$H_{\text{trans}}\subset\text{Diff}(\mathcal{M}_{\text{old}})$. I have been
unable to find any spacetime theory where this happens. In particular, when
internalizing a kinematically unreduced theory, one must be careful which
$H_{\text{trans}}$ and $H_{\text{fix}}$ are used as this could effect the end
result. We can avoid this headache and greatly simplify our analysis by making
a very slightly stronger demand of our PSTOs (i.e., that they have injection
compatibility). Recall from Sec. 2.5 that demanding that new theory is
kinematically reduced is a very weak assumption.

To close off this section, I will now prove that scope of spacetime theories
producible via externalization (i.e., the range of the ISE Methodology) is
exactly those theories with spacetime-kinematic compatibility. The above
discussion has proved one direction of this claim. To prove the other, we need
to show that any theory with spacetime-kinematic compatibility can be produced
via externalization. Given any such theory, let us take it as input into the
ISE Methodology. As Sec. 2 has shown, internalizing any such theory yields a
pair of Lie groups, $G_{\text{trans}}^{\text{(int)}}$ and
$G_{\text{fix}}^{\text{(int)}}$, acting on $S_{\text{neutral}}^{\text{kin}}$.
In Appendix A.3, I have proved that this pair of Lie groups are always PSTOs.
Moreover, I prove that by externalizing them one can recover the theory’s
original spacetime framing, up to diffeomorphism. Hence, any spacetime theory
with spacetime-kinematic compatibility can be output by the ISE Methodology
(namely, by first internalizing it and then immediately re-externalizing it).

##  4 ISE-Relatedness and the ISE-Equivalence Theorem

The previous two sections have discussed the internalization and
externalization process. In between these steps, one is supposed to search
around for good pre-spacetime translation operations to externalize. Two
example applications have been considered above in Secs. 3.3 and 3.4. An
additional two example applications of the ISE Methodology can be found in
Grimmer (2023). In principle, after internalizing any given spacetime theory
there will be a wide variety of PSTOs available for to us choose from leading
us to a wide variety of spacetime redescriptions of our theory. But how
powerful exactly is the ISE Methodology as a tool for topological
redescription?

To set the scope of ISE-relatedness, we first need to know what kind of
theories can output by the externalization process. As Sec. 3.5 has proved the
spacetime theories which can be produced by the ISE Methodology are exactly
those with spacetime-kinematic compatibility. Such theories constitute the
range of the ISE Methodology. We must next ask what kind of spacetime theories
can be input to the internalization process. As Sec. 2 has discussed, every
spacetime theory satisfying our two preliminary assumptions can be input into
the internalization process. Such theories constitute the domain of the ISE
Methodology.

It should be noted that, while this paper has been focused on theories about
scalar fields,
$\varphi_{\text{old}}:\mathcal{M}_{\text{old}}\to\mathcal{V}_{\text{old}}$,
these preliminary assumptions can be satisfied much more widely. For instance,
they might be satisfied by theories about particle trajectories,
$x_{\text{old}}:\mathcal{V}_{\text{old}}\to\mathcal{M}_{\text{old}}$, or
tensor fields, e.g.,
$\vec{\varphi}_{\text{old}}:\mathcal{M}_{\text{old}}\to\text{T}\mathcal{M}_{\text{old}}$.
Moreover, with only minor modifications the externalization process described
in Sec. 3.2 can be made to output such theories.242424See footnote 18 in Sec.
3.1. For the sake of simplicity, however, let us return our focus to theories
about scalar fields.

Having established both the domain and range of the ISE Methodology, we can
now define ISE-relatedness. Let us say that theory (X) is ISE-related to
theory (Y) when we can use the ISE Methodology to redescribe theory (X) as
theory (Y), at least up to diffeomorphism. For instance, Sec. 3.3 has shown
that the 3+1D QKG theory is ISE-related to the 1+1D QKG theory. Moreover, I
there claimed (but have not yet proved) that the reversed relation holds as
well. Similarly, Sec. 3.4 has shown that the periodic matter theory is ISE-
related to the periodic spacetime theory. As I have claimed there that the
reversed relation does not hold.

Indeed, it is not guaranteed that any given instance of ISE-relatedness is
reversible. As Sec. 3.5 has discussed every theory produced via
externalization is guaranteed to have spacetime-kinematic compatibility
whereas internalization only requires two preliminary assumptions to be
satisfied. Hence, certain spacetime theories (including the periodic matter)
can be input to the ISE Methodology but can never be output by it. While such
theories may be related to other spacetime theories (e.g., the periodic
spacetime theory) they cannot be ISE-self-related. Thus, the ISE-relatedness
is neither reflexive nor symmetric over its domain.

ISE-relatedness is, however, both reflexive and symmetric over its range. In
fact, as I will now prove, it is an equivalence relation over its range (i.e.,
when we restrict it to theories with spacetime-kinematic compatibility). To
begin, recall the discussion at the end of Sec. 3.5: Given any theory with
spacetime-kinematic compatibility, if we internalize and then immediately re-
externalize this theory then we end up back with the old theory’s spacetime
setting, up to diffeomorphism. Hence, the ISE-relatedness is reflexive over
its range.

Let us next see that ISE-relatedness is symmetric over its range. Suppose that
theories (A) and (B) have spacetime-kinematic compatibility and that we get
from theory (A) to theory (B) via the ISE Methodology. Internalizing theory
(A) yields a set of PSTOs, $G_{\text{trans}}^{\text{(A)}}$ and
$G_{\text{fix}}^{\text{(A)}}$, acting on
$\mathcal{S}_{\text{neutral}}^{\text{kin}}$. If we were to externalize these,
we would recover the theory’s original spacetime framing.252525See Appendix
A.3 for details. By assumption, however, we can also produce theory (B) by
externalizing some other set of PSTOs, $G_{\text{trans}}^{\text{(B)}}$ and
$G_{\text{fix}}^{\text{(B)}}$, acting on
$\mathcal{S}_{\text{neutral}}^{\text{kin}}$. Since theory (B) has been
produced via externalization it satisfies our preliminary assumptions and
hence can be re-internalized. After re-internalizing this theory, our original
set of PSTOs, $G_{\text{trans}}^{\text{(A)}}$ and
$G_{\text{fix}}^{\text{(A)}}$, are still available to us. By externalizing
them we can return to theory (A). Hence, we can use the ISE Methodology to get
from theory (B) to theory (A). Thus the ISE-relatedness is symmetric over its
range.

Finally, let us see that ISE-relatedness is transitive over its range.
Consider three theories (A), (B), and (C) each with spacetime-kinematic
compatibility and with (A) ISE-related to (B) and (B) ISE-related to (C).
Suppose that we have just moved from theory (A) to a spacetime-neutral
description and then into theory (B). Since theory (B) has been produced via
externalization it satisfies our preliminary assumptions (indeed, it has
spacetime-kinematic compatibility). Hence we can re-internalize it and then go
in a different direction, say to theory (C). But re-internalizing theory (B)
just takes us back to the spacetime-neutral description between (A) and (B).
Hence, we can also get from (A) to (C) using the ISE Methodology. This
concludes the proof that ISE-relatedness is an equivalence relation over its
range.

Having proved that ISE-relatedness is an equivalence relation over its range,
it is natural to try to characterize its equivalence classes. The following
theorem characterizes not only the ISE-equivalence classes, but ISE-
relatedness in general:

> ISE-Equivalence Theorem -  One spacetime theory (A) is ISE-related to
> another spacetime theory (B) if and only if the following three conditions
> hold.
>
>   1. 1)
>
> Firstly, theory (A) must satisfy the two preliminary assumptions, and theory
> (B) must have spacetime-kinematic compatibility.
>
>   2. 2)
>
> Secondly, there must exist an isomorphism
> $\mathcal{I}:\mathcal{S}_{\text{old}}^{\text{kin}}(A)\leftrightarrow\mathcal{S}_{\text{old}}^{\text{kin}}(B)$
> between their kinematically allowed states which preserves all of their
> dynamically and metaphysically relevant structures.
>
>   3. 3)
>
> Thirdly, this isomorphism,
> $\mathcal{I}:\mathcal{S}_{\text{old}}^{\text{dyn}}(A)\leftrightarrow\mathcal{S}_{\text{old}}^{\text{dyn}}(B)$,
> must also preserve the theory’s dynamics (mapping solution to solutions).
>
>

>
> Note that if theory (A) and theory (B) are to be ISE-interrelated, then the
> first condition must be strengthened to demand that both theories have
> spacetime-kinematic compatibility. Recall that, when restricted to such
> theories, ISE-relatedness is an equivalence relation. Hence, strengthening
> the first condition in this way gives an exact characterization of the ISE-
> equivalence classes.

