The structure of covtree: searching for manifestly covariant causal set dynamics

Covtree - a partial order on certain sets of finite, unlabeled causal sets - is a manifestly covariant framework for causal set dynamics. Here, as a first step in picking out a class of physically well-motivated covtree dynamics, we study the structure of covtree and the relationship between its paths and their corresponding infinite unlabeled causal sets. We identify the paths which correspond to posts and breaks, prove that covtree has a self-similar structure, and write down a transformation between covtree dynamics akin to the cosmic renormalisation of Rideout and Sorkin's Classical Sequential Growth models. We identify the paths which correspond to causal sets which have a unique natural labeling, thereby solving for the class of dynamics which give rise to these causal sets with unit probability.


Introduction
What is the role of general covariance in quantum gravity? In causal set theory (CST), where the quantum dynamics is still unknown, clues to this question come from studying the role of general covariance at the level of classical stochastic toy models. In CST, general covariance takes the form of label-independence: two causal sets are physically equivalent if they are order-isomorphic. This has a clear consequence for the observables in CST, namely that they cannot pertain to any labeling of the causal set or to the identity of the causal set elements.
But could general covariance also have direct consequences for the dynamics?
This is indeed the case in the Classical Sequential Growth (CSG) models [1,2] which satisfy the so-called Discrete General Covariance (DGC) condition. In these models a causal set (causet) grows probabilistically through a sequential birth of elements, and the order of births induces a labeling of the elements by the natural numbers. The DGC condition constrains the dynamics such that, ifC n andC n are order-isomorphic causets of cardinality n, thenC n andC n are equally likely to have been grown after the birth of the first n elements.
In CSG models general covariance enters via the choice of observables and via the choice of dynamics, but general covariance is implicit because the history space is the space of infinite labeled causets 1 and so each realisation is a labeled causet (and therefore not covariant). In contrast, in a manifestly covariant approach the history space would be the space of infinite orders, denoted by Ω. In this case general covariance need no longer be imposed at the level of the observables because each realisation is covariant, but whether general covariance plays an additional role at the level of the dynamics is still unknown. Additionally, it is hoped that a manifestly covariant approach would be more amenable to quantisation.
A first proposal for such a manifestly covariant framework for classical causet dynamics is covtree [3]. Covtree is a directed tree, each of whose nodes Γ n at level n is a set of n-orders. For each infinite directed path from the origin, P = {Γ 1 , Γ 2 , ...}, there exists at least one infinite order C whose set of n-stems, for every n > 0, is Γ n ∈ P. We call C a certificate of P. Via the correspondence between paths and their certificates, any set of Markovian transition probabilities on covtree defines a causet dynamics with the space of infinite orders, Ω, acting as the history space (i.e. a manifestly covariant dynamics).
But not every set of Markovian transition probabilities on covtree defines a physically interesting dynamics. Identifying a subset of interesting dynamics is the motivation for this current work. One challenge lies in the translation of physically desirable conditions (e.g. that manifold-like 2 orders are preferred by the dynamics) into conditions on covtree transition probabilities. Doing so requires an understanding of the relationship between paths and their certificates (e.g. which paths have manifold-like certificates). Closely related challenges include formulating a causality condition on covtree and understanding what additional constraints general covariance may impose on the transition probabilities (cf. the DGC condition in CSG models).
In addition to the relationship between paths and their certificates, an understanding of the structure of covtree is also important for constraining the dynamics.
For example, any dynamics should satisfy the Markov sum rule: the sum of the transition probabilities from a node Γ n to each of its children must equal 1. But with no knowledge of the number of children or the relation they bear to Γ n , this constraint is intractable. (In contrast, in the case of the CSG models, enough structural information is known to solve the Markov sum rule.) In addressing these challenges, one might be tempted to construct covtree explicitly. Indeed, the first three levels of covtree are given in [3], but brute force methods come up short in going to higher levels as the number of candidate nodes at level n increases rapidly as 2 |Ω(n)| − 1, where |Ω(3)| = 5, |Ω(5)| = 63 and |Ω(16)| = 4483130665195087 [4]. In this work we make progress by focusing on structural properties which are independent of level.
