Chapter 2: Axioms (Constraints)

Overview

Principles of causal structure dictate that links maintain directionality and form bounded geometries without redundancy or cycles that undermine progression. The axioms channel relations into a framework where influence propagates unidirectionally, assembling space from minimal closures while preserving order across scales.

The examination begins with the primitive link, establishing its irreflexive and asymmetric nature to ensure net advancement. Subsequent analysis reveals why antisymmetry alone permits inert structures, necessitating stricter conditions. The constructibility axiom then mandates triangulation under unique paths, with decomposition ensuring reduction to quanta. Independence confirms orthogonality, while transitive influences expose local rules' global shortfalls, culminating in the enforcement of acyclicity.

2.1 Axiom 1: The Causal Primitive

Section 2.1 Scope

The primitive link functions as the foundational unit of causality, enforcing direction without allowance for self-reference or immediate reciprocity. Isolated pairs of vertices suffice to define its properties, independent of extended paths or timestamps.

2.1.1 Axiom: The Directed Causal Link

Directed Causal Link Primitive

The directed causal link, denoted $v_i \to v_j$ , constitutes the minimal relational unit of causality. The directed causal link is irreflexive, such that $v_i \nrightarrow v_i$ holds for all $v_i$ . The directed causal link is asymmetric, such that if the link $v_i \to v_j$ exists, then the link $v_j \to v_i$ does not exist.

2.1.Z Implications and Synthesis

Axiom 1: The Causal Primitive

The directed causal link primitive establishes irreflexivity and asymmetry, the smallest unit that demands one-way propagation without allowing a vertex to loop back on itself or a pair to cancel each other out. Physically, this bars the simplest forms of inertia in the causal structure: no edge can represent a state talking to itself, and no mutual pair can pretend to advance time while standing still. Every connection must therefore contribute to a net displacement in the relational landscape, seeding the directed paths that will later define effective influence.

Analysis of these links reveals that when they chain together, local bans on loops and reciprocals do not extend automatically to longer sequences, as transitive effects introduce hidden cycles. Here emerges the first hint of scale: a single arrow points cleanly, yet a series of them might bend around to touch base unexpectedly, as paths of length greater than one open doors to influences that the primitive alone cannot police. That gap presses forward; with this basic directedness in place, the examination reaches why even the milder condition of antisymmetry proves inadequate for the physical demand of unrelenting progress, a point addressed in the theorem on the insufficiency of antisymmetry.

2.2 Antisymmetry

Section 2.2 Scope

Antisymmetry blocks mutual edges only when vertices differ, yet this mathematical condition leaves room for structures that mimic motion but deliver none. The causal primitive contrasts with this weaker relation, stemming from the need for primitives that support genuine dynamical evolution, where influences propagate without wasting potential on echoes. Antisymmetry declares that if both $u \to v$ and $v \to u$ hold, then $u = v$ , a rule sufficient for abstract orders but blind to the physical process encoded in each edge. To expose the inadequacy, lemmas reveal the pathologies self-loops introduce, followed by a theorem that ties these failures into a chain of breakdowns: cycles that halt the graph's acyclicity, null contributions to entropy, and paradoxes that undermine the very asymmetry the framework requires.

2.2.1 Theorem: Insufficiency of Antisymmetry

Insufficiency of Antisymmetry for a Physical Theory of Causality

Antisymmetry alone is insufficient for a physical theory of causality. Antisymmetry permits self-loops ( $v \to v$ ). Self-loops introduce directed cycles. Directed cycles violate the Directed Acyclic Graph (DAG) requirement. The DAG requirement is essential for Acyclic Effective Causality (§2.7.1). Self-loops create inconsistencies in the dynamics of the rewrite rule $\mathcal{R}$ .

2.2.1.1 Commentary: Argument Outline

Logical Structure of the Argument Against Antisymmetry

The argument demonstrates the inadequacy of antisymmetry by establishing the lemma on self-loops as generators of forbidden cycles. This immediately imperils the DAG structure. The DAG structure is vital for causal acyclicity. This topological breach is then deepened by the thermodynamic nullity lemma. The thermodynamic nullity lemma reveals how such loops inject zero entropic value. These loops masquerade as degrees of freedom. Yet, these loops contribute nothing to the relational ensemble's diversity or dynamical vigor. The theorem's proof integrates these insights into a quartet of cascading failures.

The first failure is the blatant breach of physical parsimony. Self-loops embody vacuous non-processes devoid of inter-event transfer.
The second failure is operational inertness. This isolates self-loops from rewrite dynamics and timestamp coherence.
The third failure is entropic barrenness. This is quantified in the lemma.
The fourth failure is systemic paradox. Reflexive influences erode the asymmetry foundational to global order.

This scaffolded argument reveals the chasm between mathematical elegance and physical exigency and underscores the irreflexive mandate's role as a prophylactic guard against the substrate's descent into stagnant redundancy. Causality ignites as a forge of genuine transformation and does not become a gallery of inert echoes.

2.2.1.2 Diagram: Ordering Constraints

Visual Comparison of Ordering Constraints: Asymmetry, Antisymmetry, and Irreflexivity.

      (A) MATHEMATICAL ANTISYMMETRY          (B) PHYSICAL IRREFLEXIVITY (Axiom 1)
          Rule: u->v & v->u => u=v.              Rule: u->u is FORBIDDEN.
          Result: Permits Self-Loops.            Result: Enforces "Arrow".

          +-------+                              u --------> v
          |   u   |--(Loop)--+                   (Strict Transfer)
          +-------+          |
              ^              |
              |______________|                   Status: ACTIVE
                                                 (Information flows u to v)
          Status: INERT
          (No information transfer)

2.2.1.3 Diagram: The Failure of Antisymmetry

Visualization of why a Self-Loop is technically allowed by mathematical antisymmetry but disastrous for physical causality.

THE INERTIA OF SELF-LOOPS
      -------------------------

      1. The Causal Link (Axiom 1)       2. The Self-Loop (Pathology)
         (Information Transfer)             (Information Stasis)

         [ State A ]----->[ State B ]       [ State A ]--+
              ^                                  ^       |
              |                                  |       |
           Effective                          Ineffective|
           Entropy > 0                        Entropy = 0|
                                                 ^       |
                                                 |_______|

      PHYSICAL VERDICT:
      State 1 drives the Sequencer forward.
      State 2 consumes a logical tick but produces no change.
      Therefore, u -> u must be explicitly forbidden.

2.2.2 Lemma: Pathology of Self-Loops

Creation of Cycles via Self-Loops

A self-loop $v \to v$ is a cycle of length 1. The self-loop $v \to v$ is a directed path starting and ending at $v$ . By definition of the Directed Acyclic Graph (DAG), Directed Acyclic Graphs contain no directed cycles of any length. Thus, self-loops violate the DAG property.

2.2.2.1 Proof: Cycle Creation

Formal Demonstration of Cycle Creation via Self-Loop Insertion

The definition of a directed cycle requires a closed walk $\pi = (v_0, v_1, \dots, v_k)$ such that $v_0 = v_k$ and the length $k \ge 1$ . A self-loop defined as $e = (v, v)$ constitutes a path where the start and end vertices are identical and the edge count is strictly one. Consequently, the self-loop satisfies the structural definition of a cycle. Since the topological requirement of the causal graph forbids all cycles of cardinality $k \ge 1$ to maintain the Acyclic property, the inclusion of a self-loop creates an immediate contradiction with the foundational DAG constraint.

Q.E.D.

2.2.2.2 Commentary: Graph Integrity

Preservation of Global Causal Integrity through the Topological Exclusion of Primordial Cycles

Within the cosmological framework, the integrity of the causal graph hinges upon strict adherence to Directed Acyclic Graph (DAG) semantics. The prohibition of cycles preserves the unidirectional cascade of influence required for emergent timestamping and precludes the recursive entanglements that fracture temporal linearity.

A self-loop constitutes a primordial infraction against this order. It functions not merely as a localized redundancy but as a topological contagion that seeds cyclic pathology. This pathology propagates transitively to vitiate effective causality computations, such as path-based influence relations, effectively arresting the engine's progression from void to structured plenum. The lemma reveals the self-loop as the atomic unit of stasis; axiomatic irreflexivity thus serves as the necessary prophylactic, guarding the substrate's evolutionary trajectory against the gravitational pull of self-referential inertia.

2.2.3 Lemma: Thermodynamic

Thermodynamic Nullity and Zero Entropy Contribution of Self-Loops

The addition of a self-loop contributes no effective entropy to the system ( $\Delta S = 0$ ). States with and without self-loops are equivalent in the relational ensemble.

2.2.3.1 Proof: Zero Entropy

Derivation of Zero Entropy Contribution in Self-Loop Configurations

Per the awareness comonad (§4.3.2), entropy $S = \ln |\Omega|$ . Here, $|\Omega|$ is the number of valid graph configurations in the microcanonical ensemble. A self-loop on vertex $v$ formally adds a binary degree of freedom. This degree of freedom is $e \in E$ or not. This increases $|\Omega|$ to $|\Omega'| = 2 |\Omega|$ per such vertex. However, self-loops are dynamically inert. Self-loops cannot participate in simple paths or the rewrite rule $\mathcal{R}$ . No compliant 2-paths involve self-loops. Thus, graphs $G$ and $G'$ are equivalent under the effective influence relation $\le$ . Graphs $G$ and $G'$ are equivalent under dynamics. Here, $G'$ is $G$ with a self-loop.

In the relational ensemble, configurations are partitioned into equivalence classes. States are identical under $\le$ in these classes. Self-loops add no new classes. Thus, the effective $|\Omega_{eff}|$ remains unchanged. The partition function includes only distinct relational states. This yields $\Delta S = 0$ for self-loops.

Q.E.D.

2.2.4 Proof: Insufficiency of Antisymmetry

Formal Proof of the Insufficiency of Antisymmetry for Causal Consistency Theorem §2.2.1

Consider a graph with one vertex $v$ and edge $e = (v, v)$ , assigned $H(v, v) = t_1$ . This configuration satisfies antisymmetry, since $v = v$ . However, the self-loop engenders four critical failures:

The first failure breaches physical parsimony and non-triviality. The self-loop embodies a non-process devoid of inter-event transfer, burdening the system with redundancy contrary to causal primitives that demand propagation.
The second failure enacts operational inertness. The self-loop isolates from the rewrite rule $\mathcal{R}$ (§4.5.1), as it precludes simple 2-paths due to vertex repetition, rendering timestamps undefined in increasing sequences for $\le$ .
The third failure imposes entropic barrenness. As shown in the lemma on thermodynamic nullity (§2.2.3), the self-loop contributes no effective entropy ( $\Delta S = 0$ ), proving nonparsimonious in the relational ensemble.
The fourth failure precipitates systemic paradox. The self-loop implies $v \le v$ via a length-1 path, eroding $\le$ 's strict asymmetry and undermining global order per the axiom on acyclic effective causality (§2.7.1).

Q.E.D.

2.2.Z Implications and Synthesis

Antisymmetry

If irreflexivity forces each link to carry forward thrust, why does antisymmetry alone permit self-loops that masquerade as activity but produce only stagnation? In the proofs above, such loops qualify as cycles of length one, violating the Directed Acyclic Graph property essential to acyclic effective causality (§2.7.1); they add formal degrees of freedom yet contribute zero to the effective relational ensemble and they cascade into operational inertness, entropic waste, and reflexive paradoxes that erode the global order the framework requires. Physically, this means antisymmetry tolerates inert decorations on the graph, burdens that dilute the substrate's capacity for true propagation and demand the stricter irreflexivity to purge them.

The deeper consequence is that directionality must embed momentum at every scale, not just locally: without this clamp, the graph risks pooling into static knots rather than flowing toward structured becoming, a risk that echoes the temporal finitude (§1.2.7). Yet even with sharpened arrows secured, a new challenge arises in assembly: how do these links close into cycles without spawning redundancies, and what minimal length allows closure while preserving sequence?

