We consider partition functions, in the form of state sums, and associated probabilistic measures for aperiodic substrates described by model sets and their associated tiling spaces. We propose model set tiling spaces as microscopic models for small scales in the context of quantum gravity. Model sets possess special self-similarity properties that allow us to consider implications on large and observable scales from the underlying (non-ergodic) dynamics. In particular we consider the implication of the underlying aperiodic substrate for the well known problem of time in quantum gravity, and propose a correspondence between small and large scales, the so-called ergodic correspondence, that addresses the emergence of matter properties and spacetime structure. In the process we find a possible bound in the mass spectrum of fundamental particles.

### About Klee Irwin

**Klee Irwin is an author, researcher and entrepreneur who now dedicates the majority of his time to Quantum Gravity Research (QGR), a non-profit research institute that he founded in 2009. The mission of the organization is to discover the geometric first-principles unification of space, time, matter, energy, information, and consciousness. **

**As the Director of QGR, Klee manages a dedicated team of mathematicians and physicists in developing emergence theory to replace the current disparate and conflicting physics theories. Since 2009, the team has published numerous papers and journal articles analyzing the fundamentals of physics. **

**Klee is also the founder and owner of Irwin Naturals, an award-winning global natural supplement company providing alternative health and healing products sold in thousands of retailers across the globe including Whole Foods, Vitamin Shoppe, Costco, RiteAid, WalMart, CVS, GNC and many others. Irwin Naturals is a long time supporter of Vitamin Angels, which aims to provide lifesaving vitamins to mothers and children at risk of malnutrition thereby reducing preventable illness, blindness, and death and creating healthier communities.**

**Outside of his work in physics, Klee is active in supporting students, scientists, educators, and founders in their aim toward discovering solutions to activate positive change in the world. He has supported and invested in a wide range of people, causes and companies including Change.org, Upworthy, Donors Choose, Moon Express, Mayasil, the X PRIZE Foundation, and Singularity University where he is an Associate Founder.**

On the Emergence of Spacetime and Matter from Model Sets

Marcelo Amaral 1 *, Fang Fang 1

, Raymond Aschheim 1

and Klee Irwin 1

Citation:

. Preprints 2022, 1, 0.

https://doi.org/

Received:

Accepted:

Published:

Publisher’s Note: MDPI stays neutral

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lished maps and institutional affiliations.

1 Quantum Gravity Research, Los Angeles, CA 90290, USA

* Correspondence: Marcelo@QuantumGravityResearch.org

Simple Summary: Starting from first principles and the mathematical structure of model set tiling spaces we

propose a framework where the emergence of spacetime and matter can be addressed. The framework relies on

a correspondence between short scale non-ergodic dynamics and large scale usual dynamics.

Abstract: We consider partition functions, in the form of state sums, and associated probabilistic measures

for aperiodic substrates described by model sets and their associated tiling spaces. We propose model set

tiling spaces as microscopic models for small scales in the context of quantum gravity. Model sets possess

special self-similarity properties that allow us to consider implications on large and observable scales from

the underlying (non-ergodic) dynamics. In particular we consider the implication of the underlying aperiodic

substrate for the well known problem of time in quantum gravity, and propose a correspondence between small

and large scales, the so-called ergodic correspondence, that addresses the emergence of matter properties and

spacetime structure. In the process we find a possible bound in the mass spectrum of fundamental particles.

Keywords: Quantum Gravity; Problem of Time; Particle Physics; Emergence; Holographic Principle; Model

Sets

1. Introduction

Emergence of spacetime geometry from underlying (Planck scale) discrete structures has been

considered recently from different approaches such as entropic gravity [1,2], spin foams [3–5],

causal dynamical triangulation [6,7], causal sets [8,9] and multiway hypergraph systems [10–13].

But no clear solution to the problem has been provided and we know little about the emergence of

matter, in terms of particle and field properties [14], especially in a unified framework.

Regarding the emergence of spacetime geometry we can distinguish at least three slightly

different approaches from the references above. First, the entropic gravity approach assumes

emergence of gravity in a standard statistic mechanic and thermodynamic setup. Newton’s law of

gravitation and Einstein’s field equations are derived from the holographic principle for arbitrary

holographic screens enclosing huge numbers of matter configurations. Second, approaches such as

causal sets and multiway hypergraphs study the appearance of the Riemannian geometry of general

relativity in the continuum limit over more general discrete structures. Third, the approaches of

spin foams (or more general loop quantum gravity) and causal dynamical triangulation look for the

classical limit of quantized geometry after quantization of general relativity. Both approaches in this

third category agree that the continuous geometric structures of general relativity should emerge at

large scale from some classical or quantum underlying granularity.

In the spirit of causal sets and multiway hypergraph approaches we consider quasicrystals

defined by the geometric cut-and-projection method in its most general form, called model sets 1 [15],

as the underlying mathematical structure under which emergence can be addressed. Nevertheless the

specific mechanism for emergence we propose is more related to the entropic gravity one. If we can

understand the emergence of matter, then the entropic gravity approach can be used to understand the

large scale gravity. In the literature of quasicrystal mathematics, model sets are usually developed as

models for quasicrystal materials with good agreement with experiment [15,17–19]. For example,

the free energy, among other observables, is shown to be invariant under translation of the window

in perpendicular space, which gives a geometric understanding of the so-called phasons modes

1 Some authors prefer the terminology of “cut and projection sets” but we agree with [16] on the priority and greater generality of the term “model sets” with regard

to aperiodic structures and will use it from now on.

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© 2022 by the author(s). Distributed under a Creative Commons CC BY license.

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[19]. Here we consider implications of the same mathematical structures on much smaller scales.

The motivation for this proposal comes from different fronts, such as a renewed interest in discrete

mathematics with regards to the quantum gravity regime as discussed in the references above, and

the problem of time, where quasicrystals provide one interesting substrate to work with, which we

will discuss later. We can also point out the fact that aperiodic structures appear in solutions of

quantum cosmology both with general relativity and modified gravity theories [20] (and references

therein) and that the grand unification program, which seeks to extend the standard model of particle

physics to higher energy scales, relies heavily on large simple Lie algebraic structures, which are

deeply related to quasicrystals through their root systems [21–23]. See also our recent work on a

slightly different approach to quantum gravity [24] and references therein. More specifically, we will

focus on quasicrystals derived from the E8 root system. If the E8 grand unified theory [25,26] can

succeed, quasicrystals are a natural outgrowth of E8 structures. Finally, self-similarity properties are

a hallmark of quasicrystals and as we discuss later are good for problems involving multiple scales.

We note that renormalization, relating physics at different scales, is an open problem in quantum

gravity [5].

Our approach relies on the construction of a probabilistic measure over the model set tiling

spaces, which allows us to compute the expectation values of observables. In statistical mechanics

(classical or quantum) one in general relies on continuous symmetry invariance to guide the con-

struction of probabilistic measures using the partition function from a Hamiltonian or a Lagrangian

of the system under consideration. For discrete contexts one builds models such that the continuous

structures can be recovered in a suitable limit. And there are also discrete symmetries which can

be preserved, and which work as a guide in the same way as continuous symmetries. For example

for lattice models one usually implements a partition function with discrete translation invariance.

These kinds of model relying on translation symmetry are very successful in solid state physics with

the well known Bloch and Floquet theories [27,28]. Quasicrystals posit a more general situation

where there is no well-known procedure with regard to continuous/classical limits. Nevertheless,

there are remarkable advances on building models implementing a particular model set scale in-

variance, which highlights the self-similar character of aperiodic structures, generalizing Bloch and

Floquet theories from periodic to aperiodic structures [15,29]. We focus on the former situation

of quasicrystals considered at small scales looking for some understanding of large scale physics

in classical/continuous limits. This more general and less explored situation allows us to relax

constraints on the construction of the partition function (and associated probabilistic measure) in

terms of the underlying Hamiltonian or Lagrangian and to propose directly the partition function

in terms of a state sum model [30,31]. State sums allow for constructing models which are an

intermediate tool between the path integral for a continuous theory in quantum gravity and a lattice

approximation of the same theory, because of the state sums independence of the discretization. Our

construction of the state sum was guided only by the model set scale invariance coupled to general

assumptions such as locality and superposition. In other words, we relax the ergodic hypothesis and

consider a time average for observables not necessarily equal to the ensemble average from Gibbs’

measures [32].

