# Reality and Super-Reality: Properties of a Mathematical Multiverse

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## Abstract

Ever since its foundations were laid nearly a century ago, quantum theory has provoked questions about the very nature of reality. We address these questions by considering the universe—and the multiverse—fundamentally as complex patterns, or mathematical structures. Basic mathematical structures can be expressed more simply in terms of emergent parameters. Even simple mathematical structures can interact within their own structural environment, in a rudimentary form of self-awareness, which suggests a definition of reality in a mathematical structure as simply the complete structure. The absolute randomness of quantum outcomes is most satisfactorily explained by a multiverse of discrete, parallel universes. Some of these have to be identical to each other, but that introduces a dilemma, because each mathematical structure must be unique. The resolution is that the parallel universes must be embedded within a mathematical structure—the multiverse—which allows universes to be identical within themselves, but nevertheless distinct, as determined by their position in the structure. The multiverse needs more emergent parameters than our universe and so it can be considered to be a superstructure. Correspondingly, its reality can be called a super-reality. While every universe in the multiverse is part of the super-reality, the complete super-reality is forever beyond the horizon of any of its component universes.

## Keywords

Reality Super-reality Multiverse Mathematical structure Emergent properties## 1 A First Approach to Reality

“the idea of an objective real world whose smallest parts exist objectively in the same sense as stones or trees exist, independently of whether or not we observe them…is impossible.”

“The doctrine that the world is made up of objects whose existence is independent of human consciousness turns out to be in conflict with quantum mechanics and with facts established by experiment”

d’Espagnat (1979)

In this paper, we are going to take the opposite view from physicist and philosopher Bernard d’Espagnat: we shall claim that self-awareness is a product of an independent reality rather than the other way around. In support of our position we shall describe heuristically how this might come about in Sect. 3.

In the meantime, to get straight to the point without being diverted by ontological and epistemological discussions (for those, a good starting point might be Leifer 2014), let us make a first attempt at a working definition of reality:

### Reality (I)

*Reality is the complete set of quantum fields extending throughout the whole of spacetime that comprises our block universe*.

Before going further, we need to unpack this definition to see if it works. The block universe may be thought of as the four-dimensional block of spacetime that encapsulates the complete past and future history of our universe. (The notion of the block universe, of course, has a respectable provenance, and, indeed, the block universe may even be demonstrated experimentally (McKenzie 2016b).) So, our definition is saying that reality is the complete past and future history of all of the quantum fields that go to make up our universe. That must include all of the particles that arise from stable excitations of the appropriate quantum fields as well as all of their interactions.

*U*on quantum fields is

*i*denotes the

*i*th quantum field,

*φ*

_{i},

*i*

_{max}is the total number of quantum fields and the curly brackets represent the set.

A consequence of our definition of reality is that it cannot be expressed more fundamentally in terms of contemporary physics. For instance, if we were instead to define reality in terms of just the fundamental particles of the Standard Model, then that definition would be open to the charge that the particles are themselves describable as excitations of the appropriate quantum fields. On the other hand, if we try to extend our definition more deeply than the quantum fields, then all we are left with, like the grin on the Cheshire cat, is the essential pattern of the quantum fields. There are straws in the wind that the patterns of quantum fields and their interactions may be further reducible into yet more fundamental patterns (see, for instance, Arkani-Hamed et al. 2014) but there is no hint of any return to *stuff*—it’s patterns all the way down!

## 2 Philosophical Aspects of Accepting a Mathematical Stratum as the Fundamental Level of Reality

Patterns, of course, are just mathematical structures, but, at least for some in the physics community, it seems to be less provocative to claim that the universe is, fundamentally, a pattern rather than to say that it is a mathematical structure.

It may be easier to accept that the fundamental level of our reality is a mathematical structure by asking how, in a Theory of Everything, we might explain our universe, particularly including space and time itself. Clearly, our explanation cannot depend upon any property of the universe that involves space or time, since that is what we are trying to explain. A pure mathematical structure satisfies this requirement. This is well illustrated, for example, in papers by physicists who attempt to model the emergence of spacetime: their approach is to start with an abstract (quantum) mathematical structure in Hilbert space and thence demonstrate the emergence of gravity. Since gravity is a property of the (general relativistic) structure of spacetime, they are effectively demonstrating the emergence of spacetime itself (see, for example, ChunJun et al. 2017; Raasakka 2017).

