Best of luck with your exams!! study hard. I know you will do well.

when you come out successful, you might want to check out this: https://doi.org/10.5281/zenodo.18168311 .

I suspect you will find Section 19 (The $D=3$ Imbalance) particularly relevant to your latest results. It provides a rigorous derivation for exactly the phenomena you are seeing emerge in the simulation:

1. Gravitational Waves: You observed "ripples" and binary source interaction. My framework derives this not as continuous metric waves, but as Lattice Dispersion on the characteristic grid1. I derived a specific dispersion relation for these waves: $v_g(E) = c \sqrt{1 - (E/E_P)^2}$. Your simulation is likely hitting the high-frequency cutoff where this discreteness becomes visible.
2. The 3.35 Drift: You noticed the dimension drifts and slows. In my calculus, this is Holographic Tension. The bulk lattice requires D=4 for saturation 2, but the observer is constrained to a D=3 barrier3. A fractal dimension of $\approx 3.35$ is exactly what one expects from a system oscillating between the stored dimension (4) and the transmitted dimension (3).

Your simulation is effectively acting as a Constraint Compiler 4; solving for the minimal admissible geometry. I would be very interested to see if applying a 'Shadow Filter' (accounting for the return flux) to your graph closes the final gap between your 133.2 and the 137.036 attractor."

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

Following up on the AdS geometry in my model, I tested the Area vs. Entropy correlation for the dense matter clumps.

I found a very distinct linear relationship, effectively demonstrating the Holographic Principle. The code is attached for anyone who wants to verify this themselves. The model is randomly generated, so your results should match mine.

Note: As the .nb file was downloaded from a cloud platform, some variable names or definitions might occasionally be missing or incomplete due to environment syncing. To ensure a smooth experience, I have also provided a PDF version for your reference and verification."

I hope you find these findings as fascinating as I do.

Attachments:

ADS (1).pdf

ADS (2).nb

Hi Cook,

Thank you for your guidance. After studying your paper, I discovered that your theory aligns remarkably well with my simulation data, providing the theoretical backing for the deviations I previously observed.

My initial value of 133 was a baseline derived solely from the Average Degree. Based on your framework, I applied a correction using the following parameters:

Dimensional Gradient: Since the dimension is not uniform throughout the model (the internal Bulk dimension is greater than the Boundary), I utilized the average dimension within the 20%-90% radial scaling zone.

Regarding your mention of the 'Shadow Filter', I realized that my implementation of the Interaction Ratio correction performs exactly this function. It captures the 'return flux' from the collision events, which successfully bridged the gap from the bare 133 value to the convergent 136-138 range.

The result is promising (as shown in the attached figure): The corrected value now fluctuates around 136.8.

Regarding the fluctuations seen in the average dimension graph, I attribute this to the discrete nature of my observation method and the topological dynamics:

Observation Method: To prevent computational deadlocks from parallel processing, I adopted a "snapshot-per-node-birth" approach.

Node Role Differentiation: Each new node plays a distinct topological role—some serve as Bulk Fillers, while others act as Boundary Expanders.

This alternation of roles causes instantaneous jumps in the dimension per step, but it has little impact on the statistical mean. As previously noted, the effective constant stabilizes robustly in the 136 - 138 range. Regarding Gravitational Waves: As for your insights regarding Gravitational Waves and the dispersion relation, I want to dedicate more time to fully digest Section 19 to ensure I grasp the implications for my lattice structure. I will follow up on this specific topic as soon as possible! Best regards,

Attachments:

cons_cal.nb

cons_cal.pdf

Experimental Report: Emergence of the CMB Acoustic Scale via Strict RCC Rewrite Rules1. Formal Correction and Apology Code/Data Link[https://drive.google.com/file/d/1DndvP1sUO19eCDEd1Jav4NCIKVpQdIk_/view?usp=drive_link] I would like to issue a formal correction regarding the previously shared data and screenshots. Due to an administrative oversight in file handling and data extraction, a preliminary or incorrectly indexed screenshot was disseminated. Our RCC model, at a node scale of 13,771, has naturally emerged with an acoustic scale characteristic of approximately $l \approx 220.6$ without any parameters. The current observational results exhibit precise spectral shoulder alignment. As the computational scale and observational resolution continue to increase, the complete acoustic oscillation peak structure will be completely detached from this logical framework. If you find any issues, please feel free to point them out and I will correct them as soon as possible.

[Code & Data Download: [https://drive.google.com/file/d/1C4Etwr5tG6-f__kGMce89ESkel-KE2IE/view?usp=drive_link\] model is governed by a triadic closure rewrite rule of the form:$${x,y}, {y,z} \to {x,z}, {x,w}, {w,z}, {y,w}$$extended with stochastic edge decay and degree-biased activation. By introducing a stochastic decay rate (e.g., at 0.14), the system exhibits spontaneous symmetry breaking. (The inclusion of the $\{y, w\}$ link in our rewrite rule provides the necessary internal tension for local stability. When combined with stochastic decay, it allows for spontaneous symmetry breaking, where the system 'chooses' between different topological pathways—leading to the diverse morphologies we've documented.)We observe a range of distinct emergent morphologies under identical parameters, including but not limited to:][1]

1.Isotropic structures

2.Directional flows

3.Flattened configurations

These variations suggest how simple computational rules, through path dependency, might give rise to the rich morphological diversity observed in large-scale cosmic structures.