Observables for cyclic causal set cosmologies

In causal set theory, cycles of cosmic expansion and collapse are modelled by causal sets with"breaks"and"posts"and a special role is played by cyclic dynamics in which the universe goes through perpetual cycles. We identify and characterise two algebras of observables for cyclic dynamics in which the causal set universe has infinitely many breaks. The first algebra is constructed from the cylinder sets associated with finite causal sets that have a single maximal element and offers a new framework for defining cyclic dynamics as random walks on a novel tree. The second algebra is generated by a collection of stem-sets and offers a physical interpretation of the observables in these models as statements about unlabeled stems with a single maximal element. There are analogous theorems for cyclic dynamics in which the causal set universe has infinitely many posts.

In causal set theory, cycles of cosmic expansion and collapse are modelled by causal sets with "breaks" and "posts" and a special role is played by cyclic dynamics in which the universe goes through perpetual cycles. We identify and characterise two algebras of observables for cyclic dynamics in which the causal set universe has infinitely many breaks. The first algebra is constructed from the cylinder sets associated with finite causal sets that have a single maximal element and offers a new framework for defining cyclic dynamics as random walks on a novel tree. The second algebra is generated by a collection of stem-sets and offers a physical interpretation of the observables in these models as statements about unlabeled stems with a single maximal element. There are analogous theorems for cyclic dynamics in which the causal set universe has infinitely many posts.
In this work, we explore a class of sequential growth dynamics that is of particular interest to cosmology: cyclic sequential growth models, in which the causal set (causet) universe goes through perpetual cycles of expansion and contraction punctuated by breaks, where a break is an ordered partition (Ã,B) of the causet that satisfies a ≺ b ∀ a ∈Ã, b ∈B. A subclass of causets with infinitely many breaks is that of causets with infinitely many posts, where a post is a single causet element with a break immediately below it and a break immediately above it. An illustration is shown in Fig.1. Cyclic models play a central role in the causal set cosmological paradigm, a heuristic that aims to explain the emergence of a flat, homogeneous and isotropic cosmos * Corresponding author: stav.zalel11@imperial.ac.uk directly from the quantum gravity era [7][8][9]. This proposal is one of several in a recent trend to develop theories of cyclic or bouncing cosmologies and determine their implications for fundamental physics. In this work, we take a step towards advancing this school of thought within causal set theory. The cyclic models which we study are those dynamics which give rise to cyclic universes and our aim is to investigate the question: what are the physical observables (covariant events) in these cyclic models?
We note that the cosmological paradigm within which we will be working pertains only to the causal set spacetime, not to any matter living on it. Whether a causal set is enough to give rise to matter degrees of freedom [1] or whether one requires additional structure such as a field living on the causal set [13,14] is still unknown. Whichever the case may be, this simplified cosmological paradigm will act as a guide to building a causal set cosmology.
In section II we review the concepts of sequential growth models, events and covariant events and known results. In section III we identify a σ-algebra R b that forms a complete set of covariant events in cyclic dynamics. We prove that R b can be constructed using a particular subcollection of the cylinder sets and discuss the implications of the result for the search for classical and quantum cyclic dynamics. However, we note that the physical interpretation of events in R b remains obscure and in section IV we identify a second σ-algebra of observables, R( S) that endows each observable in a cyclic dynamics with a physical interpretation. We prove that this physically meaningful algebra exhausts the algebra of covariant events in a well defined way. We will consider in detail the general cyclic dynamics defined above in which the infinitely many epochs (cycles) are separated by breaks. At the end of the paper, in section V we describe how our results carry over mutatis mutandis to the special case where the epochs are separated by posts. A sequential growth dynamics is a probability space (Ω,R, µ).Ω is the set of causal sets (causets, for short) on the ground-set N that satisfy x ≺ y =⇒ x < y for all x, y ∈ N, where ≺ denotes the partial ordering. LetC n denote a causet on ground-set [0, n − 1] satisfying x ≺ y =⇒ x < y. For eachC n there is the cylinder set, whereC| [0,n−1] denotes the restriction ofC to [0, n − 1]. R is the σ-algebra generated by the cylinder sets. The measure µ is the extension-via the Fundamental Theorem of Measure Theory [15]-of the measure on the semi-ring of cylinder sets given by a random walk on labeled poscau. Labeled poscau is a directed tree in which eachC n is a node andD m ≺C n ⇐⇒D m is a stem (a finite down-set) inC n (Fig.2). Then µ(cyl(C n )) = P(C n ), where P(C n ) is the probability that the random walk goes through the nodeC n [1,10]. The algebra of covariant events, R, is a subalgebra of R: where ∼ = denotes equivalence under order-isomorphism.
