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Chapter 2: Constraints (Axioms)

We find ourselves possessing a graph substrate that is topologically rich but dynamically inert. Without further instruction, a mere collection of points and lines allows for logical paradoxes, such as events that act as their own ancestors or influences that circulate endlessly without producing change. To transform this static web into a substrate capable of supporting physics, we must impose a set of inviolable rules that forbid self-reference and enforce a strict causal order. We are seeking the logical machinery that prevents the universe from collapsing into a tautology.

Our inquiry demands that we treat the causal link as a directed vector of influence, ensuring influence flows only in one direction and preventing stalling or reversing into incoherence. If we allowed a pair of events to influence each other simultaneously, we would destroy the distinction between cause and effect, collapsing the timeline into a single, undefined moment. We require a mechanism that preserves the distinctness of states, ensuring that the past is separated from the future by an impassable barrier of logic.

These constraints act as the legislative bedrock of our model, clamping down on the infinite degrees of freedom available to the graph. By forbidding loops and enforcing asymmetry, we ensure that every update pushes the system forward, converting undirected potential into a structured history. We are not just drawing shapes; we are defining the mechanical logic that permits the universe to become something new at every step. We proceed now to codify these requirements into the three fundamental axioms that will govern all subsequent evolution.

Preconditions and Goals

• Demonstrate independence through countermodels where one axiom holds and others fail.
• Verify cycle decomposition terminates at length $3$ units linearly with parallel confluence.
• Expose local primitives induce reflexive and asymmetric influences requiring global resolution.
• Confirm enforcement through local approximations with exponential error and logarithmic checks.
• Synthesize exclusions for unique constraints regarding arrows, quanta uniqueness, and strict partial order.

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2.1 The Causal Primitive

We commence the structural definition of the universe by isolating the fundamental atom of physical relation, the single causal link, which we define as a vector of inevitable influence to distinguish it from a static geometric bond. The derivation of a physics capable of supporting the concept of becoming requires a primitive that inherently distinguishes the origin of an action from its destination, thereby embedding the thermodynamic arrow of time directly into the microscopic fabric of the graph. We are searching for a directed operator that transforms the state of the universe from a condition of potentiality into a condition of realized history without relying on a pre-existing background coordinate system to provide orientation. This inquiry demands that we treat the edge as an active vector of becoming that drives the evolution of the system from one moment to the next, ensuring that the topology itself encodes the passage of time.

The assumption of a symmetric bond as the fundamental unit generates a crystalline lattice of frozen relationships where cause and effect are interchangeable variables and the evolution of state remains mathematically undefined. Such a system destroys the distinction between the past and the future because it collapses the timeline into an undirected mesh where no event can be said to precede another in any meaningful causal sense. The graph devolves into a tautological web where information propagates in stagnant circles, lacking the intrinsic gradient necessary to distinguish the source from the target and drive the computation forward. A universe built on bidirectional connections lacks the asymmetry required to support entropy or information processing, resulting in a static void incapable of supporting a history.

We resolve this foundational crisis by defining the causal primitive through the strict and inviolable constraints of irreflexivity and asymmetry to create a geometric arrow of time at the smallest possible scale of existence. Mandating that every connection must distinguish source from target while forbidding any event from influencing itself ensures that the graph evolves as a strict one-way street capable of supporting irreversible processes. This establishes a logical ratchet mechanism at the absolute foundation of reality that seeds the directionality required for the macroscopic flow of time and the eventual emergence of entropy. By enforcing this directionality at the atomic level, we guarantee that the macroscopic universe inherits a coherent temporal order that prevents the reversal of cause and effect.

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2.1.1 Axiom 1: The Directed Causal Link

Establishment of the Directed Causal Link as the Fundamental Relational Unit by Irreflexivity and Asymmetry

It is herein established that the fundamental unit of relation within the Universal State Space (§1.3.1) shall be the Directed Causal Link, denoted as the ordered pair $(u, v)$, acting upon the set of Abstract Events $V$. The validity of the edge set $E \subset V \times V$ is strictly conditioned upon the absolute satisfaction of the following two invariant properties for all elements within the domain:

1. Strict Irreflexivity: The relation shall not, under any circumstance, connect a vertex to itself. For every vertex $u$ contained within the set $V$, the edge $(u, u)$ is categorically excluded from the set $E$. This prohibition enforces the requirement that no event may serve as its own causal antecedent.
2. Strict Asymmetry: The relation shall not permit immediate reciprocity. For every distinct pair of vertices $u$ and $v$ contained within $V$, the existence of the direct edge $(u, v)$ within $E$ necessitates the absolute absence of the inverse edge $(v, u)$ from $E$. This prohibition enforces the local directionality of causal influence.

