Chapter 1: Substrate

1.3 The Causal Graph

With the manifold removed, only points and connections remain for analysis. Defining a point without a location forces a shift in perspective. It requires that an object must exist solely through its relations to others. If position is not intrinsic, then identity must be derived. We must construct a reality where location is defined entirely by connectivity. We must ask how a universe can exist before there is a physical space for it to exist in. We are effectively bootstrapping geometry from pure algebra. This requires us to abandon the comfortable intuition of spatial embedding and accept a world of pure abstraction.

Our structure must derive identity entirely from connectivity. We treat the graph as the raw material of spacetime. In this model, the edges themselves carry the burden of causality. Each link represents a transfer of influence. It is a discrete quantum of connection that binds two events together. This web of relations is not a map of the territory. It is the territory itself. It serves as the physical memory of the system. It encodes the past interactions that define the present state. Without this rigorous definition, we risk smuggling in spatial assumptions that undermine the background independence of the theory.

Analysis is restricted to a specific class of graphs to ensure physical viability. Loops where an event serves as its own ancestor are structurally unsound. Vague connections that fail to specify a direction of influence are equally problematic. We identify the necessary components as abstract events and causal links. We attach logical time to these edges to create a permanent record of creation. Constructing a structure rigid enough to preserve history yet flexible enough to permit evolution, we define the state space not as a collection of positions but as a collection of relations. This formalization allows us to treat the universe as a mathematical object that can be updated and computed.

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1.3.1 Definition: State Space and Graph Structure

Structure of the Universal State Space as a Collection of Finite Acyclic Directed Graphs

$\Omega$ comprises the set of all kinematically admissible graph configurations that satisfy the constraints of finiteness and acyclicity. Each configuration in $\Omega$ encodes an essential "moment" in the universe's history, represented by a single point $G \in \Omega$, which captures the complete relational and temporal structure at that instant without presupposing prior states or future evolutions. The finiteness constraint limits $|V| < \infty$ for every $G$, ensuring computational tractability and avoiding infinities that could undermine the discrete genesis principle, while acyclicity enforces the strict forward direction of causation, precluding loops that would imply retroactive influences or paradoxes.

$G = (V, E, H)$ constitutes the essential structural unit of $\Omega$. This triplet encapsulates the essential components of relational existence, where each element contributes to the graph's representational power: $V$ provides the discrete event basis, $E$ the primitive causal linkages, and $H$ the immutable temporal ordering.

• $V$: $V = \lbrace v_1, v_2, \ldots, v_N \rbrace$ forms a finite collection of vertices, each representing an elementary Abstract Event. These vertices serve as the raw "atoms" of existence, possessing no internal structure, spatial extent, geometric coordinates, or intrinsic properties beyond their index. The finiteness of $N = |V|$ arises from the constructive dynamics of the theory, where events emerge sequentially rather than pre-existing eternally, ensuring that the state space remains countable and free from unphysical infinities. Abstract events embody the minimal ontological primitives: they lack duration or magnitude, functioning solely as placeholders for relational intersections, which allows the theory to prioritize causality over substantival attributes.

• $E$: $E \subseteq V \times V$ collects directed edges, each representing an irreducible Causal Relation. An edge $e = (u, v)$ asserts the primitive logical proposition "$u$ precedes $v$," denoting a direct, unmediated influence from event $u$ to event $v$. Irreducibility means that no intermediate events intervene in the relation; if such mediation existed, the direct edge would decompose into a path of multiple edges, preserving the transitive closure under $\le$ without loss of expressivity. The directed nature enforces asymmetry, aligning with the irreversible arrow of time, and the subset relation $E \subseteq V \times V$ permits sparsity (Bombelli et al., 1987); (Sorkin, 2005), reflecting the vacuum's low density where most potential pairs remain unrealized until relational necessity demands them.