Within this framework for the dynamics of a discrete universe, the events in R are the physical "observables" (or beables) [3]. Each event E in R corresponds to the question "Does E happen?" to which the measure responds: "Yes, with probability µ(E)" (or "Almost surely no" if µ(E) = 0).

B. Rogues
Looking closely, one finds that a generic event in R has no obvious physical interpretation. However there exists a strictly smaller σ-algebra whose elements do have a clear physical meaning. Let C n denote an unlabeled causet (or order ), i.e. the order-isomorphism equivalence class of which the causetC n is a representative. For each C n , define the stem-set, where the union is over causetsD m that contain a stem that is order-isomorphic toC n , for all m (see Fig.3 for an illustration of stem). Let S and R(S) denote the set of stem-sets for all n and the σ-algebra generated by S, respectively. The elements of S have a comprehensible physical meaning: they correspond to countable logical combinations of statements like "the causet has a stem isomorphic to a representative of the finite order C n ." R(S) is the prime example of a physically comprehensible subalgebra of the covariant event algebra R. What physics does R(S) leave out, what physical information is not captured by the stem questions? The answer, almost by definition, is that stem questions cannot distinguish between two causets that have the same stems. We call an infinite causetŨ ∈Ω a rogue if there exists someṼ ∼ =Ũ such thatṼ ∈ stem(C n ) if and only ifŨ ∈ stem(C n ), for all stem(C n ) ∈ S. Let Θ denote the set of rogues. In [3,16] it was proved that Θ is measurable (i.e. Θ ∈ R) and indeed that Θ ∈ R(S). The main theorem of [3] is that for every event E ∈ R, there is an event E ∈ R(S) such that E E ⊂ Θ. It is in this precise technical sense that the rogues make up the difference between the covariant events and the stem events (elements of the stem event algebra R(S)).
We can go a little further and prove Lemma 0. Let E ∈ R. Then E ∩ Θ c ∈ R(S) and E ∪ Θ ∈ R(S), where the superscript c denotes complement inΩ.
Proof. Define F := E ∩ Θ c ∈ R. By the theorem mentioned above there is some F ∈ R(S) such that we have E ∩ Θ c ∈ R(S).
Thus, removing the rogues from a covariant event turns it into a stem event. And adding all the rogues to a covariant event turns it into a stem event. This motivates the defining of another algebra that will have a direct analogue when we come to discuss cyclic dynamics in the next section. Let R Θ be the the σ-algebra of all covariant events that either contain all the rogues or contain no rogues: In any dynamics in which the rogues have measure zero, i.e. µ(Θ) = 0, R Θ is then a sort of doubled physical event algebra where C. CSG models, a special class Crucially, it was also proved that in every Classical Sequential Growth (CSG) model, the most-studied class of sequential growth dynamics [1], the set of rogues Θ has measure zero [3,16]. Combined with the measure independent theorem stated above, this means that in every CSG model, for every covariant event there is a stem event such that their difference is of measure zero. Indeed, the lemma proved above says that removing the rogues from a covariant event turns it into a stem event and in dynamics satisfying µ(Θ) = 0, such as CSG models, µ(E) = µ(E ∩ Θ c ) for any covariant event E. So, for CSG models, a physically motivated class of sequential growth dynamics, the stem observables exhaust the physical observables in this well-defined sense.
Finally we can summarise the relations between the σ-algebras mentioned so far: R(S) is, in this sense, an "over-complete" set of observables in dynamics in which the set of rogues has measure zero. It is the simple physical interpretation of R(S) that makes it the most meaningful choice of physical event algebra, or its elements the most meaningful choice of physical observables. The significance of the results about R(S) is two-fold. First, as mentioned above, we can now assign a clear meaning to every dynamically relevant observable in dynamics satisfying µ(Θ) = 0: it is a logical combination of statements about which (unlabeled) stems are contained in the causet spacetime. For example, the statement "the causal set universe has a unique minimal element" corresponds to a stem event because it is equivalent to the statement "the causal set universe does not contain the 2-antichain as a stem". This event is particularly interesting for causal set cosmology, as it can be interpreted as a Big Bang event. Second, this result has led to the construction of covtree-a tree on which every random walk corresponds directly to a measure on R(S) (and vice versa)-a new tool for studying causal set dynamics [10][11][12]. More generally, a better understanding of the structure and sub-algebras of R can enable us to develop new methods through which to define a measure µ and thus to establish new avenues for seeking physically motivated dynamics.