2.3 Axiom 2: Geometric Constructibility

Section 2.3 Overview

The requirement of constructibility stipulates that space arises solely from the tightest causal closures; this compels channeling influences into indivisible units while blocking duplicate paths that would shortcut the build. The axiom divides into a positive clause that mandates closure through 3-cycles and a negative one that forbids cloning existing mediations, transforming the graph from a loose net into a triangulated lattice that expands without internal clutter.

At this stage, timestamps maintain monotonic increase and attention focuses on edge configurations; deletions pertain to dynamics. The geometric quantum appears first, justified against shorter forbidden forms, followed by the no-clone principle to eliminate extras, and a potential that tracks descent toward simplicity. Physically, this axiom channels raw edge additions into bounded areas, preparing the substrate for the decompositions that enforce stability across scales.

2.3.1 Axiom: Geometric Constructibility

The Axiom of Geometric Constructibility

The Axiom of Constructibility states that the emergence of physical geometry is defined by the set of admissible constructive tasks on the causal graph. A subgraph constitutes valid physical geometry if and only if the subgraph satisfies two complementary conditions.

The first condition is positive construction, or the primitive. Space is generated exclusively by the closure of minimal directed causal loops.

The second condition is negative constraint, or the law. It is impossible to construct a causal link that duplicates an existing minimal information path.

Formally, the state of the universe $G_{t+1}$ is a valid successor to $G_t$ only if every new relation in $G_{t+1}$ forms a Geometric Quantum (Clause A). Every new relation must do so without violating the Principle of Unique Causality (Clause B).

2.3.2 Definition: Clause A, Geometric Quantum

Clause A of Geometric Constructibility; The Primitive Geometric Quantum

The geometric quantum is the directed 3-cycle, denoted $\gamma = (u \to v \to w \to u)$ . The geometric quantum is the fundamental, indivisible unit of spatial area and topological structure in Quantum Braid Dynamics.

The geometric quantum exhibits indivisibility. A 3-cycle cannot be decomposed into smaller cycles without destroying the cycle itself.

The geometric quantum exhibits minimality. The 3-cycle is the smallest closed loop permitted by the Causal Primitive (Axiom 1). 1-cycles (self-loops) and 2-cycles (mutual edges) are explicitly forbidden.

The geometric quantum serves as a basis. All higher-order geometric structures are emergent aggregates of these fundamental 3-cycles. Examples include tetrahedra and manifolds.

2.3.2.1 Commentary: Minimality Justification

Justification of 3-Cycle Minimality in Causal Closures

The selection of the 3-cycle is not arbitrary. The 3-cycle is the unique solution to the problem of closing a causal chain without violating causality. Consider the hierarchy of loops.

Length 1 is the self-loop, $u \to u$ . The self-loop represents an event causing itself. The self-loop violates the irreflexivity required for the passage of logical time.
Length 2 is the feedback, $u \to v \to u$ . The feedback represents instantaneous mutual causality. If $u$ causes $v$ and $v$ causes $u$ , $u$ and $v$ are simultaneous. In a discrete sequence, this collapses the distinction between $u$ and $v$ . This violates the antisymmetry of the causal order.
Length 3 is the closure, $u \to v \to w \to u$ . The closure is the first structure that allows for feedback. Feedback means information returning to the source. The closure preserves temporal order. The return happens at a later logical time step.

Therefore, the 3-cycle is the "atom" of geometry. The triangle is the simplex of 2D Euclidean space. Similarly, the directed 3-cycle is the simplex of causal space. Any larger cycle ( $L > 3$ ) is dynamically reducible. The larger cycle can be "triangulated" into a sequence of 3-cycles by adding internal chords. The 3-cycle is the stable bottom of this decomposition chain.

2.3.2.2 Diagram: Loop Hierarchy

The Hierarchy of Loops Comparing Self-Loops, Feedback Loops, and 3-Cycle Closures

      -------------------------------------------------

      1. THE SELF-LOOP (Length 1)
         [ u ]--<--+       STATUS: FORBIDDEN (Axiom 1)
         |_________|       Reason: Violation of Irreflexivity.

      2. THE FEEDBACK (Length 2)
         [ u ] ------> [ v ]
         [ u ] <------ [ v ]
                           STATUS: FORBIDDEN (Axiom 3 / DAG)
                           Reason: Instantaneous Mutual Causality.
                           (Violates Antisymmetry of Influence)

      3. THE CLOSURE (Length 3)
            [ v ]
            /   \          STATUS: PERMITTED (Axiom 2)
           /     \         Reason: Smallest structure permitting
        [ u ]-----[ w ]    feedback without simultaneity.
                           "The Geometric Quantum"

2.3.3 Principle: Clause B, Unique Causality

Clause B of Geometric Constructibility; Unique Causality and the Non-Cloning of Paths (PUC).

The Principle of Unique Causality (PUC) states that it is impossible to construct a direct causal link between two events if the informational path connecting them is already degenerate.

This principle acts as a "No-Cloning Theorem" for the causal graph. The principle asserts that the universe forbids redundant causality. If there is already a defined, minimal way for information to flow from $u$ to $v$ , adding a second, identical mechanism for that flow creates a topological ambiguity. This ambiguity is a clone of the causal history. The laws of physics forbid the creation of such ambiguities. This principle filters the raw combinatorial potential of the graph. This prevents the "small-world" catastrophe. In this catastrophe, every event is connected to every other event. This forces the graph to expand into a manifold-like structure. This structure has a defined dimensionality.

2.3.3.1 Definition: Path Cloning Impossibility

Impossibility of Path Cloning and Redundant Causality

Let $u$ and $v$ be vertices in the graph $G$ . Let $\Pi(u,v)$ be the set of directed paths from $u$ to $v$ of length $L \le 2$ . The task of adding the edge $e = (u,v)$ is impossible (forbidden) if $|\Pi(u,v)| \ge 1$ , per the Principle of Unique Causality.

In the language of the rewrite rule, if a 2-path $v \to w \to u$ exists, the closing chord $(u,v)$ may only be added if no other path $v \to x \to u$ exists (where $x \neq w$ ). The closing chord may only be added if no existing direct edge $v \to u$ exists. If the path is degenerate (non-unique), the construction is blocked. This forces the system to resolve the ambiguity via thermodynamic deletion. Growth can then resume.

Pseudocode for PUC Check (Local Query for |Π|≤1; O(deg) Time):

def is_permissible(G, v, w, u):  # Check add (u,v) on 2-path v->w->u
    if G.has_edge(v, u):         # Direct path? Block
        return False
    for x in G.successors(v):    # Alt 2-path v->x->u, x≠w?
        if x != w and G.has_edge(x, u):
            return False
    return True                  # Unique: Permit

This local successor scan enforces |Π|≤1 without full BFS (for L=2). Verified via unit tests (pytest): Rejects direct/alt paths, accepts unique (e.g., isolated 2-path passes; alt 3-path ok).

2.3.3.1 Diagram: Principle of Unique Causality (PUC)

Illustration of the No Cloning rule.

      Rule: Do not clone the path.
      - If a path of length 2 exists, a chord cannot be 
        because it duplicates the causal information.

      SCENARIO: A influences C via B.

            (B)
           ^   \
          /     \
       (A)       (C)

      PROPOSAL: Add direct edge A -> C?

      [ ] If we add A->C:
          We now have TWO paths:
          1. A -> B -> C (Mediated)
          2. A -> C      (Direct)

      VERDICT: FORBIDDEN.
      Path 1 already encodes the causal relation A<=C.
      Path 2 is redundant information (Cloning).
      Result: The graph remains sparse; geometry emerges.

2.3.4 Definition: Lexicographic Potential

Lexicographic Potential Metric ( $\Phi(G)$ ) for Topological Complexity

The lexicographic potential is defined as the ordered pair $\Phi(G) = (L, N_L)$ . Here, $L=\max(\text{len}(C))$ is the length of the longest simple directed cycle in $G$ . $N_L$ is the number of distinct simple cycles in $G$ of length $L$ .

A state $G'$ is strictly simpler than $G$ if $\Phi(G') < \Phi(G)$ in lexicographic order. This means either $L' < L$ , or $L' = L$ and $N_L' < N_L$ . This defines a well-ordered measure of the graph's cyclic complexity. Both $L \le |V|$ and $N_L \le \binom{|E|}{L}$ . Thus, the lexicographic order on $\Phi(G)$ is well-founded on all finite graphs.

2.3.4.1 Commentary: Complexity Metrics

Analysis of Topological Complexity Metrics and Descent Criteria

The prevalence of triangular geometry requires formal justification. To formalize the claim that 3-cycles are the "stable" quanta, a metric of complexity must first be defined. This metric distinguishes a "simple" graph from a "complex" one. This provides the foundation for analyzing stability and decomposition. The lexicographic potential elegantly addresses this. The lexicographic potential imposes a hierarchical ordering. This ordering first minimizes the girth of the largest anomaly (via $L$ ). Secondarily, this ordering thins its multiplicity ( $N_L$ ). This forges a descent metric. The descent metric terminates precisely at the 3-cycle basin without ambiguity.

Diverging from coarser proxies like total cyclomatic genus or average girth, the lexicographic potential does not blur fine-grained instabilities. This dual-indexed gauge captures the nuanced thermodynamics of topological refinement. Protracted cycles exact disproportionate entropic penalties. These penalties arise through the propensity for chordal subdivision. This is akin to how elongated polymers in statistical mechanics favor coiled minima. The well-foundedness of the lexicographic potential on finite graphs guarantees algorithmic termination. The sensitivity of the lexicographic potential to parallel updates ensures the rewrite engine's progress remains inexorable. This holds even amidst the cacophony of concurrent local closures. This anchors the axiom's dynamical veracity in a quantifiable arrow toward simplicial purity.

2.3.Z Implications and Synthesis

Axiom 2: Geometric Constructibility

The axiom of constructibility insists that physical geometry emerges only through the directed 3-cycle as the fundamental quantum, coupled with the principle of unique causality that prohibits direct links where paths of length at most two already mediate the connection. In physical terms, this means space assembles as sheets of triangles, each closing a causal chain just long enough for feedback to return without simultaneity, while the no-clone rule ensures every influence traces a unique route, preventing the graph from collapsing into a dense tangle that shortcuts distances.

But what occurs when longer cycles slip in despite these local guards: do they persist indefinitely, or does the system possess a mechanism to subdivide them back to the quantum base? The tension arises because unchecked assemblies might form quadrilaterals or worse, bloating the topology until reductions kick in.

2.4 Decomposition

Section 2.4 Scope

Once a cycle longer than three emerges, the substrate dismantles it without scattering fragments or inflating complexity further. The reduction process for cycles beyond the quantum length channels compliant 2-paths into chords which triangulate the loop into overlapping 3-cycles, after which targeted deletions drop the potential measure while sparing the primitives. The theorem guarantees termination at length three, converting potential topological sprawl into an enforced simplicial foundation.

Chordless maximal cycles and concurrent updates that merge without conflict apply. The reasoning unfolds from confluence to ensure overlapping operations align, through chordlessness to confirm eligibility, to deletion's quantified drop, parallel steps' net descent, and induction over additions and removals. Physically, this theorem embodies the axiom's dynamical force: it reveals how local closures propel a global simplification, mirroring the discreteness observed in spatial structures by digesting irregularities into atomic tiles.

2.4.1 Theorem: General Cycle Decomposition

Finite Decomposition of General Cycles via the Universal Constructor

Let $G$ be a graph with a lexicographic potential $\Phi(G) = (L, N_L)$ . Here, $L = \maxcycle(G) \ge 4$ . The rewrite rule $\mathcal{R}$ enables the dynamic reduction of larger cycles through a two-step process. The first step consists of additions of chords. These additions decompose the large cycle into a composition of overlapping 3-cycles. The second step consists of deletions. These deletions break the large cycle structure. The result is a new graph $G'$ with a strictly smaller potential, $\Phi(G') < \Phi(G)$ . The process terminates when the maximal cycle length is $L=3$ .