The rest of this paper is structured as follows. Section 2 is dedicated to a presentation of terminology and notation. In section 3, led by the ideas of causal set cosmology, we identify the covtree paths whose certificates contain posts and breaks, prove that covtree has a self-similar structure and find the covariant analogue of the cosmic renormalisation transformation of CSG models. In section 4 we present additional structural features of covtree, as well as a toy example of how this structure can be used to constrain the dynamics. We conclude with a discussion in section 5.

Terminology and notation
Here we present a brief review of terminology and notation used in the paper. For further discussion and examples, we refer the reader to [3] and references therein.
For a recent review of causal set theory, see [5].
A labeled causet is a locally finite partial order on ground set [n] or N such that x ≺ y =⇒ x < y. A labeled causet of cardinality n is called an n-causet.
We denote labeled causets and their subcausets by capital Roman letters with a tilde, e.g.C.
If x ≺ y inC we say that y is a descendant of x or that y is above x. If x ≺ y and there is no z ∈C such that x ≺ z ≺ y we say that y is a direct descendant of x or that y is directly above x or that y is a child of x. The valency of x is the number of direct descendants of x.
An element x ∈C is in level L inC if the longest chain of which x is the maximal element has cardinality L, e.g. level 1 comprises the minimal elements.
For any x ∈C, L(x) is an integer which denotes the level of x, e.g. L(x) = 1 if x is minimal.
An order is an order-isomorphism class of labeled causets. An n-order is an order-isomorphism class of n-causets. We denote orders by capital Roman letters without a tilde, e.g. C. The cardinality of an n-order is defined to be n. We denote the cardinality of an order C by |C|.
We often (but not always) use a subscript to denote the cardinality of an ncauset or n-order, e.g.C n or C n .
A stem in a labeled causetC is a finite subcausetS inC such that if y ∈S and x ≺ y inC then x ∈S. We say a finite order, S, is a stem in order C if there exists a representative of S which is a stem in a representative of C. We say a finite order, S, is a stem in labeled causetC if the order S is a stem in the order [C]. So the meaning of stem depends on context. If a stem has cardinality n we say it is an n-stem.
An infinite order C is a rogue if there exists an infinite order D such that D = C and the two orders have the same stems.

Labeled poscau and CSG dynamics
LetΩ(N) andΩ denote the set of all finite and infinite labeled causets, respectively.
Labeled poscau is the partial order (Ω(N), ≺), whereS ≺R if and only ifS is a stem inR. 3 A poscau dynamics is a complete set of Markovian transition probabilities on labeled poscau. We denote the probability of transition fromC n to one of its childrenC n+1 by P(C n →C n+1 ). We denote the probability of a directed random walk to pass throughC n by P(C n ).
Classical Sequential Growth (CSG) models are a family of poscau dynamics which satisfy the so-called Bell Causality and Discrete General Covariance conditions of [1]. Each model in the family is specified by an infinite set of real positive coupling constants, {t k } k∈N , with t 0 > 0. The CSG transition probabilities take the form: where and m are positive integers which depend onC n andC n+1 , and Originary CSG models are a family of poscau dynamics which differ from CSG models only by the requirement that t 0 = 0.
Each poscau dynamics is equivalent to a measure space (Ω,R,μ), whereΩ is the set of infinite labeled causets,R is the sigma-algebra generated by the cylinder sets, and the measureμ is defined via µ(cyl(C n )) = P(C n ) for everyC n . The covariant sigma-algebra, R, is a sub-algebra ofR defined by

Covtree
Let Ω denote the set of infinite orders. Let Ω(n) denote the set of all n-orders for some n ∈ N + , and let Γ n denote a non-empty subset of Ω(n), i.e. a set of n-orders.