• $H$: $H: E \to \mathbb{N}$ assigns to each edge $e \in E$ a Creation Timestamp, drawn strictly from $t_L$ at the instant of the edge's formation during a dynamical tick. The codomain $\mathbb{N}$ (non-negative integers starting from 0) underscores the sequential, constructive nature of physical processes: timestamps increment monotonically ($H(e') > H(e)$ for edges formed later), recording the exact order of genesis without allowing continuous interpolation or retroactive assignment. This discreteness prevents paradoxes associated with infinite past histories or fractional times, as each edge receives its timestamp upon instantiation via the rewrite rule (§4.5.1), ensuring $H$ embeds the full temporal archive immutably.

This triplet structure ensures that each $G \in \Omega$ represents a complete, self-contained snapshot of causal reality at a logical instant, with finiteness bounding complexity, acyclicity safeguarding consistency, and the history map providing an indelible record of emergence. The choice of $\mathbb{N}$ for $H$ emphasizes the discrete genesis over continuous models, where time subdivides arbitrarily; here, the causal graph posits a punctuated history beginning from an initial empty state, avoiding logical paradoxes from pre-existing infinite chains and enabling dynamical evolution from nullity.

$H$ defines as an intrinsic attribute of the edge isomorphism class, not as a mutable data register. The timestamp is a topological invariant of the edge's existence profile. Therefore, the "record" of an edge is not a separate resource that requires storage allocation; it is a fundamental definitional component of the edge itself. To delete an edge is to alter the graph topology, but the definition of the deleted element remains mathematically distinct from a non-existent element due to its historical index.

1.3.1.1 Diagram: Causal Cone

Representation of Causal Horizons through the Emergent Growth Front

       |
       |  (Future: Potential Paths)
       |          . . . .
       |       . '        ' .
      t_L   (v4)           (v5)  <-- Emergent Horizon (Growth Front)
       |      ^ \         / ^
       |       \ \       / /
       |        \ \     / /
       |         \ (v3) /        <-- The "Now" (Focus Event)
       |          ^    ^
       |         /      \
       |        /        \
       |     (v1)        (v2)    <-- The Past (Fixed History)
       |       ^          ^
       |_______|__________|______
            Causal Foundations

1.3.2 Definition: Emergent Timestamp Assignment

Assignment of Immutable Creation Timestamps by the Global Sequencer

Time in Quantum Braid Dynamics operates as a persistent, immutable memory of creation embedded directly within the graph's structure. For any edge $e = (u, v)$ added to the graph during a dynamical tick at $t_L$, the timestamp $H(e)$ receives permanent assignment according to the current state of the Sequencer mechanism, defined in (§1.2.2):

$H(e) = t_L.$

This assignment couples the ontology of the graph to the meta-theoretical Sequencer, which tracks the cumulative count of ticks since genesis. $H(e)$ constitutes an indelible record of origin: once the edge materializes via the rewrite rule, $H(e)$ fixes irrevocably, immune to subsequent modifications or retroactive adjustments. This immutability enables the full causal order to reconstruct solely from the graph's topological data, rendering the "flow" of time an intrinsic emergent property of the relations rather than an extrinsic parameter imposed upon the structure. The natural number codomain of $H$ reinforces discreteness, with each increment marking a discrete genesis event, precluding continuous interpolation and ensuring the history forms a well-ordered sequence aligned with the theory's punctuated evolution.

1.3.1.2 Diagram: Timestamp Evolution

Illustration of Immutable Timestamp Assignment during Graph Evolution

   TICK 1 (Genesis)        TICK 2 (Growth)         TICK 3 (Merger)
   t_L = 1                 t_L = 2                 t_L = 3

    [v1]                    [v1]                    [v1]
      \                       \                       \
       \ H=1                   \ H=1                   \ H=1
        \                       \                       \
         ▼                       ▼                       ▼
        [v2]                    [v2] ── H=2 ──► [v3]    [v2] ── H=2 ──► [v3]
                                                      ^               │
                                                      │ H=3           │ H=3
                                                      │               ▼
                                                      [v4] <───────── [v5]

   RULE: H(e_new) = t_L (Current Global Logical Time)
   CONSTRAINT: H(e) is immutable once assigned.