A. Cyclic models
Our theory space is that of the sequential growth models of section II A of which CSG models are a special case. Our theorems make no use of the particular properties of CSG models but we have them in mind as a physically motivated class of models to which we can apply the theorems.
Let us call a causet with infinitely many breaks, a cyclic causet and let B ∞ denote the event that the causet is cyclic, i.e., Proof. Let event Γ n (A n ) be the event "the causet has a break with past A n " where A n is an n-order. In [11] it was proved that the event Γ n (A n ) is a stem event: whereÂ n is the covering order of A n , the (n + 1)-order formed by adding a single element that is above every element of A n , and where the intersection is over all (n + 1)-orders not equal toÂ n . Now let event Γ n be the event "the causet has a break the cardinality of whose past is n" (n > 0 by definition of break).
where the union is over all n-orders. B c ∞ is the event that the causet has finitely many breaks. A causet has finitely many breaks if there exists a k ∈ N such that it has no break whose past has cardinality greater than k. Let ∆ k be the event that the causet has no break whose past has cardinality greater than k.
And then Hence, B ∞ and its complement are elements of R(S).
Since R(S) ⊂ R, every event in R(S) is measureable in any sequential growth model. Therefore, it is a corollary of lemma 1 that B ∞ is measureable in any sequential growth model. This result enables us to define a cyclic model as follows: a cyclic model is a sequential growth model in which µ(B ∞ ) = 1.
A cyclic model may or may not be a CSG model and a CSG model may or may not be cyclic, though the bestunderstood growth dynamics-Transitive Percolationis both, since it is a CSG model in which the causet spacetime almost surely has infinitely many posts [17][18][19][20][21].
Our motivation for studying cyclic models is the key role that they play in the the causal set comological paradigm that aims to explain the emergence of a flat, homogeneous and isotropic cosmos directly from the quantum gravity era [7][8][9]. Our goal in this paper, is to classify the observables in these models. Now, in analogy to definition 4, we define R b to be the σ-algebra of all covariant events that either contain all or none of the causets which are not cyclic: where B c ∞ denotes the complement of B ∞ , i.e. the set of causets which are not cyclic. It follows from our definition of cyclic models that µ(B c ∞ ) = 0 in any cyclic model, and therefore R b exhausts the covariant events in a cyclic model in the same way that R Θ does in dynamics which satisfy µ(Θ) = 0. Note, however that the events in R b are not necessarily stem events and so do not in general have a physical interpretation. In section IV we will prove a theorem in exactly analogous form to the result of [3] and identify the covariant, physically interpretable observables in a cyclic dynamics.

B. R b is generated by principal cylinder sets
In this subsection, we prove that R b contains exactly the covariant events in the sigma algebra that is generated by a strict subset of the set of all cylinder sets.
The following terminology will be useful. We callC n a principal causet if (i) it is a causet on ground-set [0, n−1] satisfying x ≺ y =⇒ x < y, and (ii) it has a unique maximal element. If a principal causetC n is a subcauset of someC ∈Ω thenC n is a stem inC and we say that C n is a principal stem inC. We call cyl(C n ) a principal cylinder set ifC n is a principal causet. LetR b denote the σ-algebra generated by the set of all principal cylinder sets.
As we will prove below, the notion of principal causets and principal cylinder sets is closely related to the notion of observables in cyclic dynamics. At this stage, we can already develop an intuition for why this is the case. Suppose the left hand diagram of Fig.1 represents a break (Ã,B) in a cyclic causetC, and let n := |Ã|. Then element n is a minimal element inB and the causetÃ ∪ {n} is a principal causet and a principal stem inC. One can also see that every stem of cardinality n + 1 inC is isomorphic toÃ ∪ {n}. So, every cyclic causetC contains countably many principal stems -one for each break -and every stem inC is contained within some principal stem inC. The intuition is that all the physical information aboutC is encoded in the principal stems it contains. In the rest of this section, we make this intuition mathematically precise.