The proof of this claim is relatively simple. Let us first demonstrate that
any pair of ISE-related theories will have the above-listed properties.

1) In order for theory (A) to be an input to the ISE Methodology, it must
satisfy the two preliminary assumptions. Likewise, for theory (B) to be
producible by externalization it must both have spacetime-kinematic
compatibility.

2) By construction, both internalization and externalization preserve all of
the theory’s dynamically and metaphysically relevant structures,
$\mathcal{S}_{\text{old}}^{\text{kin}}\cong\mathcal{S}_{\text{neutral}}^{\text{kin}}\cong\mathcal{S}_{\text{new}}^{\text{kin}}$.
Concretely, these structures are preserved by $\mathcal{I}=J\circ\mathcal{F}$
where
$\mathcal{F}:\mathcal{S}_{\text{old}}^{\text{kin}}\leftrightarrow\mathcal{S}_{\text{neutral}}^{\text{kin}}$
mediates internalization and
$J:\mathcal{S}_{\text{neutral}}^{\text{kin}}\leftrightarrow\mathcal{S}_{\text{new}}^{\text{kin}}$
mediates externalization. N.b., the kinematic structures appearing in the
spacetime-neutral theory and the new theory are exactly those induced by the
$\mathcal{F}$ and $J$ bijections.

3) Finally, the isomorphism $\mathcal{I}=J\circ\mathcal{F}$ preserves the
theory’s dynamics since both $\mathcal{F}$ and $J$ do. N.b., the dynamics of
the spacetime-neutral theory and the new theory are defined the $\mathcal{F}$
and $J$ bijections, i.e.,
$\mathcal{S}_{\text{neutral}}^{\text{dyn}}\coloneqq\mathcal{F}(\mathcal{S}_{\text{old}}^{\text{dyn}})$
and $\mathcal{S}_{\text{new}}^{\text{dyn}}\coloneqq
J(\mathcal{S}_{\text{neutral}}^{\text{dyn}})$.262626More elegantly, the
dynamics of the spacetime-neutral theory and the new theory would be picked
out by the various kinematic structures induced in these theories. The end
result is the same, however.

To prove the converse result, let us consider two theories (A) and (B) which
have the above-listed properties. Our goal is then to show that one can get
from theory (A) to theory (B) via the ISE Methodology. Recall that
internalization proceeds via an forgetful operation, $\mathcal{F}$, which
preserves all of these theory’s dynamically and metaphysically relevant
structures. Hence, internalizing both theories we have two isomorphisms
$\mathcal{F}_{\text{(A)}}:\mathcal{S}_{\text{old}}^{\text{kin}}(A)\leftrightarrow\mathcal{S}_{\text{neutral}}^{\text{kin}}(A)$
and
$\mathcal{F}_{\text{(B)}}:\mathcal{S}_{\text{old}}^{\text{kin}}(B)\leftrightarrow\mathcal{S}_{\text{neutral}}^{\text{kin}}(B)$.
Together with our assumed isomorphism,
$\mathcal{I}:\mathcal{S}_{\text{old}}^{\text{kin}}(A)\leftrightarrow\mathcal{S}_{\text{old}}^{\text{kin}}(B)$,
we can induce an isomorphism between
$\mathcal{S}_{\text{neutral}}^{\text{kin}}(B)$ and
$\mathcal{S}_{\text{neutral}}^{\text{kin}}(A)$, namely
$\mathcal{I}_{\text{N}}:\mathcal{S}_{\text{neutral}}^{\text{kin}}(B)\leftrightarrow\mathcal{S}_{\text{neutral}}^{\text{kin}}(A)$
defined by
$\mathcal{I}_{\text{N}}\coloneqq\mathcal{F}_{\text{(A)}}\circ\mathcal{I}^{-1}\circ\mathcal{F}_{\text{(B)}}^{-1}$.

Let us next internalize theory (B). Since this theory has spacetime-kinematic
compatibility internalizing it yields pre-spacetime translation operations,
$G_{\text{trans}}^{\text{(B)}}$ and $G_{\text{fix}}^{\text{(B)}}$, acting on
$\mathcal{S}_{\text{neutral}}^{\text{kin}}(B)$. Using the isomorphism
discussed above,
$\mathcal{I}_{\text{N}}:\mathcal{S}_{\text{neutral}}^{\text{kin}}(B)\leftrightarrow\mathcal{S}_{\text{neutral}}^{\text{kin}}(A)$,
we can convert these PSTOs acting on
$\mathcal{S}_{\text{neutral}}^{\text{kin}}(B)$ into PSTOs acting on
$\mathcal{S}_{\text{neutral}}^{\text{kin}}(A)$. Namely, we can faithfully
convert $G_{\text{trans}}^{\text{(B)}}$ and $G_{\text{fix}}^{\text{(B)}}$
acting on $\mathcal{S}_{\text{neutral}}^{\text{kin}}(B)$ into another pair of
Lie groups, $G_{\text{trans}}^{\text{(A)}}$ and $G_{\text{fix}}^{\text{(A)}}$,
acting on $\mathcal{S}_{\text{neutral}}^{\text{kin}}(A)$. Note that these new
Lie groups acting on $\mathcal{S}_{\text{neutral}}^{\text{kin}}(A)$ will be
structurally identical to the old ones acting on
$\mathcal{S}_{\text{neutral}}^{\text{kin}}(B)$. Hence, they will qualify as
PSTOs. We are therefore free to pick them as our PSTOs when externalizing
theory (A). Doing so, we find a spacetime redescription of theory (A) which is
identical to our original description of theory (B), at least up
diffeomorphism. Hence, theory (A) is ISE-related to theory (B). This concludes
the proof of the ISE-equivalence theorem.

To conclude this section, let us see how this theorem applies to the two
instances of ISE-relatedness discussed in Sec. 3 and the two instances
discussed in in Grimmer (2023). Unsurprisingly, every (ordered) pair of ISE-
related theories which we have considered satisfy the theorem’s three
conditions. But are the reverse redescriptions also possible? According to the
ISE-Equivalence Theorem, the reverse redescription are possible if and only if
both of the theories in question have spacetime-kinematic compatibility. As a
simple positive example of this, consider the first case of topological
redescription discussed in Grimmer (2023), namely Fourier redescription. Since
both theories have spacetime-kinematic compatibility, this instance of Fourier
redescription is reversible.

Let us next consider the periodic matter and periodic spacetime theories
introduced in Sec. 2. As Sec. 3.4 has discussed the periodic matter theory is
ISE-related to the periodic spacetime theory. The reverse relation, however,
is not possible since the periodic matter theory lacks spacetime-kinematic
compatibility. But what is it exactly which prevents us from using the ISE
Methodology to get from the periodic spacetime theory back to the periodic
matter theory? Roughly, the ISE Methodology is supposed to build a new
spacetime setting from smooth transformations on the theory’s kinematically
allowed states. This is not possible to reconstruct the periodic matter theory
in this way because its spacetime is in some way excessive given the theory’s
kinematics. Ultimately, the ISE Methodology deems these theories inequivalent
and, moreover, gives us reason to favor the periodic spacetime theory over the
periodic matter theory.

