Gravitational molecules at minimum shape complexity
Critical points of VS = √Icm × W on the pre-shape sphere. Heavy particles red, light particles blue. 3D positions projected via PCA. N = 100 to 10,000.
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Shell structure strengthens with N.





2D + 3D, 5 mass ratios





5 mass ratios





5 mass ratios





5 mass ratios





5 mass ratios





500 starts, angle distributions, recurring motifs





Jupiter/Saturn-like flattening





Maria’s 3-body triangle vorticity





Age ordering, fossil persistence, entropy vs complexity





| Config | Equal | 1:2 | 1:10 | 1:80 | 1:2000 |
|---|---|---|---|---|---|
| N=100, 2D | 52,231 | 142,784 | 3.6×106 | 5.1×108 | 1.6×1012 |
| N=100, 3D | 43,325 | 118,835 | 3.0×106 | 4.3×108 | 1.3×1012 |
| N=500 | 2.5×106 | 7.0×106 | 1.8×108 | 2.6×1010 | 7.9×1013 |
| N=1,000 | 1.4×107 | 4.0×107 | 1.0×109 | 1.5×1011 | 4.5×1014 |
| N=2,000 | 8.2×107 | 2.3×108 | 5.8×109 | 8.6×1011 | 2.6×1015 |
| N=5,000 | 8.2×108 | 2.3×109 | 5.8×1010 | 8.5×1012 | 2.6×1016 |
| N=10,000 | 4.6×109 | — | — | — | 1.5×1017 |
2D results reproduce Maria’s methodology. 3D results projected via PCA.










Molecular segregation persists at scale.





Mass-dependent shell structure clearly visible.





Shell structure increasingly pronounced.





Five thousand particles. Clear mass-dependent spatial organisation.





z-variance penalty breaks spherical symmetry → Jupiter/Saturn-like shapes. N=200, 3D, aligned to principal axes.








3-body oriented triangle terms: VEM = √I × (α Welec + β Wmag). The magnetic term couples triangle areas to a global vorticity axis — oblate shapes from pure geometry.








500 random starts per mass ratio, keeping ALL local minima. Bond angles, asphericity, and cluster counts classify each configuration.
All 500 starts converge to one minimum. Bond angle peaks at water’s 104.5° line.


Still one minimum. Two light-particle clusters.


3 distinct minima. Dominant (434/500) has 7 light clusters.


2 distinct minima. Dominant (447/500) with pronounced segregation.


| Mass Ratio | Distinct Minima | Dominant Angle | Light Clusters |
|---|---|---|---|
| Equal | 1 | 107.9° | — |
| 1:10 | 1 | 98.1° | 2 |
| 1:80 | 3 | 86.6° | 7 |
| 1:2000 | 2 | 71.4° | — |
Bond angle decreases monotonically: 108° → 98° → 87° → 71°.
500 starts per mass ratio, keeping all local minima. Age = VS − Vmin within each family. Fossil persistence measures how much structure from a younger configuration survives in an older one.
Below 1:50, all 500 starts converge to one minimum. At 1:50 and above, two distinct minima emerge — a phase transition in the shape potential landscape.
| Mass Ratio | Distinct Minima | Age Range | Bond Angle | Interestingness |
|---|---|---|---|---|
| Equal | 1 | 0 | 107.5° | 1.80 |
| 1:2 | 1 | 0 | 101.2° | 1.80 |
| 1:5 | 1 | 0 | 93.6° | 1.74 |
| 1:10 | 1 | 0 | 96.3° | 1.70 |
| 1:50 | 2 | 26K | 90.1° | 2.52 |
| 1:200 | 2 | 982K | 79.8° | 4.42 |
| 1:2000 | 2 | 700M | 70.9° | 7.47 |
Configurations ordered by age within each mass-ratio family. Only families with 2+ distinct minima show age progression.



Persistence score (0–1) combining Procrustes alignment, angle-histogram overlap, and radial-profile similarity. Entropy increases with age.



Each point is one distinct minimum, colored by interestingness. Reference lines at water (104.5°) and tetrahedral (109.5°).



Mass ratios below 1:50 — only one minimum exists.




Negative result: face-like or household-object forms do not emerge from pure N-body at N=100. The strongest motifs are mass-shell segregation and heavy-particle scaffolds with light-particle clustering.