N-body Simulation of Star Cluster Collisions: Public Research Notes

Star Cluster Collision, B13 Weight Matrix and Verification by Controlled Experiment

Starting from negative results, isolating initial configuration, and confirming the weak effect of 4H stratification

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Attempting to simulate star cluster collision using the B13 frame

Previously, I have confirmed the coset structure Z₁₃ = {0} ∪ H ∪ 2H ∪ 4H, which is at the heart of the B13 frame, as an algebraic property.

Today's question is more direct:

Does this coset classification appear as a statistically detectable bias even within continuous dynamics like gravity?

In other words, does a group of particles labeled with B13 labels exhibit a statistically biased distribution during gravitational evolution? I will isolate this through a controlled experiment.

My prediction:

B13 Prediction: Particles belonging to the 4H coset (the forbidden zone) are more likely to be distributed in the outer layers than particles in the H/2H cosets (the 4H stratification hypothesis).

Reason: 4H is defined as an "unstable zone," and when the gravitational coupling coefficient is weakened, it is expected to be less likely to nucleate and thus be pushed to the outer layers.

I will verify this hypothesis by comparing it with a shuffled control and a random initial configuration.

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Design

N-body simulation

I use an N-body simulation with 200 particles (100 particles per star cluster).

Initial conditions:

b = 0.72   # Offset (impact parameter)
v0 = 0.90  # Initial velocity
G_cross = 2.0  # Mutual attraction coefficient
G_self  = 0.25 # Self-cluster attraction coefficient

Each particle is assigned a k0 ∈ Z₁₃, and a coset classification (H/2H/4H) is given.

B13 weight matrix

I multiply the weight for each coset pair relative to normal universal gravitation.

G_ij = G × W(c_i, c_j)

Pattern A (4H weak coupling):

Coset pair	Weight
H-H, H-2H, 2H-2H	1.00
H-4H, 2H-4H	0.90
4H-4H	0.80

Main indicators

4H stratification index (Primary): Δ_out = r_4H - r_H (average distance difference from each cluster's center of gravity)

Outer layer occupancy rate (Secondary): What percentage of 4H particles are in the top 20% radius region of all particles

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Experiment 1: 100 Shuffled Controls

First, I answer the initial question: do k0 labels affect dynamics under uniform gravity (no weights)?

I ran 100 controls with the k0 labels randomly shuffled while keeping coordinates and velocities the same, then calculated the z-score of the original data.

Results: z-score < 2.0 for all frames; zero significant frames.

frame  70: z = +1.21 (87th percentile of shuffles)
frame  71: z = +1.56 (93rd percentile of shuffles)

No frames reached the significance level (z > 2.0).

Determined negative result:

In the current implementation, k0 labels do not affect dynamics. The effect vanishes completely with shuffling.

This result confirms that "B13 is not appearing" and establishes the premises for subsequent experiments.

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Experiment 2: 4-Condition Isolation (500 Shuffled Controls)

A z=2.10 was observed for the outer layer occupancy rate even in the uniform version. I isolated whether this was due to initial configuration or dynamic effects using 4 conditions.

Condition	Initial k0 Config	Weight Matrix
Condition 1	Original config	Uniform
Condition 2	Original config	B13 (4H weak coupling)
Condition 3	Completely random	Uniform
Condition 4	Completely random	B13

I compared the z-scores of the outer layer occupancy rate with 500 shuffled controls.

Results:

Condition	z>1.96 frames	z_max
Condition 1 Original+Uniform	11/150 (7.3%)	+2.22
Condition 2 Original+B13	19/150 (12.7%)	+2.58
Condition 3 Random+Uniform	0/150 (0.0%)	+1.42
Condition 4 Random+B13	6/150 (4.0%)	+2.16

Conclusion of isolation

1. The significant bias in the uniform version is derived from the initial configuration (it disappears in Condition 3).
2. The B13 weight has an independent amplification effect (confirmed in Condition 4).
3. The maximum effect is the synergy between the initial configuration and the B13 interaction (Condition 2).

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Sensitivity Analysis: Weights 1.00→0.95→0.90→0.85→0.80

Does the z-score of the outer layer occupancy rate change monotonically when the strength of the 4H weak coupling is varied?

Weight Setting	mean_Δ_out	z>1.96 frames
Uniform (1.00)	-0.017	7.3%
w=0.95	-0.017	—
w=0.90	-0.016	—
w=0.85	-0.007	—
w=0.80	-0.009	12.7%

Weaker weights push 4H particles toward the outside. However, the average distance difference (Δ_out) itself is small, and outer layer occupancy is more effective at detecting the effect.

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What I can reliably say now

What I can say:

Introducing 4H weak coupling (weight matrix) increases significant frames for outer layer occupancy from 7.3% to 12.7%. Since this effect does not vanish even with random initial configurations (4.0% remains), the B13 weight has a weak but non-zero influence on its own.

What I cannot say:

• "B13 governs the mixing laws of the universe" — the effect is weak and limited under current weight settings.
• "4H particles separated" — it is not separation, but stratification (distortion of distribution).

Honest evaluation:

The synergy of initial configuration correlations and B13 interactions appeared as a bias in outer layer occupancy. However, the effect size is small and requires further implementation — transitions between cosets, threshold firing, etc.

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Animation

Offset collision using an N-body simulation (200 particles). Blue (cluster A) and orange (cluster B) approach, pass each other, pull back, and form a vortex with a b=72° offset. Red dots represent 4H forbidden zone particles (treated as weak coupling in the weight matrix). The time axis is displayed in 6 steps of FIB6.

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Status of the Box

Task	Status
k0 effect under uniform gravity	Negative confirmed with 100 shuffles
Isolation of initial config	Confirmed with 4-condition, 500-shuffle control. Uniform z=2.10 is initial config derived
Independent effect of B13 weight	Weak positive effect confirmed on outer layer occupancy (4.0–12.7%)
Sensitivity analysis	Monotonic change confirmed down to 0.80 weight
Inter-coset transition (firing)	Not implemented. Next step

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Postscript

This animation produces the same result no matter how many times it is played. This is natural because particle positions are just replaying an already-computed N-body coordinate table. However, particle labels (k0) and phase walking (k = (k0 + step) % 13) are updated only by integer addition, without using trigonometric functions or division, so there is no accumulation of floating-point errors.

In other words, if you use the B13phase library, you will get the same results regardless of the system on which you run it.