I have consolidated all my previous and preliminary code, along with the statistical results, into a GitHub repository. GitHub Link: https://github.com/jerry-wnag/univer_dig_cod Everything is open for replication. If you find any issues with the code logic or have suggestions for improvement, please let me know.

Emergent Double-Slit Phenomena in a Computational Universe Model: A Potential Explanatory Path Guided by the rigorous evolution of the Rule, I have serendipitously observed phenomena analogous to the double-slit experiment within this computational universe model. Through a deep analysis of technical issues such as "data sampling" during simulation, I have identified a potential new path toward explaining simulated quantum gravity effects.

1. The Inevitability of the Rule: From Topological Causality to Simulated Spacetime Emergence In this model, the Rule constitutes the most fundamental topological causal order of the universe.
2. Physical Insight: Simulated Quantum Interference under Undirected Topological Correlation What is truly intriguing is that, within the fabric woven by the Rule, introducing undirected topological correlation to calculate information wave dynamics reveals a core logic of simulated quantum gravity:

In this model, the correlation between points does not depend on the historical sequence of event generation, but on their topological hop-count within the graph network (i.e., a holistic geometric property).

1. Internet Analogy: Routing Flow vs. Network Topology We can use the Internet to intuitively understand this duality of "directed" and "undirected" structures:

Directed Causality: Similar to low-level packet routing. It must possess a clear directionality, representing the actual flow of simulated information and determining the "history" of the model’s evolution.

Undirected Correlation: Similar to the network topology of the entire web. When we measure the "topological distance" or "signal latency" between two simulated nodes, it is a holistic geometric property. It depends on the physical structure woven by the evolution rules, rather than the chronological order in which the cables were laid.

1. Conclusion: A Potential Explanatory Path Through this topological processing, I have successfully plotted a comparison of "wave-particle duality" within the model:

Cyan Solid Line (Quantum Wave Signal): Represents small "topological-ons" within the model exploring the symmetric, undirected topological grid woven by the Rule.

Orange Dashed Line (Classical Particle Trait): Represents the system collapsing back into a unidirectional, deterministic directed evolution path once simulated observation is introduced.

This provides a potential new explanatory path: In this universe model, so-called quantum coherence is essentially the all-path diffusion of simulated information across an undirected topological geometry. This perspective suggests that the reality we observe may be a topological emergence resulting from the interplay between "undirected geometric correlation" and "directed causal history" (the execution trajectory of the Rule).

For those interested in reproduction, I have attached the files for both the directed comparison experiment and the undirected comparison experiment to facilitate verification. Result:

Attachments:

Double-slit inte...nb

two-slit interfe...nb

two-slit interfe...nb

Xin, this is fascinating work. It is encouraging to see another independent derivation of cosmological ratios from discrete causal structures.I recently found that treating these values as 'geometric residues' of a 4D $\to$ 3D projection yields a Dark Matter/Baryonic ratio that matches Planck 2018 to within 0.01% (closer to 5.47 than 5.6). It seems your 'Intrinsic Logic Pulse' (1.875) is capturing the coarse-grained component of this, while the 'residue' method captures the fine structure.If your model's $D \approx 3.35$ drift stabilizes, I suspect your ratio might converge further toward the Planck value. Excellent effort in pushing the 'physics as computation' boundary."

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."

Hi Cook,Thank you for the suggestion regarding Section 19.Regarding the Lattice Dispersion and Gravitational Waves: Instead of setting up a separate binary source simulation, I decided to directly analyze the universe generated by the previous calibration code.I performed a spectral analysis (calculating the eigenvalues of the Graph Laplacian) on the static lattice structure itself. The results confirm exactly what you predicted: a distinct cutoff phenomenon appears.The Cutoff: The energy spectrum does not extend indefinitely but hits a hard wall. Across multiple runs, this cutoff ( $E_{max}$) fluctuates typically between 5.5 and 6.9, depending on the specific stochastic topology of the generated instance. This finite bound acts effectively as the instance-specific Planck Frequency ( $E_P$) in your dispersion formula $v_g(E) = c \sqrt{1 - (E/E_P)^2}$. It proves that the lattice structure itself imposes a high-frequency limit.Additional Verification:I also appended a small extra verification at the end of the code—a Holographic Page Curve test. Interestingly, the entropy peaks at 99.9% volume (rather than 50%), which seems to act as a signature for the "Holographic Tension" or Imbalance you mentioned.I have attached the resulting charts from this run. I hope you find these findings interesting!Please let me know if you spot any errors in my interpretation or the code implementation.