2.4.1.1 Commentary: Decomposition Argument

Logical Structure of the Topological Descent Argument

The argument for the General Cycle Decomposition Theorem proceeds through a structured sequence of lemmas and proofs that establish the rewrite rule's capacity to systematically reduce cycle complexity. The proof begins with a demonstration that overlapping local operations resolve consistently via the diamond property, ensuring that parallel chord additions do not lead to divergent outcomes. This local confluence is then extended to show that maximal cycles lack internal shortcuts, thereby guaranteeing the existence of eligible 2-paths for decomposition without prior interference. Next, we quantify how edge excision breaks large structures while preserving embedded 3-cycles, establishing the strict descent in the lexicographic potential. The lemma on Strict Decrease in Parallel Updates (§2.4.5) integrates these elements, showing that a full rewrite step; comprising all compliant additions followed by targeted deletions; yields a net reduction in $\Phi(G)$ , with additions non-increasing the maximum length and deletions enforcing the drop. The capstone proof synthesizes these via a two-part induction: first, additions decompose without inflating girth (proven by contradiction, leveraging path replacement theorems to rule out longer emergent cycles); second, deletions finalize the reduction, terminating at the 3-cycle basin. Verification through concrete examples (4-, 5-, and 6-cycles) and simulation data corroborates the O(k) efficiency, affirming the theorem's dynamical inevitability: larger cycles are not merely reducible but inexorably digested into the geometric quantum, forging a unidirectional arrow toward simplicial stability.

2.4.2 Lemma: Confluence of the Constructor

Confluence of Concurrent Operations in the Universal Constructor

The rewrite rule $\mathcal{R}$ is locally confluent. Overlapping 2-paths, for example $v \to w \to u$ and $w \to u \to x$ , resolve to the same graph via the diamond property. The chords $u \to v$ and $x \to w$ are independent by Principle of Unique Causality (§2.3.3) uniqueness. The rewrite rule $\mathcal{R}$ is terminating. This occurs because $\Phi(G)$ decreases. Thus, the rewrite rule $\mathcal{R}$ is confluent by Newman's lemma.

2.4.2.1 Proof: Local Confluence

Demonstration of Local Confluence via the Diamond Property of Graph Rewriting

Consider two overlapping 2-paths in $G$ . An example is $v \to w \to u$ and $w \to u \to x$ . The proposed chords are $u \to v$ and $x \to w$ . By the Principle of Unique Causality (PUC) (§2.3.3), each 2-path must be the unique path of length $\le 2$ from its start to end vertex. Examples include no direct $v \to u$ or $w \to x$ .

Applying $\mathcal{R}$ to add $u \to v$ forms the cycle $v \to w \to u \to v$ . This does not affect the 2-path $w \to u \to x$ . This is because vertices $v$ and $x$ are distinct. Similarly, adding $x \to w$ does not affect $v \to w \to u$ . Applying both chords in either order yields the same graph $G''$ . The edges of $G''$ are $E \cup \{u \to v, x \to w\}$ . This satisfies the diamond property for local confluence.

Termination follows because $\Phi(G)$ decreases with each $\mathcal{R}$ application. This decrease is proven in the lemma on decrease in parallel updates (§2.4.5). $\Phi = (L, N_L)$ is well-ordered. By Newman's lemma, local confluence and termination imply global confluence. All rewrite sequences from $G$ reach the same normal form.

Q.E.D.

2.4.2.2 Commentary: Parallel Interference

Resolution of Interference in Independent Parallel Updates

The potential for interference is analyzed in the simultaneous addition of chords in a dense graph, where one addition might invalidate the eligibility of another. PUC eligible 2-paths are defined by the absence of a closing chord. Since the addition of a chord $u \to v$ strictly affects the connectivity between $u$ and $v$ , it does not alter the adjacency relationships of disjoint vertex pairs.

In the rare event of a shared vertex leading to resource contention (e.g., a vertex participating in multiple reducible cycles), the operation is defined to act on the maximal independent set of compliant paths. Any paths temporarily blocked by contention remain eligible for the subsequent tick. Thus, while density may serialize the reduction, it does not prevent termination or alter the final normal form.

2.4.3 Lemma: Chordlessness of Maximal Cycles

Chordless Topology of Maximal Cycles in the Causal Graph

In a graph $G$ with a maximal simple directed cycle $C$ of length $L \ge 4$ , the subpath $A_C$ is chordless and simple. The subpath $A_C$ is defined as $v_{i+2} \to \dots \to v_i$ .

2.4.3.1 Proof: Chordlessness

Derivation of the Chordless Condition from Cycle Maximality

By the hypothesis, the rewrite adds chord $e = (v_{i+2}, v_i)$ based on a 2-path. This implies no prior chord exists in $C$ connecting non-adjacent vertices. Such a chord would allow a shorter cycle. Such a chord would interfere with the Principle of Unique Causality condition for the rewrite. Thus, $C$ is chordless.

Subpaths of chordless cycles are inherently chordless. Subpaths of chordless cycles are also simple. Thus, $A_C$ is simple and chordless.

Q.E.D.

2.4.4 Lemma: Reduction via Deletion

Reduction of Lexicographic Potential via Targeted Edge Deletion

The deletion of an edge in a reducible large cycle strictly reduces $\Phi(G)$ . The deletion preserves smaller cycles.

2.4.4.1 Proof: Potential Reduction

Demonstration of Strict Potential Reduction Following Deletion in Reducible Cycles

For a cycle of length $L \ge 4$ that has been made reducible by additions, deleting an edge not in all smaller cycles removes the large simple path. This drops $L$ or $N_L$ . This preserves smaller cycles if they do not use the deleted edge. Due to the local topology, the probability that an edge is shared by two distinct 3-cycles generated from disjoint 2-paths is low. Hence, deletions can be chosen to act on unshared edges almost surely.

Q.E.D.

2.4.5 Lemma: Decrease in Parallel Updates

Net Reduction of Potential under Parallel Constructor Updates

A full parallel rewrite step consists of all eligible additions followed by necessary deletions. This step strictly decreases the lexicographic potential $\Phi(G) = (L, N_L)$ . Here, $L$ is the max cycle length. $N_L$ is the number of such cycles.

2.4.5.1 Proof: Strict Decrease

Verification of Strict Potential Decrease across Synchronous Parallel Updates

The proof is structured in sub-parts.

Additions are non-increasing. By the theorem on general cycle decomposition (§2.4.1), the addition of chords to eligible 2-paths in a graph $G$ with $L \ge 4$ does not create new cycles of length $L' \ge L$ . Any new cycles formed have length $L' < L$ . The original large cycle persists but becomes decomposable. Therefore, the maximal cycle length $L$ does not increase.

Deletions are strictly decreasing. By the lemma on reduction via deletion (§2.4.4), the deletion of an edge in a reducible large cycle strictly reduces $L$ by breaking the large simple cycle form. This drops $L$ or $N_L$ . This preserves smaller cycles.

Net effect: For a non-trivial rewrite, additions decompose large cycles without increase. Deletions then break remaining large forms. Net: $\Phi(G') < \Phi(G)$ . Thus, every executed parallel rewrite strictly decreases $\Phi$ .

Q.E.D.

2.4.6 Proof: General Cycle Decomposition

Formal Proof of the Finite Decomposition of General Cycles Theorem (§2.4.1)

Part 1: Addition of Chords: The proof for additions proceeds by contradiction.

Premise: Let $G = (V, E, H)$ be a causal graph satisfying Axioms 1 and 2, with maximum simple directed cycle length $L = \maxcycle(G) \ge 4$ . Let $C = (v_1 \to v_2 \to \dots \to v_L \to v_1)$ be a simple directed cycle in $G$ achieving this maximum (so no simple directed cycle in $G$ has length exceeding $L$ ). By the lemma on chordlessness of maximal cycles (§2.4.3), $C$ is chordless: there are no edges in $E$ connecting non-adjacent vertices on $C$ (else a shorter cycle would exist, contradicting maximality). The rewrite rule $\mathcal{R}$ identifies a Principle of Unique Causality (PUC)(§2.3.3) compliant 2-path on $C$ , say $A_1 = (v_i \to v_{i+1} \to v_{i+2})$ (indices modulo $L$ ), and adds the chord $e = (v_{i+2}, v_i) \notin E$ , yielding $G' = (V, E \cup \{e\}, H')$ where $H'(e) = t_L$ exceeds prior timestamps by monotonicity (§1.3.4).

Hypothesis for Contradiction: The addition of $e$ creates a new simple directed cycle $C'$ in $G'$ with length $L' \ge L$ .

Step 1: Structure of $C'$ . Since $C'$ is new and simple, it must use the novel edge $e$ (else it existed in $G$ , contradicting maximality of $L$ ). Thus, $C'$ decomposes as $C' = (v_i \xrightarrow{P} v_{i+2} \xrightarrow{e} v_i)$ , where $P = (v_i = w_0 \to w_1 \to \dots \to w_k = v_{i+2})$ is a simple directed path in the original $G$ (length $k = \ell(P) \ge 1$ , with no self-loops by Axiom 1). By PUC, $P$ is the unique path from $v_i$ to $v_{i+2}$ of length $\le 2$ in $G$ ; since $A_1$ is the compliant 2-path enabling $e$ , any $P$ must be disjoint from $A_1$ except at endpoints (else degeneracy blocks the add).

Step 2: Length Bound. The length of $C'$ is $L' = \ell(P) + 1 = k + 1$ . By hypothesis, $k + 1 \ge L$ , so $k \ge L - 1 \ge 3$ (as $L \ge 4$ ). Note $\ell(A_1) = 2 < k$ .

Step 3: Construction of Replacement Cycle $C''$ in $G$ . Decompose the remainder of $C$ into the forward arc $A_f = (v_{i+2} \to v_{i+3} \to \dots \to v_i)$ (length $\ell(A_f) = L - 2 \ge 2$ , simple by chordlessness). Form $C'' = P \circ A_f = (v_i \xrightarrow{P} v_{i+2} \xrightarrow{A_f} v_i)$ .

Step 4: $C''$ is a Simple Directed Cycle in $G$ .

Directed Cycle: $C''$ is a closed walk from $v_i$ to itself via directed edges in $E$ (no $e$ ).
Length: $\ell(C'') = k + (L - 2) \ge (L - 1) + (L - 2) = 2L - 3 \ge 5 > L$ (since $L \ge 4$ ).
Simple (No Vertex Repetitions): Suppose for contradiction a vertex $w_j$ $w_{j}$ (for $0 < j < k$ $0 < j < k$ ) repeats in $C''$ $C^{''}$ , i.e., $w_j = v_m$ $w_{j} = v_{m}$ for some $m$ $m$ with $i+2 \le m \le i$ $i + 2 \leq m \leq i$ (modulo $L$ $L$ , on $A_f$ $A_{f}$ ). Then:
- The subpath $(v_i \to \dots \to w_j) \circ (w_j \to \dots \to v_{i+2})$ on $P$ would form a cycle in $G$ of length $< k$ (strict by simplicity of $P$ ). But chordlessness of $C$ and PUC imply no such short cycles or chords connect non-adjacent points on $C$ ; if $w_j$ on $A_f$ , this subpath shortcuts $C$ via $P$ 's prefix, yielding a cycle $(v_i \to \dots \to w_j \to \dots \to v_i)$ of length $< L$ (replacing $A_f$ 's suffix with $P$ 's prefix, shorter since $j < k$ ), contradicting maximality.
- Similarly, no internal repetition in $A_f$ (by simplicity of $C$ ). Endpoint overlap is trivial (closure). Thus, vertices in $P$ 's interior are disjoint from $A_f$ 's interior, and $C''$ has no repeats: total distinct vertices = $k + (L - 2) - 0 = 2L - 2 > L$ .

Step 5: Contradiction. $C''$ is a simple directed cycle in $G$ of length $> L$ , contradicting the premise that $L = \maxcycle(G)$ . Therefore, no such $C'$ exists: additions cannot create cycles of length $\ge L$ . They merely decompose $C$ into overlapping 3-cycles (e.g., $v_i \to v_{i+1} \to v_{i+2} \to v_i$ ), each of length 3 < L, without inflating girth. The original $C$ persists as a composite but becomes reducible.

Part 2: Deletions
For a cycle of length $L \ge 4$ that has been made reducible by additions (Part 1), a deletion of one edge in the large cycle reduces $L$ by removing the path. This preserves the smaller cycles. The process of additions (to compose) and deletions (to break) ensures $\Phi$ decreases. The process terminates at $L=3$ .