This will be one of the claims of our paper, that our reality is purely a mathematical structure. To that extent, at least, we are in agreement with Tegmark (2008). We shall see at the end of Sect. 5 that such a philosophy, which is, in essence, one of ontic structural realism (Berghofer 2018), makes it easier to accept the idea of parallel universes, because a pattern (a universe) may be duplicated without controversy.

In spite of the above, many people harbour the suspicion that, no matter how sophisticated a mathematical structure we may conjecture, that structure can never be more than a mere *description* of reality, rather than reality itself. For these people, we have not breathed fire into the equations. They find it hard to accept that their own self-awareness can be just a mathematical structure. So goes their argument, along lines reminiscent of Dr Johnson’s refutation of Bishop Berkeley’s immaterialist philosophy by kicking a stone and feeling the reverberation in his body.

The next section is essentially a response to that argument as well as to that of d’Espagnat, who, as we saw in Sect. 1, did not accept the independence of reality from human consciousness. We shall show that a simple mathematical structure can, in principle, demonstrate an awareness of itself. The structure that we use will also serve later in our discussion of super-reality in “Appendix 1”.

## 3 How a Simple Mathematical Structure Can in Principle be Self-aware

It is easy to see why many physicists reject such a statement. While they may accept that our universe can be *described* by mathematics, it is seemingly a step too far to think that it can *be* mathematics. Nevertheless, a case can be argued to make that idea more palatable by considering a version of a two-dimensional cellular automaton, Conway’s Game of Life (Gardner 1970), in which, as we shall see, there can be a structure that is “aware” of its environment. (It is sometimes helpful to use such automata rather than to appeal to our own subjective experience as self-aware beings in our own universe, with all of the associated baggage that such experience would entail.).

The main attraction of the Game of Life is that its very simple set of rules can lead to solutions of interesting complexity. The game evolves within a two-dimensional (*p*, *q*) matrix, the cells of which live or die, and it is typically played out on a computer monitor screen which displays successive generations (labelled *h*) of the matrix. The pattern in any generation (generation *h*) is transformed according to the rules, and the resulting pattern is then displayed as the next (*h* + 1) generation. Pragmatically, the matrix is finite in extent which means that boundary conditions (for instance, toroidal) must be chosen.

*h*, we can assign to each cell, \( \left( {p,q} \right) \), a state \( State\left( {p,q,h} \right) \), which takes the value 1 if the cell is live and 0 if it is dead. At the beginning of the game, when

*h*= 0, the pattern—a boundary condition—is specified for every cell in the starting matrix: \( State\left( {p,q,0} \right) \). The patterns of successive generations are then found from:

The parameter *N* is the number of live neighbours surrounding a central cell. In (3), the value of the central cell’s state is subtracted from the double summation because that value should not be included in the total, which is supposed to include only neighbouring states.

The complete pattern of the Game-of-Life block universe may be reconstructed from algorithms (2) and (3), given the state of all of the cells in the matrix at the beginning of the game. *Notice that the pattern and the algorithms are isomorphic*—*they are the same mathematical structure.*

*h*and the subsequent four generations. In each matrix, the value of

*N*is shown for each of the 49 cells. The pattern in this figure is a “glider” [discovered by the British mathematician, Richard Guy (Roberts 2015)], which progresses diagonally across the matrix, completing the cycle in four steps.

If a Game-of-Life player were to pause the execution of a program that included such a “self-aware” UTM and then re-start it at a later time, the UTM would obviously be unaware that the program had been temporarily interrupted. This is to be distinguished from the situation where we build a robot in our laboratory which is aware of its environment, switch it off and then re-start it. Our robot would be aware that it had been temporarily switched off because its environment, such as the laboratory clock, would have changed state while it was unconscious. In the scenario with the Game of Life being paused, however, it is the UTM’s *complete universe* that is temporarily halted.

This simple scenario is intended to illustrate the independence of the pattern—that is, the mathematical structure—of the Game-of-Life block universe from the computer programs that *simulate* it. It is important to appreciate that the UTM in our Game of Life does not spring into self-awareness when the program is started up: the self-awareness of the UTM is a property of the pattern of the particular Game-of-Life block universe in which it finds itself. Its self-awareness is completely independent of the simulation, which has no self-awareness.