Consider next the second example of topological redescription from Grimmer
(2023). To review, the forward redescription led us from a scalar field
$\varphi_{\text{old}}$ on a discrete spacetime
$\mathcal{M}_{\text{old}}\cong\mathbb{R}\times\mathbb{Z}$, to a bandlimited
scalar field $\varphi_{\text{new}}$ on a continuous spacetime
$\mathcal{M}_{\text{new}}\cong\mathbb{R}^{2}$. As Grimmer (2023) has
discussed, one can think of this new spacetime as arising from the following
sequence of events. 1) We note that the discrete theory has a hidden
continuous dynamical symmetry. 2) We realize at, despite being internal
transformations, this hidden dynamical symmetry is nonetheless very much like
spacetime translation operations (indeed, they are PSTOs). 3) We use the ISE
Methodology to build a new spacetime setting for the theory in which these
hidden dynamical symmetries are re-expressed as spacetime symmetries.
According to the ISE-Equivalence Theorem, the reverse redescription is also
possible (from continuous to discrete). Those familiar with the Nyquist-
Shannon sampling theory should recognize what has happened here. We have
exactly reconstructed a bandlimited signal using a sufficiently dense discrete
sampling.272727The Nyquist-Shannon sampling theory found its first application
in fundamental spacetime physics with Achim Kempf’s Kempf (1997, 2000),
ultimately leading to his claim that “Spacetime could be simultaneously
continuous and discrete, in the same way that information can be” Kempf
(2010). See Kempf (2018) for an overview of Kempf’s work on these and related
topics. My exposure to Kempf’s work was a key inspiration for my development
of the ISE Methodology.

Finally, let us turn our attention to the first example application considered
in this paper (in Sec. 3.3). There the forward redescription was a kind of
dimensional reduction. We proceeded from a single scalar field
$\varphi_{\text{old}}$ on a large spacetime
$\mathcal{M}_{\text{old}}\cong\mathbb{R}^{2}\times S^{2}$, to an infinite
collection of massive scalar fields $\\{\varphi_{\text{new}}^{(\ell m)}\\}$ on
a smaller spacetime $\mathcal{M}_{\text{new}}\cong\mathbb{R}^{2}$. Following
Grimmer et al. (2021) we might call these subfields. According to the ISE-
Equivalence Theorem, the reverse redescription is also possible, we can
collect together all of these subfields into a single field living on a larger
spacetime. Those familiar with Kaluza-Klein should recognize this move.

To see this possibility more clearly, imagine a civilization which thinks they
live on $\mathcal{M}_{\text{old}}\cong\mathbb{R}^{2}$ along with an infinite
collection of increasingly massive 1+1 dimensional fields, only to discover
the possibility of some previously hidden compactified dimensions. Such a
scenario has been discussed in Grimmer et al. (2021):282828The following quote
mentions Weyl’s law, see Weyl (1911, 1912). This law relates the spectrum,
$0<\lambda_{1}\leq\dots\leq\lambda_{j}\leq\dots$, of a Laplacian operator,
$\nabla$, and the shape of its domain $\Gamma$. Namely, Weyl’s law says that
for large $j$ the eigenvalue $\lambda_{j}$ scales as,
$\displaystyle\lambda_{j}\sim
4\pi^{2}\left(\frac{j}{V_{d}\,|\Gamma|}\right)^{2/d},$ where $d$ is the
dimension of the domain $\Gamma$, $V_{d}$ is the volume of the unit-ball in
$\mathbb{R}^{d}$ and $|\Gamma|$ is the d-dimensional volume of $\Gamma$.

> Consider a cosmology with $d=D-1$ compactified spatial dimensions [with
> cross-section $\Gamma$] and one extended spatial direction. The inhabitants
> of such a cosmos may be completely unaware of these compactified dimensions.
> Such inhabitants would likely interpret their particle physics experiments
> as indicating the existence of a finite collection of $1+1$ dimensional
> fields with some distribution of masses [$M_{j}$ indexed by $j$]. Once these
> inhabitants learn of the compactified dimensions (maybe taking inspiration
> from string theory) they can reinterpret these $1+1$ dimensional fields as
> the subfields of a single $D+1$ dimensional field. By noting the scaling of
> $M_{j}$ for high $j$ they could (by Weyl’s law) determine the number of
> compactified dimensions and the $d$-dimensional volume of $\Gamma$. By
> guessing the geometry of $\Gamma$ they would be able to predict the masses
> of new yet-to-be-discovered subfields (that they may perhaps call strings?).

One can recast this story in terms of the ISE Methodology as follows: 1) This
society notes the existence of a new dynamical symmetry (e.g., a
$\text{SO}(3)$ rotation among the subfields). 2) They realize that, despite
being internal transformations, this dynamical symmetry is nonetheless very
much like spacetime translation operations (indeed, they are PSTOs). 3) They
then use the ISE Methodology to build a new larger spacetime setting for the
theory in which these dynamical symmetries are re-expressed as spacetime
symmetries.

As this and the previous example hopefully demonstrate, it can be fruitful to
search for novel unexpected PSTOs beyond those given to us via
internalization. These might lead us to new and fruitful topological
redescriptions of our theories. Indeed, one can understand the superspace
description of theories with supersymmetry arising in this way: 1) We note a
new dynamical symmetry (here supersymmetry). 2) We note it is in some way like
a spacetime-translations. 3) We build a new spacetime setting for the theory
in which these dynamical symmetries are re-expressed as spacetime symmetries.
In principle, the ISE Methodology could be used to facilitate such a
topological redescription (although this would require a substantial technical
upgrade to the ISE Methodology: replacing Lie groups with Lie supergroups
everywhere).

##  5 Conclusion

This paper has evaluated the power and scope of the topological redescription
techniques offered to us by the ISE Methodology. As the above-proved ISE-
Equivalence theorem demonstrates, the ISE Methodology gives us access to
effectively every possible spacetime framing of a given theory’s kinematical
and dynamical content. The only bound placed on our capacity for topological
redescription is that the new theory is required to have a minimal level of
spacetime-kinematic compatibility. As I have argued in Sec. 2.5 this is a very
weak condition. Hence, the ISE Methodology is a powerful tool for discovering
and negotiating between a wide range of spacetime settings for a given
spacetime theory.

Having come to grips with the power and scope of these topological
redescription techniques, lets us recall the philosophical task which Grimmer
(2023) has developed them for. As I discussed in Sec. 1, the technical goal of
this project is to develop our capacity for topological redescription such
that it is on par with our existing capacity for coordinate redescription and
re-axiomization. Aspirationally, one could then leverage these techniques in
support of a dynamics-first view of topology (in answer to Norton (2008)’s
problem of pre-geometry). In particular, these topological redescription
techniques might be used to provide new support for a roughly Kantian view of
space and time: Rather than corresponding to some fundamental substances or
relations in the world, we can see the spacetime manifolds which routinely
appear in our best scientific theories as merely being an aspect of how we
represent the world. Such a view could be structured analogously to a roughly
Humean view of the laws and a broadly dynamics-first view of geometry: The
spacetime manifolds which routinely appear in our best scientific theories
reflect nothing metaphysically substantial in the world beyond them it being
one particularly nice way (among others) of codifying the dynamical behavior
of matter.

Unfortunately, it is beyond the scope of this paper to spell out this
Kantian/dynamics-first view of space and time in further detail. In closing,
however, I can shed some light can be shed on the following question: Is it
really true that the capacity for topological redescription offered to us by
the ISE Methodology is on par with our existing capacity for coordinate
redescription and logical re-axiomization? Note that, in each of these three
context (i.e., laws, coordinates, topology) the mechanical process of
redescription cares nothing for the metaphysical details of the theory in
question. Hence, almost all of the possible redescriptions on offer will be
horribly complicated and ill-suited to the theory in question. Importantly,
however, in each context, our capacity for redescription is so complete that
if a particularly nice description exists (however hard it may be to find)
then it is accessible to us via standard redescription techniques. In this
sense, the analogy is very strong between the topological redescription
offered to us by the ISE Methodology and our existing techniques for
coordinate redescription and logical re-axiomization.