Q.E.D.

2.4.6.1 Diagram: Digestion of Geometry

Topological Digestion of a 4-Cycle into Geometric Quanta

      (Reducing Potential L=4 -> L=3)
      ===============================

      STEP 1: The Unstable "Square"
      (A loop too large for the quantum vacuum)

           B <--------- C
           ^            ^
           |            |
           |            |
           A ---------> D

      STEP 2: The Chord Insertion (Rewrite Rule)
      (Identifying a compliant 2-path A->D->C)

           B <--------- C
           ^          / ^
           |        /   |
           |      /     |
           A --->D      |
            \           |
              \         |
                ------> D

      STEP 3: The Entropic Split
      (The chord A->C forms two 3-cycles: A-B-C and A-C-D)
      
      Result: Max cycle length drops from 4 to 3.
              Geometric Constructibility is restored.

2.4.6.2 Calculation: Worked 4-Cycle Reduction

Algorithmic Verification of the Reduction of a 4-Cycle

The following provides a concrete, step-by-step example of how a 4-cycle is dynamically reduced through iterative application of the rewrite rule $\mathcal{R}$ . The graph's state and $\Phi(G) = (L, N_L)$ are tracked. The graph is an isolated directed 4-cycle with vertices A, B, C, D and edges A → B → C → D → A.

Initial State:

Graph edges: A → B, B → C, C → D, D → A.

Cycles: The single 4-cycle A-B-C-D-A.

Maximal $L = 4, N_L = 1$ .

$\Phi(G) = (4, 1)$ .

Phase 1: Addition of Chords

The graph contains four compliant 2-paths (e.g., A → B → C, etc.). The algorithm adds all closing chords in parallel: C → A, D → B, A → C, B → D.

New graph edges: Original plus C → A, D → B, A → C, B → D.

Cycles: Now includes multiple overlapping 3-cycles (e.g., A-B-C-A, B-C-D-B, etc.), but the 4-cycle persists as a composition.

Maximal $L = 4, N_L = 1$ .

$\Phi(G') = (4, 1)$ .

Phase 2: Deletions to Break Large Cycles

The algorithm identifies the remaining 4-cycle and deletes 1 edge (e.g., B → C).

Final cycles: Remaining 3-cycles (e.g., C-D-A-C, preserved).

Max $L = 3, N_L = 1$ .

$\Phi(\text{final}) = (3, 1)$ .

This example shows additions decompose the large cycle, making it reducible as union; deletion of bridge removes large form.

2.4.6.3 Example: Worked 5-Cycle Reduction

Algorithmic Verification of the Reduction of a 5-Cycle

The following provides a concrete, step-by-step example of how a 5-cycle is dynamically reduced through iterative application of the rewrite rule $\mathcal{R}$ . The graph's state and $\Phi(G) = (L, N_L)$ are tracked. The graph is an isolated directed 5-cycle with vertices A, B, C, D, E and edges A → B → C → D → E → A.

Initial State:

Graph edges: A → B, B → C, C → D, D → E, E → A.

Cycles: The single 5-cycle A-B-C-D-E-A.

Maximal $L = 5, N_L = 1$ .

$\Phi(G) = (5, 1)$ .

Phase 1: Addition of Chords

The graph contains five compliant 2-paths (e.g., A → B → C, etc.). The algorithm adds all closing chords in parallel: C → A, D → B, E → C, A → D, B → E.

New graph edges: Original plus C → A, D → B, E → C, A → D, B → E.

Cycles: Now includes multiple overlapping 3-cycles (e.g., A-B-C-A, B-C-D-B, etc.), but the 5-cycle persists as a composition.

Maximal $L = 5, N_L = 1$ .

$\Phi(G') = (5, 1)$ .

Phase 2: Deletions to Break Large Cycles

The algorithm identifies the remaining 5-cycle and deletes 3 edges (e.g., B → C, D → E, A → B, but optimized to minimal).

Final cycles: Remaining 3-cycles (preserved as they don't rely on deleted edges).

Max $L = 3, N_L = 2$ (or similar, depending on deletions).

$\Phi(\text{final}) = (3, 2)$ .

This example shows additions create overlapping 3-cycles covering the large cycle's edges, making it reducible as union; deletions of bridges remove large form.

2.4.6.4 Calculation: Worked 6-Cycle Reduction

Algorithmic Verification of the Reduction of a 6-Cycle

The following provides a concrete, step-by-step example of how a 6-cycle is dynamically reduced through iterative application of the rewrite rule $\mathcal{R}$ . This demonstrates overlaps and confluence. The graph is an isolated directed 6-cycle with vertices A, B, C, D, E, F and edges A → B → C → D → E → F → A.

Initial State:

Graph edges: A → B, B → C, C → D, D → E, E → F, F → A.

Cycles: The single 6-cycle.

Maximal $L = 6, N_L = 1$ .

$\Phi(G) = (6, 1)$ .

Phase 1: Addition of Chords

The graph contains six compliant 2-paths (e.g., A → B → C, etc.). The algorithm adds all closing chords in parallel: C → A, D → B, E → C, F → D, A → E, B → F.

New graph edges: Original plus C → A, D → B, E → C, F → D, A → E, B → F.

Cycles: Now includes multiple overlapping 3-cycles (e.g., A-B-C-A, B-C-D-B, etc.), but the 6-cycle persists as a composition.

Maximal $L = 6, N_L = 1$ .

$\Phi(G') = (6, 1)$ .

Phase 2: Deletions to Break Large Cycles

The algorithm identifies the remaining 6-cycle and deletes 2 edges (e.g., B → C, E → F).

Final cycles: Remaining 3-cycles (preserved as they don't rely on deleted edges).

Max $L = 3, N_L = 3$ .

$\Phi(\text{final}) = (3, 3)$ .

Overlaps show confluence: Order of additions (e.g., Step2 first) yields same final via diamond (§2.4.2).

2.4.6.5 Calculation: Simulation Verification

Simulation Verification of the Cycle Reduction Algorithm

To verify the general two-phase process (Phase 1: add all chords; Phase 2: delete to break large cycles), the reduction is simulated for directed k-cycles from $k=4$ to 12. The code implements the algorithm and outputs the number of operations per phase.

import networkx as nx
import pandas as pd

def create_directed_cycle(k):
    """Creates a simple directed k-cycle."""
    G = nx.DiGraph()
    nodes = list(range(k))
    for i in range(k):
        G.add_edge(nodes[i], nodes[(i + 1) % k])
    return G

def max_cycle_length(G):
    """Finds the length of the longest simple cycle in the graph."""
    try:
        cycles = list(nx.simple_cycles(G))
        if not cycles:
            return 0
        return max(len(c) for c in cycles)
    except nx.NetworkXNoCycle:
        return 0

def find_compliant_2_paths(G):
    """Finds all PUC-compliant 2-paths (v -> w -> u)."""
    paths = []
    for v in G.nodes():
        for w in G.successors(v):
            for u in G.successors(w):
                if u == v: continue
                # PUC Check 1: No direct path v->u
                if G.has_edge(v, u):
                    continue
                # PUC Check 2: No alternative 2-path v->x->u with x != w
                other_paths = 0
                for x in G.successors(v):
                    if x != w and G.has_edge(x, u):
                        other_paths += 1
                # PUC Check 3: The closing chord u->v does not already exist
                if other_paths == 0 and not G.has_edge(u, v):
                    paths.append((v, w, u))
    return paths

def add_all_chords(G):
    """
    Phase 1: Add all PUC-compliant chords in parallel.
    Returns the number of additions.
    """
    ops = 0
    paths_to_add = find_compliant_2_paths(G)
    for v, w, u in paths_to_add:
        G.add_edge(u, v)
        ops += 1
    return ops

def delete_to_break_large_cycles(G):
    """
    Phase 2: Sequentially remove edges from large cycles
    until max L <= 3.
    Returns the number of deletions.
    """
    ops = 0
    current_max_len = max_cycle_length(G)
    while current_max_len > 3:
        cycle_found = None
        for cycle in nx.simple_cycles(G):
            if len(cycle) > 3:
                cycle_found = cycle
                break

        if cycle_found:
            # Delete the first edge of this cycle
            edge_to_remove = (cycle_found[0], cycle_found[1])
            if G.has_edge(*edge_to_remove):
                G.remove_edge(*edge_to_remove)
                ops += 1
            current_max_len = max_cycle_length(G)  # Re-check
        else:
            break

    return ops

def total_reduction_steps(k):
    """
    Calculates the total steps to reduce a k-cycle using the
    two-phase (add-all, then-delete) algorithm.
    """
    if k <= 3:
        return 0, 0

    G = create_directed_cycle(k)
    # Phase 1: Add all k chords (in simple k-cycle, all compliant)
    add_ops = add_all_chords(G)
    # Phase 2: Delete until max L <= 3
    del_ops = delete_to_break_large_cycles(G)
    return add_ops, del_ops

# --- Main execution ---
results = []
for k in range(4, 13):
    adds, dels = total_reduction_steps(k)
    results.append({
        "Cycle Length (k)": k,
        "Add Operations (Phase 1)": adds,
        "Delete Operations (Phase 2)": dels,
        "Total Reduction Steps": adds + dels
    })

# Format and print the results as a Markdown table
df = pd.DataFrame(results)
print(df.to_markdown(index=False, floatfmt=".0f"))

Simulation Results:

Cycle Length (k)	Add Operations (Phase 1)	Delete Operations (Phase 2)	Total Reduction Steps
4	4	1	5
5	5	3	8
6	6	2	8
7	7	3	10
8	8	3	11
9	9	3	12
10	10	3	13
11	11	3	14
12	12	3	15

This table shows the deterministic two-phase reduction steps for k-cycles. For k-cycle, Phase 1 adds k chords (one per 2-path), and Phase 2 deletes until max $L \le 3$ (typically 2-3 deletions for these k, varying by overlap). The total scales roughly as k + 3 for larger k, as deletions stabilize around 3 (sufficient to break the original cycle while preserving 3-cycles). This confirms the theorem's efficiency: O(k) steps to reduce to 3-cycles.

2.4.Z Implications and Synthesis

Decomposition

Any graph harboring a maximal cycle of length four or more succumbs to the rewrite rule, which first overlays chords to shatter it into a cover of 3-cycles (§2.4.1), then excises edges to fracture the remnants, yielding a strictly lower lexicographic potential (§2.4.5) and terminating uniquely at the quantum basin. In physical interpretation, this means the substrate cannot sustain bloated loops; they break down inexorably under local pressures, with confluence (§2.4.2) resolving overlaps and chordlessness (§2.4.3) ensuring clean access, much as thermal fluctuations favor compact configurations over elongated ones.

The broader ramification is self-correction toward dimensionality: unchecked growth would yield amorphous webs, but decomposition enforces lateral spread in triangulated layers, hinting at emergent flatness; yet does this process rely on the axioms' separation, or could one collapse without the other? That question of orthogonality draws the examination to the theorem on independence of axioms 1 and 2, where countermodels isolate each constraint's domain.

2.5 Independence

Section 2.5 Overview

A system enforces directed links without compelling triangular geometry, as the quantum's closure does not presuppose irreflexive arrows. The logical independence of the first two axioms appears through graphs where one holds amid the other's breach, such as a chordless square that directs cleanly but defies quanta, or an isolated triangle paired with a self-loop that geometrizes locally yet reflexively. To flag these mismatches, the geometric syndrome serves as a local charge detector.

Timestamps and rewrites remain outside this static probe of entailment. The strategy builds Model A to uphold causality absent constructibility, Model B to realize quanta despite reflexivity, then synthesizes non-derivation. Physically, this independence underscores distinct foundations: one carves the arrow's thrust, the other the tile's boundary, their combination the sparsest base for directed space without overlap.

2.5.1 Theorem: Independence of Axioms 1 and 2

The Logical Independence of the Causal Primitive and the Geometric Primitive

Axiom 1 (The Causal Primitive) and Axiom 2 (The Geometric Primitive) are logically independent.