We can now state our theorem: and we spend the rest of this subsection proving it. We will use the following terminology when discussing breaks. Given a break (Ã,B),Ã andB are called the past and the future of the break, respectively. If a causet C contains more than one break we order its breaks by the cardinality of their pasts: A segment (or epoch) is a subcauset that lies between two consecutive breaks (i.e. the segments ofC are . IfC contains exactly k breaks where 0 < k < ∞ then B k is also a segment, the infinite final epoch. IfC contains no breaks thenC itself is the only segment inC. LetF ⊂Ω denote the set of all infinite causets that contain finitely many principal causets as stems (i.e.C ∈ F ⇐⇒ ∃ m ∈ N such that, for all n > m,C| [0,n] contains at least two maximal elements).
Proof. Note thatF ⊂ B c ∞ , since a causet with infinitely many breaks necessarily contains infinitely many principal causets as stems. Therefore, if there exists somẽ D ∈F such thatD ∼ =C thenC ∈ B c ∞ . To prove the converse, letC ∈ B c ∞ and let N be the cardinality of the past of the last break inC. SupposeC contains infinitely many principal causets as stems and consider the infinite sequence, We now construct a bijection g :C → N and definẽ D ∈Ω to be the infinite causet in which x ≺ y ⇐⇒ g −1 (x) ≺ g −1 (y) inC. The definition ofD ensures that D ∼ =C and our construction of g ensures thatD ∈F. This proves the claim.
Proof. For each principal causetC n let S(C n ) denote the set of principal causets with cardinality greater than n whose restriction to [0, n − 1] isC n , and define ΓC n is the set of infinite causets that (i) contain the principal causetC n as a stem and (ii) contain no principal causet of cardinality greater than n as a stem. We will use the following properties: 1. Each ΓC n is an atom ofR b (i.e. the elements of ΓC n cannot be separated by the principal cylinder sets).

The collection of all the ΓC
n is a partition ofF (since everyC ∈F is contained in some ΓC n and for any two principal causetsC n =D m we have ΓC n ∩ ΓD m = ∅). Given a principal causetC n , letXC n ∈Ω denote the infinite causet whose restriction to [0, n − 1] isC n and in which all elements m ≥ n are unrelated to all others. LetX C n denote a causet isomorphic toXC n in which the element 0 is unrelated to all others. ThenXC ∈ E for all principal causetsC n . Since E is covariant, XC n ∈ E for every principalC n . Hence, by property 1, ΓC n ⊂ E for every principalC n . By property 2,F ⊂ E and by corollary 4, B c ∞ ⊆ E. Now, consider any E ∈R b ∩R for which E ∩B c ∞ = ∅ and letC ∈ E ∩ B c ∞ . Then there exists someD ∼ =C and somẽ C n such thatD ∈ ΓC n and therefore ΓC n ⊂ E. Hencẽ XC n ∈ ΓC n . Since E is covariant,X C n ∈ E. Therefore ΓC 1 ⊂ E, which completes the proof.
Proof. Consider the collection ofC n that contain no breaks. We enumerate these causets using the label i ∈ N, so thatC ni is the i th causet that contains no breaks and its cardinality is n i . We will use the stringC ni 1 ...C ni k−2C ni k−1 to represent the finite principal causet which has k segments and whose j th segment is (canonically order-isomorphic to)C ni j for 1 ≤ j ≤ k − 1.
For each causetC ni 1 that contains no breaks, define the set, (16) where the union is over all sequences (i 2 , ...., i k ) of natural numbers. Note that ifC ∈ B 1 (C ni 1 ) thenC ni 1 is the first segment inC, and thereforeC contains at least one break. Additionally, ifC ∈ B ∞ andC ni 1 is the first segment inC thenC ∈ B 1 (C ni 1 ).