Notice, too, that the pattern for the block universe is unique: simulations of it by different computers are different simulations, but they are simulations of a *unique* mathematical structure. In the same way, there may be different simulations of a sphere on different computers, but the *structure* that they are simulating (*x*^{2} + *y*^{2} + *z*^{2} = *constant*) is unique.

## 4 Emergent Parameters in a Simple Mathematical Structure

It may be argued that the structure for the Game-of-Life block universe and that for a sphere do not themselves create the three dimensions of the worlds in which their patterns operate. In other words, some may argue that these mathematical structures can only exist by virtue of our own universe, which provides the requisite geometry, and that these mathematical structures are therefore not independent of our universe. However, that perception arises because we used the labels *h*, *x*, *y* and *z*, which are suggestive of our own geometry, for convenience. Fundamentally, there are no labels, only patterns.

This pattern lends itself to simple descriptions such as “digits in the 7 × 5 matrix are “1” when they are on the perimeter and “0” otherwise”. So, in a Kolmogorov-complexity sense (Li and Vitanyi 2008) it singles itself out as special. Hence, even in this crude example, without using labels, a dimensionality emerges from within the structure, rather than the other way around. We call the parameters that define the dimensionality (in this case, seven rows and five columns) *emergent parameters*—these are defined in “Appendix 1”.

With the above discussion in mind, our definition of reality may now be refined:

### Reality (II)

*Our reality is the complete mathematical structure of our block universe.*

## 5 Fundamental Uncertainty Implies a Parallel-Universe Model

Let us now look at some of the patterns within our block universe. We may be particularly struck by the outcomes of identical double-slit experiments, each using a single electron. We can arrange for these experiments to be distributed throughout our block universe, including experiments separated by space-like intervals (taking place at the same moment in widely separated laboratories) as well as experiments performed serially in the same laboratory. In each experiment, only a single electron is used, and we note the position of its interaction on the detector screen in every case.

When we look at the outcomes of a very large, but finite, number of experiments, it is clear that, while the position of the interaction on the screen in any single experiment could not have been predicted (it is apparently random), there is nevertheless a pattern that connects the ensemble of outcomes. When the outcomes of each experiment are collected together and plotted on a frequency distribution graph across the detecting screen, the result is a sinc-squared term multiplied by a cosine-squared term: \( \left( {\frac{\sin \beta }{\beta }} \right)^{2} \cos^{2} \alpha \), where *α* and *β* are both functions of the distance across the detecting screen. We notice that this pattern matches the absolute square of the value of the probability amplitude along the detecting screen, which we can calculate from the Lagrangians of the quantum electron field summed over time for all of the different trajectories from the slits to the screen.

We notice further that there is always one—and never more than one—interaction between the quantum electron field and the detecting screen. This is surprising because points on the screen are generally space-like separated: after all, if the screen is sufficiently far from the double slits, then the detecting area would have to be kilometres wide. Our surprise stems from the fact that we already noticed that disturbances in quantum fields transmit at a finite speed, so that there is no way for information about an interaction at one point on the screen to be relayed by the quantum field to the other parts of the screen to prevent a duplicate interaction.

*here*rather than

*there*.

There appear to be only two ways in which such experiments, which we have distributed both in space and in time, can be unpredictable individually and yet be connected through a common pattern, such as the sinc-squared–cosine-squared term in this example. The first way is that, in each experiment, the point on the screen where the electron is detected is determined randomly, but with the randomness being weighted according to the absolute square of the probability amplitude derived from summing the Lagrangians of the quantum electron field over time for all possible trajectories between the electron gun and the detecting screen (see Fig. 4). Of course, this is a conventional formulation of quantum mechanics.

However, since this is a probabilistic explanation, it requires a mechanism that effectively throws the dice for each experiment. The difficulty with this explanation is that no mathematical algorithm or random-number generator can supply the necessary unpredictability, because, by definition, such algorithms are ultimately predictable. Truly random sequences exist in mathematics, of course, such as the digits in a Chaitin halting probability, but such sequences are unknowable in principle—they cannot be computed—and so cannot be candidates for a random-number generator. The only exception is random-number generation using quantum processes (Herrero-Collantes and Carlos Garcia-Escartin 2017), but that would be a logical circularity: you cannot fundamentally explain quantum randomness by saying that it is based upon quantum randomness!

