find $\|Y\|=1+\varepsilon$. Therefore
$\|X\|\|Y\|=(1-\varepsilon)(1+\varepsilon)=0$, as required, and so satisfies
the norm. However, the individual norms $\|X\|,\|Y\|$, are not scalars because
they do not satisfy the orthogonality constraint.

Reviewer # 4

The author has satisfactorily responded to the reviewers’ comments and
suggestions and appropriately modified the manuscript. I recommend the comment
for publication.

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

Author action: No action.

Reviewer # 5

With respect to sec. 2.4, I criticized material from the core of the section.
I objected to a quotation from Christian as implying that a certain claim is
both true and false. The quotation runs:

> “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.”

Christian has now cancelled some of the material (“mathematically correct”,
“fallacious”) that led to trouble.

Author response: I have not “cancelled” any material. I had never used the
word “fallacious” in the manuscript, for example, so there is no question of
“cancelling” it. I only improved the wording and presentation of Section 2.4
to make my argument clearer.

Author action: I have further improved the wording and presentation of Section
2.4 in the revised manuscript.

The revised quotation runs:

> “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 as the critique notes, 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 are noncommuting, as in the Bell-test experiments.”

However, the text still contains the following sentence:

> “[…] while the sum of expectation values is mathematically the same as the
> expectation value of the sum […], and while this assumption is valid in […
> theory A …], it is not valid for [… theory B.]”

To understand this sentence, we require clarification. Strictly speaking, a
claim (an assumption or proved identity) cannot be valid or invalid but only
true or false. Is the present claim (“the sum of expectation values is
mathematically the same as the expectation value of the sum”) an assumption or
a proved identity? Christian clearly means the latter (“is mathematically the
same”) despite his reference to “assumption”. Hence, his claim is that a
proved identity ‘is valid in theory A, not valid in theory B’ or, more
exactly, ‘is true in theory A, false in theory B’. This is absurd. Assume
alternatively that the present claim is an assumption that we just have made,
aware of the fact that we do not know whether it is true or false (= a mere
assumption). Then, Christian says that the present claim ‘is a mere assumption
that is true in theory A, false in theory B’. This is also absurd.

Hence, Christian cannot dispel the original concern that his presentation
implies that a certain claim is both true and false.

Author response: As I explained in the previous round of reviews, the argument
I have presented in Section 2.4 of the manuscript is not the one I have
invented myself. It cannot be brushed aside as “absurd”, 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. It was also recognized by Grete Hermann around the same time. 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. As I noted with a list of references
in the previous round of reviews, the same problem with von Neumann’s theorem
is also discussed by several other well known experts in foundations of
quantum mechanics. Apart from the historical precedence of the argument I have
presented in Section 2.4, there is a near universal agreement among the
experts in foundations of quantum mechanics on the invalidity of the
assumption of additivity of expectation values for hidden variable theories
[18]. One has to adapt double standards – one standard for von Neumann’s
theorem and another for Bell’s theorem – to deny the invalidity of the
assumption of additivity of expectation values for hidden variable theories.

Author action: I have revised Section 2.4 to explain the problem and the above
points more clearly with the help of the example of Stern-Gerlach magnet Bell
gives in Chapter 1 of his book [17].

With respect to sec. 2.5, I have criticized that it is unclear from what
theory a violation of the Bell-CHSH inequality is derivable. Originally,
Christian wrote:

> “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 S3 , instead of a flat Euclidean space, or IR3
> […]. But once the physical space is modelled as S 3 instead of IR3 […], the
> correlation between the results Aa and Bb […] inevitably turns out to be
> E(AaBb) = -a·b […]”.

To repeat my earlier criticism: the first sentence seems to claim that two
things are “[jointly?] necessary to support Bell’s theorem”, the failure of
eq. 1 for “operators [that] are non-commuting” and a mutable spacetime. But
the last sentence (“[…] once the physical space is modelled as S 3 …], the
correlation […] inevitably turns out to be […the quantum-mechanical one]”)
suggests that introducing S 3 alone is sufficient to derive the same
correlation.

Christian does not address this problem. He responds to the criticism as
follows:

> “[…] 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].”