We now generalise (16) to any l ≥ 1. Given a string C ni 1 ...C ni l of finite causets that contain no breaks define the set where the union is over all sequences (i l+1 , ...., i k ) of natural numbers. IfC ∈ B l (C ni 1 ...C ni l ) thenC ni 1 , ...,C ni l are the first l segments inC (and thereforeC contains at least l breaks). Additionally, ifC ∈ B ∞ andC ni 1 , ...,C ni l are the first l segments inC thenC ∈ B l (C ni 1 ...C ni l ). Define, where the union is over all sequences (i 1 , ..., i l ). For each l, B l ⊃ B ∞ and ifC ∈ B l thenC contains at least l breaks. Therefore, Proof. First, we show that if E ∈ R and E ⊆ B ∞ then E ∈R b ∩ R. Let E ∈ R and E ⊆ B ∞ . Then Θ ∈ E c and therefore E ∈ R(S). Moreover, E is in the restriction of R(S) to B ∞ , where the restriction is defined by, and is countably generated by the collection, Using definition (3) we can write, where the union is over all causetsD m for all m that contain a stem that is order-isomorphic to a representative of C n . Now, note that any causet in cyl(D m ) ∩ B ∞ contains infinitely many breaks and therefore has a principal stem of cardinality > m that containsD m as stem. Therefore, we can restrict the domain of the union in (22) to the set of principal causetsD m for all m that contain a stem that is order-isomorphic to a representative of C n . Combined with lemma 7, this proves that every set of the form (22) is contained inR b and hence E ∈R b . Since E is covariant by assumption,

C. Cyclic models as random walks up a new tree
In the context of random walks, the upshot of theorem 2 is that one can conceive of the cyclic models that satisfy µ(B ∞ ) = 1 as random walks up a new tree which we dub reduced poscau (Fig. 5).
One can think of reduced poscau as obtained from labeled poscau by merging groups of nodes into one, so that in reduced poscau each node is a collection of causets. Principal causets are never merged with others, so each principal causet is contained in a node on its own. Given a principal causet, all the causets that are directly above it in labeled poscau (except for the one that is itself a principal causet) are merged into one node. Thus, in reduced poscau each principal causet has exactly two nodes directly above it. A node that contains r causets has r+1 nodes above it: r nodes, each of which contains a single principal causet (formed by adding a single element above each of the r causets), and an additional node that contains all other causets that in labeled poscau are directly above any of the r causets. The meaning of each node is "one of these causets is a stem in the growing causet".
There is a correspondence between the nodes of reduced poscau and sets ofΩ. A node that contains a principal causetC n corresponds to the principal cylinder set cyl(C n ). A node that contains non-principal causets corresponds to a set that is constructed recursively from principal cylinder sets as follows. Let a be a non-principal node directly above the node b and letC 1 n , ...,C r n be the principal causets contained in the remaining nodes directly above b. Then the set corresponding to a is equal to the set corresponding to b take away r i=1 cyl(C i n ). If b is a non-principal node then the set corresponding to it can be constructed in the same manner. One works recursively down the path and the process ends when reaching a principal node.
It follows that the σ-algebra generated by reduced poscau via this correspondence is equal toR b . The collection of all sets corresponding to nodes in reduced poscau is a semi-ring, and therefore each random walk on reduced poscau induces a probability measure onR b , where the measure of each set in the semi-ring is equal to the probability of reaching the corresponding node. Thus, reduced poscau can be used to define cyclic dynamics.
Note that the set of principal cylinder sets alone does not form a semi-ring and therefore we cannot work with it directly to define a measure onR b . The sets that correspond to non-principal nodes complete the collection into a semi-ring and allow to define a measure.
Note that each walk on reduced poscau induces a walk on labeled poscau (the proof is analogous to that of lemma 4.9 in [10]) so that the walks on labeled poscau and on reduced poscau yield the same class of probability measures on the covariant R b . However, the different structures of the two trees could lead to different formulations of physical constraints in terms of transition probabilities and hence to identifying different classes of physically interesting dynamics.