Fundamentally, the problem with the probabilistic explanation comes back to the block universe: if you picture our universe as a block of spacetime embedded throughout with outcomes of quantum interactions, what is it about the underlying pattern of our universe that determines which of the several or many possible outcomes appears at each embedded interaction? Simply to put it all down to probability might at least suggest that the model is incomplete.

The only other way for a pattern to emerge across the individually unpredictable outcomes of identical experiments widely spread in time and space is through a parallel-universe model. A rudimentary example of the particular model used in this paper is described in “Appendix 2” as a “Toy Multiverse”. The three key features of the parallel-universe model are (1) there is a finite number of discrete, parallel block universes, in contrast to the branching structure of the Many Worlds Interpretation (MWI); (2) the universes are independent of each other (they do not interact); and (3) their numbers are distributed according to the Born probabilities of the quantum outcomes that they each contain.

Hence, for example, in the case of a Stern-Gerlach experiment with two possible outcomes, one with a 75% chance of occurring and the other with a 25% chance of occurring, there are three times as many universes in the multiverse containing the high-probability result than there are universes that contain the low-probability result. If the same experiment is repeated a million times, this will mean that, in the multiverse, most of the universes which feature the sequence of a million experiments will contain approximately 750,000 of the high-probability results and 250,000 of the low-probability results.

While such a hypothesis may be regarded by some as preposterous in its plethora of universes, others who accept that our universe is ultimately purely a pattern see no fundamental reason why the pattern may not be repeated, albeit on an unimaginable (but always finite) scale.

## 6 Super-Reality Within a Mathematical Superstructure

Our definition of reality applies equally to all of the parallel universes. Since each universe is independent of the others, each reality is confined to its own universe. Hence, in Fig. 4, reality in the block universe containing the quantum electron field configuration shown in experiment (a) includes the electron being detected to the right of the screen, whereas reality in the block universe containing the identical quantum electron field configuration at the moment of detection in experiment (b) includes the electron being detected near the centre of the screen.

A definition of reality that encompasses parallel block universes is:

### Reality (III)

*Reality within a parallel, block universe is the complete mathematical structure of that parallel, block universe.*

Equally, since each universe is itself a mathematical structure:

### Reality (IV)

*Reality within a mathematical structure is the complete definition of that structure.*

To clarify this final statement, an incompletely defined structure would, for example, be one in which the value of a function of a parameter can be, say, either one or zero, with the actual value being left unspecified. The parallel with quantum uncertainty is clear: there is a unique, definite outcome for every quantum event in our block universe, as may be seen when viewing the event in retrospect. The uncertainty arises because such an outcome cannot be determined from the mathematical structure: it is unpredictable.

From the above example using parallel universes to account for the outcomes of the Stern-Gerlach experiment, the complete pattern, or mathematical structure, of any given block universe is apparently not unique. Indeed, since each block universe is full of events with two or more possible outcomes, there will be, in general, many exact copies of any given block universe. This may be seen by inspecting Eq. (6) and Fig. 13. In that figure, each of the block universes has several exact duplicates, and the numbers of such duplicates will increase with the total number of events and outcomes in the universes.

As a concrete example of this, consider the three block universes of type *i* = 2 shown in Fig. 13, containing outcomes A_{2}, C_{1}, B_{1}. In Fig. 10, the ratios of probabilities of outcomes C_{1}:C_{2}:C_{3} are given as 1:3:1. This might lead us to think that there is one solitary universe with outcome C_{1} and three universes containing outcome C_{2}. However, we see that there are, in fact, three universes of type *i* = 2 in Fig. 13, rather than one solitary universe. If we try to reduce the number of universes in the Toy Multiverse by a factor of three, so that the population of type *i* = 2 is reduced to one solitary universe, then the total number in the A_{2} branch would also have to be reduced from 30 to 10. However, by the same token, the total number in the A_{1} branch would need to come down from 10 to \( 3{\raise0.5ex\hbox{$\scriptstyle 1$} \kern-0.1em/\kern-0.15em \lower0.25ex\hbox{$\scriptstyle 3$}} \), which fails the requirement that the number of universes is necessarily integral. So, in general, there must be a (very large) number of indistinguishable copies of any block universe in the multiverse.