Christian leaves it unclear whether A and B are (jointly?) necessary for C or
whether B alone is sufficient for C. Asking which argument exactly Christian
wants to present is not the same as mixing up arguments that should be kept
separate. Moreover, in the present context it is of great interest to know
from what assumptions precisely a violation of the Bell-CHSH inequality
follows.

Author response: As I noted in my response to the above comments in the
previous round of reviews, the current manuscript is a response to a comment
paper [2] on my 2018 paper [1]. 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], [5], [6], [7], [8], [9], [10], [15], [18], and [22] cited
in the manuscript. My discussion in Sections 2.4 and 2.5 of the manuscript are
a direct response to two of the issues raised in the comment paper [2]. Among
these issues, 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 modelof 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 in [1].

Had the reviewer read my original paper [1] instead of opting not to read it
before reviewing the manuscript, he/she would have noticed Section 3.3 of the
paper entitled “Derivation of Tsirel’son’s bounds on the correlation strength”
in which I explicitly derive violations of Bell inequalities within the
7-sphere framework presented therein, thereby precisely reproducing the
quantum mechanical predictions. Unfortunately, the reviewer has instead opted
to argue abstractly in terms of ‘theory A’ and ‘theory B’ without substance
and without specifying what these theories are.

Author action: No action.

Neither response to the two criticisms is convincing. Moreover, they are
representative for Christian’s style of argumentation. In a response to Gill’s
criticisms, these criticisms should be properly addressed, not dismissed in
favor of more text concerning other topics and ideas.

I still do not recommend the publication of this response.

Author response: Both issues raised by the reviewer are somewhat tangential to
the main goal of the manuscript, which is to address the specific issues
raised in the critique [2]. The reviewer has focused only on Sections 2.4 and
2.5 of the current manuscript and acknowledged (in the previous round) that
he/she has not read or intend to read either the original paper [1] or its
critique [2]. It is therefore not surprising that the reviewer’s comments lack
any substantive context of the details of the constructive model presented in
the original paper [1] or its critique in [2].

The reviewer has instead commented on my style of argumentation. However, in
my view the substantive scientific issues I have elucidated in Sections 2.4
and 2.5 are independent of my style of argumentation or other idiosyncrasies.
I therefore contend that the reviewer’s recommendation for the rejection of my
manuscript is not justified.

Author action: No action.

Reviewer # 6

My comments relate to the author’s responses to my own first round comments as
a referee (number 6 in the list).

There has been a fractional step forward here, in that the author now
recognises that what he is saying does not work for the standard definition of
norm.

Author response: The above statement is not accurate. It gives an incorrect
impression of what I actually said. What I explained, with demonstration, in
the previous round of reviews is that the standard definition of norm in
Geometric Algebra that picks out the scalar part of the geometric product in
an ad hoc manner (as in Eq. (2.8) of [1]) is not applicable to an equation
such as $\|XY\|=\|X\|\,\|Y\|$ that involves geometric product at least on one
of its sides, because it leads to mixing of two different product rules of the
two sides of the equation. That amounts to introducing inconsistency. That is
what the reviewer’s alleged counterexample did. It inadvertently introduced
inconsistency in the equation to derive the alleged contradiction. One can
always draw a contradiction from any mathematical equation by employing two
different rules on the two sides of the equation (in our case product rules).

The reviewer’s statement gives the incorrect impression that I have recognized
something new “now”, thanks to the reviewer’s comments in the previous round,
or that I have changed my mind about something in the light of the critique
[2]. That is not at all the case. I have not invented new definitions just for
my response to the critique [2]. My arguments were already very clearly stated
in the original paper in 2018. Nothing new has transpired since.

Author action: No action.

In particular, he says the norm he quotes in equation (2.8) in the original
RSOS paper (2018) (which is the standard one), is not the one he is using by
the time he gets to Section $2.5$ of that paper.