The existence of an edge $e = (u, v)$ constitutes the physical encoding of the proposition that event $u$ acts as the necessary causal antecedent of event $v$ within the local reference frame.

2.1.2 Commentary: The Physics of Directionality

Derivation of Temporal Directionality from the Topological Rejection of Inertia and Simultaneity

The selection of a strictly directed and irreflexive primitive constitutes the foundational requirement for modeling a universe of becoming (dynamic evolution) rather than a universe of being (static existence). This distinction aligns directly with the Causal Set program initiated by (Bombelli et al., 1987), which posits that the causal order is the primary structure of spacetime, antecedent to metric geometry. However, while Causal Set Theory often assumes the partial order as a given, QBD constructs it mechanically from the edge primitive. In classical crystallography or standard network theory, an undirected edge $\{u, v\}$ signifies a mutual and persistent bond; a state of structural equilibrium where the relationship exists simultaneously for both nodes. However, a theory of fundamental causality requires a mechanism to drive the system strictly out of equilibrium. If the fundamental relations were symmetric, the system would settle into a static lattice; by enforcing directionality, we compel the system to compute its own future.

The Rejection of Inertia (Irreflexivity) Irreflexivity serves as the topological enforcement of fundamental change. A reflexive link $u \to u$ represents a "closed loop of zero length"; a pathological process wherein the output of an event instantaneously feeds back into its own input without traversing any distance in the causal graph. Such a structure models a state of pure inertia or solipsism; it decouples the event from the rest of the relational web. In a universe governed by information transfer, a state that only communicates with itself is thermodynamically indistinguishable from a state that does not exist. By axiomatically forbidding $u \to u$, the theory mandates that existence requires interaction with the external. An event cannot sustain itself through internal recurrence; it must derive its existence from a distinct antecedent and contribute its influence to a distinct consequent. This constraint effectively "hard-codes" the flow of time into the topology; the system must move to persist.

The Rejection of Simultaneity (Asymmetry) Asymmetry serves as the microscopic seed of the macroscopic arrow of time. If the substrate permitted symmetric relations (where $u \to v$ and $v \to u$ coexist), the distinction between "cause" and "effect" would vanish within that local neighborhood. This would collapse the temporal separation between $u$ and $v$ into a single simultaneous cluster; effectively reducing the causal graph to a rigid and undirected lattice akin to a spatial crystal. The imposition of strict asymmetry creates a local potential gradient. It ensures that every elementary interaction acts as a "ratchet"; permitting influence to propagate in only one direction. This atomic directionality, resonating with (Sorkin, 2005) definition of discrete gravity, prevents the system from stagnating in reversible loops and provides the necessary thrust for the emergence of a global and irreversible causal order.

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2.1.Z Implications and Synthesis

Axiom 1: The Causal Primitive

The directed causal link establishes the fundamental asymmetry of the universe, enforcing that influence propagates as an irreversible vector rather than a static bond. Irreflexivity prohibits events from causing themselves, eliminating the possibility of causal stagnation, while asymmetry ensures that no pair of events can influence each other simultaneously. These constraints physically encode the arrow of time at the atomic level, mandating that every connection contributes to a net displacement in the relational landscape.

This shifts the ontology from a lattice of "being" to a network of "becoming," where the structure of the graph itself enforces the distinction between past and future. By forbidding instantaneous loops and self-reference, we ensure that the system cannot become trapped in tautological states, compelling it to evolve through interaction with distinct elements. This mechanism prevents the universe from freezing into a crystalline block, guaranteeing that history is a dynamic process of accumulation rather than a static arrangement.

The imposition of strict directionality drives the system relentlessly forward, ensuring that every update advances the causal order without the possibility of reversal. This microscopic irreversibility is the root of all macroscopic thermodynamics, establishing that the universe is not a reversible machine but a generative process that consumes logical potential to produce history. By locking the arrow of time into the definition of the edge itself, we render the concept of a "rewind" physically meaningless, as the topological structure that defines the present exists only as a consequence of the directed momentum of the past.

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