An order C is a certificate of Γ n if Γ n is the set of all n-stems in C. If C is a certificate of Γ n and there exists no stem D = C in C which is a certificate of Γ n , then we say that C is a minimal certificate of Γ n . A labeled causetC is a labeled certificate of Γ n if it is a representative of a certificate C of Γ n .
For any n and any Γ n , the map O − takes Γ n to the set of (n − 1)-stems of elements of Γ n : Let Λ denote the collection of sets of n-orders, for all n, which have certificates: Covtree is the partial order (Λ, ≺), where Γ n ≺ Γ m if and only if n < m and If Γ n ∈ Λ, we say that Γ n is a node in covtree. If Γ n is a node in covtree and Γ n contains a single n-order, we say that Γ n is a singleton. If Γ n is a node in covtree and Γ n contains exactly two n-orders, we say that Γ n is a doublet.
..} be a path in covtree. An infinite order C is a certificate of path P if Γ n ∈ P is the set of n-stems in C, for every n. Every path has at least one certificate, and every infinite order is a certificate of exactly one path.
A covtree dynamics is a complete set of Markovian transition probabilities on covtree. We denote the probability of transition from Γ n to one of its children Γ n+1 by P(Γ n → Γ n+1 ). We denote the probability of a directed random walk to pass through Γ n by P(Γ n ). We denote a covtree dynamics by {P}.
The certificate set, cert(Γ n ), of some node Γ n is the set of infinite orders which are certificates of cert(Γ n ): cert(Γ n ) = {C ∈ Ω|C is a certificate of Γ n }.
Each covtree dynamics, {P}, is equivalent to a measure space (Ω, R(S), µ), where Ω is the set of infinite orders, R(S) is the sigma-algebra generated by the certificate sets 4 and µ is a measure defined via µ(cert(Γ n )) = P(Γ n ) for all Γ n .
Equivalently, we can conceive of each certificate set as a set of labeled causets: In this formulation, R(S) is a set of subsets ofΩ. In fact, R(S) ⊂ R ⊂R and therefore every poscau dynamics induces a covtree dynamics via a restriction of the measureμ fromR to R(S) . We say that a covtree dynamics is a CSG dynamics if it is the restriction of a CSG dynamics.

Covtree and causal set cosmology
In the heuristic causal set cosmology paradigm proposed in [7], the cosmos emerges from the quantum gravity era sufficiently flat, homogeneous and isotropic to explain present-day observations (without the need for a period of inflation). Within the context of CSG models, the fine-tuning problem is that of choosing a CSG dynamics which displays this behavior almost surely. The need for fine-tuning is overcome by a "cosmic renormalisation" associated with cycles of expansion and collapse, punctuated by Big-Crunch-Big-Bang singularities.
At least heuristically, posts and breaks are the causal set structures which underlie Big-Crunch-Big-Bang singularities [8,9]. LetC be a labeled causet. A post is an element x ∈C which is related to every other element inC. A break inC is an ordered pair, (Ã,B), of nonempty subsets ofC such that  Smolin and others that the parameters of nature are modified as the universe is "squeezed through" a singularity [10][11][12][13].
In this cosmological paradigm the fine-tuning problem is resolved by an evolutionary mechanism. It rests on the hypothesis that there exist stationary points of the renormalisation transformation which give rise to the desired cosmological features, and that the basin of attraction of these stationary points is large and contains an abundance of dynamics each of which gives rise to an infinite sequence of Big-Crunch-Big-Bang singularities with unit probability [8,[14][15][16][17]. Given this, no fine-tuning is required for our universe to be governed by a dynamics in the basin of attraction which gives rise to an infinite sequence of singularities. At each singularity the couplings undergo a renormalisation, and in this way a flow towards the stationary point is generated in the space of couplings. It is then only a matter of time until our universe displays the desired behaviour.
This narrative acts as guidance as to which covtree dynamics we should be seeking, namely: We begin section 3.1 by recasting the definitions of post and break in covariant form 6 . We identify which covtree paths correspond to orders with posts and breaks and write down the defining feature of covtree walks which belong to family (a).