Accepting that the underlying model set’s discrete structures does not necessarily follow the

ergodic hypothesis allows us to postulate a correspondence between observables computed by usual

quantum mechanics ensemble averages and the model set time averages under coarse graining. We

propose that in the observation of physical properties such as the mass of particles, there is a limit

in which the ergodic hypothesis should be recovered. Quantum theory such as the quantum field

theory of the standard model of particle physics requires some parameters, like the mass, to be

fixed by observation, which motivates the proposed correspondence, allowing us to address the

emergence of some of those quantum observables such as the particles mass. In other words, if the

short scale model set tiling space is seen as an internal structure of usual point like excitations, then

time averaging the internal tiling space properties corresponds, in the limit of large numbers of tiling

space points, to the ensemble average of the large scale system. The correspondence allows to get

intrinsic properties from the model, instead of as external parameters.

Another implication to be considered by having different probabilistic measures for large and

small scales is to use the small scale model set tiling spaces as a background for relativistic quantum

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evolution. This background allows us to address the so-called problem of time in quantum gravity

[33–36], which is the problem of implementing unitary quantum evolution in the absence of an

absolute time parameter. We follow the usual program of considering relational evolution where

some physical degree of freedom evolves relative to others. In this scenario we can distinguish three

necessary notions of time, the usual background time parameter, the global monotonic relativistic

one and a new notion of a clock variable [37]. We show that internal model set variables can play

the role of a clock variable for relational evolution. A last related implication to be discussed

is with regard to the emergence of spacetime structures such as inertia and the correspondence

principle. We already mentioned the entropic gravity approach where geodesic motion of particles

is understood as a result of an entropic force, a key element being the relationship between an

accelerated frame of reference and temperature. We address the connection between acceleration and

temperature for a given particle by proposing that a deviation from ergodicity leads to variation on

emergent temperature, while at the same time leading to acceleration through the proposed ergodic

correspondence.

This paper is organized as follows: in Section 2 we introduce the minimum concepts and

definitions of model sets necessary for our discussion. In Section 3 we present the proposed state

sum model, with associated probabilistic measure and expectation values, from the model set tiling

spaces structures, which show the possibility of more general non-ergodic dynamics. In subsection

3.2 we present a novel method of model set embedding in a periodic lattice, which enables us to

provide some explicit computations. The novel method is discussed in an explicit construction of a

three-dimensional model set derived from the E8 lattice. We present the implications in terms of

the correspondence between quantum mechanical ensemble average observables and the model set

time average ones, under coarse graining, and discuss the model set substrate playing the role of

(internal) mass, inertia and clock for relational evolution in Section 4. We present our conclusions in

Section 5. For self-consistency, we collect some results of model set theory, including the notion of

tiling spaces, in Appendix A, and in Appendix B we present additional details on the novel method

to describe model sets.

2. Model Sets Definitions

We start with some mathematical preliminaries for quasicrystals [15] relevant to our discussion.

A cut-and-project scheme (CPS) is a 3-tuple G =

(

Rd, G,L

)

, where Rd is a real euclidean space,

G is some locally compact abelian group (in general it can be any topological group) and L is a

lattice in Rd ×G, with the two natural projections π:Rd ×G → Rd and π⊥:Rd ×G → G, subject to

the conditions that π(L) is injective and that π⊥(L) is dense in G. E = Rd ×G is the embedding

space, the space Rd is called the parallel or physical space (the space of the model set) and G is the

perpendicular or internal space. Together with a given CPS we need a non-empty relatively compact

subset K ⊂ G called the window or the coding set. With L = π(L), for a given CPS π is a bijection

between L and L. Then this scheme has a well-defined map, called star map, ? : L→ G :

x

7→ x? B π⊥(π−1L (x)),

(1)

where π−1L (x) is the unique point in the set L∩ π

−1(x). The ?-image of L is denoted L?. From now

on we will restrict the internal space to be another real euclidean space G = Rd′ .

For a given CPS G and a window K, model sets can be generated by setting two additional

parameters: a shift γ ∈ Rd ×Rd′/L with γ⊥ = π⊥(γ), and a scale parameter λ ∈ R. The projected

set

4λγ(K) B

{

x ∈ L | x? ∈ λK + γ⊥

}

=

{

π(y) | y ∈ L, π⊥(y) ∈ λK + γ⊥

}

,

(2)

is called a model set. Note that we would have 4λγ(K) = 4λγ′(K) if and only if γ − γ

′ ∈ L.

Some properties are representative of model sets 4, which we review in Appendix A. Within

a model set we can have different tiling configurations T4 as reviewed in Appendix A.2. In this

case, given an xi ∈ 4 and a tiling T4, one can associate to xi different configurations of prototiles

around it, called vertex types VTi. A space-filling tiling will be given by a finite set of VTs. For

completeness and to set the notation for the following presentation, we give a general outline in

Appendix A for model sets and their associated patterns, tilings and symmetry.

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Model sets and associated tilings don’t retain discrete translational symmetry of the lattice in

the embedding space, but they exhibit new discrete symmetries [38]. We will consider the scaling

symmetry (self-similarity) associated to the scale parameter λ and the locally indistinguishable (LI)

property associated to the shift parameter γ. Within a model set for some fixed γ a position xi can

be used as a fixed point, the similarity (or homothetic) center (κ), for a local inflation-deflation

symmetry (LIDS) transformation (see A.3). A simple way to construct tilings with LIDS is by

re-scaling the window by an appropriately scale parameter λ ∈ R. Case by case we need to find a λ

such that T4(K) and T4(λK) are mutually locally derivable, T4(K)

MLD

∼ T4(λK), which clearly can

be inverted by re-scaling the window by λ−1. We illustrate the procedure in Figure 1 for the one

dimensional Fibonacci chain quasicrystals derived from the L = Z2 lattice. In this case the parallel

space Rd is rotated from the Z2 lattice vector by arctan(φ−1), with scale parameter λ = φ, where

φ =

√

5+1

2

is called the golden ratio. The set of parameters which give a good MLD equivalence

class can be written as λn = λno where λo, which works as a fundamental scaling ratio, is called an

inflation multiplier. A simple geometric derivation of the inflation multiplier for the Fibonacci chain

is given also in Figure 1. Note that when the window scales to a larger one than the original we call

it a deflation (projected points become denser), and when the window scales to a smaller one we call

an inflation (projected points become sparser).

Figure 1. Three scaled Fibonacci chains in parallel space at the bottom generated by three scaled windows,

K1 in red, K2 = φK1 in blue and K3 = φ2K1 in gray. The inflation multiplier is shown to be λo = φ. The

red window thickness is sin(θ) + cos(θ). The blue first deflation window thickness is sin(θ) + 2 cos(θ). The

black second deflation window thickness is 2 sin(θ) + 3 cos(θ) and the deflation at level n will have window

thickness Fib(n) sin(θ) + Fin(n + 1) cos(θ), with Fib given the Fibonacci sequence. It can be checked that

the ratio between the level n window thickness and level (n − 1) should be φ.

The relative frequency of VT s in some tiling Tλγ of 4λγ (VTTλγ ), given from Eq. (A1) in terms

of ratios of window polytope volumes, is

f r4λγ

(VTTλγ) =

Vol(KVTTλγ )

Vol(Kλγ )

,

(3)

where KVTTλγ is the window necessary to generate the cluster VT

Tλγ . KVTTλγ is contained in the

window Kλγ = λK + γ. This also applies to more general patterns in Tλγ, which can generate in

most cases non-space-filling tilings (see Appendix A.2, Remark A2). To get the absolute frequency

of the vertex type, VTTλγ , we take dens(4λγ) f r4λγ(VT

Tλγ). Note that the absolute frequency depends

on the scale (lg) of G, but it is the same for tilings related by specific scale transformations generated

by inflation multipliers λo, being an MLD invariant – it is invariant under LIDS.

Consider a general model set 4i = 4(Ki) where Ki can be Kλiγi or some other specification of

K, such as a vertex type window KVTTγλ . Then, given two model sets, 4i and 4 j, the overlap of

their point sets we call hits 4Hi j . Their window overlap is given by Ki j = Ki ∩ K j. The hits is then

given by

4Hi j B 4(Ki j) =

{

x ∈ (4i ∩ 4 j)

}

=

{

π(y) | y ∈ L, π⊥(y) ∈ Ki j

}

,

(4)

A measure of the global overlap of the model sets 4i and 4 j inside one larger 4k, (4i ∪4 j) ⊂ 4k

in the stack, is given in terms of their absolute frequency

Hi j = f a4k (4Hi j) = dens(4k)

Vol(Ki j)

Vol(Kk)

.