D. Complex Transitive Percolation
The formulation of causal set dynamics in terms of probability measures and random walks is a precursor to the fully quantum dynamics to be expressed as a decoherence functional. In [22] it was proposed that a decoherence functional can be obtained from a complex measure onR, itself derived as follows: replace the real transition probabilities on labeled poscau by complex transition amplitudes A(C n →C n+1 ) that satisfy the sum-rule, where i labels the nodes directly aboveC n . Denote the amplitude of reachingC n by A(C n ). Then A defines a complex function (a "pre-measure") on the cylinder sets whose extension to a complex measure onR-if it exists-is the desired complex measure from which the decoherence functional can be obtained. The following criteria for the existence of an extension to a complex measure onR were given in [23]. Let ζ be the following real function on the ground-set of labeled poscau, where the equality follows from (23). We now apply this technology to the discussion of Complex Transitive Percolation, a family of cyclic models defined by where p ∈ C, q = 1 − p, and L and R are the number of links and relations inC n , respectively. In these models, an extension to a measure onR exists if and only if p ∈ [0, 1] (i.e. when the amplitude A reduces to a real probability) [22,23]. But, with our new understanding of R b as the algebra of covariant events in this model, we can ask whether there exists an extension to the strictly smallerR b when p ∈ [0, 1]. (Note that this question is equivalent to asking whether there are measures onR b that do not extend to a measure onR. When the measures are real probability measures, the answer is no: every measure onR b extens to a measure onR. However, here we are concerned with complex measures, and it is unknown to the authors whether an analogous theorem holds in this case.) To this effect, we can apply equations (23-25) and the above criteria for extension by re-placingR withR b and theC n with the nodes of reduced poscau. This generalisation is possible since the results of [23] rely only on the fact that labeled poscau is a finitevalency tree with no maximal elements, see [24] for further discussion. The transition amplitudes on reduced poscau are fixed by requiring that (26) holds for principal causets and by imposing (23). We solved numerically for ζ min n as a function of p for n = 2, 3, 4. Our results (Fig.6) suggest that for any p, there is a level m above which the function ζ takes its minimum on the nodes that contain principal causets, i.e. ζ min n = |p| + |1 − p| − 1 for all n > m. If this is borne out then ∞ n=1 ζ min n = ∞ when p ∈ [0, 1] and no extension toR b exists, meaning that Complex Transitive Percolation does not give rise to a well-defined quantum dynamics in this framework. Whether other classical cyclic dynamics can give rise to quantum dynamics via this formalism is an open question.

IV. OBSERVABLES FROM PRINCIPAL STEM-SETS
In section III we identified R b as a complete set of observables in cyclic dynamics, but it is unclear what is the physical meaning of each observable in this set. In this section, we identify a second algebra that forms a complete set of observables in cyclic dynamics and in which each observable has a clear physical interpretation.
We begin with terminology. We say that an order C is a principal order if its representative is a principal causet. We say that stem(C) is a principal stem-set if C is a principal order. We define S ⊂ S to be the set of principal stem-sets and write R( S) to denote the σ-algebra that they generate. We can now state our theorem: Theorem 9. Given E ∈ R, there exists some E ∈ R( S) such that E E ⊂ B c ∞ . In perfect analogy to the result of [3], the upshot of lemma 9 is that R( S) exhausts the set of observables in any cyclic dynamics, since the measure of any E ∈ R is fixed by the measure of some E ∈ R( S) via µ(E) = µ(E ).
Importantly, each event in R( S) is equivalent to a logical combination of statements about which principal orders are contained as stems in the growing causet, giving the observables in cyclic dynamics a clear physical interpretation. We will make the notion of "an order contained as a stem" precise after lemma 11.
We now prove theorem 9 through a series of lemmas.
Lemma 10. Let X denote a set of points and let ∼ denote an equivalence relation on X. Let (X, R) denote a standard Borel space. Let (X, T ) denote the Borel space derived from (X, R) via the "covariance" property: Let F ⊂ T denote a countable family of sets that separates X up to equivalence under ∼, i.e. for any a ∼ b ∈ X there exists a set in F that contains a but not b. Then F generates T .
Proof. Let X denote the set of equivalence classes of elements of X under ∼ and let φ : X → X denote the projection that maps each a ∈ X onto its equivalence class. Given (X, R), the quotient Borel space induced by φ is denoted by (X , R ), where R is the set of all The quotient Borel space (X , R ) is analytic whenever it is countably separated (second theorem on p.74 in [25]). In an analytic Borel space, any countable separating family is a generating family (corollary on p.73 in [25]). Therefore, any countable separating family in (X , R ) is a generating family.
Since T is derived from R via the "covariance" property, φ induces a bijection between T and R . Under this bijection, F ⊂ T is mapped onto a countable separating family F in R . Since F generates R , and since bijections preserve countable set operations, F generates T .
We will need the following terminology. Given a finite order C and a (finite or infinite) causetD, we say that C is a stem inD if there exists a stem inD that is orderisomorphic to a representative of C. If C is a stem inD andD is a representative of D, then we say that C is a stem in D. The cardinality |C| of an order C is defined to be equal to the cardinality of a representative of it. We use the term n-stem to mean a stem of cardinality n. We say than an order contains a break if its representative contains a break. The covering causet ofC is the principal causet formed fromC by placing an element above all x ∈C. The covering order C of C is the order whose representative is a covering causet of some representative of C.