In section 3.2 we show that covtree has a self-similar structure. In section 3.3 we use covtree's self-similarity to solve for the covtree walks which belong to family (b). We conclude with a discussion of open questions, including a proposal for a causality condition for covtree dynamics.

Certificates with posts and breaks
In solving for the covtree walks which belong to family (a), a question arises: which paths correspond to orders with posts and breaks? To pose this question more precisely, let us extend the definitions of posts and breaks to orders. LetC andÃ be representatives of orders C and A, respectively. We say that an order C contains a break (post) with past A ifC contains a break (post) with pastÃ. If order C contains a break (post) with past A we say that C contains an A-break (A-post).
Our question then becomes: which paths have certificates with posts and breaks?
To answer this question we introduce the concept of the covering causet. If C is a labeled causet of cardinality n then its covering causet is the labeled causet of cardinality n + 1 which is formed by putting the element n above every element ofC, and we denote it by putting a hat on: C . IfC and C are representatives of orders C and C, respectively, then we say that C is the covering order of C. An example is shown in figure 2.
We will show that: 1. C contains an A-break if and only if { A} is a node in P. To prove theorem 3.1 we will need the following lemma about labeled causets which contain breaks:

C contains an
LetC andÃ be labeled causets. The following statements are equivalent: (i)C contains anÃ-break, Proof. LetÃ be a stem inC. Let x denote a minimal element inC \Ã. Let a denote an element inÃ.
(i) =⇒ (ii) It follows from the definition of anÃ-break that every (|Ã| + 1)-stem inC is of the formÃ ∪ {x} and that each such stem is isomorphic to Ã .
(iii) =⇒ (i) By assumption (iii), x a for all x and a, and hence (by definition of break) C contains anÃ-break.
The following covariant statement is a corollary: 3. An order C contains an A-break if and only if A is its unique We can now prove theorem 3.1: by definition, the set of n-stems of C is the node Γ n in P.
To prove part 1, suppose C contains an A-break. Then by corollary 3.3, the set Then by definition of certificate, { A} is the set of (|A| + 1)-stems of C. By corollary 3.3, C contains an A-break. An illusration is shown in figure 3.

Covtree self-similarity
To identify covtree dynamics which fall into families (b) and (c) we must first understand how the renormalisation transformation is manifest on covtree. This turns out to be inextricably linked to covtree's self-similar structure. In this section we identify this self-similarity.
Let us begin by defining what we mean by a self-similar structure of a partial order. Let Π and Ψ be partial orders. We say that Ψ contains a copy of Π if there exists a convex sub-order Π ⊆ Ψ which is order-isomorphic to Π. If Ψ contains infinitely many copies of itself we say that Ψ is self-similar.  which contain a covering order. A certificate C of P contains a break whose past is the 1-order (corresponding to the node at level 2) and a post whose past is the 4-order q q q A q (corresponding to the node at level 6). That C contains this post implies that C also contains a break whose past is the 4-order q q q A q (corresponding to the node at level 5).
Let us denote the covtree partial order by Λ. For any finite order A, let Λ A ⊂ Λ be the convex sub-order of covtree which contains the singleton { A} and everything above it. We will show that: For any finite order A, Λ A is a copy of covtree.
The following theorem is a corollary: We will need the following definition:  Proof of lemma 3.4: We will show that, for any finite order A, and the result follows. Note that, by definition 3.6, Λ is a strict subset of the domain of G A . Here we use G A to denote the restriction of the map to Λ. The use of G A should be clear from the context.
To prove part (i), we first show that G A (Λ) ⊆ Λ A . Let Γ n ∈ Λ and let the morder C m be a certificate of Γ n . Let D k denote the order of cardinality k = m + |A| which contains a break with past A and future C m . Then D k is a certificate of Second, we show that Λ A ⊆ G A (Λ). Let Γ n ∈ Λ A and let the p-order E p be a certificate of Γ n . Necessarily, E p contains a break with past A and some future B. Let Γ l denote the set of l-stems of B, where l = n − |A|. Then Γ l ∈ Λ and To prove part (ii), we use the commutativity of the operations O − and G A to show that G A : Λ → Λ A is order-preserving. Suppose Γ n ≺ Γ n+1 , then by definition of covtree we have that Γ n = O − (Γ n+1 ), and therefore Covtree contains countably many copies of itself, each with ground set Λ A and root { A} (where we can think of Λ itself as Λ ∅ ). An illustration is shown in figure 5.