(5)

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3. Geometric State Sum Models from Model Sets, and Expectation Values

The object of interest here is analogous to the partition function in the context of statistical

mechanics and thermodynamics, or the path integral in quantum mechanics. But in a discrete setting

we move from integration to sums and, without a prior notion of Hamiltonian or Lagrangian (and

associated Gibbs measures), we directly specify the weights or amplitudes. With a more general

measure for the model set structures we will later discuss limits where one recovers measures with a

usual Hamiltonian.

Usual state sums (see [30] for a recent review) aim to construct (quantum) invariants of a

manifold from its triangulations or more general cell complex decompositions. The main point

of those constructions is the state sum’s independence of the choice of discretization. A physical

application of state sums usually considered is as an intermediate tool between the path integral for a

continuous theory in quantum gravity and a lattice approximation of the same theory. The basic idea

that appears in many state sum constructions is to have a labeling scheme for some discretization,

leading to a number which is shown to be an invariant, independent of the original discretization. In

[31] we proposed the idea of labeling schemes for discrete models based on the discrete geometry

of the discretization itself, which is straightforward to implement for general lattice-like models,

in particular quasicrystals. At first glance, a labeling scheme based on the discretization itself

seems to conflict with the desired goal of discretization independence, but this may be resolvable in

structures that possess ‘self’ properties such as being self-modeling or self-contained. The tiling

spaces of quasicrystals have such properties, so they are good candidates for this kind of scheme.

If one chooses to model some system with some discrete mathematical structure, in particular in

quantum gravity, the internal structure of the discrete system should provide the necessary elements

for dynamics without need for additional external structures such as abstract labels or parameters –

we call this geometric realism.

The tiling spaces associated to model sets are well understood, see [39] for a short review

on the subject. Given a CPS G and a window K one has model sets 4λγ and tilings Tλγ. Then a

specific tiling Tλγ is a point in the associated tiling space Xλγ of the given CPS. By varying γ one

can get more model sets and associated tilings, which are new points on this space. The tiling

space is in general very complicated but it will have nice properties, such as compactness, if the

tilings have nice properties, such as those presented in Appendix A.2 (discussed there for model sets

with the name of geometric hull, which can be continuous or discrete). Such are the tilings under

consideration here, where we will focus more on the tilings’ properties rather than the full space

properties. So a point on the Xλγ space has internal structure given by the model sets and associated

tilings’ discrete structures, allowing for the notion of geometric realism. A natural invariance on

this space is the LIDS. Given a point or tiling Tλγ of 4λγ, there is a finite number of VTs where their

relative frequencies f r4λγ

(VTTλγ), Eq. (3), are invariant under transformation to a new tiling Tλ′γ

with Tλγ MLD

∼ Tλ′γ (the mutually locally derivable relation MLD

∼

is explained in Appendix A.2).

Another object that retains LIDS invariance (under re-scaling in parallel space, which we will

discuss later) is the hits measure, Eq. (5). The relative frequencies and hits measure will be our main

building blocks in the construction of the probabilistic measure below. Given a tiling space Xλγ

for a given G and window K, we focus on one point in this space, which represents one of many

specific tiling configurations. We can then construct a partition function or state sum model for the

internal structure of this point. To specify the configuration of states and associated weights we also

consider some general principles:

1- Superposition: Given a finite set (from a finite number of translations of the shift γ) of

different tiling configurations (space-filling or not) T γ4 ⊂ T4, with T4 = {Tλγ} for fixed λ, there are

weights WT γ4 associated with them and we consider the dynamic state sum

WT4 =

∑

T γ4⊂T4

WT γ4 .

(6)

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2- Locality: Weight mainly encodes local information. Tilings allow global regularity to result

from the structure of local configurations

WT γ4 =

∏

VTi j@T γ4

WVTi j ,

(7)

where the pair of vertex types (i j) provides information on the connectivity of space-filling tilings,

and in the case of non-space-filling tilings we consider a ball of radius R (as in Appendix A, Remark

A1) with respect to the subset T γ4 . The symbol @ means that the vertex type is a valid fragment of

the tiling T γ4 .

3- Geometric realism: The symmetry of the local weights should reflect the symmetry of the

underlying discrete geometry as discussed above (see also [31,40]). Implementing the symmetry

at the level of the weight itself has the bonus that the implementation of superposition of local

configurations by the imposition of the first principle above must take into account the admissible

geometric configurations of the tiling configuration space. A signature of the local configuration is

given by the relative frequency of the local VT and its hits interactions with neighbors, Eq. (5), with

4k = 4λγ, Ki = KVTTγλ

i

, K j = KVTTγλ

j

and Kk = Kλγ , which we denote by H

λ

i j = dens(4

λ

γ)

Vol(Ki j)

Vol(Kλγ )

.

This leads to the definition of the local weights,

WVTi j = F[ f

r

4λγ

(VT i) f r4λγ

(VT j)Hλ

i j],

(8)

where the function F, which could give the usual Boltzmann weights, here will be taken as

F[ f ] =

1,

f = 0,

f , otherwise,

(9)

Theorem 1 (Geometric State Sum LIDS Invariance). For a given CPS with model sets of fixed

density, the state sum WT4 is invariant under LIDS transformation T

γλ

4 → T

γλ′

4 with T

γλ

4

MLD

∼ T γλ

′

4

.

PROOF. The frequencies Eq. (3) are ratios of volumes, which are scaled by the same inflation

parameter λn = λ′/λ, so that they are invariant under LIDS. The hits overlap with all neighbors can

also be quantified by the polytope overlap of their windows, which means it defines a locally finite

specific cluster Ci j with window KCi j such that hits reduces to the relative frequency of Ci j multiplied

by the density of 4. The density, Eq. (A3), remains constant if we re-scale the parallel space by the

same inflation multiplier. Due to the properties of locally finite structure and finite local complexity,

Remark A1, WT κ4n can always be written as a product of a finite number of the relative frequencies

of clusters Ci with windows KCi , where these clusters include the VTs and the ones generated by

the hits (Ci j). But in the larger N limit, the number of a specific cluster Ci in the general product

over a tiling must approach its frequency, so that we can write WT γ4 =

∏n′

i

(

f r4λγ

(Ci)

) f r

4λγ

(Ci)N

, with

n′ the number of clusters (Cis). As the relative frequencies are invariants, WT γ4 is invariant.

This means that WT4 is a sum over the LIDS invariants, and by fixing λ it counts just one

element of each LIDS MLD equivalence class. We note that there are two scales involved, one being

the inflation scale fixed by λ and another the scale lg for the CPS G, which sets the scale in parallel

space. Including the scale lg maintains the invariance at each fixed lg. So we assume that in WT4 , 4

has encoded the scale lg.

A probability measure on the finite tiling space is given from

µT γ4 (T

γ

4 ) =

1

WT4

WT γ4 ,

(10)

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and the expectation value of a measurable function f : T γ → R by

E4[ f ] B µT γ4 ( f ) =

∑

T γ4⊂T4

f (T γ4 )

1

WT4

WT γ4

(11)

3.1. Dynamics—Computational Aspects of Tiling Spaces

The state sum and probabilistic measure constructed above aim to describe one point in the

tiling space. To describe dynamics on the tiling space, such as the transition WT4(xi) → WT4(x j),

with xi, x j being points of the tiling space Xλγ, one would need to study its topology and attach

a metric in this space. Here we will avoid that by continuing to focus on the tiling properties

themselves.

The probabilistic measure built above allows us to define random walks on Xλγ. Basically, if

we sample a path on Xλγ with N points, tl (l = 1, ..., N), between xi and x j, each point in this path

corresponds to one tiling from the sum Eq. (6) with a certain probability. We can restrict the set of

tilings in the tiling space, and thus in each path, by considering that the initial tiling at xi have a

specific VT at the origin (let us call it κi) and requiring that the tilings in the path tl and in x j are in

the Z-module of κi, Z(T κi4 ), Eq. (A9). Then each point in the path can have a tiling from a subset

of γ translations that is in Z(T κi4 ), T

κi

4 ⊂ T4. Associated to this path there is a ordered set of tiling

configurations (an assignment of a tiling for each point), which we call an animation Atκi

A

cκi

t =

{

T κi4tl | l = 1...N

}

,

(12)

with cardinality N and cκi representing a specific set of tilings that are translations of κi. By repeating

this procedure M times we generate a set of animations Aκi

t =

{

A

cκi

t

| cκi = 1...M

}

.