Author response: That is correct, but it is by no means a new statement. It
should not come as a surprise to anyone who has read the original RSOS paper
[1], even if they are unaware of my repeatedly pointing out to this fact in
the discussion thread of the paper and in my replies [3] and [15] to the
critique [14]. Let me reproduce the relevant paragraph form Section 2.5 of [1]
with yellow highlights for convenience in case the reviewer has missed it:

> Now the normalization of ${{\mathbb{Q}}_{z}}$ in fact necessitates that
> every ${{\bf q}_{r}}$ be orthogonal to its dual ${{\bf q}_{d}}$:
>
>
> $\hbox{\pagecolor{yellow}$\displaystyle||{\mathbb{Q}}_{z}||=\sqrt{2}\,\varrho\;\;\Longrightarrow\;\;{\bf
> q}_{r}\,{\bf q}^{\dagger}_{d}+{\bf q}_{d}\,{\bf q}^{\dagger}_{r}=0$}\,,$
> (2.54)
>
> or equivalently, ${\langle\,{\bf q}_{r}\,{\bf
> q}^{\dagger}_{d}\,\rangle_{s}=0}$ (i.e., ${{\bf q}_{r}\,{\bf
> q}^{\dagger}_{d}}$ is a pure quaternion). We can see this by working out the
> product of ${{\mathbb{Q}}_{z}}$ with ${{\mathbb{Q}}^{\dagger}_{z}}$ while
> using ${\varepsilon^{2}=+1}$, which gives
>
> ${\mathbb{Q}}_{z}\,{\mathbb{Q}}^{\dagger}_{z}\,=\;\left({\bf q}_{r}\,{\bf
> q}^{\dagger}_{r}\,+\,{\bf q}_{d}\,{\bf
> q}^{\dagger}_{d}\right)\,+\,\left({\bf q}_{r}\,{\bf
> q}^{\dagger}_{d}\,+\,{\bf q}_{d}\,{\bf
> q}^{\dagger}_{r}\right)\,\varepsilon\,.$ (2.55)
>
> Now, using the definition of ${\bf q}$ in (2.16), it is not difficult to see
> that ${{\bf q}_{r}\,{\bf q}^{\dagger}_{r}={\bf q}_{d}\,{\bf
> q}^{\dagger}_{d}=\varrho^{2}}$, reducing the above product to
>
>
> ${\mathbb{Q}}_{z}\,{\mathbb{Q}}^{\dagger}_{z}\,=\;2\,\varrho^{2}\,+\,\left({\bf
> q}_{r}\,{\bf q}^{\dagger}_{d}\,+\,{\bf q}_{d}\,{\bf
> q}^{\dagger}_{r}\right)\,\varepsilon\,.$ (2.56)
>
> It is thus clear that for ${{\mathbb{Q}}_{z}\,{\mathbb{Q}}^{\dagger}_{z}}$
> to be a scalar ${{\bf q}_{r}\,{\bf q}^{\dagger}_{d}+{\bf q}_{d}\,{\bf
> q}^{\dagger}_{r}}$ must vanish, or equivalently ${{\bf q}_{r}}$ must be
> orthogonal to ${{\bf q}_{d}}$.
>
> But there is more to the normalization condition ${{\bf q}_{r}\,{\bf
> q}^{\dagger}_{d}+{\bf q}_{d}\,{\bf q}^{\dagger}_{r}=0}$ then meets the eye.
> It also leads to the crucial norm relation (2.40), which is at the very
> heart of the only possible four normed division algebras associated with the
> four parallelizable spheres ${S^{0}}$, ${S^{1}}$, ${S^{3}}$, and ${S^{7}}$
> (cf. appendix A).

In the rest of the Section 2.5 I then sketch the proof of the norm relation
(2.40), leaving details for exercise.

Author action: No action.

This seems contradicted by explicit statements in Section $2.4$ of the paper,
which is directly connected to Section 2.5, and contains very explicitly the
key result in terms of norms, which the author spends a lot of time seeking to
prove, namely equation (2.40). In Section (2.4), it introduces the expansion
of general quantities in the even subalgebra in terms of coefficients
multiplied by basis elements (equations (2.36) and (2.37)). Then it says that
using (2.8) then after normalisation the square of their norms is given by the
expressions in (2.38), which read

$\|X\|^{2}=\sum_{\mu=0}^{7}X_{\mu}^{2}=1$ (1.1)

with a similar expression for $Y$, and then this result is repeated in (2.41)
for their product $Z$.