1.3.3 Definition: Abstract Event

Identity of the Abstract Event Vertex as a Purely Relational Nexus

An Abstract Event is a vertex $v \in V$. The identity of $v$ is determined strictly by its relational connectivity within $E$. The vertex possesses no intrinsic properties, coordinates, or internal structure independent of these relations. It is a structureless point of intersection for causal influences.

1.3.3.1 Commentary: Relational Justification

Justification of Pre-Geometric Event Identity through Diffeomorphism Invariance

This definition resolves the background dependence paradoxes inherent in classical physics by locating identity strictly within the links rather than the nodes. The abstract event diverges fundamentally from a "point" in classical or Riemannian geometry. A geometric point derives identity from extrinsic coordinates embedded within a pre-existing background manifold, which serves as the substantive stage upon which dynamics unfold. In contrast, the abstract event in Quantum Braid Dynamics admits no such background. Its identity emerges purely relationally, defined exhaustively by the directed edges incident to it: outgoing edges designate it as cause, incoming as effect, with the degree sequence and timestamp offsets providing the sole descriptors.

For instance, in a minimal universe comprising two connected events $A \to B$, event $A$ acquires no absolute position or intrinsic marker. Event $A$ manifests relationally as "the direct cause of $B$," while event $B$ manifests as "the direct effect of $A$." The absence of self-attributes ensures that physics originates from the topology and dynamical evolution of the relations interconnecting them. This relational ontology aligns the foundational structure with the background-independent imperatives of quantum gravity theories, where spacetime arises as a derived construct from causal sets or spin networks rather than a primitive arena. The explicit exclusion of coordinates precludes substantivalism, enforcing diffeomorphism invariance at the discrete level: relabeling vertices preserves the causal skeleton, with isomorphism classes under edge-preserving maps defining equivalence. This shift from substantive objects to relational structures not only evades the hole argument but also embeds the theory's discreteness, where events nucleate via edge additions, inheriting timestamps and influences solely from predecessors.

1.3.4 Theorem: Monotonicity of History

Strict Monotonicity of Causal Timestamp Sequences enforced by Recursive Assignment

The assignment of timestamps ensures that $H$ induces a well-founded partial order on $E$. Specifically, for any newly created edge $e = (u, v)$, the timestamp satisfies the local recurrence relation:

$$H(e) = 1 + \max\left( \lbrace H(e') \mid e' = (w, u) \in E \rbrace \cup \lbrace0\rbrace \right)$$

where the maximum ranges over all edges $e'$ incoming to the source vertex $u$. If $u$ admits no incoming edges (i.e., the set is empty, as occurs for isolated vertices in the initial vacuum state), the convention $\max(\emptyset) = 0$ applies, guaranteeing that primordial edges receive $H(e) = 1$. This recurrence enforces strict monotonicity of causality: no effect precedes its cause in the timestamp ordering, preserving the forward arrow of logical time across all transformations (Lamport, 1978).

1.3.4.1 Proof: Monotonicity

Formal Proof of Order Preservation from Inductive Stability

I. The Timestamp Assignment Algorithm

Let $\mathcal{C}$ be the constructor function responsible for edge creation. For any new edge $e = (u, v)$, the constructor assigns a timestamp $H(e)$ based on the strict causal history of the source vertex $u$. We define the set of incoming edges to $u$ as $\text{In}(u) = \{ e' \in E \mid e' = (w, u) \}$. The assignment rule is defined recursively:

$$H(e) = 1 + \max \left( \{ H(e') \mid e' \in \text{In}(u) \} \cup \{0\} \right)$$

II. The Irreflexivity Condition (Proof by Stability Analysis)

We test the stability of the timestamp assignment for a hypothetical self-loop edge $e_{self} = (u, u)$.