Now we are in position to state a fourth constraining principle, for the state sums under

consideration, to deal with dynamics in the tiling space.

4- Principle of efficient language (PEL) [41–43]: This aims to implement a notion of efficiency

in discrete or computational systems, which are codes or languages. The random walks with their

associated animation sets Aκi

t above can be interpreted as a form of look-ahead algorithm (so-called

look-savings-ahead algorithm in [31]) that allows us to define two computational functions that can

be coupled to the geometric state sum model. First we consider the so-called hit potential, Y , which

takes values on the natural numbers including zero, Y : T λ4 → N. We start with the initial condition

of a tiling at xi with κi at the origin, which generates the subset of translations T κi4 ⊂ T4 from which

each point of one animation will be sampled. So at the initial tiling for each κ j = VT j position,

which is a valid position for κi, the hit potential Y j can be defined by the number of tilings on the

full set of animations that uses that position κ j

Y j = card

({

T κi4t ⊂ (∪A

cκi

t ) | κ j @ T

κi

4t

})

.

(13)

The second function considered is the called savings potential, S : T k → N. In each animation

A

cκi

t we count the number of vertex type κi overlaps in the union of all tilings steps

S (A

cκi

t ) = card

({

κi u (T κi4tl ⊂ A

cκi

t ), l = 1, ..., N

})

.

(14)

In this way we can consider transitions on the tiling space

WT4(x j) = WT γ4 (xi)WA

κi

t

,

(15)

with

WAκi

t

=

∑

A

cκi

t ⊂A

κi

t

S (A

cκi

t )

∑

T k⊂Acκi

t

WT k ,

(16)

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and a new state sum can be defined for transitions

WT4(x j) =

∑

T γ4⊂T4

WT γ4 (xi)WA

κi

t

,

(17)

with WVTi j = F[ f

r

4λγ

(VTi) f r4λγ

(VT j)Hλ

i jYiY j] in WT γ4 .

Now a recursive look-savings-ahead algorithm can be designed to implement dynamics such

that the use of computational resources are minimized. For example one can generate one initial

set of animations from some initial tiling. Then a second set of animations can be generated by

using the first animations to compute the hit and savings potentials, changing the probabilistic

measure through Eq. (17), and so on. The algorithms can be exploited to generate dynamics which

maximize the savings potential, which allow for minimizing the resources for handling the large set

of animations, while at the same time favoring local connectivity through the hit potential.

A last point in this section is the possibility to define expectation values. One natural expectation

value of interest is for the tiling state itself, WT γ4 ,

E4[WT γ4 ] =

1

WT4

∑

WTγ4

@WT4

W2T γ4

WT γAκi

.

(18)

Another expectation value that can be considered is one for the autocorrelation function Eq. (

A4),

E4[γ4] =

1

WT4

∑

T γ4⊂T4

γκi4WT γ4WA

κi

t

,

(19)

(20)

where γκi4 is the autocorrelation on the inital tilings with regards to the translations of κi. This

expectation value allows one to define an order parameter, called the hit average H,

H4 =

E4[γ4]

N

.

(21)

3.2. Model Set Examples

The model sets of interest here are the ones for d = 3. The most studied family of model sets

are those for which the relative frequency, Eq. (A1), of their VT s are elements of the ring of integers

[15], here called Dirichlet integers,

Z[φ] = {m + nφ |m, n ∈ Z},

(22)

where φ is again the golden ratio. This family includes the 3-dimensional icosahedral model sets

[15]. The two main examples are projected from the Z6 lattice, GZ6 = (R3,R3, Z6), and from the

D6 lattice, GD6 = (R3,R3, D6). The GZ6 model set has 24 VTs and GD6 has 36 VTs, and tables

with explicit values for the relative frequencies can be found for example in [44].

For these two model sets the scale invariant part—basically, cluster frequencies—of the state

sums, probabilistic measures and expectation values defined in the previous section, are elements of

Z[φ], as its powers are again in Z[φ] due to φ2 = φ+ 1.

The GZ6 and GD6 model sets give space-filling tilings. Possible λn-inflation non-space-filling

tilings in these cases can be re-scaled to space-filling ones. Their VT s are not regular polytopes, but

are deformed projections from groups of hyper-dimensional regular polytopes in the embedding

lattice. Other examples are derived from the E8 lattice [45], which are non-space-filling tilings,

so-called compound quasicrystals (CQC) or n-component model sets (see [15] for an example of a

3-component model set describing the Danzer’s ABCK tiling). E8 has a well known 4-dimensional

model set implementation, the so-called Elser-Sloane model set [46], which can be naturally

extended to 3-dimensional model sets [21]. The 3-dimensional E8-CQC uses a different approach

by considering 6-dimensional sub-spaces of E8. GnE8 = (R

3,R3, L6n ⊂ E8). One can employ the

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usual CPS on the 6-dimensional sub-spaces by considering the canonical window from the Voronoi

polytope there, but as we don’t need space-filling tilings, the spherical approximation here can

be taken as the window defining the CPS. The construction relies only on one window K and

λn-inflations given by powers of φ. This construction makes it possible to combine periodic and

aperiodic order by having model sets with tilings with regular VT s, which are tetrahedrons or groups

of tetrahedrons, where the groups considered are multiples of groups of four tetrahedrons, up to 20

(n4G, n = 1, 2, 3, 4, 5), that can occupy the same center position κi and generate convex hulls of

crystallographic objects such as the cuboctahedron.

This procedure gives the relative frequencies for λn-inflation tilings to be elements of Z[φ−1],

and to be the same for all n4Gs. The relative frequencies are therefore the same for the different

n4Gs, and the important part is the scale dependent density dens(4n4G).

In the next section we provide a novel method for constructing the E8-CQCs, the so-called

texture model set scheme (TMS), to do the explicit computations by embedding in a particular lattice,

a set which is isomorphic and very close to the original model set.

3.2.1. Explicit construction of the E8-CQC

An analytical method to build the Z[φ]-related model sets and the CQCs, which gives analytical

foundations for the densities and vertex type frequencies, has been recently developed. We’ll

describe the analytic construction of a model set, in lattice coordinates, with a lattice of the target

space, not of the embedding space. This makes for a huge dimension reduction and accelerates the

computation, without any loss of information of the model set.

The union of five model sets of Gn

E8

, compounded together in a unique 3-dimensional space

embedded in R3, is a discrete set in D3, the cartesian product of 3 instances of the so-called Dirichlet

ring, see equation (22). Our new description of this model set considers a discrete texture of the

target space embedded in a lattice (Z3 in 3-dimensions), and a family of coordinate transformations

between the most compact set and the exact model set in D coordinates.

Using the same notation of section 2 we define a TMS as a 3-tuple G′ =

(

Zd, G′,L

)

, where

Zd is a euclidean cubic lattice, G′ is a bounded region, the image of a unit cube under a nonlinear

transformation of Zd, and L is a lattice in Zd ×G′, with the two natural projections π:Zd ×G′ → Zd

and π⊥:Zd ×G′ → G′. With L = π(L), π is a bijection between L and L, and π⊥(L) is dense in G′.

E = Zd ×G′ is again the embedding space, the space Zd is the parallel space and G′ is the virtual

perpendicular or virtual internal space.

This scheme has also the well-defined star map applied to G′, ? : L→ G′:

? (x) = x/φ − ⌊x/φ⌉.

(23)

The star cube ?(Z3) is a cube of unit edge length, centered at point O = {0, 0, 0}. The virtual

spherical cut window is inside of this cube but has a smaller diameter φ−1 and is centered on κi.

Restricting to points that are centroids of a specific VT corresponds to operating on a subset of

the star cube. For the VT of a group of 4 tetrahedrons, whose convex hull is a cuboctahedron, the

restriction in the star cube is simply a smaller O-centred star cube of diameter 3φ−7. We’ll define in

Appendix B a function fV(k) such that all model set vertices are in fV(Z)3 and 2| f (Z)| < φ−1, and

a function f4G(k) such that all 4G-centroids are in f4G(Z)3 and 2| f4G(Z)| < 3φ−7.

More details on the CQC implementation, such as how to get the precise coordinates, are

shown in Appendix B.