Author response: There is no contradiction in what I have written in Section
2.4 and Section 2.5 of the paper. The norm relation (2.40) holds for all
elements of the algebra ${\cal K}^{\lambda}$, without exception. I have proved
(2.40) in [1] and in Refs. [3], and [25] cited in the manuscript. And now
Reviewer # 3 has also independently verified my proof of (2.40). As evident
from Eq. (2.54), the scalar values for the norms obtained by using (2.8) and
(2.54) are identical:

$\sqrt{\,\langle\,{\mathbb{Q}}_{z}\,{\mathbb{Q}}_{z}^{\dagger}\rangle_{s}}\;\equiv\sqrt{\left({\mathbb{Q}}_{z}{\mathbb{Q}}^{\dagger}_{z}\right)\\!\Big{|}_{\,{\bf
q}_{r}\,{\bf q}^{\dagger}_{d}+{\bf q}_{d}\,{\bf q}^{\dagger}_{r}\,=\,0}\,}\;.$

Therefore all the statements made in Sections 2.4 and 2.5 of the paper are
correct and compatible with Eq. (2.54).

Author action: No action.

There are only two ways these expressions can apply to general non-zero
elements. The first is that the definition (2.8) is being used (as in fact
stated by the author at this point in the paper) - this involves taking the
scalar part of e.g. $X\tilde{X}$ before taking the square root. The second is
to work with quantities $X$ of a type which when we form $X\tilde{X}$ then
this product does not contain non-scalar parts. In the $4\mathrm{~{}d}$
Euclidean Clifford algebra, the extra part which could be present in this
product is a grade- 4 pseudoscalar, so a quantity of this form would have to
satisfy

$\langle X\tilde{X}\rangle_{4}=0$ (1.2)

in order for (2.38) to follow, and similarly for $Y$.

Author response: The reviewer’s equation (1.2) is the normalization condition
I have used in my original paper [1].This is evident from Eqs. (2.54) to
(2.56) in the yellow highlighted paragraph I have quoted above from the paper.
It is not difficult to appreciate that both definitions (2.8) and (2.54) in
[1] give the same scalar value for the norm.

Author action: No action.

Such an object is called a scaled rotor in Geometric Algebra, and the
assertion made in the key equation (2.40) that

$\|XY\|=\|X\|\,\|Y\|$ (1.3)

does indeed follow for this type of object, and corresponds to the fact that
the composition of rotors is another rotor.

However, the author claims that (2.40) is true for any even multivectors $X$
and $Y$,

Author response: Yes, the norm relation (2.40) holds for any multivectors $X$
and $Y$ within the algebra ${\cal K}^{\lambda}$, without exception. I have
proved the norm relation (2.40) in Refs. [1], [3], and [25] cited in the
manuscript, and now Reviewer # 3 (whose calculations I have reproduced above
verbatim) has independently verified my proof.

Author action: In Section 2.8 of the revised manuscript I have summarized the
proof again for convenience.

and I think we both now agree that for the norm which is apparently, and also
stated to be used at this point, then this statement is false (as demonstrated
by the counterexamples) - it only works (in a general sense) for the scaled
rotors.

Author response: No, I do not agree with the above claims. The norm relation
(2.40) holds for any multivectors $X$ and $Y$ within the algebra ${\cal
K}^{\lambda}$, without exception. No counterexample to (2.40) can or does
exist. In the last round of reviews, in [3], and in the manuscript, I have
comprehensively refuted the alleged counterexample to (2.40).

Author action: No action.

At points in the current reply by the author, there is a recognition that the
product rule for norms only works for the special case where $\langle
X\tilde{X}\rangle_{4}=0$, and the equivalent for $Y$, and is not general, but
this is coupled with an insistence that this restriction does not limit the
type of object considered. It does, to scaled rotors.

Author response: I do not agree with the above claim. The confusion in it
stems from a failure to appreciate that $\langle X\tilde{X}\rangle_{4}=0$ is a
normalization condition (cf. Eq. (2.54) of [1] quoted above). Normalization
conditions do not impose any restrictions on the algebra itself. This is not
difficult to understand. But since the confusion has persisted over four
years, let me give an elementary example from geometry. Consider a vector
space $\mathrm{I\\!R}^{3}$ spanned by vectors of the form ${\bf v}=x\,{\bf
e}_{x}+y\,{\bf e}_{y}+z\,{\bf e}_{z}$ from the origin. Then a 2-sphere of a
fixed radius $\rho$ embedded in $\mathrm{I\\!R}^{3}$ can be constructed as a
surface $S^{2}$ by normalizing the vectors in a familiar manner:

$\rho=\sqrt{{\bf v}\cdot{\bf v}}=\sqrt{x^{2}+y^{2}+z^{2}}.$

Now this normalization condition is a constraint equation that restricts the
length of the vectors to $\rho$. But it by no means puts any restriction on
the space $\mathrm{I\\!R}^{3}$. Similarly, the normalization condition
$\langle X\tilde{X}\rangle_{4}=0$ I have used to construct $S^{7}$ in my paper
may seem unorthodox, but it by no means puts any restriction on the algebra
${\cal K}^{\lambda}$ itself.

Author action: No action.

The remedy currently proposed by the author, in order to rescue the statement
that (2.40) applies to all objects in the algebra, is to enlarge the concept
of ‘norm’ to be the square root of $X\tilde{X}$ even when this product
contains non-scalar parts.

Author response: The fact that the norm relation (2.40) holds for all elements
of the algebra ${\cal K}^{\lambda}$ does not require either rescue or remedy.
It is a proven fact. No counterexample to (2.40) can or does exist. In the
previous round of reviews the present reviewer attempted to allege a
counterexample to (2.40). However, I demonstrated that the counterexample
fails, as it must, because (2.40) has been rigorously proven by me and now
also by Reviewer # 3. On the other hand, the contradiction in the alleged
counterexample was achieved by the present reviewer and the critique [2] by
employing two different product rules on the two sides of the equation
$\|XY\|=\|X\|\,\|Y\|$, and also by mixing up two different product rules on
its left-hand side. Contradiction can be drawn from any mathematical equation
by employing different rules on its two sides. Thus the alleged counterexample
was both logically and mathematically flawed. Since no counterexample to the
norm relation (2.40) can or does exist, and since explicit proof of it exists,
no remedy or rescue is needed for its validity. It holds for all elements $X$
& $Y$ of the algebra ${\cal K}^{\lambda}$.

In the alleged counterexample that failed, the reviewer considered two-
dimensional objects $X=\varepsilon-1$ and $Y=\varepsilon+1$ within ${\cal
K}^{\lambda}$ by setting the coordinates of the remaining six dimensions of
${\cal K}^{\lambda}$ to zero. That amounted to the real and dual quaternions
not being orthogonal to each other — i.e., that amounted to setting ${\bf
q}_{r}\,{\bf q}^{\dagger}_{d}+{\bf q}_{d}\,{\bf q}^{\dagger}_{r}\not=0$, and
thereby violating the normalization condition (2.54) I have used in my papers
to reduce the values of the norms to scalar quantities. This violation of
orthogonality or normalization condition in the reviewer’s counterexample is
also noticed by Reviewer # 3 whose calculations I reproduced above. It is then
not surprising that the two-dimensional objects considered by the reviewer
cannot be normalized to scalar quantities. The reviewer’s strategy therefore
was to simply pick the scalar parts of the geometric products $XX^{\dagger}$
and $YY^{\dagger}$ and substitute their square roots as norms into
$\|XY\|=\|X\|\,\|Y\|$ to derive contradiction. But that strategy inevitably
depends on applying two different product rules on the two sides of the
equation $\|XY\|=\|X\|\,\|Y\|$ because $XY$ on its left-hand side is
inevitably a geometric product and not a scalar product. Thus contradiction
can be achieved only by introducing inconsistency.

It is also important to note that word “currently” in the above sentence by
the reviewer again gives the wrong impression that I have invented a new
strategy “now” amounting to “remedy” or “rescue” of (2.40), in the light of
the critiques [2] and [14], or during this review process of the current
manuscript. But it should be evident from the yellow highlighted paragraph I
have quoted above from my original 2018 paper that that impression is not
correct.

Author action: No action.

This does not have a justification with what is done for complex numbers,
quaternions and octonions, despite what the author says. For the first two,
the quantity $X\tilde{X}$, is automatically a scalar with no restriction being
necessary. For octonions, the conjugate octonion is constructed so that its
octonionic product with the original octonion, returns a positive scalar. It
is for this reason that restrictions are not necessary in these cases.