Proof. LetD,Ẽ ∈ B ∞ . Let D i denote the unlabeled past of the i th break inD, so D i is an order that is a stem iñ D. Note that D i may or may not be principal. Define E i similarly forẼ. Note that D i and E i each contain i − 1 breaks and therefore either D i = E i or D i = E k ∀k ∈ N.
LetD ∼ =Ẽ. Then there exists an n ∈ N such that D i = E i for all i ≥ n.
Without loss of generality, let |D n | ≥ |E n |. We will show thatẼ / ∈ stem( D n ). Suppose for contradiction that D n is a stem inẼ. Then E n is the only | E n |-stem in D n =⇒ E n is the only | E n |-stem inD (since D n is the only | D n |-stem inD, and "not a stem in any stem is not a stem") =⇒ E n is an unlabeled past of a break inD =⇒ there exists some i ∈ N such that D i = E n , which is a contradiction.
Corollary 12. The countable family of sets, separates B ∞ up to equivalence under orderisomorphisms, i.e. for anyD ∼ =Ẽ ∈ B ∞ there exists a set in S ∩ B ∞ that containsD but notẼ.
Proof. We will show that both the LHS and the RHS are generated by the family S ∩ B ∞ , and the result follows. We begin with the RHS. Note that (Ω,R) is a Polish space (lemma 6 in [3]), and therefore its subspace (B ∞ ,R ∩ B ∞ ) is a standard Borel space (definition 1 on p.71 in [25]). The Borel space derived from (B ∞ ,R∩B ∞ ) via the "covariance" property (27) with respect to equivalence under order-isomorphism is equal to (B ∞ , R∩B ∞ ) and contains the countable family of sets S ∩ B ∞ . Together, corollary 12 and lemma 10 imply that the family S ∩ B ∞ generates (B ∞ , R ∩ B ∞ ).
For the LHS, note that if a Borel space (X, S) is generated by a family F then its Borel subspace (Q, S ∩ Q) is generated by the family F ∩ Q.
Proof of theorem 9. Consider some E ∈ R.
In the special case when the epochs are separated by posts, the dynamics satisfy µ(P ∞ ) = 1, where P ∞ ∈ R is the event that the causet spacetime contains infinitely many posts, and the σ-algebra, exhausts the covariant events, by analogy to the discussion of (4) and (14). The strictly stronger analogue of theorem 2 is, Here,R p denotes the σ-algebra generated by the cylinder sets associated with those principal causetsC n whose restrictionC n | [0,n−2] is itself a principal causet. We call such a causetC n , and the cylinder set, order and stemset associated with it, doubly principal. The analogue of theorem 9 is, Theorem 15. Given E ∈ R, there exists some E in the σ-algebra generated by the doubly principal stem-sets such that E E ⊂ P c ∞ .
Let us define a principal break to be a break whose past is a principal stem. Then there is a post if and only if there is a principal break: the post is the maximal element of the past of the principal break. The proof of theorems 14 and 15 can be obtained from the proofs of theorems 2 and 9 respectively, mutatis mutandis i.e. by replacing "break" with "principal break" and "principal stem" with "doubly principal stem". So, for example, a segment is now defined to be the portion of the causet between two principal breaks and is always a principal causet.
Measures onR p correspond to walks up doubly-reduced poscau whose description is obtained from the description of reduced poscau (see section III C) by replacing "principal" with "doubly principal" (so only doubly principal causets are contained in nodes of their own). Thus, cyclic models in which the epochs are separated by posts can be conceived of as random walks on this novel tree. Complex Transitive Percolation is such a cyclic model, and so we can extend the discussion in section III D by asking whether there exists an extension of the measure to the strictly smaller σ-algebraR p ⊂R b . We do so by modifying equations (23)(24)(25) and the criteria for extension by replacingR withR p and theC n with the nodes of doubly-reduced poscau, but their application to Complex Transitive Percolation is inconclusive. On the one hand, we cannot prove that an extension exists since ζ is equal to |p| + |1 − p| − 1 on every doubly principal causet so ζ max n ≥ |p| + |1 − p| − 1 > 0 for all n when p ∈ [0, 1]. On the other hand, we can cannot rule out an extension, since every level n > 1 in doubly-reduced poscau contains nodes with valency equal to 1 and on these nodes ζ vanishes and therefore n ζ min n = 0. We now amend the extension criteria to provide further scrutiny in the special case where there are nodes with valency equal to 1. Let T denote a finite-valency directed tree that contains no maximal elements and let T n ⊂ T denote the set of nodes at level n. Let D n denote a node at level n.