The relative frequencies for tilings are computed from the distance in the perpendicular space

between the star maps of two similarity centers, κi and κ j, whose square is a Dirichlet integer.

d2(?(κi),?(κ j)) = (

−−−−→

?(κi) −

−−−−→

?(κ j))

2

(24)

The relative volume of the intersection of two balls of radius 1, whose centers are distant by d, with

respect to the enclosing ball of radius φ is (see Eq. 5)

Vol(Kκi j)

Vol(K)

=

1

16φ3

(2 − d)2(4 + d).

(25)

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If d = m + nφ the powers of φ here can be reduced to give

Vol(Kκi j)

Vol(K)

= (2φ − 3) + 3

4

((3m − 2n) + φ(−2m + n))

+

1

16

((−3m3 + 6m2n − 3mn2 + n3) + φm(2m2 − 3mn + 3n2)).

(26)

When i = j we have the relative frequency of the inflation window given by (2φ − 3) = φ−3,

which is the volume of a ball of radius one relative to a ball of radius φ. For |d| < 2/φ,

Vol(Kκi j )

Vol(K) <

φ−3(1 − 2φd/5) which is a good linear approximation.

4. Implications

In physical terms, the quasicrystal structures described by the higher dimensional model set

mathematics discussed here are positioned at large scales or low energy, such as the quasicrystal

materials built upon fundamental particles over the geometry of spacetime. The agreement of the

model set theory with experiments involving quasicrystal materials is remarkable as we discussed in

the introduction. Here we’ll discuss implications to push the model set structures to short distances,

such as the Planck scale, as a candidate for the unification structure for quantum gravity and particle

physics.

The approach of entropic or emergent gravity [1,2] considers a particle of mass m approaching

a holographic screen from the side where spacetime has already emerged. The holographic principle

is applied to assume the encoding of the microscopic configurations behind the screen, Nc, in

the area As of the screen, Nc ∝ As. Then the law of inertia, the equivalence principle, Newton’s

gravitational law, or Einstein equations can be derived. In this paradigm, gravity is an entropic force

emergent from the underlying configuration of quantum matter. One thus transfers the problem of

emergence of spacetime to the problem of emergence of quantum matter. Once one has mass and

energy distributions and their dynamics, gravity will emerge. Essentially, the holographic principle

and the associated covariant entropy bound are linked to regimes where spacetime can be classically

well approximated with sufficient matter content. Nevertheless, the linking of the bound to the area

and not to the volume of the region gives strong evidence in favor of unitarity of the underlying

quantum field theory rather than locality of quantum fields, because quantum evolution preserves

information. It is noteworthy that the locality principle in section 3 refers to the state sum weights

of the tiling space structures, which is about the local topology given from those structures. We

will require in the construction below emergent quantum evolution unitarity without need to refer to

emergent locality in a relational description.

So, how does one define evolution without a prior notion of spacetime, regarding gravity as

the geometry of spacetime? The problem is finding a unification of general relativity with quantum

mechanics in a theory of quantum gravity, usually referred to as the problem of time in quantum

gravity [33–36]. One approach, called internal time, is to consider relational evolution where some

physical degree of freedom evolves relative to others, with the dynamics governed by more general

Hamiltonian constraints [47]. This works well if the degree of freedom playing the role of internal

time behaves monotonically, which is generally not the case. The physical system’s internal time can

be a quantum field, such as a scalar field in cosmology, or some underlying structure. In general, the

choice of internal time or clocks is local and dynamic, which is the same problem as gauge fixing in

the quantization of gauge theories [37,48,49]. This leads to the conclusion that the problem of time

is a special case of the so-called Gribov problem in general gauge theories. In other words, how does

one define global evolution with respect to some underlying oscillating clock system with turning

points? One simple model system solution was given recently [37], where one distinguishes three

necessary notions of time, the usual background time parameter, the global monotonic relativistic

one and a new notion of a clock variable. One of the implications of having the model set tiling

spaces as the underlying structure is that it can play the role of the underlying clock, providing the

global monotonic time and the clock variable for relativistic unitary quantum evolution.

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4.1. An Ergodic Correspondence: Internal Mass, Clock and Inertia

In usual model sets relative frequencies are uniformly distributed, so one expects to have

ergodicity [15,29], but here the savings and the hit potentials allow for dynamics that steer probabil-

ities away from uniform distributions. Another feature that can lead away from ergodicity is the

more general possibility of less homogeneous non-space-filling tilings. A further aspect is that we

don’t want to use the model set framework to model fundamental particles here; rather, we want

to maintain the usual description at the particle scale, but make it compatible with a hypothetical

model set tiling space structure at a smaller (perhaps Planck-level) scale. So let us say we have

a quantum system in the usual equilibrium situation with a valid measure given by a usual path

integral measure [32], µPI , implementing physics laws such as momentum and energy conservation.

Now if this system has an underlying structure given by µT λ4 and we have some expectation value of

some function computed with the µPI( f ) measure, we can stipulate a correspondence with µT λ4 ( f )

by identifying a good regime where the correspondence could apply. This leads us to propose the

following function,

f h4 [ f ] = δ(µT λ4 ( f ) − µPI( f )),

(27)

which aims to connect different ensemble and time averaging measures. If we locate the quantum

system in atomic or sub-atomic scales and the tiling spaces at the Planck scale we can coarse grain

the tiling space to infer intrinsic properties of the quantum system. The function Eq. (27) allows us

to establish a correspondence between quantities in both probabilistic measures. Due to Theorem 1,

frequencies of clusters in one point (tiling) of a tiling space are natural conserved quantities under

re-scaling to be mapped to physical quantities such as mass and energy.

According to Theorem 1, the limit that is of interest for the point averaging of tiling space Xλγ

is taking a large number of tiles, N. This can be implemented for a fixed λn by just extending the

parallel space to a larger and larger size for each different γ tiling, which entails projecting more

and more points from the embedding lattice, with a corresponding higher density in the window in

the perpendicular space. Now, coarse graining can be achieved by scaling the window for specific

values of λ, which preserves LIDS, λn = λno, with an inflation multiplier λo. For the Fibonacci chain

and the examples of section 3.2 we can set the inflation multiplayer to be λo = φ−1. In this large N

regime the state sum works as a partition function, which, from the proof of Theorem 1, reduces

to a product of cluster frequencies

∏n′

i

f fiN

i

. Asymptotically this has the exponential expression

eN(λ

n)(

∑

i filn( fi)), where the number of tiles grows with the number of inflations of the window. The

new quantity that appears in the exponential we identify with the information entropy of the tiling

state,

ITk = −

∑

i

filn( fi).

(28)

In terms of the partition functions we can re-state the proposed correspondence from Eq. (27) as

∑

T γ

e−N(λ

n)ITγ −

∑

i

e−βEi = 0

(29)

which allows us to establish a correspondence between tiling state probabilities and large scale

(emergent) energy state probabilities, with N(λn) corresponding to inverse temperature, and cluster

frequencies (or their information entropy) corresponding to energy.

Consideration of explicit hadronic states [50] with Enl ∼ M

3

2

h (2n + l −

1

2 )

2

3 gives a direct map

of hadronic masses Mh and IT γ . Interestingly, for a given tiling space the values IT γ are bounded

from above. This is easily seen for Eq. (28), where ignoring hits reduces the VT frequencies, which

we compute for GZ6 as shown in Figure 2. Including hits, they are ratios of volumes of associated

clusters’ windows over a fixed larger window. Using the spherical approximation for the windows

and considering that for the model sets of interest the radius is a number in Z[φ−1] or Z[φ] such that

the radius can be written as (m + nφp) where m, n ∈ N and p ∈ Z, allows us to get IT γ from Eq. (

28). The general form of Eq. (28) is invariant over the variations of the parameters and is shown in

Figure 3 where, without normalization, it has a maximum of Euler’s number, e−1, but with most

states being on lower scales away from this maximum, and with visible gaps. An expanded view of

a smaller range is shown in Figure 4.

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Figure 2. Ik for each of the 24 VT s of GZ6 .

Figure 3. Sorted Ik for different windows with radius (m + nφp) in a large window (mmax + nmaxφpmax ).

Figure 4. Expanded view at an intermediary scale of sorted Ik for different windows with radius (m + nφp) in

a large window (mmax + nmaxφpmax ).