### Acknowledgements

The authors thanks James Read, Tushar Menon, Nick Huggett, Harvey Brown,
Oliver Pooley, Adam Caulton, and Neils Linneman for their helpful feedback.

### Declarations

The author did not receive support from any organization for the submitted
work. The author has no competing interests to declare that are relevant to
the content of this article.

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##  Appendix A Several Proofs in Support of the Main Text

This appendix will prove several results in support of the main text. Firstly,
in Sec. A.1 a useful lemma will be proved regarding how the quotient of two
Lie groups behaves under a Lie group homomorphism. Sec. A.2 will then use this
result to prove that in order for a theory’s spacetime manifold to faithfully
and robustly survive internalization requires only a weakened version of
spacetime-kinematic compatibility. Indeed, I will there prove that this
condition is both necessary and sufficient. Finally, Sec. A.3 will prove that
the Lie group actions produced by the internalization process (namely,
$G_{\text{trans}}^{\text{(int)}}$ and $G_{\text{fix}}^{\text{(int)}}$) are
PSTOs. Moreover, I will prove that by externalizing these PSTOs one can
immediately return to the theory’s old spacetime setting, at least up to
diffeomorphism.

###  A.1 Proof of the Quotient of Images Lemma

This section will prove the following lemma for later use.

> Quotient of Images Lemma: Let $F\subset T\subset G$ be Lie groups with $F$ a
> closed subgroup of $T$. Let $\phi:G\to H$ be a Lie group homomorphism. Let
> $K\coloneqq\text{ker}(\phi)\cap T$. Note that since $F$ is closed in $T$ it
> follows that $\phi[F]$ is closed in $\phi[T]$. Hence, we can construct two
> smooth homogeneous manifolds, $\mathcal{M}\coloneqq T/F$ and
> $\mathcal{M}_{\phi}\coloneqq\phi[T]/\phi[F]$. Up to diffeomorphism, the
> quotient of the images, $\mathcal{M}_{\phi}$, is a certain quotient of
> $\mathcal{M}$, namely:
>
>
> $\displaystyle\mathcal{M}_{\phi}\coloneqq\frac{\phi[T]}{\phi[F]}\cong\frac{T}{F\,K}\cong\frac{T/F}{K}=\frac{\mathcal{M}}{K}$
> (38)
>
> We will have $\mathcal{M}\cong\mathcal{M}_{\phi}$ if and only if $K\subset
> F$ (such that all of $K$’s orbits on $\mathcal{M}$ are trivial).

Proof: Note that $\phi$ being a Lie group homomorphism implies that
$\text{ker}(\phi)$ is normal in $G$. This in turn implies that
$K\coloneqq\text{ker}(\phi)\cap T$ is normal in $T$. Similarly we have that
$\text{ker}(\phi)\cap F=K\cap F$ is normal in $F$. By the first isomorphism
theorem of group theory we then have,

$\displaystyle\phi[T]\cong T/(\text{ker}(\phi)\cap T)$ $\displaystyle=T/K$
(39) $\displaystyle\phi[F]\cong F/(\text{ker}(\phi)\cap F)$
$\displaystyle=T/(K\cap F).$

Applying these results to the numerator and denominator of
$\mathcal{M}_{\phi}$ we have,

$\displaystyle\mathcal{M}_{\phi}\cong\frac{T/K}{F/(K\cap F)}.$ (40)

The elements in this quotient set are $F/(K\cap F)$-cosets in $T/K$, namely
$tKF(K\cap F)$. Recall, however, that $K$ is normal in $T$ such that $K$ can
move past $F\subset T$. We thus have,

$\displaystyle tKF(K\cap F)=tFK(K\cap F)=tFK,$ (41)

since $K(K\cap F)=K$. Hence, we have $\mathcal{M}_{\phi}\cong T/(FK)$ as
claimed in the lemma.292929Note that it follows from $K$ being closed and
normal in $T$ that the Frobenius product, $FK=KF$, is a closed Lie subgroup of
$T$. Hence, the quotient $T/(FK)=T/(KF)$ is a smooth manifold.

To see the other diffeomorphic representations of $\mathcal{M}_{\phi}$ we will
again use the fact that $K$ is normal in $T$. As such we have,

$\displaystyle tFK=tKF=KtF.$ (42)

Hence we can see the quotient of images, $\mathcal{M}_{\phi}$, as the quotient
of $\mathcal{M}=T/F$ by the orbits of $K$ acting on either the left or the
right.

We will have $\mathcal{M}\cong\mathcal{M}_{\phi}$ if and only if every
$K$-orbit on $\mathcal{M}=T/F$ is trivial, namely $KtF=tF$. Since we have $K$
is normal in $T$ we can pass the $K$ through the $t$ here. Canceling the
factor of $t$ on the left we then have $KF=F$ which is equivalent to $K\subset
F$.

###  A.2 Robust and Faithful Manifold Survival Requires A Weakened Spacetime-
Kinematic Compatibility

Sec. 2 showed that given a theory with spacetime-kinematic compatibility, its
spacetime manifold, $\mathcal{M}_{\text{old}}$, will faithfully and robustly
survives internalization as follows:

$\displaystyle\mathcal{M}_{\text{old}}\cong\mathcal{M}_{\text{H}}\coloneqq\frac{H_{\text{trans}}}{H_{\text{fix}}}\xrightarrow[\text{int.
as}]{\text{survives}}\mathcal{M}_{\text{G}}\coloneqq\frac{G_{\text{trans}}}{G_{\text{fix}}}\cong\frac{H_{\text{trans}}}{H_{\text{fix}}}\cong\mathcal{M}_{\text{old}}.$
(43)

Surviving robustly here means that, up to diffeomorphism, the final result
$\mathcal{M}_{\text{G}}\cong\mathcal{M}_{\text{old}}$ is independent of which
$H_{\text{trans}}$ and $H_{\text{fix}}$ we have used to reconstruct
$\mathcal{M}_{\text{old}}$. This section will prove a stronger result:
Firstly, a slightly weaker version of spacetime-kinematic compatibility is
already sufficient to guarantee the faithful and robust survival of
$\mathcal{M}_{\text{old}}$. Moreover, this weakened version of spacetime-
kinematic compatibility is also necessary for the faithful and robust survival
of $\mathcal{M}_{\text{old}}$. Hence, I will here provide a necessary and
sufficient condition for $\mathcal{M}_{\text{old}}$ to survive internalization
faithfully and robustly.

The weakened version of spacetime-kinematic compatibility is as follows.

> Definition: (Weakened) Spacetime-Kinematic Compatibility -
> A spacetime theory has the weakened version of spacetime-kinematic
> compatibility just in case:
>
>   1. 1.
>
> it satisfies our two preliminary assumptions (namely, the modal assumption
> and the smooth lifting assumption), and
>
>   2. 2.
>
> it is kinematically homogeneous, and
>
>   3. 3.
>
> it is either kinematically reduced, or else failing this…
>
>   4. $3^{\prime}$.
>
> Let $H_{\text{trans}}\subset\text{Diff}(\mathcal{M}_{\text{old}})$ range
> over all finite-dimensional and kinematically-allowed which act smoothly and
> transitively on $\mathcal{M}_{\text{old}}$. Note that at least one such
> $H_{\text{trans}}$ exists given kinematic homogeneity. Recall that
> $H_{\text{kin}}$ is the theory’s group of kinematically trivial
> diffeomorphisms. If the theory in question is not kinematically reduced
> (i.e, if it has $H_{\text{kin}}\neq\\{\mathbbm{1}\\}$) then it must instead
> have $H_{\text{kin}}\cap H_{\text{trans}}=\\{\mathbbm{1}\\}$ for every such
> $H_{\text{trans}}$.
>
>

Note that the definition of spacetime-kinematic compatibility appearing in
Sec. 2.3 is simply $1.\land 2.\land 3.$ whereas the present definition is
strictly weaker, namely $1.\land 2.\land(3.\lor 3^{\prime}.)$. Note also that
$3.\implies 3^{\prime}.$

This section is organized as follows: First I will argue that points 1. and 2.
listed in the above definition are both necessary conditions for the faithful
and robust survival of $\mathcal{M}_{\text{old}}$. I will then argue that
given points 1. and 2. we will have the faithful and robust survival of
$\mathcal{M}_{\text{old}}$ if and only if point $3.\lor 3^{\prime}.$ is
satisfied. The logic is then as follows:

$\displaystyle\qquad\ \ P\implies(1.\land 2.)$ (44)
$\displaystyle\underline{(1.\land 2.)\implies(P\Longleftrightarrow(3.\lor
3^{\prime}.))\ \,}$ $\displaystyle\vdash\quad\ \,P\Longleftrightarrow(1.\land
2.\land(3.\lor 3^{\prime}.))$

where $P$ denotes the old theory’s spacetime manifold,
$\mathcal{M}_{\text{old}}$, faithfully and robustly surviving internalization.