Define, S n := Dn∈Tn |A(D n )|, and note that an extension exists if and only if sup n S n < ∞ [23].
Let T n ⊂ T n be the set of level n nodes that have valency greater than 1, and define, (1 + ζ min n−r+i ) .
Thus, if the right hand side of (31) diverges with n then no extension exists. Applying this improved criterion to Complex Transitive Percolation on doubly-reduced poscau, our numerical solutions for ζ min n as a function of p for n = 2, 3, 4 ( Fig.7) suggest that for any p ∈ C, there exists a level m above which the function ζ restricted to the valency > 1 nodes takes its minimum value on the doubly principal causets, i.e. ζ min n = |p| + |1 − p| − 1 for all n > m. If this is borne out then, when p ∈ [0, 1], the first term on the right hand side of (31) diverges as n → ∞. Whether sup n S n = ∞ depends on the behaviour of S v=1 n−r in the second term. We note that, since S v=1 n−r is the sum over absolute values of amplitudes of reaching a doubly principal causet by stage n − r − 1, in future it could be computed using techniques similar to those used to obtain the probability of a post [17].

in Complex
Transitive Percolation models.

VI. CONCLUSION
In this work, we considered how the set of covariant observables can be distilled to a smaller exhaustive set of observables under the family of cyclic dynamics. This interplay between the kinematic constraint of covariance and the dynamic restriction to on-shell configurations (cyclic causets) gives rise to interrelated systems of sets with rich mathematical structure. In particular, we identified both R b and R( S) as exhaustive observable σ-algebras, and it is natural to ask how the two are related. First, note that both are sub-algebras of R(S), since neither separates the set of rogues. Additionally, definition (14) of R b and theorem 9 combine to imply that: for any E ∈ R b , there exists some E ∈ R( S) such that E E ⊂ B c ∞ , and vice versa, for any E ∈ R( S), there exists some E ∈ R b such that E E ⊂ B c ∞ . Therefore, under cyclic dynamics the two algebras of observables are equal up to sets of measure zero and can be considered equivalent. However, at the level of the kinematics the two σ-algebras are very different since the only events that they share are the unit and the empty set, as we now prove.
Lemma 16. Let E ∈ R(S). If E is contained in both R b and R( S) then E is either the empty set or the unit elementΩ.
Proof. Consider some causetC ∈ B ∞ and let C i denote the unlabeled past of the i th break inC, so C i is an order which is a stem inC. DefineC to be the causet which is some labeling of the disjoint union of the C i 's. The event i stem( C i ) is the smallest event in R( S) that containsC, in the sense that ifC ∈ E ∈ R( S) then i stem( C i ) ⊂ E. The event i stem( C i ) is also the smallest event in R( S) that containsC . Therefore any E ∈ R( S) contains either both or neither ofC andC .
Suppose E ∈ R b . By definition of R b , either E ⊆ B ∞ or B c ∞ ⊆ E. We show that in both cases, E ∈ R( S), which completes the proof.
Case(i): E ⊆ B ∞ and E = ∅. Note that E contains someC ∈ B ∞ . Therefore, if E ∈ R( S) then E contains C ∈ B ∞ . Contradiction. Case(ii): B c ∞ ⊆ E and E =Ω. Note that there exists some causetC ∈ B ∞ that is not contained in E and theforeC ∈ E. Contradiction.
In particular, lemma 16 implies that B ∞ ∈ R( S). Therefore one cannot tell from the measure on R( S) whether the dynamics is cyclic, however if one knows the dynamics is cyclic then R( S) is an exhaustive set of observables. On the other hand, since B ∈ R b , knowing the measure on R b is sufficient to determine whether the dynamics is cyclic.
Finally, our motivation for studying cyclic dynamics has been the key role that they play in the the causal set comological paradigm that aims to explain the emergence of a flat, homogeneous and isotropic cosmos directly from the quantum gravity era [7][8][9]. But what form do these models take and do they occupy a significant volume in theory space? The Transitive Percolation family of cyclic CSG model [1,18] has served as a starting point for searches of cyclic models and some progress has been made in this direction: a class of cyclic dynamics in which epochs are separated by posts has been identified in [20] and a conjecture of another class of such models has been put forward in [21]. But these formal results are yet to be fully understood and implemented (e.g., via computer simulations) and the characterisation of cyclic dynamics remains an important open question.