So far we have been concerned with one point on a tiling space and intrinsic properties of

large scale quantum systems. Fixing a CPS and a window K fixes the possible clusters’ relative

frequencies and associated information entropy, and hence a possible spectrum, such as the one in

Figure 3. To illustrate how a CPS could be used through the ergodic correspondence to predict the

hadronic spectrum we show a plot of the hadron masses (with the large mass normalized to e−1)

in Figure 5 and include the known fundamental particles such as quarks and the massive bosons

in Figure 6. Similar features to Figure 3, such as gaps, a bound on higher mass states and a dense

distribution from below can be noted, which motivates more research in this direction.

The fact that the energies Ei come from the Hamiltonian of the system suggests that we can

state this correspondence in terms of a Hamiltonian constraint δ(HT − H), with H a Hamiltonian

for the system and HT providing new variables for relational evolution from the internal averaging

over tiling space.

Consider now a specific large scale system with a canonical pair of degrees of freedom (q, p)

and some Hamiltonian H(p, q). We propose to extend the large and small (internal) correspondence

to the transitions in tiling space Eq. (15) or the new non-ergodic state sum Eq. (17). In this way a

transition in the canonical conjugate pair (q, p) from H should be accompanied by a suitable new

pair of variables coming from expectation values from the underlying tiling space with a suitable

emergent Hamiltonian HT . Now we want to synchronize transitions from an initial state qi to a final

state q j, qi → q j, with the transition between two points xi and x j in the internal tiling space Xλγ.

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Figure 5. Hadronic masses M in MeV/c2 normalized by the largest mass going to e−1. We used the

ParticleData function from the Mathematica software.

Figure 6. Hadronic and other fundamental particles masses M in MeV/c2 normalized by the bigger mass

going to e−1. We used the ParticleData function from the Mathematica software.

We re-state the correspondence Eq. (27) for the Hamiltonians HT and H as δ(HT − H), suggesting

the emergence of Hamiltonian constraints (C) for this kind of dynamics, C = HT − H = 0. To

specify a pair of of degrees of freedom for HT , we consider, as in section 3.1, that the tilings in

the steps xi → x j are sampled from a subset, which are translations of the initial tiling at xi, with

a specific vertex type κi at the origin. Note that a specific vertex type has some specific relative

frequency in a given CPS with window K, and so a specific Ik corresponding to a specific rest mass.

When sampling a path with Eq. (17), the natural variable to parameterize a path in tiling space is

the possible translations t of κi, in the parallel space of the CPS. Position in the parallel space has a

natural conjugate variable, which is the projection of the dual lattice to parallel space, and which

arises on the Fourier domain of functions in parallel space. See the variable k ∈ L = π(L), L

the dual lattice of L, in Eq. (A5) and Eq. (A6). So we have a natural pair of effective degrees of

freedom, which is the translation in parallel space t and its dual, from a Fourier transform, here called

pt, (t, pt). This permits a more explicit expression for the Hamiltonian constraint, which in the

relativistic case will be given by C = H2T (t, pt) −H

2(q.p) = 0 (for example C = p2 + m2 − p2

t − t

or C = p2 + m2 − p2

t − t2). This kind of system leads to an action principle

S =

∫ τj

τi

dτ(pq̇ + pt ṫ − NC),

(30)

where N is an auxiliary variable whereby the variation imposes the constraint C = 0. Then relational

evolution can be employed by a gauge fixing t = f (τ), which selects t as the internal time variable

[37] leading to the associated transition amplitudes given by

(qitiτi|q jt jτ j) =

∫

Dm exp

(

i

h̄

τ j

∫

τi

dτ(pq̇ + pt ṫ − NC)

)

,

(31)

with Dm an appropriate path-integral measure, and paths restricted so that q(τi) = qi, q(τ j) = q j,

t( f (τi)) = ti and t( f (τ j)) = t j. The new time variable t depends on the underlying dynamics

of transitions on tiling space, and if it has good monotonic behaviour with respect to τ, unitary

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evolution is straightforward. Even if it is an oscillating function, different pieces of evolution can

be composed together to give well defined quantum unitary evolution with the framework of [37].

Another aspect of the tiling space variable t is that it comes together with its star map t∗, where

linear translations on parallel space correspond to oscillations in perpendicular space, which can be

seen from Eq. (23) (see also [51]). With the star map, the gauge fixing t = f (τ) allows for unitary

evolution by means of three notions of time – the usual global background time, and the new pair of

time parameters, with a clock variable t∗, and a possible global monotonic t parameter, which is the

subject of our current research in progress.

To finish we will just touch on the implications for more general dynamics related to the

emergence of spacetime structures—essentially, how the underlying tiling spaces and associated

partition functions and measures relate to the emergence of inertia and the equivalence principle.

Going back to the discussion on emergent gravity and the holographic principle at the beginning

of this section, the idea is to understand geodesic motion of particles as a result of an entropic

force. Following [1] one introduces a static background with a global timelike Killing vector ξa, and

considers a generalization of Newton’s potential ϕ = 12 log(−ξaξa). Then one can use ϕ to define a

foliation of space, and put general holographic screens at surfaces of constant redshift to consider

a force on a particle of mass m close to one screen. The velocity ua and the acceleration ab of the

particle can be expressed in terms of ξa and ϕ, and in particular the acceleration can be expressed as

the gradient ab = −∇bϕ.

The local temperature T on the screen, using the Davies–Unruh temperature [1], can be defined

by

T =

h̄

2π

eϕNb∇bϕ

(32)

where Nb is the outward pointing vector normal to the screen and to ξb, as the acceleration is

perpendicular to the screen. The redshift factor eϕ is because the temperature is measured with respect

to the reference point at infinity. Leaving the details aside, Eq. (32) brings together thermodynamics

on one side (through the temperature) and gravitational force on the other (through the potential),

leading to a relativistic analogue of the law of inertia. The right side of Eq. (32) can be interpreted,

with the equivalence principle, either in terms of a gravitational field or an accelerated frame of

reference, and both views can be seen as emergent phenomena. Our point with this short review

of the framework of emergent gravity is to point out the key role of the (somewhat mysterious)

correlation between temperature and acceleration (and so gravity), where Eq. (32) is interpreted as

telling us the temperature needed to cause a certain acceleration ab and not the other way around.

This picture finds a natural implementation with the short-scale non-ergodic tiling-space-dynamics

background. First one considers a transition in tiling space given by Eq. (15) and a probabilistic

measure, Eq. (10), from the state sum Eq. (17), for a tiling space of translations of some κi vertex type,

where the look-savings-ahead algorithm dynamics is such that the ergodic correspondence holds –

there is some associated number of clusters dependent on the inflation scale N(λn), and from Eq. (

31) there is a gauge connecting the tiling translation parameter and the relativistic time parameter

t = f (τ). Now consider the same transition, where the savings potential from a look-savings-ahead

algorithm leads to a deviation from ergodicity. As the savings potential can give different probability

for different paths, it can change t and N(λn). Specially, we linked N(λn) with temperature such

that the deviation from ergodicity can be connected with a delta in temperature and then, following

the entropic gravity approach, with a non-ergodic entropic force. So we have correlated Eq. (32)

with a deviation from ergodicity, the quantitative analyses of which we will discuss elsewhere.

5. Conclusions

In this paper we have presented the idea of more general small-scale (time average) probabilistic

measures implemented by quasicrystal structures modeled by the model set framework. This

procedure is general such that the model set structure can be replaced by other mathematical discrete

structures. Our main point on the correspondence given by Eq. (29) is that the small-scale structures

must have some imprint on large scale quantum observables to be a meaningful scientific proposal.

Our specific construction leads to the possibility of having a specific CPS predicting masses on

particle physics, and with a limit on the highest mass that can be measured. This should be well

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below the Planck mass, considering the model set structure closer to the Planck scale. This is because

in the ergodic correspondence proposed, in the short-distance tiling space side of the correspondence,

ITγ is bounded from above.

In summary, the measure µT k4 allows one to establish a correspondence between tiling space

invariants and intrinsic properties of fundamental physical systems such as the mass discussed above.

It can also provide, through the internal tiling space structures, a clock for relational evolution,

as well as a delta in temperature for the emergence of inertia and the correspondence principle in

the framework of entropic gravity. Gravity as an emergent entropic force, which requires a large

number of constituents according to the usual definition of emergence, normally suggests it would

be only a classical phenomenon without need for quantization—an aggregate result of microscopic

interactions of quantum matter. Here, by contrast, temperature and hence an entropic force can arise

for even a single particle with regard to its internal tiling space, making its quantization meaningful.