1. Pre-computation: The constructor queries the current history of $u$. Let the maximum existing timestamp be $T_{max}$.

   $$T_{max} = \max \left( \{ H(e') \mid e' \in \text{In}(u)_{\text{pre}} \} \cup \{0\} \right)$$

   The calculated timestamp for the new edge is:

   $$H(e_{self}) = T_{max} + 1$$

2. State Update: If the edge $e_{self}$ is added to the graph, the set of incoming edges updates:

   $$\text{In}(u)_{\text{post}} = \text{In}(u)_{\text{pre}} \cup \{ e_{self} \}$$

3. Stability Constraint: For the assignment to be valid, the rule must hold for the edge after it is added to the set.

   $$H(e_{self}) > \max_{k \in \text{In}(u)_{\text{post}}} H(k)$$

4. Substitution: The maximum of the new set includes the edge itself.

   $$\max_{k \in \text{In}(u)_{\text{post}}} H(k) = \max(T_{max}, H(e_{self}))$$

   Since $H(e_{self}) = T_{max} + 1$, the maximum is $H(e_{self})$. Substituting back into the stability constraint:

   $$H(e_{self}) > H(e_{self})$$

5. Contradiction: The inequality $x > x$ is false for all real numbers. Thus, no stable timestamp can be assigned to a self-loop. The operation creates a logical contradiction and is rejected by the constructor.

III. Transitive Order Preservation (Inductive Step)

We prove that for any causal path $\pi = (v_0, v_1, \dots, v_k)$, the sequence of edge timestamps is strictly increasing.

1. Path Definition: Let $e_i$ be the edge connecting $v_{i-1}$ to $v_i$. Let $H(e_i) = t_i$.

2. Adjacency Relation: For any step $i$ where $1 \le i < k$: The edge $e_i$ terminates at $v_i$. Therefore, $e_i \in \text{In}(v_i)$. The edge $e_{i+1}$ originates at $v_i$.

3. Application of Assignment Rule: The timestamp $t_{i+1}$ for edge $e_{i+1}$ is calculated relative to $\text{In}(v_i)$.

   $$t_{i+1} = 1 + \max \left( \{ H(k) \mid k \in \text{In}(v_i) \} \cup \{0\} \right)$$

4. Inequality Derivation: Since $e_i \in \text{In}(v_i)$, it follows that:

   $$\max \left( \{ H(k) \mid k \in \text{In}(v_i) \} \right) \ge H(e_i) = t_i$$

   Substituting this into the assignment rule:

   $$t_{i+1} \ge 1 + t_i$$ $$t_{i+1} > t_i$$

IV. Conclusion

The timestamp function $H$ enforces a strict total ordering on all causal chains.

$$t_1 < t_2 < \dots < t_k$$

This monotonicity guarantees that the causal graph is a Directed Acyclic Graph (DAG), as any cycle would require the contradiction $t_i < t_i$.

Q.E.D.

1.3.Z Implications and Synthesis

The Causal Graph

A network of relations has replaced the coordinate system. The timestamp functions as a permanent label. It freezes the moment of creation for every link and embeds the arrow of time directly into the topology. This creates a static skeleton. It is a record of events and their causes that stands independent of any observer. We have successfully translated the abstract concept of causality into a concrete, countable structure. This graph is the absolute floor of reality. Beneath this graph there is no sub-structure. There is only the logic of the code itself.

This structure provides the memory of the system. It encodes the past interactions that define the present state. In this ontology, space is not a pre-existing container that events happen within. Space is the relationship between events. If two particles are far apart, it is not because there is a lot of empty void separating them. It is because the graph distance is large. The graph distance is the sheer number of causal links one must traverse to get from one to the other. This is a background-independent description of reality that does not require an external ruler or grid. By embedding the timestamp $t_L$ onto the edges, we ensure that the graph is not just a spatial web. It is a spacetime history. It is a growing block of causal connections where the past is preserved in the topology of the present.

Our inquiry now turns to the dynamics. Defining the specific operations allowed to transform this graph from one moment to the next is the next logical step. We have the object, but we do not yet have the motion. We must determine how this static web becomes a living, evolving universe. A graph that sits eternally unchanged is not a physics. It is a painting. To breathe life into this structure, we must define the legal moves that can alter it. This leads us to the definition of the task space.

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