I will now re-trace the discussion in Sec. 2 regarding how topological
information about $\mathcal{M}_{\text{old}}$ can survive internalization. The
key difference here is that spacetime-kinematic compatibility is not assumed.
To begin note that in order for $\mathcal{M}_{\text{old}}$ to survive the
internalization process, we must be able to begin the internalization process.
For this to happen the two preliminary assumptions defined in Sec. 2.1 must be
satisfied (i.e., the modal assumption and the smooth lifting assumption). This
establishes that $P\implies 1.$

Given these preliminary assumptions, the discussion presented in Sec. 2.2 can
proceed as before. Individual kinematically-allowed diffeomorphisms,
$d\in\text{Diff}_{\text{kin}}(\mathcal{M}_{\text{old}})$, survive
internalization as follows:

$\displaystyle d^{*}\xrightarrow[\text{int.
as}]{\text{survives}}G(d)\coloneqq\mathcal{F}\circ
d^{*}|_{\text{kin}}\circ\mathcal{F}^{-1}.$ (45)

If $\text{Diff}_{\text{kin}}(\mathcal{M}_{\text{old}})$ is to survive
internalization (now, potentially unfaithfully) as a Lie group then $G$ must
be a Lie group homomorphism. This will happen exactly when
$\text{ker}(G)\eqqcolon H_{\text{kin}}$ is a closed and normal subgroup of
$\text{Diff}_{\text{kin}}(\mathcal{M}_{\text{old}})$. To see that
$H_{\text{kin}}=G^{-1}(\\{\hat{\mathbbm{1}}\\})$ is closed, note that it is
the pre-image of the singleton, $\\{\hat{\mathbbm{1}}\\}$. If follows from $G$
being continuous and the singleton being a closed set, that $H_{\text{kin}}$
is a closed set as well.

Next let us see that $H_{\text{kin}}$ is normal in
$\text{Diff}_{\text{kin}}(\mathcal{M}_{\text{old}})$. That is, let us see that
for all $d\in\text{Diff}_{\text{kin}}(\mathcal{M}_{\text{old}})$ and all $h\in
H_{\text{kin}}$ we have $d^{-1}\,h\,d\in H_{\text{kin}}$. To see this, note
that for all $\varphi_{\text{old}}\in S_{\text{old}}^{\text{kin}}$ we have,

$\displaystyle(d^{-1}\,h\,d)^{*}(\varphi_{\text{old}})$
$\displaystyle=d^{*}{}^{-1}\,h^{*}\,d^{*}(\varphi_{\text{old}})$ (46)
$\displaystyle=d^{*}{}^{-1}\,d^{*}(\varphi_{\text{old}})$
$\displaystyle=\varphi_{\text{old}}.$

The first line follows from $*_{\text{lift}}$ being a group homomorphism.
Namely, it follows from $(a\,b)^{*}=a^{*}\,b^{*}$ and
$(a^{-1})^{*}=(a^{*})^{-1}$. Next, note that since
$d\in\text{Diff}_{\text{kin}}(\mathcal{M}_{\text{old}})$ we know that
$d^{*}(\varphi_{\text{old}})\in S_{\text{old}}^{\text{kin}}$ is a
kinematically allowed state. The second line follows from the fact that $h\in
H_{\text{kin}}$ acts trivially on all such states. Thus, we have that
$H_{\text{kin}}\lhd\text{Diff}_{\text{kin}}(\mathcal{M}_{\text{old}})$ is a
normal and closed subgroup. Therefore $G$ is a Lie group homomorphism.

Given that $G$ is a Lie group homomorphism, we therefore have that:

$\displaystyle\text{Diff}_{\text{kin}}(\mathcal{M}_{\text{old}})\xrightarrow[\text{int.
as}]{\text{survives}}G(\text{Diff}_{\text{kin}}(\mathcal{M}_{\text{old}}))\cong\text{Diff}_{\text{kin}}(\mathcal{M}_{\text{old}})/H_{\text{kin}}.$
(47)

That is, up to isomorphism,
$\text{Diff}_{\text{kin}}(\mathcal{M}_{\text{old}})$ survives internalization
as a Lie group by being quotiented by the theory’s kinematically trivial
group, $H_{\text{kin}}$.

The above discussion can be easily generalized to analyze the survival of an
arbitrary subgroup of the kinematically allowed diffeomorphisms,
$H\subset\text{Diff}_{\text{kin}}(\mathcal{M}_{\text{old}})$. Restricting the
$G$ map to this subgroup as $G|_{H}$ yields a Lie group homomorphism if and
only if restricting $H_{\text{kin}}$ to $H$ yields a closed and normal
subgroup of $H$. One can easily check that
$\text{ker}(G|_{H})=H_{\text{kin}}\cap H$ is both a normal and a closed
subgroup of $H$. In fact, the above arguments used in the
$H=\text{Diff}_{\text{kin}}(\mathcal{M}_{\text{old}})$ case apply unchanged.
Thus, for every kinematically allowed Lie group,
$H\subset\text{Diff}_{\text{kin}}(\mathcal{M}_{\text{old}})$, we have that,

$\displaystyle H\xrightarrow[\text{int. as}]{\text{survives}}G(H)\cong
H/(H_{\text{kin}}\cap H).$ (48)

That is, up to isomorphism,
$H\subset\text{Diff}_{\text{kin}}(\mathcal{M}_{\text{old}})$ survives
internalization as a Lie group by being quotiented by $H_{\text{kin}}\cap H$,
its kinematically trivial subgroup.

Thus we have that some diffeomorphisms can survive internalization (albeit
potentially unfaithfully). Moreover, recall from Sec. 2 that some smooth
manifolds can be reconstructed in terms of their diffeomorphisms (i.e., the
homogeneous manifolds can, $\mathcal{M}_{\text{old}}\cong
H_{\text{trans}}/H_{\text{fix}}$). Putting these possibilities together it
seems conceivable that the spacetime manifolds of some theories might also
survive internalization (potentially unfaithfully). Concretely, the theory’s
spacetime manifold might survive internalization as a quotient of its
surviving diffeomorphisms.