Author response: The above comments by the reviewer are also mistaken. To
begin with, contrary to what the reviewer claims, any normalization process is
inevitably a restriction on the coefficients of the multivector such as $X$,
regardless of whether that multivector is a complex number, quaternion,
octonion, or an element of ${\cal K}^{\lambda}$. In one of my comments above I
gave a simple example of ordinary 3-vectors normalized to construct $S^{2}$
embedded in $\mathrm{I\\!R}^{3}$.

Any multivector $X$ in general, regardless of its algebra, would not be a
scalar quantity. The usual strategy to infer its “length” is to consider an
entirely different multivector $X^{\dagger}$, which is the “conjugate” or
“reverse” of $X$. It is then noted that their geometric product $XX^{\dagger}$
is a scalar quantity, say $\varrho^{2}$, at least in the case of complex
numbers, quaternions, and octonions, so that we can say the “length” of $X$ is

$\|X\|=\sqrt{XX^{\dagger}}=\varrho.$

But this is a constraint equation. Contrary to what is claimed by the
reviewer, it puts a restriction on the coefficients of $X$, just as the
restriction on the coefficients of the 3-vectors in the case of $S^{2}$
embedded in $\mathrm{I\\!R}^{3}$ I considered above.

Thus what I have used in my original paper to infer the length of $X$ is no
different in principle from this strategy. For arbitrary multivector
${\mathbb{Q}}_{z}=\,{\bf q}_{r}+{\bf q}_{d}\,\varepsilon$ in ${\cal
K}^{\lambda}$, in [1] I have used the following definition for a scalar-valued
norm:

$||{\mathbb{Q}}_{z}||:=\sqrt{\left({\mathbb{Q}}_{z}{\mathbb{Q}}^{\dagger}_{z}\right)\\!\Big{|}_{\,{\bf
q}_{r}\,{\bf q}^{\dagger}_{d}+{\bf q}_{d}\,{\bf
q}^{\dagger}_{r}\,=\,0}\,}\;=\varrho,$

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 [1]). This is
explained also with more detail in [25]. As explained in Eqs. (2.54) to (2.56)
in [1], the quadratic form in the above definition with split-complex image is

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

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 this definition
gives a scalar value for the norm as the Pythagorean distance from the origin
in eight dimensions,

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

with the notation ${\bf q}_{r}=g+I_{3}{\bf u}$ and ${\bf q}_{d}=h+I_{3}{\bf
v}$, and therefore the norms so defined are also positive definite:

$||{\mathbb{Q}}_{z}||=0\iff{\mathbb{Q}}_{z}=0.$

Finally, note that while both definitions of normalization discussed above —
the traditional one and the one I have used in my paper — are constraint
equations and therefore put restrictions on the coefficients of the
multivectors $X$ to deduce their lengths, they do not put any restrictions on
the corresponding algebras itself. Thus the claim by the reviewer that the
definition of norm I have used in my paper puts a restriction on the algebra
${\cal K}^{\lambda}$ is not correct.

Author action: In Section 2.8 of the manuscript I have added a proof of the
positive definiteness of the norm.

Returning to the $4\mathrm{~{}d}$ Euclidean Clifford algebra, if norms really
were to be taken as being the square roots of scalar+pseudoscalar quantities,
a further point is that we would need to worry about ‘branch cuts’ in the
definition of the square roots, in an analogous way to what we need to worry
about in taking the square root of a complex number. This doesn’t seem to be
worried about by the author, maybe because of his belief that the restriction
of the type contained in his equation (2.54) (in the RSOS paper), which is the
same as my restriction $\langle X\tilde{X}\rangle_{4}=0$ above, somehow
doesn’t in fact restrict the quantities being considered, and we can apply it
in general, and thereby rule out of consideration the quantities
$1\pm\epsilon$ used in the counterexample.

Author response: That is correct. As I noted above, normalization conditions
do not put restrictions on algebra.

Author action: No action.

So my summary so far, is that the author is being disingenuous in saying that
he was using the definition $\|X\|=\sqrt{X\tilde{X}}$ in Section $2.5$ of the
original paper. This is because Section $2.4$ (in particular equation (2.38))
clearly shows he was using (2.8) at that time, and also because of the fact
that he felt it necessary to restrict objects by imposing the condition (2.54)
(whilst simultaneously claiming this was not a restriction).