2.5.1.1 Commentary: Independence Argument

Logical Structure of the Mutual Non-Entailment Argument

The proof of independence between Axiom 1 and Axiom 2 follows the standard model-theoretic approach: each axiom is shown not to entail the other by constructing explicit countermodels where one holds while the other fails. The argument commences with Model A, a sparse directed 4-cycle graph that satisfies Axiom 1's irreflexivity and asymmetry (no self-loops or mutual edges) but violates Axiom 2's constructibility by harboring an unreduced cycle longer than 3, defying the geometric quantum's minimality and the mandate for triangulation. This model's syndrome analysis reinforces the breach: local triplets signal precursors (+1) yet permit global profligacy, underscoring Axiom 1's silence on topological granularity. Symmetrically, Model B inverts the violation: a disjoint union of a pristine 3-cycle (affirming Axiom 2's core primitive) and an isolated self-loop (breaching Axiom 1's irreflexivity), where the geometric excitation (-1 syndrome) thrives independently of the causal degeneracy (0 syndrome), proving Axiom 2's topological focus cannot enforce universal directionality. The concluding synthesis integrates these countermodels into the theorem's capstone: mutual non-entailment cements the axioms as orthogonal foundations; one sculpting causal arrows, the other geometric quanta; each indispensable yet irreducible, their interplay the bedrock of the framework's coherentist edifice without derivational hierarchy.

2.5.1.3 Diagram: Independence Matrix

Logical Independence Matrix Contrasting Axiom Satisfaction across Countermodels

      ------------------------------------------
      We demonstrate independence by constructing two universe models
      where one axiom fails while the other holds.

      | MODEL      | STRUCTURE                | AXIOM 1      | AXIOM 2      |
      |            |                          | (Causal)     | (Geometric)  |
      |------------|--------------------------|--------------|--------------|
      | CASE A    | A 4-cycle with           | SATISFIED    | VIOLATED     |
      |            | no chords.               | (No loops,   | (Contains    |
      |            | (A->B->C->D->A)          |  Directed)   |  unreduced   |
      |            |                          |              |  L=4 cycle)  |
      |------------|--------------------------|--------------|--------------|
      | CASE B    | A 3-cycle disjoint       | VIOLATED     | SATISFIED    |
      |            | from a self-loop.        | (Contains    | (Geometry    |
      |            | ({A->B->C->A} U {X->X})  |  reflexive   |  exists as   |
      |            |                          |  X->X)       |  3-cycle)    |
      |------------|--------------------------|--------------|--------------|

2.5.2 Lemma: Independence Case A

The Existence of Causal Validity amidst Geometric Violation in the Chordless 4-Cycle Model

Case A satisfies Axiom 1 but violates Axiom 2.

2.5.2.1 Proof: Verification of Case A

Formal Verification of the Independence of the Causal Primitive from Geometric Constructibility

The countermodel $G_A$ consists of a single connected component forming a directed cycle of length four. The vertex set $V$ contains four distinct elements $\{A, B, C, D\}$ . The edge set $E$ connects these elements in a strict sequence $A \to B \to C \to D \to A$ . The graph contains no additional edges; specifically, the graph lacks the internal chords $A \to C$ or $B \to D$ that would triangulate the structure.

The analysis first tests the model against Axiom 1. The edge set $E$ contains no edges of the form $(v, v)$ , satisfying the requirement of irreflexivity. Furthermore, for every directed edge $(u, v)$ present in $E$ , the inverse edge $(v, u)$ is absent from the set. This absence satisfies the requirement of asymmetry. Consequently, the model $G_A$ strictly adheres to the definitions of the Causal Primitive.

The analysis next tests the model against Axiom 2. Axiom 2 mandates that valid physical geometry arises exclusively from the closure of minimal directed causal loops, defined as 3-cycles. The graph $G_A$ contains a cycle of length four. Due to the absence of internal chords, this 4-cycle admits no decomposition into constituent 3-cycles. The structure persists as an irreducible topological unit that exceeds the defined geometric quantum. This irreducibility directly contradicts the Constructibility Axiom, which requires that all spatial structures derive from the fundamental triangular basis.

The model $G_A$ therefore demonstrates a state that is causally valid but geometrically invalid. This establishes that the constraints of the Causal Primitive do not logically entail the constraints of Geometric Constructibility.

2.5.3 Lemma: Independence Case B

Existence of Geometric Validity amidst Causal Violation in the Disjoint Reflexive Model

Case B satisfies Axiom 2 but violates Axiom 1.

2.5.3.1 Proof: Verification of Case B

Formal Verification of the Independence of Geometric Constructibility from the Causal Primitive

The countermodel $G_B$ comprises two disjoint subgraphs. The first subgraph $C_1$ forms a directed 3-cycle involving vertices $\{A, B, C\}$ . The second subgraph $C_2$ forms an isolated vertex $X$ possessing a single reflexive edge $(X, X)$ . The complete state is the union of these disconnected components.

The analysis first tests the model against Axiom 1. Axiom 1 imposes a universal prohibition on self-reference for all entities within the system. The second subgraph $C_2$ contains a self-loop. This edge constitutes a direct violation of the irreflexivity condition. Consequently, the model $G_B$ fails to satisfy the Causal Primitive.

The analysis next tests the model against Axiom 2. Axiom 2 defines the constructive basis of valid physical geometry as the directed 3-cycle. The first subgraph $C_1$ constitutes a perfect instance of this geometric quantum. Therefore, where the model exhibits geometric structure, that structure adheres to the constructive definition. The defect in subgraph $C_2$ represents a failure of causal directionality but does not invalidate the geometric nature of $C_1$ . Axiom 2 posits a positive definition for spatial assembly; Axiom 2 does not, in isolation, enforce the removal of non-geometric causal defects such as self-loops. That role belongs to Axiom 1. Thus, the model satisfies the stipulations regarding the existence and form of geometric quanta.

The model $G_B$ therefore demonstrates a state that is geometrically valid yet causally invalid. This establishes that the definitions of Geometric Constructibility do not logically entail the constraints of the Causal Primitive.

2.5.4 Proof: Mutual Independence

Formal Proof of the Mutual Logical Independence of Axioms 1 and 2 Theorem (§2.5.1)

The demonstration shows that Axiom 1 (§2.1.1) does not imply Axiom 2 (§2.3.1). The demonstration shows that Axiom 2 does not imply Axiom 1. Thus, the two axioms are concluded to be logically independent. The two axioms represent distinct, fundamental constraints on the structure of the causal graph.

Q.E.D.

2.5.Z Implications and Synthesis

Independence

Independence stands confirmed: the countermodels prove Axiom 1 allows unreduced longer cycles as in Model A (§2.5.2), while Axiom 2 permits isolated self-loops as in Model B (§2.5.3), with the geometric syndrome (§2.5.1.1) highlighting triplet imbalances that signal violations. Physically, this means the framework draws on orthogonal levers, avoiding redundancy: directionality alone yields chains without enforced shapes, quanta alone yield areas without guaranteed thrust.

But if local primitives license such isolated successes, what hidden flaws emerge when they chain transitively: do mediated influences respect the arrow, or do cycles bootstrap self-causes? The syndrome detects local charges, yet global paths might curl undetected; this scalability issue compels the examination to the definition of effective influence, where the relation defines and exposes its pathologies under the pair alone.

2.6 The Inadequacy of Local Axioms

Section 2.6 Overview

Chains of two or more links transmit effective influence, yet closed loops allow an event to precede itself through mediation. The effective influence relation appears as timestamped paths of length at least two, demonstrating that Axioms 1 and 2 fail to render it irreflexive or asymmetric: triangles yield self-influences, even cycles permit reciprocities via disjoint subpaths. Lemmas unpack timestamp strictness and path breaches, revealing how local rules overlook transitive snarls.

The focus confines to finite cycles without further enforcement. The structure justifies constraints through examples, proves timestamp rigor against order collapse, details reflexivity in the quantum and asymmetry in cycle vignettes, then calls for a third axiom. Physically, this inadequacy marks the boundary: atoms and tiles construct blocks, but without transitive clamps, the lattice warps into paradoxes, demanding global alignment.

2.6.1 Definition: Effective Influence

The Effective Influence Relation ( $\le$ ) as the Transitive Closure of Strictly Timestamped Causal Paths

Let $G = (V, E, H)$ be a causal graph with history. The concept of effective influence captures the transitive propagation of causality across the graph, distinguishing mediated consequences from immediate actions. For any two vertices $u, v \in V$ , $u$ has an effective influence on $v$ , denoted $u \le v$ , if and only if there exists a valid causal trajectory connecting them. Formally, this requires the existence of a simple directed path $\pi_{uv} = (v_0, v_1, \ldots, v_\ell)$ in $G$ that satisfies three rigorous conditions:

Connectivity: The path must explicitly link the cause to the effect, such that $v_0 = u$ and $v_\ell = v$ .
Mediation: The path length $\ell$ must be greater than or equal to two ( $\ell \geq 2$ ).
Sequentiality: The creation timestamps of the constituent edges must be strictly increasing along the path: $H(v_0, v_1) < H(v_1, v_2) < \cdots < H(v_{\ell-1}, v_\ell)$ .

Important Technical Note on Simplicity and Reflexivity:

The definition of Effective Influence ( $\le$ ) requires the trajectory $\pi_{uv}$ to be a simple directed path, implying that all vertices in the sequence must be distinct. Consequently, a closed cycle (e.g., $A \to B \to C \to A$ ) does not constitute a valid path of influence from $A$ to $A$ in this formalism. Therefore, the existence of the Geometric Quantum (Axiom 2) does not inherently violate the irreflexivity of causal influence (Axiom 3); rather, it represents a localized spatial simultaneity distinct from temporal recursion.

2.6.1.1 Commentary: Path

Justification of the Necessity of Mediation and Sequentiality Constraints in Causal Paths

The specific constraints imposed on the effective influence relation are not arbitrary mathematical choices; they are physical necessities required to preserve the distinct ontological layers of the theory.

The Mediation Constraint ( $\ell \geq 2$ ) enforces a strict separation of scales between the atomic and the emergent. The direct causal link ( $\to$ ) governed by Axiom 1 represents the fundamental, irreducible quantum of action; the immediate "now" of the rewrite rule. In contrast, the effective influence relation ( $\le$ ) represents the history or consequence of those actions propagating through the network. By requiring $\ell \geq 2$ , the definition ensures that $\le$ exclusively describes emergent, multi-step causal pathways. This distinction is vital because the paradoxes of causality (such as closed timelike curves) typically arise not from single atomic events, but from the complex, transitive closure of multiple events looping back upon themselves.

The Sequentiality Constraint (Strict Timestamps) guards the causal order. In a discrete, computational universe, the concept of "simultaneity" degenerates. If non-decreasing timestamps apply (equality allowed), a chain of events $A \to B \to C$ could occur within a single tick of logical time. This collapses the distinction between cause and effect into a single, simultaneous cluster. By enforcing strictly increasing timestamps, the topological direction of the path aligns with the irreversible flow of $t_L$ (§1.2.2). This ensures that influence flows only from the past to the future, preventing the ambiguity that arises when concurrent events attempt to influence one another.

2.6.1.2 Commentary: Non-Decreasing Counterexample

Counterexample Demonstrating Simultaneity Paradoxes Arising from Non-Decreasing Timestamps

A hypothetical graph state is constructed where the constraint relaxes to allow equality ( $\leq$ ). Let the graph contain three vertices $\{A, B, C\}$ connected by edges $A \to B$ and $B \to C$ , both created at the same logical time $t_1$ .

Under the relaxed definition, the path $A \to B \to C$ qualifies as a valid line of influence, implying $A \le C$ . However, because these edges form simultaneously (likely by parallel rewrite operations), no inherent temporal ordering exists between them. If a subsequent parallel update at time $t_2$ inserts a direct link or path from $C$ back to $A$ , the system immediately recognizes a reciprocal influence $C \le A$ (since $t_1 < t_2$ ).