Covariant cosmic renormalisation
In this section we give a brief review of cosmic renormalisation in CSG models, Remarkably, the effective dynamics is itself a CSG model and one can think of the coupling constants as undergoing a renormalisation: where {t where G A is the mapping introduced in definition 3.6. For a generic covtree dynam- Now, choose E = cert(Γ n ) for some node Γ n . This sets G A (E) = G A (cert(Γ n )) = cert(G A (Γ n )). Then, use the relations µ(cert(Γ m )) = P(Γ m ) and µ A (cert(Γ m )) = P A (Γ m ) to rewrite condition 12 in terms of probabilities. Finally, we can use induction to reach transformation 11.
The transformation R A acts directly on transition probabilities, not on a set of couplings. To emphasis this, we call R A a similarity transformation rather than a renormalisation transformation. If (as one hopes) in future we are able to characterise a covtree dynamics by a set of couplings, it may be possible to write the similarity transformation as a renormalisation transformation which acts on the couplings directly.
Similarly, we reserve the term stationary point for a set of couplings which is mapped onto itself by the renormalisation transformation. If a covtree dynamics is mapped onto itself by a similarity transformation R A , we say that it is self-similar with respect to R A . A covtree dynamics {P} is self-similar with respect to R A if and only if it satisfies the condition for every n and every transition Γ n → Γ n+1 . Constructing a self-similar dynamics is simple: assign any set of transition probabilities to the transitions which lie outside Λ A , and then use equality 13 to set the transition probabilities in Λ A .
It is possible to use this procedure to fix the transition probabilities in Λ A for every A simultaneously, thus constructing a dynamics which is self-similar with respect to R A for all A. We call such dynamics maximally self-similar.
A dynamics cannot be self-similar with respect to a unique transformation R A .
If a dynamics is self-similar with respect to some transformation R A then it is also self-similar with respect to (R A ) n , for any positive integer n. But (R A ) n is itself a similarity transformation: (R A ) n = R A n , where we define A n to be the order of cardinality n|A| which is a stack of n copies of A separated by breaks.
Therefore there exists no dynamics which is self-similar with respect to a unique transformation. We say that a dynamics {P} is minimally self-similar if there exists a unique order A such that {P} is only self-similar with respect to (R A ) n for all n. There is an alternative formulation of the effective dynamics after a post [8]. In this alternative formulation, the post is considered a part of the future rather than the past. The future is therefore constrained to have a unique minimal element (the post itself) but is otherwise dynamical. The effective dynamics is not a CSG model. Instead it is an originary CSG model, ensuring that every new element is born above the post. The coupling constants renormalise as: where m = |Ã|. The renormalisation transformation depends onÃ only via its cardinality, as signified by the label m on the transformation, S m , and on the renormalised couplings, t (m) k . A derivation of transformation 14 can be found in [8]. The stationary points of S m , for any m, are the Originary Transitive Percolation (OTP) models: where t is any positive real number.
An originary formulation exists also in the covariant case. We say that a covtree dynamics {P} is originary if P(Γ 1 →{ q q }) = 1. In the originary viewpoint of the A-post condition, a (generic) covtree dynamics {P} is mapped onto an originary covtree dynamics {P A } via the transformation 7 : While our success in adapting the cosmic renormalisation to the covtree framework bodes well for a covariant causal set cosmology, our results will remain purely formal until we are able to identify a class of covtree dynamics to work with. Having said that, these results could be used to advance the search for physical covtree dynamics. In the remainder of this section, we present directions for further study.