Author response: All mathematical statements made is Sections 2.4 and 2.5 of
the original paper [1] are correct, unambiguous, and mutually consistent. In
particular, Eq. (2.38) explicitly considers unit multivectors. In other words,
multivectors with unit scalar-valued lengths. It is quite clear from the
discussion in the highlighted paragraph of Section 2.5 reproduced above that
the value of the length deduced using the normalization condition (2.54) in
Section 2.5 would be identical to that deduced using the traditional
normalization condition (2.8) used in Section 2.4:

$\sqrt{\,\langle\,{\mathbb{Q}}_{z}\,{\mathbb{Q}}_{z}^{\dagger}\rangle_{s}}\;\equiv\sqrt{\left({\mathbb{Q}}_{z}{\mathbb{Q}}^{\dagger}_{z}\right)\\!\Big{|}_{\,{\bf
q}_{r}\,{\bf q}^{\dagger}_{d}+{\bf q}_{d}\,{\bf q}^{\dagger}_{r}\,=\,0}\,}\;.$

Once again, Eq. (2.54) is a normalization condition. It by no means puts any
restriction on the algebra ${\cal K}^{\lambda}$ itself.

Author action: No action.

Leaving this all aside, we can provisionally accept that the author now wants
to work with the definition $\|X\|=\sqrt{X\tilde{X}}$. The relation $\|XY\|=$
$\|X\|\|Y\|$ is true for this definition, since we can write

$\|XY\|^{2}=(XY)\widetilde{(XY)}=XY\tilde{Y}\tilde{X}=(X\tilde{X})(Y\tilde{Y})$
(1.4)

where we can move the $\tilde{X}$ through the $Y\tilde{Y}$ since the
$4\mathrm{~{}d}$ pseudoscalar commutes with all even elements.

Author response: I once again object to the adjective “now” in the first
sentence of the above comment, unless by “now” the reviewer means the year
2018. As I noted before, I am not inventing new strategies of defence “now.”

Author action: No action.

However, this definition will not allow the author to say anything about
octonions (since this isn’t how octonion norms work).

Author response: I do not intend to say anything about octonions. My paper
concerns an associative algebra ${\cal K}^{\lambda}$,which is the even
subalgebra of the Clifford algebra $\mathrm{Cl}_{4,0}$, not the Cayley algebra
with octonionic product rules.

Author action: No action.

Moreover, we do not need this to be the definition in order to prove that the
product of scaled rotors is another scaled rotor, since this works using the
original (2.8) definition.

Author response: I disagree with the above comment. To begin with, the
traditional 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
$||XY||=||X||\,||Y||$. If we 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 [2] and [14]. Thus definition
(2.8) has limited applicability.

Author action: No action.

Finally, we need to be clear that this definition of norm does not agree with
the usual mathematical one, since in mathematics, a norm takes us from a space
of elements of some kind, to the non-negative real numbers, and the new ‘norm’
now favoured by the author, fails to do this.

Author response: I disagree with this claim. It is evident that the
normalization condition (2.54) I have used in the paper [1] also takes us from
a space ${\cal K}^{\lambda}$ of eight-dimensional multivectors to the non-
negative real numbers.”

Author action: No action.

A possible way forward, which would allow the author to continue using the
correct norm (2.8), is if the author is willing to admit that the entire
mathematical development through to the end of Section $2.5$ of the RSOS paper
amounts to the assertion that the product of scaled rotors is another scaled
rotor. We would then have nothing more to argue about on this point.

Author response: As I explained above, definition (2.8) is not universally
applicable. By now it should be evident that there are very good reasons why I
have instead introduced the correct normalization condition (2.54) in [1].

Author action: No action.

This would remove anything of interest connected with octonions of course,
since the preservation of the (true) norm would then only apply to a selected
subset of elements, and thus remove the claim of having found an associative
version of the octonion algebra, which would have to work for all elements,
but it may be sufficient mathematical equipment to be able to carry on with
what the author needs for his purposes in the second part of the paper. This
has yet to be determined.

Author response: Nowhere in [1] have I made a claim that I have found an
associative version of octonion algebra. My paper does not concern octonions
or Cayley algebra. It concerns the eight-dimensional associative algebra
${\cal K}^{\lambda}$ (which is the even subalgebra of the Clifford algebra
$\mathrm{Cl}_{4,0}$) and the corresponding 7-sphere embedded within ${\cal
K}^{\lambda}$.