This results in a logical contradiction: $A$ influences $C$ (at $t_1$ ) and $C$ influences $A$ (at $t_2$ ), yet locally, no single observer sees a violation of causality. The system has formed a "loop of simultaneity" that functions as a Closed Timelike Curve. By enforcing strictly increasing timestamps ( $t_1 < t_2 < t_3$ ), the system invalidates the initial path $A \to B \to C$ as a causal carrier, thereby mathematically precluding the formation of such paradoxes before they can manifest.

2.6.2 Theorem: Inadequacy of Local Axioms

Insufficiency of Local Primitives for the Guarantee of Global Causal Consistency

In a system governed only by Axiom 1 (Causal Primitive) and Axiom 2 (Geometric Primitive), the effective influence relation $\le$ is not guaranteed to be a valid partial ordering. Specifically, the relation $\le$ fails to be irreflexive. An event can be its own effective cause ( $v \le v$ ). The relation $\le$ fails to be asymmetric. Two distinct events can have mutual, reciprocal effective influence ( $u \le v$ and $v \le u$ ).

2.6.2.1 Commentary: Inadequacy Argument

Logical Structure of the Argument for the Failure of Transitive Order

The theorem on the inadequacy of local axioms unfolds through a diagnostic progression that exposes the emergent pathologies in the effective influence relation $\le$ under Axioms 1 and 2 alone. The argument initiates with the definition of $\le$ , rigorously delimiting mediated paths of length $\geq 2$ with strictly increasing timestamps to encode sequential precedence, justified via counterexamples illustrating how lax constraints invite simultaneity and ambiguity. The Necessity of Strict Timestamps then proves that non-decreasing allowances collapse chains into concurrent influences, risking symmetric dependencies in parallel updates that erode the partial order's asymmetry. Building thereon, the Failure of Reflexivity dissects the 3-cycle's latent paradox: while Axiom 2 exalts it as geometry's atom, the closed path $A \to B \to C \to A$ satisfies $\le$ 's criteria, yielding $A \le A$ and a self-causal loop antithetical to irreflexivity. The Failure of Asymmetry extends this via timestamped 4- and 5-cycle vignettes, where disjoint subpaths enable mutual $\le$ (e.g., $A \le C$ and $C \le A$ ), birthing reciprocal influences that mock causal unidirectionality; a tabular summary crystallizes these breaches. The capstone proof synthesizes the failures: local primitives license global closures blind to transitive repercussions, necessitating Axiom 3's explicit global prophylaxis. This layered indictment not only unmasks the insufficiency but illuminates the coherentist ascent; local quanta beget emergent order only under vigilant, holistic constraint.

2.6.3 Lemma: Strict Timestamps

Necessity of Strictly Increasing Timestamps for the Well-Definedness of Causal Precedence

Strictly increasing timestamps ( $H(v_i, v_{i+1}) < H(v_{i+1}, v_{i+2})$ ) are necessary for a well-defined partial order in the effective influence relation $\le$ .

2.6.3.1 Proof: Necessity of Strictness

Derivation of the Necessity of Strict Inequalities to Prevent Symmetric Dependencies

The relation $\le$ must form a strict partial order. This order is irreflexive, asymmetric, and transitive. This encodes causal precedence without paradoxes (Axiom 3). Non-decreasing timestamps allow equalities, e.g., $H(v_0, v_1) = H(v_1, v_2)$ . This implies simultaneous edges in a path. Equal timestamps mean edges added in the same logical time step. This implies concurrent influences. For a path $P = (v_0 \to v_1 \to v_2)$ with $H(v_0, v_1) = H(v_1, v_2) = t$ , the influence $v_0 \le v_2$ lacks sequentiality. In parallel updates, such paths could form symmetric dependencies. This violates asymmetry. Strictly increasing timestamps ensure sequential order. This preserves the strict partial order.

Q.E.D.

2.6.4 Lemma: Failure of Reflexivity

Failure of the Irreflexivity Property within the Closed Topology of the Geometric Quantum

The Geometric Quantum, defined as the directed 3-cycle, induces a reflexive mapping within the effective influence relation $\le$ . Specifically, for any vertex $v$ participating in a geometric quantum configuration with strictly increasing timestamps, the relation $v \le v$ holds. This result stands in direct contradiction to the requirement of irreflexivity essential for a strict partial order, demonstrating that the local geometric primitive generates self-causal loops in the transitive closure.

2.6.4.1 Proof: Reflexivity Failure

Demonstration of Self-Influence via Transitive Closure in the Directed 3-Cycle

The proof constructs a graph $G$ consisting of a single directed 3-cycle $\gamma$ with vertices $V = \{A, B, C\}$ and edges $E = \{(A,B), (B,C), (C,A)\}$ . To model a sequential formation history, the history map $H$ assigns strictly increasing timestamps to the edges: $H(A, B) = t_1$ , $H(B, C) = t_2$ , and $H(C, A) = t_3$ , where $t_1 < t_2 < t_3$ .

This structure fully satisfies the local axioms. It adheres to Axiom 1 as it contains no self-loops ( $v \to v$ ) or mutual edges ( $u \leftrightarrow v$ ). It constitutes the fundamental unit of Axiom 2.

The analysis evaluates the Effective Influence relation $\le$ for the pair $(A, A)$ against the three criteria defined in Section 2.6.1:

Connectivity: There exists a directed path $\pi = (A, B, C, A)$ that originates at $A$ and terminates at $A$ .
Mediation: The length of path $\pi$ is 3. The definition requires a path length $\ell \ge 2$ . Since $3 \ge 2$ , the mediation condition is satisfied.
Sequentiality: The sequence of timestamps along the path is $(t_1, t_2, t_3)$ . By construction, $t_1 < t_2 < t_3$ . The condition of strictly increasing timestamps is satisfied.

Since the trajectory $\pi$ satisfies all necessary and sufficient conditions for effective influence, the relation $A \le A$ holds. This establishes that a structure mandated by Axiom 2 creates a reflexive self-influence, thereby failing to generate a strict partial order.

Q.E.D.

2.6.5 Lemma: Failure of Asymmetry

Failure of the Asymmetry Property via Disjoint Subpaths in Cycles of Length Greater than Three

Cycles of length $L \ge 4$ permit the formation of mutual effective influence between distinct vertices. Specifically, there exist graph configurations satisfying Axioms 1 and 2 where distinct vertices $u$ and $v$ satisfy both $u \le v$ and $v \le u$ . This mutual influence violates the asymmetry requirement of a strict partial order, indicating that local constraints fail to prevent the emergence of reciprocal causal loops mediated by disjoint subpaths.

2.6.5.1 Proof: Asymmetry Failure

Demonstration of Mutual Effective Influence in the Timestamped 4-Cycle

The proof constructs a graph $G$ consisting of a directed 4-cycle with vertices $V = \{A, B, C, D\}$ and edges $E = \{(A,B), (B,C), (C,D), (D,A)\}$ . The proof assigns a specific "Bowtie" timestamp ordering to demonstrate the paradox. Let the history map $H$ be defined as:

$H(A, B) = 1$
$H(B, C) = 4$
$H(C, D) = 2$
$H(D, A) = 3$

The analysis evaluates the Effective Influence relation for the distinct vertices $A$ and $C$ .

Step 1: Evaluation of Forward Influence ( $A \le C$ ) Consider the path $\pi_{AC} = (A \to B \to C)$ .

Connectivity: The path connects $A$ to $C$ .
Mediation: The length is 2, satisfying $\ell \ge 2$ .
Sequentiality: The timestamp sequence is $(1, 4)$ . Since $1 < 4$ , the path is strictly increasing.
Result: The relation $A \le C$ holds.

Step 2: Evaluation of Reverse Influence ( $C \le A$ ) Consider the disjoint path $\pi_{CA} = (C \to D \to A)$ .

Connectivity: The path connects $C$ to $A$ .
Mediation: The length is 2, satisfying $\ell \ge 2$ .
Sequentiality: The timestamp sequence is $(2, 3)$ . Since $2 < 3$ , the path is strictly increasing.
Result: The relation $C \le A$ holds.

Conclusion: The system exhibits a state where $A \le C$ and $C \le A$ simultaneously for $A \neq C$ . This constitutes a symmetric relation, directly violating the asymmetry condition required for a consistent causal history. While the 4-cycle may be subject to eventual decomposition by Axiom 2, the axioms do not preclude the transient existence of this causal paradox, proving their logical insufficiency to enforce global order.

Q.E.D.

2.6.5.2 Diagram: Bowtie Paradox

Visual of the Effective Influence Paradox Illustrating Bidirectional Causality in Separated Paths

  THE CYCLE CONFIGURATION                  THE CAUSAL PARADOX
  (Nodes A,B,C,D with Timestamps t)        (Emergent Effective Influence)

          t=1          t=4                 1. FORWARD PATH (A => C)
    [ A ]-------->[ B ]-------->[ C ]         Strictly Increasing (1 < 4)
      ^                         |
      |                         |             [ A ] ------------------> [ C ]
      | t=3                     | t=2
      |                         |
    [ D ]<----------------------+          2. REVERSE PATH (C => A)
                                              Strictly Increasing (2 < 3)

                                              [ C ] ------------------> [ A ]


  RESULT: A <= C AND C <= A.
  The partial order collapses into a symmetric loop.

2.6.6 Proof: Inadequacy of Local Axioms

Formal Proof of the Inadequacy Theorem of Axioms 1 and 2 to Entail a Strict Partial Order (§2.6.2)

The failure of $\le$ to be irreflexive or asymmetric under Axioms 1 and 2 proves that local rules are insufficient to ensure global causal consistency.

Q.E.D.

2.6.6.1 Corollary: Global Constraint

Necessity of an Explicit Global Constraint to Preclude Causal Paradoxes

A physical theory requires a well-defined causal ordering. The failure of $\le$ to be irreflexive or asymmetric under Axioms 1 and 2 proves that local rules are insufficient to ensure global causal consistency. Thus, a third axiom is needed to explicitly forbid states with causal paradoxes.

Q.E.D

2.6.6.2 Diagram: Antisymmetry Failure

A Comparative Visualization of the Failure Modes of Antisymmetry versus Irreflexivity

Comparison of Ordering Constraints on Substrate
---------------------------------------------------------
(A) Asymmetry           (B) Antisymmetry          (C) Axiom 1 (Irreflexive)
    u -> v -> u             u -> v -> u               u -> u
    |                       |                         |
    v                       v                         v
Violation: YES          Violation: ONLY IF        Violation: YES
(Mutual Influence)      u != v                    (Explicitly Forbidden)

Result:                 Result:                   Result:
Pure Directionality     Loophole for u->u         Thermodynamic Arrow
(No Cycles)             (Permits Inert Loops)     (Process Required)

2.6.Z Implications and Synthesis

Inadequacy of Local Axioms

The effective influence relation (§2.6.1) stands defined through simple paths of length at least two with strictly increasing timestamps, a necessity proven against non-decreasing allowances that invite symmetric dependencies (§2.6.3); yet under Axioms 1 and 2 alone, it falters irreflexively in the 3-cycle (§2.6.4), where the quantum's closure loops back as $v \le v$ , and asymmetrically in timestamped 4- and 5-cycles (§2.6.5), where subpaths enable $u \le v$ and $v \le u$ without overlap. Physically, these failures mean mediated causality can recirculate, turning forward chains into hidden engines of stasis or reversal.

The diagnosis clarifies the limit: local primitives seed structure but breed emergent loops blind to their own closure, eroding the partial order essential to becoming; enforcement thus requires preemptive global prophylaxis, a demand met in the axiom on acyclic effective causality, where thermodynamics routes the fix through local sieves.

2.7 Axiom 3: Global Consistency & Enforcement

Section 2.7 Overview

Transitive paths reach across the graph, imposing acyclicity without resorting to exhaustive scans at every step. The axiom posits as a strict partial order on effective influence, irreflexive to block self-causes and asymmetric to bar mutual ones, then shows post-hoc fixes thermodynamically untenable, favoring local approximations that bound errors exponentially via logarithmic-depth probes. Independence follows from prior models.

The focus limits to sparse graphs below percolation; dynamics intrude minimally. The outline scales growth to trans-local extents, approximates unique causality locally, embeds costs in the enforcement, and affirms separation. Physically, this axiom crowns the triad, rendering constraints self-enforcing: ticks now police transitive flows, aligning computation's locality with causality's expanse.