Attachments:

cons_cal (1).nb

cons_cal_cur.pdf

data/code : [https://drive.google.com/drive/folders/13MEkyQ_g5Ry-d8PMdsPG-leqc8QIQbnx?usp=drive_link] When constructing this network generation model based on RCC (Recursive Causal Computation), we compared the experimental data from the cloud (high concurrency/random environment) with that from the local environment (serial/deterministic environment), and observed significant differences in the system's evolution path under different conditions and its eventual convergence. The following is an algorithmic review of the evolutionary logic from "no memory" to "steady-state spiral": 1. Baseline Comparison: Environmental Differences in a Memoryless Environment (The Memoryless Baseline) In the initial experiment, we set the system to completely discard "historical memory" (that is, each round of iteration clears the old connections and only retains the latest topological relationships). Under this extreme condition, the output of the cloud and the local system exhibited significantly different geometric characteristics: Cloud performance: Random "dynamic trends" Phenomenon: In the cloud environment, due to concurrent computing and random point selection, a large amount of randomness at the computational level is introduced. The data points exhibit a blurry and somewhat directional distribution in the coordinate system (that is, the blurry spiral shape you see). Cause: This is not that the system has memorized anything, but rather the instantaneous manifestation of the fundamental rule of "adding/eliminating edges". In each tiny calculation step, the increase in node density will instantly change the local topological structure. The randomness of the cloud superimposes these countless "rule reactions" together, forming a macroscopic dynamic trend. Local performance: Strict "value locking" Phenomenon: The data obtained through local operation exhibits extremely strong regularity, forming a clear and rigid curve. Cause: Local computation is highly deterministic. The traversal order of nodes and the generation logic of edges are fixed, without any random noise interference. Therefore, the algorithm rules directly reveal their underlying mathematical logic, manifesting as an unshakable operational rigidity. Phase summary: When there was no memory involved, the cloud displayed the statistical trend of the rules under random perturbations, while the local display showed the arithmetic determinism of the rules themselves. 2. Introduction of Variables: Forgetfulness Rate To observe the performance of the system over a long period of evolution, we introduced a "memory" mechanism, which is quantified as the "forgetfulness rate". Definition: It represents the proportion of existing connections that are retained during the iterative process of the network. Function: It introduces the "lagging effect of the 'time dimension'" into the system. The current topology is no longer solely determined by this millisecond operation, but is also influenced by the residual structure from the previous round. 3. Evolutionary Convergence: Why do both the cloud and the local systems exhibit a "spiral" pattern after the introduction of memory? When a specific forgetting rate parameter (such as 0.45) is introduced, we observe an interesting phenomenon: the randomness of the cloud and the rigidity of the local system eventually converge to the same convergent spiral (Convergent Spiral) form. Smooth effect (for the cloud): The memory parameters act like a "filter". They filter out the high-frequency computational noise, forcing the originally chaotic random points to develop along the "inertia" direction of the network connections, thereby making the trajectories clearer. Lag effect (for the local area): The memory parameters disrupt the originally straightforward linear growth. Because the "breakage of old connections" occurs later than the "generation of new nodes", the system cannot maintain a simple linear growth but is forced to undergo damped oscillations (manifested as a spiral) around a balance point. Phase summary: The spiral form is a geometric representation of the interaction and eventual balance between the two forces of "node growth (doing addition)" and "connection forgetting (doing subtraction)" on the timeline. 4. Parameter Tuning and Safety Threshold: 0.14 vs 0.45 During the parameter exploration process, we discovered that the selection of the forgetting rate value is crucial for the stability of the system: 0.14 risk (data overflow): In the earlier 0.14 configuration, the system "forgot" too slowly. The accumulation of old connections far exceeded expectations, while new nodes were increasing exponentially. As shown, the system generated thousands or even tens of thousands of nodes in an instant at Step 3. This explosive growth would cause the computing memory to overflow instantly, preventing us from observing the complete evolutionary process. 0.45 Adjustment and Throttling: In order to observe the final state of the system while protecting the computing equipment, we adopted the "high forgetting + strict limitation" strategy: Increase the forgetting rate to 0.45: Accelerate the speed of clearing old connections to prevent data from accumulating too quickly. Limiting growth step size: By strictly restricting the number of nodes generated in each step, the previously instantaneous "explosion" is elongated into a slow "evolutionary shot". Final result: Through this control, we successfully captured the complete trajectory of the system from growth to stability, and ultimately rendered a stable topological structure consisting of 2386 nodes with self-organizing characteristics.

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.