Author action: No action.

(Note there are several more parts of the author’s reply, but these basically
amount to statements, without evidence, that he is right, so it is not really
necessary to reply to these.

Author response: This claim by the reviewer is not correct. It can be easily
verified that I have not made any statements in my replies in the previous
round without providing evidence and reasons for my disagreements.

Author action: No action.

An example would be what I pointed out last time concerning the span of two
sets of elements in fact being the same even though they are given different
symbols. The sets of elements in question are the same except for signs, and
the span of a set of elements consists of all possible linear sums of them,
with arbitrary coefficients. This means that two sets that differ only in the
signs of the elements must have the same span, but the author continues to
deny this. I appreciate fully that he is trying to talk about orientations,
but what he has written are spans, and hence are certainly the same.)

Author response: As I pointed out in the previous round of reviews, what is
raised in the above comments is a non-issue. As I have explained in the
manuscript, there is no error in [1] in this regard. Nowhere have I claimed
that ${\cal K}^{+}$ and ${\cal K}^{-}$ span different vector spaces. They
differ only in orientation $\lambda$, and this orientation plays a role of a
Bell-type hidden variable representing the initial state of the physical
system in the 7-sphere model. I have already explained this adequately in
Section 2.3 of my original paper [1] and in Section 2.6 of the current
manuscript. In the last round, Reviewer # 1 also made the following
observation 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. That is why ${\cal K}^{+}$ and ${\cal
K}^{-}$ are specified with two different symbols. This is not difficult to
understand.

Author action: For clarity, at the end of Section 2.6 in the revised
manuscript I have added the following sentence: “Therefore ${{\cal K}^{+}}$
and ${{\cal K}^{-}}$ are physically not identical within the 7-sphere
framework proposed in [1].”

Reviewer # 7

I believe that the usual and well accepted judgements issued with regard to a
mathematical inequality—whose origins can be traced to Boole in 19th century
as the author has reminded his readers—on mostly metaphysical questions
relating to the nature of physical reality, locality etc., can and should be
subjected to questioning and further investigation—particularly in view of the
understanding of the nature of space itself that is provided by the power
mathematical tools offered by geometric algebra and Einstein’s theory of
gravity.

It appears to me that a quest for a better understanding of the nature of
space is directly connected to addressing the well accepted conclusions, and
wisdom, from Bell’s theorem regarding physical reality and locality that are
presented as the most logical, direct, and doubtless outcomes of the theorem.

I see Christian’s work from the past many years as such a quest that has
raised hope for more satisfactory answers to difficult questions at the heart
of foundations of quantum mechanics.

I believe that while respecting the mathematical core of Bell’s theorem, often
presented as a system of inequalities, the usual conclusions, and wisdom,
regarding the nature of physical reality, locality, etc. can and needs to be
subjected to questioning and further investigation. If an improved description
of space, as offered by geometric algebra or the powerful mathematical tools
of general relativity, can help us in revising these usual conclusions then
why should there be any objection or resistance to that and why such an effort
should be viewed as a scientific heresy as seems to be the case.

Reviewer #1 recommends removing the discussion of a mathematically valid and a
physically valid equation.

Author response: I have dealt with that recommendation by Reviewer # 1 in my
response to it above.

Author action: No action.

Reviewer #5 considers the author’s discussion on this in Section 2.4 to be
“incoherent argumentation” that should not be published in a scientific
context. Referring to a number of earlier mentions of this problem in
literature on the foundations of quantum mechanics, the author has kept this
discussion in his manuscript, and has further clarified it in his revision,
and to which I agree.

Reviewer #2 has objected to the author’s use of geometry of space-time as
offered by general relativity (GR) relevant for explaining Bell’s experiments?
I do not see why such an investigation can or should be criticized in
particular if it can offer a hope in answering what many find as the troubling
questions, usual implications and accepted wisdom, regarding Bell’s theorem.

Reviewer #6 has made a list of technical remarks/objections to which the
author has given what I see as satisfactory responses.

I believe that he has presented convincing and careful replies, with attention
to details, to all comments and objections from referees and this manuscript
should be accepted in its present form.

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

Author action: No action.