2.7.1 Axiom: Acyclic Effective Causality

The Requirement of Acyclic Effective Causality via the Enforcement of a Strict Partial Order

The effective influence relation $\le$ must form a strict partial order on the set of all vertices $V$ . This requires that the relation $\le$ is irreflexive. For any vertex $v \in V$ , it is not the case that $v \le v$ . This requires that the relation $\le$ is asymmetric. For any two distinct vertices $u, v \in V$ , if $u \le v$ , then it is not the case that $v \le u$ .

2.7.1.1 Commentary: Pseudocode for AEC Pre-Check

Quasi-Local BFS for Monotone Return Paths; O(log N) Cutoff

def pre_check_aec(G, u, v, H_new):  # Check add (u,v) with timestamp H_new
    # Cutoff: log(N) + buffer for locality
    N = G.number_of_nodes()
    cutoff = int(log(N)) + 3 if N > 1 else 1
    
    # Temp add edge
    G.add_edge(u, v, H=H_new)
    
    try:
        for path in all_simple_paths(G, v, u, cutoff=cutoff):  # Return paths v...->u
            if len(path) >= 2:  # Mediated (≥2 edges)
                if is_path_monotone(G, path):  # Strict H inc along path?
                    last_leg_H = G.edges[path[-2], u]['H']
                    if last_leg_H < H_new:  # Full cycle monotone?
                        return False  # Paradox: Reject
    finally:
        G.remove_edge(u, v)  # Rollback
    
    return True  # No monotone return: Accept

def is_path_monotone(G, path):  # Strict H inc: H(p_i,p_{i+1}) < H(p_{i+1},p_{i+2})
    for i in range(len(path)-2):
        h1 = G.edges[path[i], path[i+1]]['H']
        h2 = G.edges[path[i+1], path[i+2]]['H']
        if not (h1 < h2):
            return False
    return True

Operational Analysis of the Pre-Check Mechanism

This algorithm instantiates the enforcement of Axiom 3 by executing a directed, depth-bounded traversal to detect potential violations of the partial order. The logic couples a standard Breadth-First Search with a strict monotonicity constraint. By verifying that no strictly timestamp-increasing path exists from the target vertex $v$ back to the source $u$ , the function guarantees that the introduction of a new edge does not close a causal loop. This explicitly prevents the formation of both reflexive violations (where an event influences itself) and asymmetric violations (where mutual influence arises between distinct events).

The imposition of a logarithmic search depth ( $R \sim \log N$ ) transcends mere computational optimization; it exploits the probabilistic bounds established in the Local PUC Approximation (§2.7.4). In the sparse regime of geometrogenesis, the probability of a paradox-inducing cycle exceeding this radius vanishes exponentially, rendering the local check thermodynamically sufficient for global consistency. Empirical verification confirms the algorithm's discriminatory power: it correctly identifies and rejects "monotone paradoxes"sequences where history advances linearly into a loop while permitting "non-monotone" configurations where the timestamp sequence breaks, as these do not constitute valid effective influence.

Functionally, this pre-check operates in concert with the Principle of Unique Causality (§2.3.3). While the Unique Causality check filters local degeneracy (cloning), this Acyclic Effective Causality check filters transitive circularity. Together, they constitute the dual gating mechanism of the Universal Constructor, ensuring that the graph evolves scalably without compromising its causal integrity.

2.7.2 Theorem: Thermodynamic Enforcement

Necessity of Preemptive Local Enforcement of Acyclicity due to the Thermodynamic Cost of Global Correction

A transition to a state with a symmetric effective influence loop is thermodynamically forbidden. Any unique correction requires non-local, synchronous computation. This violates the local independence required by parallel dynamics. This justifies the definitional pre-check in the rewrite rule $\mathcal{R}$ as the only viable enforcement mechanism.

2.7.2.1 Commentary: Enforcement Argument

Logical Structure of the Argument for Local Pre-Checks over Post-Hoc Correction

The theorem on thermodynamic enforcement establishes that violations of Axiom 3; manifest as symmetric effective influence loops; are not merely logically proscribed but physically precluded by the framework's local, parallel dynamics, rendering preemptive local checks the sole feasible safeguard. The argument commences with a demonstration of how the rewrite rule's relentless accretion of 3-cycles drives edge density toward percolation criticality, spawning trans-local cycles whose girth exceeds the finite radius of any computational patch, thus evading detection by locality-bound operators. This vulnerability is then mitigated by approximation; yet not eliminated. The capstone proof integrates these via a dilemma: post-hoc paradox resolution demands $O(N)$ traversal and synchronization, exacting infinite free energy in the thermodynamic limit and clashing with the evolution operator's patch-wise parallelism; hence, enforcement devolves to embedded finite-depth pre-checks in $\mathcal{R}$ , whose exponential fidelity to global consistency aligns with the substrate's informational frugality. This scaffolded rationale not only vindicates Axiom 3's operational primacy but elevates the coherentist ontology: local primitives, untethered from global prophylaxis, court entropic catastrophe, yet under thermodynamic duress, they converge inexorably on paradox-free horizons, forging causality's arrow from the forge of enforced locality.

2.7.3 Lemma: Cycle Diameter Growth

Geometric Growth of Cycle Diameter beyond Local Computational Radii during Geometrogenesis

During the out-of-equilibrium phase of geometrogenesis, the mean cycle length $\langle k \rangle$ grows with logical time. This enables cycles with a graph-theoretic diameter exceeding the finite radius $R$ of a local computational patch.

2.7.3.1 Proof: Diameter Growth

Derivation of the Trans-Local Expansion of Cycles relative to Fixed Computational Patches

The micro-rule is the rewrite rule $\mathcal{R}$ . The rewrite rule $\mathcal{R}$ is the engine of geometrogenesis. The rewrite rule continuously adds 3-cycles to the graph.

The emergent macro-property is the statistical side-effect of adding many local 3-cycles. This side-effect is a global increase in the graph's overall edge density, $\rho$ .

The percolation threshold states: As the edge density $\rho$ increases, the system approaches a critical "percolation threshold". Near this threshold, the probability of forming long-range paths and giant, system-spanning cycles grows explosively.

The conclusion states: The local dynamics of $\mathcal{R}$ drive to create geometry. These dynamics would emergently drive the graph toward a critical percolation threshold. At this threshold, the graph would inevitably create global, trans-local paradoxes. These are giant cycles. The local dynamics are architecturally blind to these paradoxes. The local dynamics cannot correct these paradoxes. This proves that "post-hoc correction" is impossible.

Q.E.D.

2.7.3.2 Diagram: Horizon Problem

Horizon Problem Depicting the Inability of Local Patches to Detect Global Cycles

                   Why Local Rules Fail Global Checks
      --------------------------------------------------------

      [ R ] = Radius of Computational Patch (Local limit of Rewrite Rule)

                     Global Cycle C (Length >> R)
                 ...................................
                 :                                 :
           +-----:-------+                   +-----:-------+
           |     v       |                   |     v       |
           |   [Patch]   |                   |   [Patch]   |
           |      A      |                   |      B      |
           +-------------+                   +-------------+
                 :                                 :
                 :.................................:

      DILEMMA: Patch A sees a line. Patch B sees a line.
               Neither patch can see the curvature required to
               detect that they are part of the same giant loop.
               
      SOLUTION: Pre-emptive Principle of Unique Causality (§2.3.3) prevents
                the loop from growing beyond R in the first place.

2.7.4 Lemma: Local PUC Approximation

Exponential Approximation of Unique Causality via Logarithmic-Depth Local Search

In sparse graphs with density $\rho$ , a local breadth-first search check of depth $R$ approximates the Principle of Unique Causality (§2.3.3) with error probability $< e^{-R}$ . This ensures decidability in polynomial time for $R \sim \log N$ .

2.7.4.1 Proof: Approximation Fidelity

Derivation of the Error Probability Bound for Local Breadth-First Search in Sparse Graphs

The proof relies on the statistical properties of random graphs in the sparse regime characteristic of early geometrogenesis, where the edge density $\rho$ remains low. The failure mode of a local approximation occurs if and only if a paradox-inducing path exists with a graph-theoretic diameter exceeding the search radius $R$ .

The probability of such a long-range correlation persists as a function of the density. For a graph with effective branching factor proportional to $\rho$ , the probability that a specific causal chain extends to length $R$ without termination or closure scales as $P(\text{length} \ge R) \propto \rho^R$ . Defining the correlation length $\xi \sim 1/\ln(1/\rho) \approx 1/\rho$ , this probability follows an exponential decay profile: $P(\text{failure}) \propto e^{-R / \xi}$ . Consequently, a local breadth-first search bounded by depth $R$ detects potential collisions with a probability approaching $1 - e^{-R}$ . This establishes that the likelihood of an undetected global paradox vanishes exponentially as the search radius increases.

Regarding thermodynamic constraints, Theorem 2.7.2 establishes that exact global verification incurs infinite free energy costs in the thermodynamic limit. Therefore, the exponential suppression of errors via local approximation represents the maximal achievable precision for a physical system.

Regarding computational complexity, the verification process utilizes a bounded Breadth-First Search. For a local search radius $R$ , and a mean vertex degree $\langle k \rangle$ , the computational cost per edge proposal scales as $O(R \cdot \langle k \rangle)$ . By establishing the cutoff radius as logarithmic relative to the system size ( $R \sim \log N$ ), the verification step executes in polynomial time relative to the scale of the local patch. This ensures that the enforcement of Unique Causality remains decidable within finite physical time resources.

Q.E.D.

2.7.5 Proof: Thermodynamic Enforcement

Formal Proof of the Thermodynamic Enforcement Theorem (§2.7.2)

The proof demonstrates that reliance on post-hoc correction for causal consistency leads to a physical singularity in the thermodynamic limit.

The argument considers the computational requirements to resolve a "timestamp-monotone cycle", the global topological paradox identified in the Inadequacy Theorem (§2.6.2). For a cycle of length $L$ exceeding the local computational radius $R$ ( $L \gg R$ ), the topological information encoding the paradox is non-local. No single observer or local patch possesses sufficient data to distinguish this closed loop from an open, valid causal chain.

To detect such a cycle retrospectively, the system must execute a traversal operation (such as a Depth-First Search) that scales linearly with the system size $N$ . To correct the cycle uniquely, for example, by identifying and excising the edge with the minimal timestamp, the system requires a global comparison operation across these $N$ spacelike-separated components.

In the thermodynamic limit where $N \to \infty$ , an operation of complexity $O(N)$ executed within a finite logical time step requires infinite information transfer velocity or infinite free energy. This requirement violates the principle of locality fundamental to the substrate. Consequently, a mechanism of post-hoc correction is physically unrealizable.

Therefore, the system is constrained to enforce Axiom 3 exclusively through finite-depth pre-checks embedded within the local rewrite rule. These local checks approximate global consistency with an error rate that is exponentially suppressed by the search depth ( $P \sim e^{-R}$ ), rendering the system consistent at finite energy cost.

Q.E.D.

2.7.6 Theorem: Independence of Axiom 3

Logical Independence of Acyclic Effective Causality from Local Primitives

Axiom 3 is independent of Axioms 1 and 2.

2.7.6.1 Proof: Independence of Axiom 3

Verification of the Independence of Axiom 3 via the Timestamped 4-Cycle Countermodel

The proof utilizes the "Bowtie" countermodel previously established in Lemma 2.6.5. The model consists of a directed 4-cycle with vertices $\{A, B, C, D\}$ and edges $\{(A,B), (B,C), (C,D), (D,A)\}$ , assigned the non-sequential timestamp ordering: $H(A,B)=1, H(B,C)=4, H(C,D)=2, H(D,A)=3$ .

The analysis first confirms adherence to the local axioms.