We note that the time average probabilistic measures here considered play a role similar to that

of fast variables in [14], or the neural network variables in [52,53]. As future research, model

sets may provide one way to reconcile discrete, finite systems with invariance under continuous

symmetry groups, addressing the problem "How can discrete, finite systems such as fields on a

lattice, display invariance under continuous symmetry groups?" [54]. The novel possibility arises, in

terms of spacetime symmetry, by realizing that the discrete and continuous can coexist over LIDS

transformations. And in terms of charge gauge symmetry, by generalizing the group G in the CPS G

to more general groups such as the Lie groups necessary for the standard model of particle physics,

S U(3), S U(2) and U(1).

As a final note, a non-ergodic substrate for quantum gravity, with an associated non-ergodic

causal entropic force, is in line with the ontological hypothesis about the status of reality presented

in [43].

Acknowledgments: We thank and acknowledge Dugan Hammock for providing the base code used to test

some of the concepts discussed in this paper, such as the CQC model set derived from E8.

Appendix A. Review of Model Sets

Some properties are representative of model sets 4, and are necessary to derive the concept of

a tiling space. For completeness we review them here, following mainly [15,55].

Appendix A.1. Model Set Properties

Remark A1 (Model set properties). 1.

Uniformly discrete: There is a radius r > 0 such that

each ball of radius r contains at most one point of 4;

2.

Relatively dense: There is radius R > 0 such that each ball of radius R contains at lest one

point of 4;

3.

Locally finite: For all compact Λ ⊂ Rd, the intersection C = Λ∩ 4 (called cluster) is a finite

or empty set;

4.

Finite local complexity (FLC): The collection

{

(t + Λ) ∩ 4 | t ∈ Rd

}

, usually represented by

translations 4 − 4, contains only finitely many clusters up to translations.

5.

Uniform distribution: A theorem proved in [15], which says that given some ordering sequence

(xi)i∈N of the points on 4, the sequence (x?

i )i∈N is uniformly distributed in K. The star map,

?, is given in Eq. (1).

Point sets with properties 1 and 2 above are called Delone Sets. A characteristic property of a

point set in Rd is the average number of points per unit volume. We are interested in the relative

frequency f r and absolute frequency f a of sub-sets or clusters of a model set. Due to the uniform

distribution property, the relative frequency of a cluster can be computed as a ratio of window

volumes

f r4(C) =

Vol(KC)

Vol(K)

,

(A1)

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where KC is the compact window or coding region of C ⊂ 4, given by KC B ∩x∈C(K − x?), and

Vol gives the volume of the window. The absolute frequency is given by

f a4(C) = dens(4) f r4(C),

(A2)

where dens(4) is the density of the model set, given by

dens(4) = limr→∞

card(4r)

Vol(Br(0))

,

(A3)

where for model sets the limit inferior and superior coincide. The open ball of radius r around x is

given by Br(x), and card is the set cardinality.

A hallmark function widely used to characterize a model set is the autocorrelation function

γ4 =

∑

t∈4−4

dens(4)Vol(K ∩ (K − t

?))

Vol(K)

δt,

(A4)

from which the diffraction spectrum can be computed from the Fourier transform

γ̂4 =

∑

k∈L

I(k)δk.

(A5)

Here L = π(L), with L the dual lattice of L, and the coefficients I(k) = |A(k)|2, which in the case

of internal space Rd is given by

A(k) =

dens(4)

Vol(K)

∫

K

e2πi(yk

?)dy.

(A6)

In practical application it is useful to use the spherical approximation of the window K, a ball

BRw = BRw(0) of radius Rw, in spherical coordinates

A(k) =

dens(4)

Vol(BRw)

∫

BRw

e2πi(yk

?)dy = dens(4)

Γ( d2 + 1)

(|k?|πRw)

d

2

J d

2

(2π|k?|Rw),

(A7)

where Γ is the gamma function, J a Bessel function of the first kind, and the radius of the approxi-

mating spherical window is given by

Rw =

(

vol(K)

π

d

2

Γ(

d

2

+ 1)

) 1

d

.

(A8)

Appendix A.2. Model Set Patterns and Tilings

A pattern T in Rd (T @ Rd) is a non-empty set of non-empty subsets of Rd. The elements of

T are the fragments of the pattern T . For example, a locally finite point set such as 4 is naturally

turned into a pattern as T = T4 = {{x} | x ∈ 4}.

A tiling in Rd is another example of a pattern with T = {Ti

| i ∈ I} @ Rd, where I is a

countable index set, and such that the fragments Ti of T are non-empty closed sets in Rd subject to

the conditions

Remark A2 (Tiling conditions). 1.

∪i∈ITi = Rd,

2.

int(Ti) ∩ int(T j) = Ø for all i , j and

3.

Ti being compact and equal to the closure of its interior Ti = int(Ti).

If we release condition 1 above we have a non-space filling tiling as another example of a

pattern. The Ti are called regular tiles of the tiling and their equivalence class up to congruence

are called prototiles. Recall the locally finite property of point sets from Remark A1. This extends

naturally to the pattern T , as does the definition of a cluster C = ΛuT of T .

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Two locally finite patterns T and T ′ are locally indistinguishable (LI), T LI∼ T ′, when any

cluster of T occurs also in T ′ and vice versa. This means that there are translations t, t′ ∈ Rd such

that T uΛ = (−t′ +T ′) uΛ and T ′ uΛ = (−t +T ) uΛ. The translation of a tiling is understood

as t + T = {t + Ti | i ∈ I}. Locally indistinguishability is an equivalence on the class of patterns,

written as LI(T ).

A Z-module for a given Λ and T , ZΛ(T ), for some x ∈ Rd, is defined as

ZΛ(T ) =< t | T u (x + Λ) = (−t + T ) u (x + Λ) >Z,

(A9)

which is the Z-module generated by all translations between occurrences of some Λ-cluster in T .

When Λ ⊂ Λ′, one has ZΛ′(T ) ⊂ ZΛ(T ) and ZΛ∪Λ′(T ) = ZΛ(T ) ∩ZΛ′(T ). The limit translation

module (LTM) Z(T ) is then defined as the inductive limit of the ZΛ(T ) over all Λ ⊂ Rd, ordered

according to inclusion. The Z(T ) is an invariant of LI(T ).

A pattern T ′ @ Rd is locally derivable (LD) from a pattern T , T LD∼ T ′, when a compact

neighborhood Λ ⊂ Rd of 0 exists such that whenever (−x + T ) u Λ = (−y + T ) u Λ holds for

x, y ∈ Rd, one also has (−x + T ′) u {0} = (−y + T ′) u {0}. This extends to LI classes of patterns.

A class LI(T ′)is called LD from LI(T ), LI(T ) LD∼ LI(T ′), when patterns T1 ∈ LI(T ) and

T ′1 ∈ LI(T

′) exists such that T1 LD∼ T ′1 . This also applies to the point set itself: two model sets

obtained from the same CPS, but with different windows K1 and K2, satisfy 4(K1) LD∼ 4(K2) if and

only if K2 can be expressed as a finite union of sets each of which is a finite intersection of translates

of K1, with translations from L?.

Two patterns T1 and T2 (similarly for two LI classes) are called mutually locally derivable

(MLD) from each other when T1 LD∼ T 2 and T2 LD∼ T 1. MLD is an equivalence relation on patterns

(or LI classes), T1 MLD

∼ T2. It is straightforward to show that with T LD∼ T ′ one has Z(T ) ⊂ Z(T ′)

and with T MLD

∼ T ′ one has Z(T ) = Z(T ′). Then the LTM Z(T ) of T is an invariant of the entire

MLD class of LI(T ).

A pattern T is translationally repetitive when, for every compact Λ there is a compact Λ′ ⊂ Rd

such that for every x, y ∈ Rd, the relation T u (x + Λ) = (−t +T )u (y + Λ) holds for some t ∈ Λ′.

The set Λ′ quantifies the local search space to locate arbitrary Λ-clusters of T . For T LD∼ T ′, if T is

repetitive, then so is T ′.

