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]