The cosmic transformations can be used to study the relationship between covtree dynamics and CSG dynamics: 2. The action of Q m,r on the couplings t k with k > 0 can be factorised as If {P} arrives at a self-similar dynamics after N applications of T A , expression 17 simplifies to: The N = 1 case is of special interest to us. The Transitive Percolation (TP) models are a 1-parameter family of CSG models, defined by t 0 = 1, t k = t k , t ∈ R + . It is easy to show that, under an application of S m , a TP model with parameter t is maped onto the OTP model with the same t value. Does this mean that for a covtree dynamics {P} to be a TP model it must satisfy condition 18 with N = 1?
Finally, covtree is an opportunity to uncover new dynamics with physical features such as: 6. Infinitely many breaks or posts: it is known that OTP gives rise to infinitely many posts with unit probability [18]. Is this property related to that the fact that it is a stationary point? Do self-similar covtree dynamics give rise to an infinite sequence of posts or breaks? Could this be a feature of the maximally self-similar dynamics?
7. Causality: when the transformation R A factorises as R A = R |A| , the effective dynamics is independent of the causal structure of the past. Therefore, could the condition that R A factorises be interpreted as a causality condition on covtree dynamics? break post non-originary originary

Further structure of covtree
A pair of challenges on the path to physical covtree dynamics are understanding the structure of covtree and understanding the relationship between paths and their certificates. In this section we present further properties of covtree and its certificates, and illustrate with a toy example how an understanding of the structure of covtree could be a useful tool for constraining covtree dynamics.

Property 7. Only singletons can have exactly one direct descendant in covtree.
To prove this, consider the node Γ n = {A 1 , ..., A k } with k ≥ 2, and assume for contradiction that Γ n+1 is the only direct descendant of Γ n .
First, we show that Γ n+1 contains the covering order A i of every A i ∈ Γ n .
Suppose for contradiction that Γ n+1 does not contain the covering order A i of some A i ∈ Γ n . Let C m be an m-order that is a certificate of Γ n+1 . Then there exists an (m + 1)-order D m+1 which contains C m and A i as stems (a representative of D m+1 can be constructed by taking a representative of C m and adding an element above the stem which is isomorphic to a representative of A i ). D m+1 is a certificate of This contradicts the assumption that Γ n+1 is the only direct descendant of Γ n . Hence Γ n+1 contains the covering order A i for Now we show that, since Γ n+1 contains the covering order A i of some A i ∈ Γ n , it cannot be the only direct descendant of Γ n . Let the p-order E p be a minimal certificate of Γ n . Therefore Γ n ≺ Γ n+1 ≺ {E p } and hence E p is a certificate of Γ n+1 .
Then there exists a (p − 1)-order F p−1 such that F p−1 is a stem in E p and A i is not a stem in F p−1 (a causet isomorphic to a representative of F p−1 can be constructed by taking a representative of E p and removing the element which is maximal in the stem isomorphic to a representative of A i ). Then F p−1 is a certificate of Γ n , which contradicts the assumption that E p is a minimal certificate of Γ n . Therefore, only singletons can have exactly one direct descendant in covtree.
Additionally,   Let us explain the pattern shown in figure 6. Each order C n k has cardinality n k = 4k − 1. A representativeC n k of C n k , contains n k −1 2 elements in level 1, n k −1 2 elements in level 2, and a single element in level 3. Each element in level 1 is below two elements in level 2. Each element in level 2 is above two elements in level 1.
The element in level 3 is above all but one of the elements in level 2.
We construct the singleton descendants of {C n k } as follows. Consruct a new causet fromC n k by adding a new element directly above all but one of the level 2 elements subject to the constraint that no level 2 element is maximal in the resulting causet. There are 2k − 2 ways to do this, leading to a collection of 2k − 2 causets. One can show that each of these causets is order-isomorphic to exactly one other in the collection, and taking the corresponding orders gives k − 1 distinct (n k + 1)-orders whose only n k -stem is C n k . A singleton containing each of these orders is directly above {C n k }. The additional singleton descendant is { C n k }.
Similarly,  Figure 7 shows the first three doublets in the sequence and their direct singleton descendants.