Let us next see that kinematic homogeneity is a necessary condition for the
theory’s spacetime to survive internalization simpliciter (i.e., faithfully or
not). Firstly, in order for us to be able to reconstruct the old theory’s
spacetime manifold as $\mathcal{M}_{\text{old}}\cong
H_{\text{trans}}/H_{\text{fix}}$ it is necessary that
$\mathcal{M}_{\text{old}}$ is homogeneous. In particular, there must exist
some finite-dimensional Lie group of diffeomorphisms,
$H_{\text{trans}}\subset\text{Diff}(\mathcal{M}_{\text{old}})$, which acts
smoothly transitively on $\mathcal{M}_{\text{old}}$. Secondly, in order for
$\mathcal{M}_{\text{old}}$ to survive internalization as a quotient of its
surviving diffeomorphisms the Lie group appearing in the numerator,
$H_{\text{trans}}$, must itself survive internalization as a Lie group. That
is, $H_{\text{trans}}$ must be kinematically allowed. Putting these two
requirements together, we have the following result: For a theory’s spacetime
to survive internalization (faithfully or not), the theory must be
kinematically homogeneous. This establishes that $P\implies 2.$ and hence that
$P\implies(1.\land 2.)$

Assuming points 1. and 2. we can continue the internalization process as
follows. Kinematic homogeneity guarantees the existence of some finite-
dimensional kinematically-allowed transitively-acting Lie group,
$H_{\text{trans}}\subset\text{Diff}_{\text{kin}}(\mathcal{M}_{\text{old}})$.
Using this Lie group and one of its stabilizer subgroups $H_{\text{fix}}$, we
can reconstruct $\mathcal{M}_{\text{old}}\cong
H_{\text{trans}}/H_{\text{fix}}$ as a quotient and then internalize it,

$\displaystyle\mathcal{M}_{\text{old}}\cong\frac{H_{\text{trans}}}{H_{\text{fix}}}\xrightarrow[\text{int.
as}]{\text{survives}}\mathcal{M}_{\text{G}}\coloneqq\frac{G(H_{\text{trans}})}{G(H_{\text{fix}})}\cong\frac{\mathcal{M}_{\text{old}}}{H_{\text{kin}}\cap
H_{\text{trans}}}$ (49)

where the last step follows from the Quotient of Images Lemma proved in Sec.
A.1. Thus, assuming only kinematic homogeneity the theory’s old spacetime
survives internalization, although it has been quotiented by the orbits of
$H_{\text{kin}}\cap H_{\text{trans}}$. Note that $\mathcal{M}_{\text{old}}$ to
will survive faithfully if and only if the orbits of $H_{\text{kin}}\cap
H_{\text{trans}}$ in $\mathcal{M}_{\text{old}}$ are trivial. As noted in the
Quotient of Images Lemma, this will happen exactly when we have
$H_{\text{kin}}\cap H_{\text{trans}}\subset H_{\text{fix}}$. This is a
necessary and sufficient condition for faithful survival of
$\mathcal{M}_{\text{old}}$ given our choice of $H_{\text{trans}}$ and
$H_{\text{fix}}$ (and given points 1. and 2.).

If we moreover want faithful and robust survival, then we need to demand that
we have $\mathcal{M}_{\text{G}}\cong\mathcal{M}_{\text{old}}$ no matter which
$H_{\text{trans}}$ and $H_{\text{fix}}$ we used to reconstruct
$\mathcal{M}_{\text{old}}\cong H_{\text{trans}}/H_{\text{fix}}$ as a quotient.
Namely, we will have $\mathcal{M}_{\text{old}}$ surviving faithfully and
robustly if and only if we have $H_{\text{kin}}\cap H_{\text{trans}}\subset
H_{\text{fix}}$ for all $H_{\text{trans}}$ and $H_{\text{fix}}$. Here
$H_{\text{trans}}$ ranges over all finite-dimensional, kinematically-allowed,
transitively-acting Lie groups,
$H_{\text{trans}}\subset\text{Diff}_{\text{kin}}(\mathcal{M}_{\text{old}})$.
Moreover, $H_{\text{fix}}$ is any one of $H_{\text{trans}}$’s stabilizer
subgroups (i.e., ranging over all $p_{0}\in\mathcal{M}$). As I will now prove
(given points 1. and 2.) this demand for faithful and robust survival will be
satisfied if and only if the spacetime theory in question satisfies either
points 3. or $3^{\prime}$. Namely, $(1.\land
2.)\implies(P\Longleftrightarrow(3.\lor 3^{\prime}.))$.

Let us first show the sufficiency of $3.\lor 3^{\prime}$. If we have point 3.
(namely, $H_{\text{kin}}=\\{\mathbbm{1}\\}$) then $H_{\text{kin}}\cap
H_{\text{trans}}\subset H_{\text{fix}}$ follows immediately for all
$H_{\text{trans}}$ and $H_{\text{fix}}$. Moreover, if we have point
$3^{\prime}.$ (namely, $H_{\text{kin}}\cap H_{\text{trans}}=\\{\mathbbm{1}\\}$
for all $H_{\text{trans}}$) than we automatically have $H_{\text{kin}}\cap
H_{\text{trans}}\subset H_{\text{fix}}$ for all $H_{\text{trans}}$ and
$H_{\text{fix}}$.

Let us next show the necessity of $3^{\prime}.$, and hence the necessity of
$3.\lor 3^{\prime}$. Let $p_{0}\in\mathcal{M}_{\text{old}}$ be the point at
which $H_{\text{fix}}$ is $H_{\text{trans}}$’s stabilizer subgroup. Note that
$H_{\text{kin}}\cap H_{\text{trans}}\subset H_{\text{fix}}$ implies that
$H_{\text{kin}}\cap H_{\text{trans}}$ maps $p_{0}$ to itself. If this happens
for all $H_{\text{fix}}$ then it must happen for all $p_{0}$. Hence if we want
faithful and robust survival then we must have $H_{\text{kin}}\cap
H_{\text{trans}}$ acting trivially everywhere. The only diffeomorphism which
acts trivially everywhere is the identity. Hence, if we want faithful and
robust survival then we must have $H_{\text{kin}}\cap
H_{\text{trans}}=\\{\mathbbm{1}\\}$. Namely, $H_{\text{kin}}$ has to have a
trivial intersection with every finite-dimensional transitively-acting Lie
group,
$H_{\text{trans}}\subset\text{Diff}_{\text{kin}}(\mathcal{M}_{\text{old}})$.
But this is just what point $3^{\prime}.$ demands. This completes the proof.

In sum: For a theory’s spacetime manifold to faithfully and robustly survive
internalization, the theory must satisfy the two preliminary assumptions
(point 1.), and it must be kinematically homogeneous (point 2.). Given these
two necessary conditions, the faithful and robust survival of
$\mathcal{M}_{\text{old}}$ becomes equivalent to demanding something slightly
weaker than that the theory in question is kinematically reduced (namely,
point $3.$ vs. $3.\lor 3.^{\prime}$). It follows from this that the weakened
version of spacetime-kinematic compatibility defined at the top of this
section is a necessary and sufficient condition for the faithful and robust
survival of $\mathcal{M}_{\text{old}}$.

###  A.3 Internalizing a Theory with Spacetime-Kinematic Compatibility Yields
PSTOs

As Sec. 2 has discussed, internalizing any theory with spacetime-kinematic
compatibility yields a pair of Lie groups, $G_{\text{trans}}^{\text{(int)}}$
and $G_{\text{fix}}^{\text{(int)}}$, acting on
$S_{\text{neutral}}^{\text{kin}}$. I will now prove that this pair of Lie
group actions automatically qualify as PSTOs. Moreover, by externalizing these
PSTOs one can always return to the theory’s original spacetime setting.

To begin, recall that internalization first reconstructs the old spacetime,
$\mathcal{M}_{\text{old}}$, as a quotient of its kinematically allowed
diffeomorphisms, $\mathcal{M}_{\text{old}}\cong
H_{\text{trans}}/H_{\text{fix}}$. Specifically,
$H_{\text{trans}}\subset\text{Diff}_{\text{kin}}(\mathcal{M}_{\text{old}})$ is
some Lie group which acts transitively on $\mathcal{M}_{\text{old}}$ and
$H_{\text{fix}}$ is any of the stabilizer subgroups of $H_{\text{trans}}$.
Since the theory in question has spacetime-kinematic compatibility, every
kinematically allowed diffeomorphism,
$d\in\text{Diff}_{\text{kin}}(\mathcal{M}_{\text{old}})$, survives
internalization as $G(d)\coloneqq\mathcal{F}\circ
d^{*}|_{\text{kin}}\circ\mathcal{F}^{-1}$. In particular, we have
$H_{\text{trans}}$ and $H_{\text{fix}}$ faithfully surviving internalization
becoming Lie groups acting on $S_{\text{neutral}}^{\text{kin}}$, namely
$G_{\text{trans}}^{\text{(int)}}\coloneqq G(H_{\text{trans}})\cong
H_{\text{trans}}$ and $G_{\text{fix}}^{\text{(int)}}\coloneqq
G(H_{\text{fix}})\cong H_{\text{fix}}$. Hence, the old theory’s spacetime
manifold survives internalization as
$\mathcal{M}_{\text{old}}\cong\mathcal{M}_{\text{G}}\coloneqq
G_{\text{trans}}^{\text{(int)}}/G_{\text{fix}}^{\text{(int)}}$.