Now, the block universe is a *unique* mathematical structure, just as we observed earlier that a sphere, or, for that matter, the sequence of the first hundred prime numbers, is a unique mathematical structure. Indeed, the adjective “unique” is superfluous in such a context. If we remember that the block universe is purely a pattern—a mathematical structure—then it is just as wrong to regard it as having an indistinguishable copy as it is to think of the sequence of primes from 2 to 97 as having an indistinguishable copy. The structure is the structure!

However, the above paragraph is clearly in conflict with the preceding one which states that there must generally be many indistinguishable copies of any block universe in the multiverse. The difficulty arises because we regard each block universe as an independent mathematical structure. Deutsch (1997) can claim that the double-slit pattern arises from interference between many universes just after branching/splitting, because quantum interference is a feature of Everett’s MWI formulation. However, as stated earlier, the topology of a block universe rules out such branching in our model: the mathematical structures of a multiverse of block universes never overlap, and are therefore independent of each other, and this includes groups of those that are indistinguishable from each other.

The only way for the mathematical structures of the block universes to be unique and yet allow for indistinguishable copies of such structures is for all of the structures to be embedded within a mathematical *superstructure*. “Appendix 1” clarifies our terminology of mathematical structures and superstructures and shows how they are related.

*φ*

_{i}, is a function of location in the block universe, and we may make this dependence explicit by writing it as \( \varphi_{i} \left( {x,y,z,t} \right) \). From Eq. (1), our universe

*U*is dependent upon the same parameters: \( U\left( {x,y,z,t} \right) \). Using

*M*to represent the mathematical superstructure of our multiverse, its pattern may then be written symbolically as

*N*, is the total (finite) number of parallel universes in our multiverse. The reason for considering

*N*as finite is discussed in the second paragraph of “Appendix 2”.

As in the Toy Multiverse of “Appendix 2”, there must be at least one parameter *θ* that determines the quantum–mechanical distribution of the *N completely independent* universes, *U*_{n}, embedded in the superstructure of our multiverse.

Equation (4) is effectively the same as Eq. (5) that describes the Toy Multiverse of “Appendix 2”, although, in our multiverse, the number *N* is, of course, unimaginably greater than the 40 universes of the Toy Multiverse. The case for assigning a unique set of parameters, \( \left\{ {x_{n} ,y_{n} ,z_{n} ,t_{n} } \right\} \), to each universe, *U*_{n}, is also more subtle than the one we used for the Toy Multiverse. Indeed, the alternative, where all of the universes in our multiverse would share a common set of parameters, \( \left\{ {x,y,z,t} \right\} \), was actually put forward by Aguirre and Tegmark (2011).

They suggested that the parallel universes (corresponding to our *U*_{n}) in their Level III Multiverse are similar to, or identical copies of, our own Hubble volume, distributed far across the cosmos. Their model fails because, as McKenzie argues (2017), the eigenstate of any given parallel universe would extend throughout the whole cosmos—that is, including regions of space that are receding from each other at superluminal speeds. Since this applies to all parallel universes, many of which are in mutually orthogonal eigenstates, the scenario of a common set of parameters is ruled out because of this potential clash of eigenstates.

Notice that, since the universes *U*_{n} are embedded within the superstructure, *M*, it is permissible to have duplicate—that is, identical—universes. Duplicate universes are identical to each other when viewed from within each one—that is, using only the unique set of emergent parameters belonging to each individual universe. (The emergent parameters may be regarded as the set of parameters, \( \left\{ {x_{n} ,y_{n} ,z_{n} ,t_{n} } \right\} \), upon which each universe, *U*_{n}, depends.) However, at the level of the superstructure, they are distinguishable by their “position” in the superstructure (formally, by the parameter(s) *θ* and the index *n* that applies to every universe *U*_{n}).

From the Reality (IV) statement, since each universe, *U*_{n}, has its own, unique set of emergent parameters, \( \left\{ {x_{n} ,y_{n} ,z_{n} ,t_{n} } \right\} \), each universe also has its own, unique reality. Because each set of emergent parameters is unique to each universe, the realities within each universe cannot ever “overlap” in any way. However, since the superstructure *M* contains all of these parallel universes, they are all part of its reality. In order to distinguish between (1) the separate realities of the individual parallel universes, *U*_{n}, of which ours is one; and (2) the reality of the superstructure, *M*, which is our multiverse, it is convenient to use the term *super*-*reality* for reality within the superstructure.

