Emergent SOC, Chaos, and Fractal Dynamics in Bounded-Rate Paradox-Referential Graph Processes: Numerical Verification of the Trinity Graph Process

Title

Emergent SOC, Chaos, and Fractal Dynamics in Bounded-Rate Paradox-Referential Graph Processes: Numerical Verification of the Trinity Graph Process

Authors

Jia, Baolong (贾宝龙)

Description

Can self-organized criticality, chaos, and fractal structure emerge from pure topological dynamics on graphs, without any pre-defined spatial lattice?

This paper presents the first large-scale numerical verification of the Trinity Graph Process (TGP) — a dynamical system on directed graphs governed by four bounded-rate parameters (v₁, v₂, v₃, v₄) and three interacting mechanisms: Paradox-Referential selection (PR), Entity-Relation substrate (ER), and Lazy Evaluation conflict resolution (LE). Unlike prior graph-rewriting models that operate on fixed lattices or use externally imposed rules, TGP generates its spatial topology dynamically — nodes and edges are created and destroyed according to rules that read the graph's own structure.

Key results from six-scale simulations (N = 100 to N = 3,611, a 36× range):

1. Self-Organized Criticality (HIGH confidence): Power-law avalanche distributions (α ≈ 2.5–3.1) with strong finite-size scaling (γ_max = 0.80, γ_mean = 1.01), verified across six system sizes. The system self-tunes to critical edge density E = N·v₃.

2. Deterministic Chaos (HIGH confidence): Positive Lyapunov exponents (λ ≈ 0.63–0.91) that increase with system size as λ ∝ N^0.16.

3. Fractal Structure (HIGH confidence): Box-counting dimension D ≈ 1.33 invariant across the full 36× scale range, with Hurst exponents H ≈ 0.56–0.69 confirming long-range correlations.

4. Self-Repairing Structures (MEDIUM-HIGH confidence): 28.7% triangle reappearance rate after destruction, persistence up to 30,000 ticks (surviving multiple complete graph turnovers), and mobile patterns enriched 4.4× above random expectation — demonstrating Phase 2 of the predicted evolutionary hierarchy.

5. Self-Organized Topology: TGP spontaneously forms a single connected component with logarithmic diameter (ℓ ∝ log N), PR-driven clustering, and precisely self-regulated edge density E = N·v₃ (CV < 0.001).

The conservative cascade mechanism (100% edge redistribution with v₄ boundary dissipation) provides an exact topological analog of the Manna sandpile model. The power-law exponent α ≈ 2.6 matches the mean-field prediction for SOC on complex networks, placing TGP in the network universality class.

This is the third paper in a series formalizing the JiaBaolong Axiom System's Trinity Universe Theory, following the foundational axiom system paper (Zenodo 19440952, "The JiaBaolong Universal Axiom System: Two Axioms, One Universe") and the cosmology formalization paper (Zenodo 19469833, "Mediumless Topological Cosmology: A Turing-Transcendent Ontological Generator"). It provides the first computational verification of the theoretical predictions.

Simulation code (Rust, ~1,200 lines) and analysis scripts (Python) are included with this upload. Results are fully reproducible with seed = 42.

Keywords

self-organized criticality, SOC, Trinity Graph Process, TGP, paradox-referential, bounded-rate graph dynamics, topological chaos, fractal dimension, Lyapunov exponent, finite-size scaling, self-repairing structures, conservative cascade, Manna model, emergent topology, logarithmic diameter, JiaBaolong Axiom

中文关键词

自组织临界、SOC、三元图过程、TGP、悖论自指、有界速率图动力学、拓扑混沌、分形维度、Lyapunov指数、有限尺寸标度、自修复结构、守恒级联、Manna模型、涌现拓扑、对数直径、贾宝龙公理