Abstract / 摘要 This repository documents a "computational accident" discovered by a Computer Science undergraduate. By modeling the universe purely as a path-optimization problem on a causal graph, the system spontaneously generates macroscopic behaviors strikingly analogous to General Relativity, Quantum Mechanics, Electromagnetism and Holographic Thermodynamics.
本项目记录了一名计算机专业本科生的“计算意外”。通过将宇宙建模为因果图上的路径优化问题，系统自发涌现出了与广义相对论、量子力学、电磁学和全息热力学惊人相似的宏观行为。
This project explores a simple question: What if the laws of physics are just the statistical side-effects of a universe trying to optimize its causal paths?
Unlike traditional physics simulations, no physical laws (e.g., gravity, Schrödinger equation) are pre-programmed here. Everything emerges from a single recursive rule applied to a discrete topological network.
The model operates on a dynamic graph using a "Tri-Chord" Rewriting Rule, incorporating a Causal Freezing Mechanism to establish the arrow of time.
-
Shortcut (
$x, z$ ): Represents Gravity/Tunneling (Space contraction). -
Insertion (
$w$ ): Represents Space Expansion (Dimensional growth).
-
Active Interaction (
$x \leftrightarrow y$ ): Edges capable of rewriting (The Quantum "Now"). -
Causal Fixation (
$x \Rightarrow y$ ): Once a path is optimized, it freezes into an inert state. This phase transition generates an irreversible Arrow of Time and forms the stable backbone of spacetime. - union or the other relevant deduplication function:Pauli exclusion principle
Without manual fine-tuning, the long-term evolution of the graph exhibits statistical properties that remarkably mirror fundamental physical constants and phenomena.
We invite physicists and mathematicians to help verify if these are mere mathematical coincidences or signs of a deeper computational isomorphism.
The system's topological ratios and coupling strengths tend to stabilize around values perilously close to known physical constants:
-
Fine-Structure Constant:
$\alpha \approx 1/137$ -
The Golden Ratio:
$\phi \approx 1.618$ (Observed in pulse dynamics) -
Euler's Number:
$e \approx 2.71$ (Average dimensional scaling)
In simulations tracking the "evaporation" of high-density topological clusters (black hole analogues):
- The Entanglement Entropy follows an Inverted V-shape curve, accurately reproducing the Page Curve and suggesting information conservation (Unitarity).
-
Area Law: A linear relationship is observed between the boundary area of a subgraph and its internal information content (
$S \propto A$ ).
-
Chirality: At
$t \approx 2000$ steps, neutral topological knots spontaneously develop a preferred handedness (Left/Right asymmetry). - Particle Capture: Complex "Knots" (Mass) are observed to spontaneously capture and confine surrounding active edges, mimicking Quark Confinement or Accretion.
- Gravity Wells: The grid naturally curves and contracts around high-connectivity nodes.
- Gravitational Waves: Movement of dense clusters generates ripple-like disturbances that propagate through the causal fabric.
- Orbital Mechanics: Test nodes exhibit emergent elliptical and helical trajectories around central clusters.
- Quantum & Topological Constraints：
Pauli Exclusion Principle: No two nodes can share identical topological neighborhoods; any "address collision" triggers an automatic State-Merge (Conflict Resolution) to maintain graph uniqueness.
Topological Uniqueness: The system enforces a fundamental distinguishability where duplicate informational states are compressed, manifesting as the emergent "repulsion" between fermions.
State Compression: The use of Union logic in graph updates ensures that redundant connectivity is pruned, effectively preventing identical quantum states from occupying the same causal coordinate.
Fermionic Pressure: High-connectivity clusters generate a "topological pressure" to maintain distinct connectivity signatures, ensuring the structural stability of emergent matter.
Deductive Predictions
- Gravity as Path Optimization: Gravity is the pressure of the graph seeking the shortest computational path. G is the ratio between Node Expansion (Birth of w) and Path Collapse (Degree Reduction).