It is natural to ask what this super-reality might “look like”. Since we are constrained by the emergent parameters of our own universe, then our own universe is the only part of the super-reality that we can explore in any depth. There may be some limited insight to be gained by asking why super-reality appears to be structured along quantum–mechanical lines rather than those of any other paradigm. However, that approach might just turn out to be equivalent to asking why we were born in this century and not 200 years ago.

In the final analysis, no picture of the higher reality can ever be verified by checking it directly. By definition, the vast structure of super-reality that is inaccessible to our own universe exists only for that super-reality, lying forever hidden beyond our own horizon.

## 7 Conclusion

In this paper, we support the case proposed by others that our universe is, fundamentally, a pattern, a mathematical structure. This suggests a relatively simple definition: reality within a mathematical structure is the complete definition of that structure. The idea that our universe is ultimately a pattern may also make it easier to imagine the pattern being extended to include a whole multiverse of parallel universes (we argue that only such a multiverse can account for the absolute randomness of quantum outcomes).

However, such a picture presents us with a dilemma: in order to account for observed probabilities of quantum outcomes, there must be many identical copies of universes with particular outcomes. But this means that there must be many identical copies of particular mathematical structures. The difficulty is that every mathematical structure is unique—there cannot be two identical mathematical structures any more than there can be two identical sets of the first ten prime numbers—there is only one such set. There can be duplicate *representations* of a set or of a mathematical structure, but every structure is unique.

The resolution of the dilemma is that the structure of every parallel universe must be embedded within—must be part of—a larger pattern, which we call a mathematical superstructure. This means that there can be groups of identical universes which can be distinguished from each other (since they occupy different “parts” of the pattern of the superstructure). Thus, each universe is unique from the perspective of the superstructure (and so the requirement that every mathematical structure is unique is not violated) and yet the internal descriptions of each of these universes will be identical, which is one of the requirements of the multiverse explanation of measured quantum probabilities.

In “Appendix 1”, we note how sets of parameters can emerge from the process of defining a mathematical structure in the simplest possible terms—we call these *emergent parameters*. Each of the parallel universes in the mathematical superstructure is defined in terms of its own unique set of emergent parameters. The uniqueness of the emergent parameters in each universe means that there can be no interference or overlap between universes. However, the superstructure contains all of the unique sets of emergent parameters of these individual universes, and so every universe is accessible to the superstructure. In addition, the superstructure must contain one or more emergent parameters that are not common to any universe. Such parameter(s) allow all of the universes to be distinguished individually, and they also, presumably, account for the universes being distributed numerically in such a way that leads to the expected ratios of quantum outcomes.

Since the superstructure is a mathematical structure, then the same definition of reality applies to it as to the individual parallel universes embedded within it. This reality, however, is different from the realities within individual universes. While the realities of individual universes can never overlap (since the universes each have a different set of emergent parameters), the reality of the superstructure includes the reality of every embedded universe (since the superstructure includes all of the different sets of emergent parameters of the embedded universes). For the reality of the superstructure, we use the term *super*-*reality*.

In the end, we seem to have made a philosophical conundrum for ourselves. We have deduced that a mathematical multiverse containing parallel universes must exist as a mathematical superstructure. However, since our universe lacks most of the parameters (including the emergent parameters of the other universes) that are intrinsic to this superstructure, then the vast majority of this superstructure cannot be part of our reality. If we regard the concept of reality as synonymous with that of existence, then we have effectively proved that the superstructure does not exist, despite our earlier conclusion that it does!

Of course, it is a false conundrum, and it may be resolved by adopting a wider viewpoint. In Fig. 9, a UTMA is running in the program of each of the five UTMs, namely UTM1—UTM5. Each UTMA will deduce, from its narrow viewpoint, that the superstructure Game of Life, which is running the five UTMs, does not exist. However, from the viewpoint of the superstructure Game of Life, all five of the Games of Life embedded within its structure most certainly do exist. From within our own universe, the superstructure of the multiverse will remain forever hidden below our horizon, and the super-reality will be no more than a metaphysical curiosity. If we ever wish to glimpse and understand the super-reality that lies beyond our horizon, then we shall have to elevate our perspective accordingly.

## Notes

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