A representativeC m k of C m k has cardinality m k = 4k + 5 and partial ordering as described below property 8 (with n k replaced by m k ). A representative of D m k has these same properties, except that the element in level 3 is above all but two of the elements in level 2. The two elements missed out must have a common element in their pasts.
We construct the singleton descendants of {C m k , D m k } as follows. Consruct a new causet fromC m k by adding a new element directly above all but two of the level 2 elements subject to the constraints that no level 2 element is maximal in the resulting causet and that the two elements which are missed out have a common element in their pasts. In this way, one generates a collection of 2k causets. One can show that each of these causets is isomorphic to exactly one other in the collection, and taking the corresponding orders gives k distinct (m k + 1)-orders whose set of m k -stems is {C m k , D m k }. This proves property 9.
A key hurdle in the construction of covtree is understanding which sets of norders are covtree nodes. The following property gives a necessary condition in the case of doublets: Property 10. {A n , B n } is a doublet in covtree only if there exists an (n − 1)-order S which is a stem in both A n and B n .
To prove property 10, letẼ be a labeled minimal certificate of {A n , B n }. Let A n andB n be stems inẼ which are isomorphic to representatives of A n and B n , respectively. DefineS :=Ã n ∩B n . We will show that |S| = n − 1 and property 10 follows.
Property 11 is a corollary: Property 11. If Γ n is a doublet in covtree then all minimal certificates of Γ n are (n + 1)-orders.
Therefore, if Γ n is a doublet in covtree and Γ n ≺ Γ n+1 then Γ n+1 contains some minimal certificate of Γ n . It is a corollary of properties 9 and 11 that for any integer k ≥ 1 there exists a doublet in covtree with k minimal certificates.

Paths
In this section, we present properties of certain covtree paths and their certificates.
Property 12. In covtree, there are infinite upward-going paths from the origin in which every node is a singleton.
We call the subset of covtree which contains exactly all these paths singtreea tree of singletons. Figure 8 shows the first three levels of singtree. Figure 8: The first three levels of singtree.
To discuss singtree we will need the concept of the Newtonian order. A Newtonian causet is a causet in which every element in level k is above every element in level k − 1. A Newtonian order is an order whose representatives are Newtonian.
In a Newtonian causet, every pair of elements which are unrelated have the same past and the same future, alluding to a notion of a Newtonian global time, hence its name 8 . A Newtonian causet is a "stack of antichains", and for any natural number N , the union of the first N levels is a past of a break. The local finiteness condition implies that every level whose elements are not maximal must be finite.
To characterise the nodes of singtree and the certificates of singtree paths we will need the following lemma about Newtonian orders: The following properties of a finite or infinite order, C, are equivalent: (i) C has a unique representative, (ii) for every natural number n ≤ |C| there is a unique n-order which is a stem in C, (iii) C is Newtonian.
Proof. LetC denote a representative of C. Recall that L(x) denotes the level of element x.
(i) =⇒ (ii) Suppose for contradiction that C has two n-stems, C n = C n , for some n ≤ |C|.
Then there is a representative of C whose restriction to the interval [n − 1] = {0, 1, ..., n − 1} is a representative of C n , and similarly for C n . Hence C has at least two representatives. Contradiction.
(ii) =⇒ (iii) Suppose for contradiction that C is not Newtonian, i.e. there exist some x, y ∈C such that L(x) > L(y) and x y. LetS be the union of the inclusive past of x with levels 1, 2, ..., L(y). Note thatS ⊂C is a stem inC.
(iii) =⇒ (i) Let f :C →C be an order-isomorphism between two representatives of C.
The Newtonian condition restricts the action of f on each level inC to be a permutation, and therefore f must be the identity map. Hence C has a unique representative.
The equivalence of statements (ii) and (iii) in lemma 4.1 implies that: Property 13. A singleton {C n } is in singtree if and only if C n is Newtonian.