Prediction: G scales with the average degree of the graph.
- Proton-Electron Ratio (1836.15): This is the stable ratio between High-Degree Clusters (Protons) and Low-Degree Chiral Chains (Electrons).
Prediction: At N→∞, the sum of degrees in stable 3D-clusters vs. 1D-chains converges to 1836.15.
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Dark Matter: Dark Matter is Redundant Connectivity. Nodes that contribute to the Sum of Degrees (Gravity) but fail to synchronize into Chiral Loops (Electromagnetism).
-
Time as State-Commit: Time is the discrete count of graph-rewrite operations. The "Arrow of Time" is the irreversible trend of path merging in a finite-resource system.
Calculate redshift and wave packets
While primarily a cosmological model, the discrete causal dynamics demonstrated here share deep mathematical isomorphism with several complex systems. These theoretical mappings constitute prior art for applied algorithms in:
- Chemical Reaction Networks (CRN): Modeling molecular recombination events as graph rewriting rules.
- Artificial Intelligence: Gradient-free topology evolution (Dynamic Neural Architecture).
- Financial Microstructure (Theoretical): The emergent non-Gaussian fluctuations in the causal graph serve as a candidate model for stochastic volatility in discrete order flows (High-Frequency Trading dynamics).
- Distributed Consensus (Theoretical): The strict partial ordering of the causal set provides a mathematical framework for conflict-free ordering in DAG-based Blockchains without global clocks.
All logic is contained within the Wolfram Language notebooks. The raw data plots and simulation logs are available for independent verification.
core_source_code/: Contains the primary algorithms implementing the Tri-Chord rule and Freezing mechanism.Result_PDF_or_snapshot/: (Must See!) Contains visualizations of the Page Curve, Gravitational Wave propagation, and detailed PDF derivations.
- Cook, Charles Rodbourn. (2026).
Coherence Calculus: Finite Observability via Horizon Projections and Projection-Induced Defects (Version 0.4).
- Source: Zenodo Archive
- Note: This theoretical framework provides the mathematical derivation for the "Projection Defects" that spontaneously emerge in our TCCT simulations.
Special thanks to Charles Rodbourn Cook.
During our exchange regarding my TCCT model, he brought his work Coherence Calculus to my attention. I am grateful for this recommendation, as it provided a rigorous theoretical language to describe the phenomena that had already emerged in my independent simulations (specifically the relativistic time dilation corrections).
The convergence of my computational results with his theoretical predictions serves as a robust cross-validation for both frameworks.
I am a Computer Science student, not a theoretical physicist. This model was born from a desire to apply Dynamic Programming principles to the structure of reality.
The results—specifically the spontaneous emergence of the Page Curve and the "1/137" ratio—are either a testament to the power of the Law of Large Numbers or a hint that the universe fundamentally computes.
For more details or discussions, please refer to my post on the Wolfram Community:
👉 Tri-Chord Causal Theory Discussion
Disclaimer: Due to my limited personal level, errors are inevitable. I warmly welcome everyone to correct me or provide feedback in the comments or Issues tab. (由于我个人水平有限，难免会有错误，欢迎大家指正。)
As I am approaching graduation, I am officially designating this repository and its underlying logic as core research material for my further education applications.
At the same time, I also hope that everyone can communicate with me, because my understanding of physics is also very limited and there might be some errors in expression or even logical errors in some aspects. I sincerely hope that everyone can point them out. I would be very grateful. However, the "Continuum Limit" of this graph-rewriting rule remains an open question for me. > I would be deeply grateful if researchers with access to higher computational bandwidth could test this rule at a larger scale . I am eager to learn if the topological properties I've observed hold true as the system scales. Any insights, corrections, or data sharing from the community would be an invaluable part of my learning journey Email:xinw12424@gmail.com
Distributed under the AGPL-3.0 License. AGPL-3.0 License
Copyright (c) 2026 Wang Xin