A Percolating Cluster Network at n=100,000 (Rapidity)
Under hyperbolic rapidity integration, 24,521 particles - 24.5% of a 100,000-particle swarm - ended up connected through a chain of local bonds spanning the entire expanded volume as a shell, not gathered in one place.
● the identified 24,521-particle connected structure, shown growing across the run's full history - not just its final state
Run Parameters
| Total particles | 100,000 (50,000 posons + 50,000 negons) |
| Initial diameter | 126.0 |
| Integration mode | Hyperbolic Rapidity |
| Force computation | Brute force (exact, not Barnes-Hut approximated) |
| Signals enabled | Crowding, proximity, acceleration, velocity-matching, cluster identification, cluster event capture |
| Steps completed | 56,000 of 56,000 (full run, ~14.8 hours) |
Observations
- The largest connected structure found anywhere in the run was 24,521 particles - 24.5% of the entire swarm - by far the largest of any run in this project so far, and tracked with exact, live particle IDs throughout.
- This is not a compact cluster. The swarm itself expanded enormously over the run - median particle distance from center grew roughly linearly from ~50 units at the start to over 50,000 by the end, consistent with particles simply flying outward at close to their natural speed. This structure's 24,521 members are spread almost isotropically across that entire expanded volume (91% of them sit in the outer third of the swarm's radius, with no preferred direction), connected end-to-end by a continuous chain of locally-bonded pairs - some as close as ~11 units apart, others over 1,000 units apart but still locally consistent with the bonding rule. It reads as a single giant connected network threading through the expanding swarm, not one localized blob.
- The structure was still growing when the run ended: its peak size was recorded just 81 steps before the very last step of the entire 56,000-step run. This number is a floor, not a settled result - a longer run would likely show it larger still, or reveal where it actually stops.
Read honestly: "24.5% of the swarm in one structure" is a real, exact number from live engine tracking - but don't read it as "a quarter of the swarm huddled together." It's a connectivity result: a long, unbroken chain of locally-bonded pairs that happens to span nearly the whole expanded volume. Whether this kind of giant connected network is a real, physically meaningful structure, or an artifact of how generously the union-find bonding radius links up sparse, spread-out particles, is an open question this page doesn't settle. The clip above shows the honest picture - starting as a single point and growing into a shell filling the frame - not a cropped-in view designed to look more dramatic than it is.
Go Deeper
The full analysis - exact cluster tracking, the radial/isotropy check behind the "shell, not blob" reading, and how this compares to the two earlier Direct Normalization runs - is available as a rendered notebook.