Finally it is possible to derive a concept of tiling spaces. For simplicity we define the local

topology for two FLC sets 4, 4′, but it is straightforward to generalise to patterns. So two FLC sets 4,

4′ are ε-close when one has 4∩ B1/ε(0) = (−t + 4′) ∩ B1/ε(0) for some t ∈ Bε(0). The topology

is generated by the possible neighborhoods with all ε > 0 sufficiently small. This topology permits

the concept of a continuous tiling space X(4) given by X(4) =

{

t + 4 | t ∈ Rd

}

, with the closure

in the local topology. The discrete tiling space, X0(4), is given by X0(4) =

{4′ ∈ X(4) | 0 ∈ 4′}.

Appendix A.3. Model Sets Inflation-Deflation Symmetry

To go beyond the "classic" symmetries, one needs an extension of other invariance properties

to discrete structures. A discrete structure 4 is said to have a local inflation-deflation symmetry

(LIDS) relative to a linear map L if 4 and L(4) are MLD, 4 MLD

∼ L(4). When L(x) = λox (or

L(x) = λoRx in general, with R in the orthogonal group, R ∈ O(d,R)), the number λo is called the

inflation multiplier of the LIDS. A necessary condition for L to define an LIDS is Z(L(4)) = Z(4).

There are different methods to generate LIDS tilings. On concrete method is to consider an

inflation rule in one direction, which consists of the homothetic mappings

λTi

7→ ∪nj=1T j + Ai j

(A10)

with finite sets Ai j ⊂ Rd, subject to the mutual disjointness of the interiors of the sets on the right hand

side and to the (individual) volume consistence condition vol(Ti) =

∑n

j=1 vol(T j)card(Ai j), both

for each 1 ≤ i ≤ n. In the other direction the inverse map is a consequence of local recognisability

[15,56]. For more details on the construction of self-similar aperiodic tilings using inflation/deflation

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composition rules see for example [56]. Another method to generate LIDS tilings is by re-scaling

the window by an appropriated inflation multiplier as discussed in the main text. Many concrete

examples are discussed in [15], see for example [15, Proposition 6.3] for the Penrose–Robinson

tiling, [15, Corollary 6.9] for the Danzer’s ABCK tiling, [15, Remark 7.7] for the Ammann–Beenker

point set and [15, Remark 9.10] for the silver mean chain. Remarkably, if a tiling possesses a LIDS,

then the tiling is non-periodic, [15, Theorem 6.3].

Appendix B. 3-Dimensional Compound Quasicrystal from the E8 Lattice

The canonical coordinate system of the E8-CQC (Compound Quasicrystal) can be given using

an enumeration function. An enumeration function is an odd and growing function in and on Z

fV : Z→ Z|(∀x ∈ Z, fV(x) = − fV(−x)) and (∀x, y ∈ N2, fV(x + y) ≥ fV(x)),

(A11)

and the canonical enumeration function is

fV(x) = bxφe.

(A12)

The values taken by the function fV(Z) are given by the integer sequence A007067 in the on-line

encyclopedia of integer sequences (OEIS) [57] and its first 20 values are: {0, 2, 3, 5, 6, 8, 10, 11,

13, 15, 16, 18, 19, 21, 23, 24, 26, 28, 29, 31}. The density of fV(Z) in Z, or the probability of an

integer to be in the image of f , is δ( f ) = φ−1. The interval function, or discrete derivative d fV(x),

is the palindrome Fibonacci word encoded with Long=2 and Short=1, given by the integer sequence

A006340 in the OEIS [57]

d fV(x) = fV(x + 1) − fV(x) =

⌊

(x + 1)φ

⌉ − bxφe.

(A13)

The canonical coordinate system of the E8-CQC is the image of Z3 by the canonical enumeration

function: fV(Z3). In the following we will need the Elser-Sloane conditions:

1.

The canonical planar elimination function h(x, y) partially encodes the fact that the E8-CQC

is constructed from specific slices of an Elser-Sloane projection of the E8 Lattice:

h(x, y) = hx8(x, y) ≡ 0 (mod 4) or hy8(x, y) ≡ 0 (mod 4),

(A14)

where

hx8(x, y) =(−x + 2y − 3

⌊

x/φ

⌉ − 3⌊y/φ⌉) mod 8,

(A15)

hy8(x, y) =(−3x − 4y − 3

⌊

x/φ

⌉

+ 3

⌊

y/φ

⌉

) mod 8.

(A16)

2.

The canonical volumetric elimination function h(x, y, z) fully encodes the fact that the E8-CQC

is constructed from specific slices of an Elser-Sloane projection of the E8 Lattice, including

the constraints of the planar elimination function as a subset. (Formally, the planar function is

redundant, but the volumetric function is more computationally intensive. On large data sets,

therefore, it is more efficient to first apply the planar function, and then apply the volumetric

function to those vertices that remain.) The volumetric function is given by

h(x, y, z) =

−1

2

0 −3 −3

0

−1 −4

2

1

1

2

−3 −4

2 −3

1 −4

−1 −4 −4

1 −1 −2

−1

2

2 −1 −3

2

1

2

2

3 −3

0

−3 −4

0 −3

3

0

−3

2 −4 −3 −3 −2

x

y

z

⌊

x/φ

⌉

⌊

y/φ

⌉

⌊

z/φ

⌉

≡ (0||4) (mod 8),

(A17)

where (0||4) means a vector whose components are all 0 or all 4, i.e., the 8 components of the

resulting vector are either all congruent to 0 modulo 8, or all congruent to 4 modulo 8. The

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matrix given in Eq. (A17) is deduced from the orientation of the six dimensional subspace of

E8 projected by the Elser-Sloane projection to the selected 3-dimensional slice in which the

point (x′, y′, z′) lives.

3.

The canonical spherical elimination function s(x, y, z) encodes the fact that the star map image

is restricted to a sphere of radius φ − 1

s(x, y, z) = ?(x)2 + ?(y)2 ? (w)2 < r2.

(A18)

Finally, the model set coordinates x′, y′, z′ in D3, and the coordinates x”, y”, z” in Z3, here

called the crystal proxy, are deduced from the set {x, y, z} by the following bijections between x, x′

and x” (and the same for y and z):

x′(x) =10x −

√

20? (x)

(A19)

x(x′) =

⌊

x′/10

⌉

(A20)

x”(x) =10x −

⌊√

20? (x)

⌉

(A21)

x(x”) =

⌊

x”/10

⌉

(A22)

x′(x”) =10

⌊

x”/10

⌉ − √20? (⌊x”/10⌉)

(A23)

x”(x′) =

⌊

x′

⌉

.

(A24)

From the vertices found in the canonical coordinates Eq. (A12), the E8-CQC is comprised of those

selected by Eqs. (A14,A17,A18).

There are further restrictions of possible interest. As a first step, the E8-CQC-T is a selection

from the E8-CQC of only those vertices forming regular tetrahedrons in the slice (corresponding

to regular tetrahedrons in E8). One way to implement this is an elimination function, applied on

fV(Z3), but it is proved that they don’t use the full set fV(Z), so it is more efficient to compose

the enumeration function fV with a second enumeration function gT , where the composition gives

fT . At the next step, the E8-CQC-4G is formed by keeping only the “4-groups”, the tetrahedrons

which meet as a group of four at one common vertex such that their convex hull is a cuboctahedron.

We know that they will correspond to the equator of a 24-cell in the E8 Lattice. Again, one way to

implement this is an elimination function, applied now on fT (Z3), but it is more efficient to compose

the enumeration function with a new enumeration function g4G, to give f4G

g4G(x) = ((x mod 3) > 0)

⌊

φ(x mod 3)−3(5(

⌊

φ(

⌊

x/3

⌋

+ 1)

⌉ − ⌊φ⌊x/3⌋⌉) + 3)⌉

+5

⌊

φ

⌊

x/3

⌋⌉

+ 3

⌊

x/3

⌋

,

(A25)

f4G(x) = ( fV ◦ g4G)(x) = fV(g4G(x)),

f4G(x) = ((x mod 3) > 0)

⌊

φ(x mod 3)−3(8(

⌊

φ(

⌊

x/3

⌋

+ 1)

⌉ − ⌊φ⌊x/3⌋⌉) + 5)⌉

+8

⌊

φ

⌊

x/3

⌋⌉

+ 5

⌊

x/3

⌋

.

(A26)

Note that if we substitute 5, 3, 5 for 2 and 3, 2, 3 for 1 in the word Eq. (A13), we obtain

dg4G(Z):

dg4G(x) = g4G(x + 1) − g4G(x).

(A27)

Details on python and Wolfram language implementations will be discussed elsewhere.

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