If {C n } is a node in singtree then it has exactly two direct descendants in singtree: { C n } and {D n+1 }, whereD n+1 is the Newtonian order whose representative is constructed from a representative of C n by adding a new element to its maximal level. If {C n } is a node in singtree then it has exactly three direct descendants in covtree: its singtree descendants, { C n } and {D n+1 }, and the doublet A second corollary of lemma 4.1 is: An infinite order C is Newtonian if and only if it is a certificate of a singtree path.
Given property 14, it is now a simple matter to solve for the family of covtree dynamics in which the set of non-Newtonian orders is null: it is the set of covtree walks in which the walker stays in singtree with probability 1, i.e. P(Γ n ) = 0 ∀ Γ n not in singtree. (19) While this family of dynamics is not physically interesting, it acts as a proof of principle, illustrating how an understanding of covtree could allow one to solve for a dynamics with particular features.

Discussion
Taking our cue from [3], in this work we studied the structure of covtree and its certificates as a first step to constructing physically well-motivated covtree dynamics.
We made progress on the cosmological front. We identified covtree's self-similar structure as well as which nodes correspond to posts and breaks. This allowed us to write down a similarity transformation between covtree dynamics, akin to the cosmic renormalisation of CSG models. As well as being a starting point for a paradigm of covariant causal set cosmology, these developments provide us with a concrete arena in which to investigate which covtree dynamics are the physically interesting ones, e.g. by allowing us to propose a causality condition on covtree transition probabilities.
We also presented a glimpse of covtree's intricate structure. For example, we saw that there is no upper bound on the valency of covtree nodes, even in the simplest case of singletons. After a study of singtree and its certificates, we solved for the class of dynamics in which the set of Newtonian orders has measure 1, thereby providing a toy example of how an understanding of the structure of covtree could be utilised in writing down dynamics with desired features. But, since these Newtonian dynamics are not physically interesting, this is very much a case of "looking under the lamp-post".
Where are we to look if not under the lamp-post? One avenue for exploration is to ask: what role, if any, do rogues play in the physics of covtree walks?
Since in CSG models the set of rogues is null [6], identifying covtree dynamics which possess this property is a step towards understanding what form CSG dynamics take on covtree. Moreover, if following [6] we are to choose R to be our sigma-algebra of observables then -unless the covtree measure on R(S) has a unique extension to R -one is faced with ambiguities both in interpretation and calculation. It is sufficient that the set of rogues be null for there to exist a unique extention, and therefore rogue-free dynamics are compatible with this approach.
Finally, rogue-free dynamics are cosmologically interesting. A rogue causet contains an infinite level and as a result cannot contain an infinite sequence of posts or breaks. Therefore, that the dynamics is rogue-free is a necessary condition for the dynamics to be relevant for our cosmological paradigm.
One can draw an analogy between the condition that the set of rogues is null and the condition that the set of non-Newtonian orders is null: the former is the condition that the set of paths with more than one certificate is null, the latter the condition that the set of paths with more than one labeled certificate is null.
However, while we were able to solve for the latter, solving for the former poses a new challenge because it is a limiting condition: at no finite stage of the covtree walk can the claim that the growing order is a rogue be verfied or falsified. This is because for every node in covtree there exist both an infinite certificate which is a rogue and an infinite certificate which is not a rogue.
This means that there is no rogue analogue to singtree. Instead, we must look for other ways to obtain rogue-free dynamics. Pursuing the strictly stronger condition that the dynamics gives rise to infinitely many posts or breaks with unit probability is a promising route. We already know the defining feature of these covtree walks: that, with unit probability, they pass through infinitely many nodes of the form { A}. The challenge ahead is to formulate this feature in terms of covtree transition probabilities.
Acknowledgments: Parts of this work are a result of collaboration with Fay Dowker, Amelia Owens and Nazireen Imambaccus and have previously appeared in [19,20]. The author thanks the Perimeter Institute and the Raman Research Institute for hospitality while this work was being completed. The author is partially supported by the Beit Fellowship for Scientific Research and by the Kenneth Lindsay Scholarship Trust.
A Table of