Statistical Emergence of a Physical Constant ( $\alpha^{-1} \approx 133.2$) in a Computational Universe Model1. Context: Large-Scale Time Series Stress TestingI have been conducting large-scale stress tests on a graph evolution model based on the rewriting rule: $\{\{x, y\}, \{y, z\}\} \to \{\{x, z\}, \{x, w\}, \{w, z\}\}$. My simulations ranged from 500 steps up to 20,000 steps.Since the code employs Monte Carlo sampling to maintain efficiency with large-scale graph structures, the specific evolutionary path of each run is stochastic. However, surprisingly, across multiple independent iterations and varying seeds, certain core topological parameters exhibited a remarkable degree of statistical invariance.2. Core Observation: A Stable Numerical Range (130-134)The data suggests that regardless of the evolutionary stage (whether at 500, 1,500, or 20,000 steps), the values derived from the model's topological structure consistently stabilize within a narrow range of 130-134.This stability implies that the system reaches a Topological Equilibrium, where the following parameters converge:Geometric Coupling ( $8\pi \cdot K$): The product of the average connectivity ( $K$) and spherical flux shows high consistency.Spin-Corrected Value ( $\approx 133.2$): By incorporating the non-zero net chirality observed in dense clusters, the derived constant stabilizes around 133.2.While this shows a deviation of approximately 2.8% from the experimental value of the fine-structure constant ( $\approx 137.036$), it presents a compelling first-order approximation derived purely from geometric origins.3. On the Emergence of ChiralityA notable feature of this model is the spontaneous emergence of chirality. The rewriting rule induces an asymmetric nesting of triangle loops, which generates local topological torsion.In the final step of the code shared below, I implemented a Chirality Search. The empirical data confirms that dense clusters ("protons") exhibit a non-zero net chirality. This serves as a computational analog to symmetry breaking, providing the necessary spin correction factor for the calculation.4. Theoretical Rationale: A Heuristic Geometric Approach(inspired by GR flux but adapted for discrete topology

)The derivation of the factor $8\pi$ is currently a heuristic based on the model's 3D mapping requirements:$4\pi$ (Isotropy): Represents the geometric baseline for a point source radiating into the full solid angle.$2$ (Spin Freedom): Derived from the observed bidirectional information flow and non-zero chirality.$K$ (Background Density): Represents the connectivity of the discrete vacuum grid.The product $8\pi \cdot K$ essentially measures the geometric propagation efficiency of an interaction within this discrete spacetime.5. Interpretation: Mass as Fractal Path LengthBy comparing path measurements, I observed that within high-mass clusters, the Rule's recursive application causes paths to be infinitely subdivided. Following the principle that "new information takes new paths," the number of Hops a photon must take to traverse a high-mass region increases.If we assume $c=1$ (1 hop/tick), this offers a geometric interpretation for gravitational time dilation: light isn't slowing down; the topological distance inside the mass is simply becoming longer due to fractalization.Closing Thoughts & Future Work:The stability of this value up to 20,000 steps suggests that the fine-structure constant might fundamentally be a geometric invariant of a graph rewriting system at equilibrium.Current limitations include the static nature of the measurement (a snapshot at step $N$). I welcome any feedback on the code and the theoretical framework.

Attachments:

Constant_ca.nb

Hi Charles,Thank you for your inspiring guidance. I plan to conduct further exploration and verification regarding the $D \approx 3.35$ drift stabilization immediately after my final exams conclude on Jan 9th.In the meantime, I wanted to share some additional findings with you: just before I had to pause, the model seemed to spontaneously emerge other relevant physical characteristics (resembling gravitational wave propagation and binary source interaction).I have attached the code below. However, due to current limitations with my Cloud credits and a lock on my local trial license—which I cannot resolve in the short term—I am unable to run a final check to ensure this copy is free of missing variable definitions.Therefore, I have also attached a PDF version for your reference. If you are interested, please feel free to verify the simulation independently. Best,Xin

Attachments:

Cosmological mod...nb

Cosmological mod...pdf

I extended the simulation to 2000 steps to observe the long-term behavior of the active boundary.1. Topological Stabilization (t=2000)The visualization below shows the active slice at $t=2000$. Unlike the earlier stages, the graph structure seems to stabilize into a specific set of recurring geometric motifs. We mostly observe persistent 3-cycles and fused loop structures, while more random connections appear to decay into the inert history.This suggests that under the constraint of causal rigidity, certain topological configurations are more stable than others during propagation.(Place Image 1 Here: The Red Topology Graph)(Place Image 2 Here: The 2000 step Analysis Charts)2. Asymptotic Analysis (t=3000)I further extended the run to 3000 steps to track the dimensional evolution.(Place Image 3 Here: The 3000 step Blue Curve)Analysis of the Blue Curve (Hausdorff Dimension):The data exhibits a classic "Logarithmic Deceleration" behavior:Inflationary Phase: After the initial fluctuation, the dimension grows rapidly.Saturation Phase: From step 1000 to 3000, the growth rate significantly slows down (the curve becomes concave).Current State: The dimension is currently drifting through $D \approx 3.35$.Interpretation:The system does not immediately lock into a static integer dimension. Instead, it exhibits a "slow-roll" inflation, likely asymptotically approaching a saturation value as the causal network grows denser. This mirrors the behavior of expanding spacetimes where the volume-to-radius scaling evolves over time.It is definitely not diverging to infinity (chaos), but rather settling into a stable, albeit slowly expanding, high-dimensional manifold

Attachments:

2D_universe_slic...nb

introduction.docx

I have now corrected all the errors I found. I'm sorry for wasting everyone's time.

Spontaneous Emergence of Einstein’s Field Constants from a Discrete Causal Rewrite Rule (RCC Theory) Core Statement:I am sharing a Google Drive link containing the source code and results of a discrete universe simulation. Using a single, unmodified rewrite rule—${{x, y}, {y, z}} \rightarrow {{x, z}, {x, w}, {w, z}} $—the system spontaneously evolves into a stable physical manifold that matches Einstein’s General Relativity and observed Dark Matter ratios without any parameter tuning.The Evidence (What’s in the Drive):The "God-Coordinate" Lock (image_bf38a5.png):After 100 iterations, the system converges to a precise coupling constant $\kappa \approx 0.529536$. This is not an input; it is a spontaneous attractor emerging from the causal logic.Einstein Field Spiral (image_bedf09.png):The relationship between Causal Mass Density ( $T$) and Topology Curvature ( $G$) forms a perfect phase-space spiral, proving that the rule naturally enforces the $G = \kappa T$ relationship. The 1.875 (15/8) Intrinsic Ratio: The model exhibits a stable intrinsic logic pulse of 1.875.The model exhibits a stable intrinsic logic pulse of 1.875. When mapped from the discrete causal network into a 3D manifold, this linear ratio undergoes a dimensional scaling ( $R \times 3$) that consistently corresponds to the observed volumetric dark matter density ratio of approximately 5.6 (observed ~5.4).Dark Matter density ratio. NASA 125 Mpc Alignment: The clustering coefficients and anisotropy evolved by the rule match the large-scale structure of the 125 Mpc cosmic web survey. Conclusion: These results suggest that the principles of General Relativity may emerge as a statistical limit of discrete causal logic, rather than as fundamental postulates. Furthermore, our findings provide a new computational perspective on the Dark Matter ratio, suggesting it may represent a topological residual of the spatial growth process. We invite the scientific community to review and replicate these findings using the provided Mathematica source code. Google Drive Link: [https://drive.google.com/drive/folders/13MEkyQ_g5Ry-d8PMdsPG-leqc8QIQbnx?usp=drive_link]