I will now prove that $G_{\text{trans}}^{\text{(int)}}$ and
$G_{\text{fix}}^{\text{(int)}}$ are PSTOs. Firstly, note that it follows from
$H_{\text{fix}}$ being the stabilizer subgroup of
$H_{\text{trans}}\subset\text{Diff}(\mathcal{M}_{\text{old}})$ that
$H_{\text{trans}}$ and $H_{\text{fix}}$ have faithful construction
compatibility (see the Homogeneous Manifold Characterization Theorem proved in
an appendix of Grimmer (2023)). It follows immediately that
$G_{\text{trans}}^{\text{(int)}}$ and $G_{\text{fix}}^{\text{(int)}}$ have
faithful construction compatibility (because $H_{\text{trans}}$ and
$H_{\text{fix}}$ do). For brevity, the (int) superscripts will now be dropped.

Thus, all that remains to show is that this pair of Lie group actions have
Injection Compatibility. Demonstrating this requires that we construct an
injective map, $J_{\text{G}}:S_{\text{neutral}}^{\text{kin}}\to
S_{\text{G}}^{\text{all}}$, with certain properties.303030Namely, it must have
the two properties listed in the definition of injection compatibility in Sec.
3.1 with the following replacements:
$(G_{1},G_{2},S_{\text{N}})\mapsto(G_{\text{trans}},G_{\text{fix}},S_{\text{neutral}}^{\text{kin}})$.
Before this, however, we must construct the set $S_{\text{G}}^{\text{all}}$ of
all states definable on $\mathcal{M}_{\text{G}}$. This appendix is focused
only on the scalar field case,
$\varphi_{\text{old}}:\mathcal{M}_{\text{old}}\to\mathcal{V}_{\text{old}}$. In
principle, one is free to use the ISE Methodology to switch between theories
about different types of states (e.g., the new theory could be about a
particle trajectory,
$x_{\text{new}}:\mathcal{V}_{\text{old}}\to\mathcal{M}_{\text{old}}$, or about
a tangent vector field,
$\vec{\varphi}_{\text{new}}:\mathcal{M}_{\text{old}}\to\text{T}\mathcal{M}_{\text{old}}$).
Our present goal, however, is to return to the theory’s original spacetime
setting. Hence, the new theory will be about a scalar field,
$\varphi_{\text{new}}$. In light of this, we first need to pick a value space,
$\mathcal{V}_{\text{G}}$, so that we can define $S_{\text{G}}^{\text{all}}$ as
the set of all $\mathcal{V}_{\text{G}}$-valued functions definable over
$\mathcal{M}_{\text{G}}$. Since our goal here is to return to the old theory’s
spacetime setting we will naturally pick the value space,
$\mathcal{V}_{\text{G}}=\mathcal{V}_{\text{old}}$.

Once we have defined $S_{\text{G}}^{\text{all}}$, we then need to pick an
injective map, $J_{\text{G}}:S_{\text{neutral}}^{\text{kin}}\to
S_{\text{G}}^{\text{all}}$, with the following two properties: Firstly, we
must have,

$\displaystyle\forall g\ \forall\bm{\varphi}\ \ \Big{(}(J_{\text{G}}\circ
g)\,\bm{\varphi}=(\bar{g}^{*}\circ J_{\text{G}})\,\bm{\varphi}\Big{)},$ (50)

and, secondly, for all diffeomorphisms,
$d\in\text{Diff}(\mathcal{M}_{\text{G}})$, we must have,

$\displaystyle\Big{(}\forall g\ \forall\bm{\varphi}\ \
d^{*}\,J_{\text{G}}(\bm{\varphi})=J_{\text{G}}(\bm{\varphi})\Big{)}\Longleftrightarrow
d=\mathbbm{1}.$ (51)

In both of these expressions (and throughout this section) the ranges of $g\in
G_{\text{trans}}$ and $\bm{\varphi}\in S_{\text{neutral}}^{\text{kin}}$ will
be suppressed for notational convenience. Similarly, the range of
$\varphi_{\text{old}}\in S_{\text{old}}^{\text{kin}}$ will be suppressed.

To find a map, $J_{\text{G}}$, satisfying Eq. (50) and Eq. (51), let us first
identify the diffeomorphism,
$R:\mathcal{M}_{\text{G}}\to\mathcal{M}_{\text{old}}$, which effectively
undoes the internalization process. Recall that the internalization process
reconstructed $\mathcal{M}_{\text{old}}$ as
$\mathcal{M}_{\text{H}}=H_{\text{trans}}/H_{\text{fix}}$ and then
$\mathcal{M}_{\text{G}}=G_{\text{trans}}/G_{\text{fix}}$. Hence, we are
looking for a map $R$ which sends $G_{\text{fix}}$-cosets in
$G_{\text{trans}}$ back onto spacetime points in $\mathcal{M}_{\text{old}}$.
For notational convenience, let $[g^{\prime}]=g^{\prime}G_{\text{fix}}$ denote
the $G_{\text{fix}}$-cosets in $G_{\text{trans}}$ containing $g^{\prime}\in
G_{\text{trans}}$. The $R:\mathcal{M}_{\text{G}}\to\mathcal{M}_{\text{old}}$
map acts on points $[g]\in\mathcal{M}_{\text{G}}$ as follows,

$\displaystyle R([g])\coloneqq G^{-1}(g)(p_{0})$ (52)

where $p_{0}\in\mathcal{M}_{\text{old}}$ is the point at which
$H_{\text{fix}}$ is the stabilizer subgroup of $H_{\text{trans}}$. One might
worry that this definition of $R$ depends on which representative of $[g]$ we
pick. Fortunately, it does not. Recall from above that the $G$ map is the
isomorphism relating $H_{\text{trans}}$ and $G_{\text{trans}}$. The $G^{-1}$
map thus relates each $G_{\text{fix}}$-coset in $G_{\text{trans}}$ (i.e., each
point $[g]\in\mathcal{M}_{\text{G}}$) to an $H_{\text{fix}}$-cosets in
$H_{\text{trans}}$ (i.e., to a point $[h]\in\mathcal{M}_{\text{H}}$). It
follows from the orbit-stabilizer theorem that every element of an
$H_{\text{fix}}$-coset maps $p_{0}$ to the same place. Hence the above
definition of the $R$ map is well-defined.

In addition to this diffeomorphism, $R$, relating $\mathcal{M}_{\text{G}}$ and
$\mathcal{M}_{\text{old}}$ we also have another nice relationship between
these smooth manifolds. Namely, we can think of $G_{\text{trans}}$ as being
faithfully represented as diffeomorphisms on both $\mathcal{M}_{\text{G}}$ and
$\mathcal{M}_{\text{old}}$. Firstly, we have that $G_{\text{trans}}$ is
faithfully represented as diffeomorphisms on $\mathcal{M}_{\text{old}}$,
namely $H_{\text{trans}}\subset\text{Diff}(\mathcal{M}_{\text{old}})$.
Secondly, it follows from $G_{\text{trans}}$’s faithful construction
compatibility with $G_{\text{fix}}$ that it is faithfully represented as
diffeomorphisms on $\mathcal{M}_{\text{G}}$, namely
$\overline{G_{\text{trans}}}\subset\text{Diff}(\mathcal{M}_{\text{G}})$.
Recall that we have $\overline{G_{\text{trans}}}\cong G_{\text{trans}}$ where
$\bar{g}([g^{\prime}])\coloneqq[g\,g^{\prime}]$.