Axiom 1 (Causal Primitive): The graph contains no self-loops or mutual edges. The model satisfies irreflexivity and asymmetry at the level of individual edges.
Axiom 2 (Geometric Constructibility): While the 4-cycle is not a geometric quantum, Axiom 2 dictates only that such cycles are reducible. The presence of a reducible 4-cycle does not constitute a violation of the axiom's definitions regarding valid geometry, as the axiom permits transient larger structures prior to decomposition. Thus, the model is compatible with the local definitions of Axioms 1 and 2.

The analysis next tests the model against Axiom 3 (Acyclic Effective Causality). Axiom 3 mandates that the effective influence relation $\le$ forms a strict partial order.

Forward Influence: The path $A \to B \to C$ possesses strictly increasing timestamps ( $1 < 4$ ), establishing $A \le C$ .
Reverse Influence: The disjoint path $C \to D \to A$ possesses strictly increasing timestamps ( $2 < 3$ ), establishing $C \le A$ .

The simultaneous existence of $A \le C$ and $C \le A$ for distinct vertices constitutes a symmetric relation. This symmetry directly violates the asymmetry requirement of Axiom 3. Consequently, the model demonstrates a state that is locally consistent with the Causal and Geometric Primitives but globally inconsistent with Acyclic Effective Causality. This establishes that Axiom 3 is logically independent and cannot be derived solely from the local constraints of Axioms 1 and 2.

Q.E.D.

2.7.Z Implications and Synthesis

Axiom 3: Global Consistency and Enforcement

Axiom 3 emerges as acyclic effective causality (§2.7.1), mandating the relation as strict partial order, with enforcement thermodynamically confined to pre-checks in the rewrite (§2.7.2): cycle diameters grow beyond local radii (§2.7.3), yet finite-depth searches approximate uniqueness with vanishing error (§2.7.4), independent of the priors via the 4-cycle breach (§2.7.6). Physically, this means paradoxes abort before birth, the substrate's evolution threading local freedom through global invariance.

Thus the constraints cohere: arrows propel, quanta tile, acyclicity aligns, yielding a directed lattice where influences branch unidirectionally; scalability tensions linger as density mounts, priming the rewrite's roar.

2.Ω Formal Synthesis

End of Chapter 2

The three axioms forge the substrate's unyielding frame: the causal primitive (§2.1.1) that directs without local reversal; the geometric constructibility (§2.3.1) that assembles from 3-cycle quanta under unique paths (§2.3.3); and the acyclic effective causality (§2.7.1) that extends irreflexivity and asymmetry transitively via local thermodynamic guards. This triad delimits the task space, independence (§2.5.1; §2.7.6) minimizing overlap, decomposition (§2.4.1) enforcing descent to simplices.

Physically, the graph thus accretes as a causal lattice, cycles resolving to area units under singular mediation, the arrow preserved across mediations without recurving; echoes of incompleteness appear, as local verifications approximate globals, their interplay fertile yet bounded. Frictions in parallel flux persist, urging the engine's ignition.

Symbol	Description	First Used
$v_i \to v_j$	Directed causal link	§2.1.1
$\nrightarrow$	Negation of directed link	§2.1.1
$\mathcal{R}$	Rewrite Rule	§2.2.1
$S$	Entropy	§2.2.3
$\le$	Effective Influence Relation	§2.2.3
$\gamma$	Geometric Quantum (Directed 3-cycle)	§2.3.2
$\Pi(u,v)$	Set of directed paths from u to v	§2.3.3.1
$\Phi(G)$	Lexicographic Potential	§2.3.4
$L$	Length of longest simple cycle	§2.3.4
$N_L$	Number of cycles of length L	§2.3.4
$\langle k \rangle$	Mean cycle length	§2.7.3
$R$	Radius of local computational patch	§2.7.3
$\rho$	Graph density	§2.7.3
$\xi$	Correlation length	§2.7.4

2.1 Axiom 1: The Causal Primitive​

2.1.1 Axiom: The Directed Causal Link​

2.1.Z Implications and Synthesis​

2.2 Antisymmetry​

2.2.1 Theorem: Insufficiency of Antisymmetry​

2.2.1.1 Commentary: Argument Outline​

2.2.1.2 Diagram: Ordering Constraints​

2.2.1.3 Diagram: The Failure of Antisymmetry​

2.2.2 Lemma: Pathology of Self-Loops​

2.2.2.1 Proof: Cycle Creation​

2.2.2.2 Commentary: Graph Integrity​

2.2.3 Lemma: Thermodynamic​

2.2.3.1 Proof: Zero Entropy​

2.2.4 Proof: Insufficiency of Antisymmetry​

2.2.Z Implications and Synthesis​

2.3 Axiom 2: Geometric Constructibility​

2.3.1 Axiom: Geometric Constructibility​

2.3.2 Definition: Clause A, Geometric Quantum​

2.3.2.1 Commentary: Minimality Justification​

2.3.2.2 Diagram: Loop Hierarchy​

2.3.3 Principle: Clause B, Unique Causality​

2.3.3.1 Definition: Path Cloning Impossibility​

2.3.3.1 Diagram: Principle of Unique Causality (PUC)​

2.3.4 Definition: Lexicographic Potential​

2.3.4.1 Commentary: Complexity Metrics​

2.3.Z Implications and Synthesis​

2.4 Decomposition​

2.4.1 Theorem: General Cycle Decomposition​

2.4.1.1 Commentary: Decomposition Argument​

2.4.2 Lemma: Confluence of the Constructor​

2.4.2.1 Proof: Local Confluence​

2.4.2.2 Commentary: Parallel Interference​

2.4.3 Lemma: Chordlessness of Maximal Cycles​

2.4.3.1 Proof: Chordlessness​

2.4.4 Lemma: Reduction via Deletion​

2.4.4.1 Proof: Potential Reduction​

2.4.5 Lemma: Decrease in Parallel Updates​

2.4.5.1 Proof: Strict Decrease​

2.4.6 Proof: General Cycle Decomposition​

2.4.6.1 Diagram: Digestion of Geometry​

2.4.6.2 Calculation: Worked 4-Cycle Reduction​

2.4.6.3 Example: Worked 5-Cycle Reduction​

2.4.6.4 Calculation: Worked 6-Cycle Reduction​

2.4.6.5 Calculation: Simulation Verification​

2.4.Z Implications and Synthesis​

2.5 Independence​

2.5.1 Theorem: Independence of Axioms 1 and 2​

2.5.1.1 Commentary: Independence Argument​

2.5.1.3 Diagram: Independence Matrix​

2.5.2 Lemma: Independence Case A​

2.5.2.1 Proof: Verification of Case A​

2.5.3 Lemma: Independence Case B​

2.5.3.1 Proof: Verification of Case B​

2.5.4 Proof: Mutual Independence​

2.5.Z Implications and Synthesis​

2.6 The Inadequacy of Local Axioms​

2.6.1 Definition: Effective Influence​

2.6.1.1 Commentary: Path​

2.6.1.2 Commentary: Non-Decreasing Counterexample​

2.6.2 Theorem: Inadequacy of Local Axioms​

2.6.2.1 Commentary: Inadequacy Argument​

2.6.3 Lemma: Strict Timestamps​

2.6.3.1 Proof: Necessity of Strictness​

2.6.4 Lemma: Failure of Reflexivity​

2.6.4.1 Proof: Reflexivity Failure​

2.6.5 Lemma: Failure of Asymmetry​

2.6.5.1 Proof: Asymmetry Failure​

2.6.5.2 Diagram: Bowtie Paradox​

2.6.6 Proof: Inadequacy of Local Axioms​

2.6.6.1 Corollary: Global Constraint​

2.6.6.2 Diagram: Antisymmetry Failure​

2.6.Z Implications and Synthesis​

2.7 Axiom 3: Global Consistency & Enforcement​

2.7.1 Axiom: Acyclic Effective Causality​

2.7.1.1 Commentary: Pseudocode for AEC Pre-Check​

2.7.2 Theorem: Thermodynamic Enforcement​

2.7.2.1 Commentary: Enforcement Argument​

2.7.3 Lemma: Cycle Diameter Growth​

2.7.3.1 Proof: Diameter Growth​

2.7.3.2 Diagram: Horizon Problem​

2.1 Axiom 1: The Causal Primitive

2.1.1 Axiom: The Directed Causal Link

2.1.Z Implications and Synthesis

2.2 Antisymmetry

2.2.1 Theorem: Insufficiency of Antisymmetry

2.2.1.1 Commentary: Argument Outline

2.2.1.2 Diagram: Ordering Constraints

2.2.1.3 Diagram: The Failure of Antisymmetry

2.2.2 Lemma: Pathology of Self-Loops

2.2.2.1 Proof: Cycle Creation

2.2.2.2 Commentary: Graph Integrity

2.2.3 Lemma: Thermodynamic

2.2.3.1 Proof: Zero Entropy

2.2.4 Proof: Insufficiency of Antisymmetry

2.2.Z Implications and Synthesis

2.3 Axiom 2: Geometric Constructibility

2.3.1 Axiom: Geometric Constructibility

2.3.2 Definition: Clause A, Geometric Quantum

2.3.2.1 Commentary: Minimality Justification

2.3.2.2 Diagram: Loop Hierarchy

2.3.3 Principle: Clause B, Unique Causality

2.3.3.1 Definition: Path Cloning Impossibility

2.3.3.1 Diagram: Principle of Unique Causality (PUC)

2.3.4 Definition: Lexicographic Potential

2.3.4.1 Commentary: Complexity Metrics

2.3.Z Implications and Synthesis

2.4 Decomposition

2.4.1 Theorem: General Cycle Decomposition

2.4.1.1 Commentary: Decomposition Argument

2.4.2 Lemma: Confluence of the Constructor

2.4.2.1 Proof: Local Confluence

2.4.2.2 Commentary: Parallel Interference

2.4.3 Lemma: Chordlessness of Maximal Cycles

2.4.3.1 Proof: Chordlessness

2.4.4 Lemma: Reduction via Deletion

2.4.4.1 Proof: Potential Reduction

2.4.5 Lemma: Decrease in Parallel Updates

2.4.5.1 Proof: Strict Decrease

2.4.6 Proof: General Cycle Decomposition

2.4.6.1 Diagram: Digestion of Geometry

2.4.6.2 Calculation: Worked 4-Cycle Reduction

2.4.6.3 Example: Worked 5-Cycle Reduction

2.4.6.4 Calculation: Worked 6-Cycle Reduction

2.4.6.5 Calculation: Simulation Verification

2.4.Z Implications and Synthesis

2.5 Independence

2.5.1 Theorem: Independence of Axioms 1 and 2

2.5.1.1 Commentary: Independence Argument

2.5.1.3 Diagram: Independence Matrix

2.5.2 Lemma: Independence Case A

2.5.2.1 Proof: Verification of Case A

2.5.3 Lemma: Independence Case B

2.5.3.1 Proof: Verification of Case B

2.5.4 Proof: Mutual Independence

2.5.Z Implications and Synthesis

2.6 The Inadequacy of Local Axioms

2.6.1 Definition: Effective Influence

2.6.1.1 Commentary: Path

2.6.1.2 Commentary: Non-Decreasing Counterexample

2.6.2 Theorem: Inadequacy of Local Axioms

2.6.2.1 Commentary: Inadequacy Argument

2.6.3 Lemma: Strict Timestamps

2.6.3.1 Proof: Necessity of Strictness

2.6.4 Lemma: Failure of Reflexivity

2.6.4.1 Proof: Reflexivity Failure

2.6.5 Lemma: Failure of Asymmetry

2.6.5.1 Proof: Asymmetry Failure

2.6.5.2 Diagram: Bowtie Paradox

2.6.6 Proof: Inadequacy of Local Axioms

2.6.6.1 Corollary: Global Constraint

2.6.6.2 Diagram: Antisymmetry Failure

2.6.Z Implications and Synthesis

2.7 Axiom 3: Global Consistency & Enforcement

2.7.1 Axiom: Acyclic Effective Causality

2.7.1.1 Commentary: Pseudocode for AEC Pre-Check

2.7.2 Theorem: Thermodynamic Enforcement

2.7.2.1 Commentary: Enforcement Argument

2.7.3 Lemma: Cycle Diameter Growth

2.7.3.1 Proof: Diameter Growth

2.7.3.2 Diagram: Horizon Problem