These two faithful representations of $G_{\text{trans}}$ as diffeomorphisms
(on $\mathcal{M}_{\text{old}}$ and $\mathcal{M}_{\text{G}}$) are naturally
related by the $R:\mathcal{M}_{\text{G}}\to\mathcal{M}_{\text{old}}$ map
introduced above. Namely, for each $g\in G_{\text{trans}}$ we have
$\bar{g}\in\overline{G_{\text{trans}}}$ and $h=G^{-1}(g)\in H_{\text{trans}}$
related as follows:

$\displaystyle\forall g\ \ h\circ R=R\circ\bar{g}.$ (53)

This is easy to confirm. Evaluating the right-hand-side at a point
$[g^{\prime}]\in\mathcal{M}_{\text{G}}$ yields,

$\displaystyle(R\circ\bar{g})([g^{\prime}])=G^{-1}(g\,g^{\prime})(p_{0}).$
(54)

Likewise, evaluating the left-hand-side at a point
$[g^{\prime}]\in\mathcal{M}_{\text{G}}$ yields,

$\displaystyle(h\circ
R)([g^{\prime}])=(G^{-1}(g)\,G^{-1}(g^{\prime}))(p_{0}).$ (55)

But these are identical since $G$ is a group isomorphism,
$G^{-1}(g\,g^{\prime})=G^{-1}(g)\,G^{-1}(g^{\prime})$.

As I will now discuss, we can leverage this relationship between
$\overline{G}_{\text{trans}}\subset\text{Diff}(\mathcal{M}_{\text{G}})$ and
$H_{\text{trans}}\subset\text{Diff}(\mathcal{M}_{\text{G}})$ to find the
desired $J_{\text{G}}$ map. First, note that we can rewrite Eq. (53) as,

$\displaystyle\forall g\ \ h^{-1}\circ R^{-1}=R\circ\bar{g}^{-1}.$ (56)

For any scalar field state in the old theory, $\varphi_{\text{old}}\in
S_{\text{old}}^{\text{kin}}$, we can apply these diffeomorphism on the right
yielding,

$\displaystyle\forall g\ \forall\varphi_{\text{old}}\ \
\varphi_{\text{old}}\circ h^{-1}\circ R=\varphi_{\text{old}}\circ
R\circ\bar{g}^{-1}.$ (57)

Using the definition of the $*_{\text{lift}}$-map (namely,
$d^{*}\varphi_{\text{old}}\coloneqq\varphi_{\text{old}}\circ d^{-1}$) we then
have,

$\displaystyle\forall g\ \forall\varphi_{\text{old}}\ \
(h^{*}\varphi_{\text{old}})\circ R=\bar{g}^{*}(\varphi_{\text{old}}\circ R).$
(58)

Recall that we have a one-to-one correspondence,
$\mathcal{F}:S_{\text{old}}^{\text{kin}}\to S_{\text{neutral}}^{\text{kin}}$,
relating $\mathcal{F}:\varphi_{\text{old}}\mapsto\bm{\varphi}$. Hence, we
have,

$\displaystyle\forall g\ \forall\bm{\varphi}\ \
(h^{*}\mathcal{F}^{-1}(\bm{\varphi}))\circ
R=\bar{g}^{*}(\mathcal{F}^{-1}(\bm{\varphi})\circ R).$ (59)

Recalling that $g$ and $h=G^{-1}(g)$ are related as $g=G(h)=\mathcal{F}\circ
h^{*}\circ\mathcal{F}^{-1}$ we then have,

$\displaystyle\forall g\ \forall\bm{\varphi}\ \
\mathcal{F}^{-1}(g\,\bm{\varphi})\circ
R=\bar{g}^{*}(\mathcal{F}^{-1}(\bm{\varphi})\circ R).$ (60)

This proves one of the desired properties, namely Eq. (50), for the injective
map $J_{\text{G}}(\bm{\varphi})=\mathcal{F}^{-1}(\bm{\varphi})\circ R$.

Similarly, one can prove Eq. (51) for
$J_{\text{G}}(\bm{\varphi})=\mathcal{F}^{-1}(\bm{\varphi})\circ R$ as follows.
Given that the old theory is kinematically reduced, we have
$H_{\text{kin}}=\\{\mathbbm{1}\\}$. From this it follows that for all
$d_{\text{old}}\in\text{Diff}(\mathcal{M}_{\text{old}})$ we have,

$\displaystyle\Big{(}\forall\varphi_{\text{old}}\ \ \varphi_{\text{old}}\circ
d_{\text{old}}=\varphi_{\text{old}}\Big{)}\ \Longleftrightarrow\
d_{\text{old}}=\mathbbm{1}.$ (61)

Taking
$\varphi_{\text{old}}=\mathcal{F}^{-1}(\bm{\varphi})=J_{\text{G}}(\bm{\varphi})\circ
R^{-1}$ and composing with $R$ on the right we then have,

$\displaystyle\Big{(}\forall\bm{\varphi}\ \ J_{\text{G}}(\bm{\varphi})\circ
R^{-1}\circ d_{\text{old}}\circ R=J_{\text{G}}(\bm{\varphi})\Big{)}\
\Longleftrightarrow\ d_{\text{old}}=\mathbbm{1}.$ (62)

The diffeomorphisms, $d_{\text{old}}$, on $\mathcal{M}_{\text{old}}$ are in
one-to-one correspondence with diffeomorphisms, $d_{\text{G}}=R^{-1}\circ
d\circ R$, on $\mathcal{M}_{\text{G}}$. Consequently, we have that for all
$d_{\text{G}}\in\text{Diff}(\mathcal{M}_{\text{G}})$,

$\displaystyle\Big{(}\forall\bm{\varphi}\ \ J_{\text{G}}(\bm{\varphi})\circ
d_{\text{G}}=J_{\text{G}}(\bm{\varphi})\Big{)}\ \Longleftrightarrow\
d_{\text{G}}=\mathbbm{1}$ (63)

as desired. This completes the proof that $G_{\text{trans}}^{\text{(int)}}$
and $G_{\text{fix}}^{\text{(int)}}$ are PSTOs.

Let us next see where externalizing these PSTOs with this choice of
$J_{\text{G}}$ takes us. The map which is used in externalization,
$J:S_{\text{neutral}}^{\text{kin}}\to S_{\text{new}}^{\text{all}}$ is related
to the map $J_{\text{G}}:S_{\text{neutral}}^{\text{kin}}\to
S_{\text{G}}^{\text{all}}$ discussed above as follows,
$J(\bm{\varphi})=J_{\text{G}}(\bm{\varphi})\circ\mathcal{F}_{\text{man}}^{-1}$.
Recall from Sec. 3.2 that
$\mathcal{F}_{\text{man}}:\mathcal{M}_{\text{G}}\to\mathcal{M}_{\text{new}}$
is the diffeomorphism relating the quotient of our PSTOs,
$\mathcal{M}_{\text{G}}$, to the theory’s new spacetime manifold,
$\mathcal{M}_{\text{new}}$. Picking the map $J_{\text{G}}$ map discussed
above, we have the field states in the new theory, $\varphi_{\text{new}}$,
related to field states in the old theory, $\varphi_{\text{old}}$, as follows:

$\displaystyle\varphi_{\text{new}}=(J\circ\mathcal{F})\,\varphi_{\text{old}}=\varphi_{\text{old}}\circ
R\circ\mathcal{F}_{\text{man}}^{-1}$ (64)

where
$R\circ\mathcal{F}_{\text{man}}^{-1}:\mathcal{M}_{\text{new}}\to\mathcal{M}_{\text{old}}$
is a diffeomorphism between $\mathcal{M}_{\text{new}}$ and
$\mathcal{M}_{\text{old}}$. Hence, this choice of $J$ directly transplants the
old theory’s states (and their dynamics) onto the new spacetime manifold. We
have thus recovered the old spacetime theory up to diffeomorphism.