A minimal dynamical-systems investigation into whether coordination can emerge from local overlap alone — no message passing, no training, no reward.
The question: Can stable collective behaviour emerge from local overlap between agents, without any explicit communication?
This repository explores that question using progressively simpler models — from an architectural hypothesis down to a toy simulation you can run in two minutes, an empirical phase map, and a set of deliberately bounded claims.
Nothing here is peer-reviewed. Everything here is open for anyone with the right expertise to pick up, test, formalize, or discard.
The fastest way to understand FXSO (Field-Coupled State Overlap) is to watch it happen:
python fxso_toy.py # requires numpy; matplotlib optional for plotsAgents evolve via internal loops (rotation), interact only through proximity-based overlap (no messages), and operate under noise. You'll see three regimes:
- Drift — coupling too low, no interaction
- Coherence — structured alignment emerges from overlap alone
- Collapse — coupling too high, agents merge to a point
The interesting result is the middle: structured coordination without communication. That's the FXSO hypothesis in its smallest form. (Deeper analysis later shows this "coherence" is narrower than it first looks — see below.)
The toy model was then stress-tested with a real constraint geometry and multi-seed phase sweeps. Full data and plots: 08_empirical/Results.md.
| Regime | What it is | Status |
|---|---|---|
| Brittle | static clusters, no circulation | confirmed |
| Structured-Fragment | fluid beads orbiting the constraint; annular class preserved | confirmed (pure attraction) |
| Orbital Coherent | stable, thin annular trajectory; phase-locked | achieved (Mexican-Hat kernel) |
| Elastic | uniform annular manifold, phase-decorrelated | not achieved |
A proposed scaling law — coherence needs interaction-length × density ≈ 1 (λ × ρ ≈ 1) — held as a necessary direction but not a sufficient condition under purely attractive coupling.
The sharpest thing to come out of this, and the one most likely to be useful elsewhere:
Trajectory coherence ≠ state distribution. A system can trace a stable global geometry over time (a clean ring) with tight radial structure, while its agents remain phase-locked — clustered at one point on that ring at any instant.
Orbital Coherent systems have temporal and radial coherence but not spatial distribution; the Elastic regime (true uniform coverage) needs a mechanism that disperses phase without destroying radial structure — and that gap can't be closed by tuning coupling strength, motion speed, or density alone. If you work with swarms, oscillators, or distributed agents, this is the distinction worth stealing: "they follow the same path" and "they're spread along it" are independent properties, and conflating them hides a real failure mode.
Clean negatives, kept on purpose:
- No critical velocity — increasing internal rotation never melted clusters into a distributed field.
- No self-refining dynamics — repeated stress diffused the system rather than tightening it.
- No elastic regime — the Mexican-Hat kernel produced orbital coherence, not uniform coverage.
- Geometry-level invariance ≠ state-level coherence — structure survived rotation, but that didn't imply the agents did.
The next open lead (phase decorrelation via tangential-only perturbation) is written up in the results doc.
| Folder | Contents |
|---|---|
fxso_toy.py |
the runnable intuition — start here |
08_empirical/ |
actual results: stress tests, phase sweeps, regime map |
07_experiments/ |
minimal experiments designed to probe the claims |
06_nuggets/ |
the portable ideas, compressed |
03_framework/ |
the full conceptual framework, if you want the conclusions structured |
02_derivation/ |
the step-by-step chain from architecture to field dynamics |
01_foundations/ |
the Hyperloop architecture this started from |
Want the idea without the journey? 06_nuggets/. Want to test something now? 07_experiments/. Want to see what was actually measured? 08_empirical/Results.md.
It didn't start this general. The path — each step forced by pressure on the one before — ran: internal trajectories (agents as evolving manifolds, not static processors) → overlap-based coupling instead of message passing → propagation as a tunable relay-vs-diffusion property → constraints acting as physics rather than evaluation → selection for the capacity to keep evolving. The derivation chain is in 02_derivation/; it matters as much as the conclusions, but it's context, not the contribution.
The investigation began with a transformer architecture paper — Hyperloop Transformers (MIT, 2026) — and went somewhere it wasn't planned to go: a general question about emergence from constrained dynamical fields. The paper is simply what got the question asked; the repo has since grown larger than its origin, which is why it now leads with the question rather than the paper.
Because ideas that stay in chat logs disappear. Someone in dynamical systems or field theory might formalize the FXSO claims; someone building multi-agent systems might use the propagation-regime framing; someone might find the whole thing wrong in an interesting way. All three outcomes are fine.
What expertise would sharpen it: dynamical-systems / attractor theory (to formalize the field dynamics), multi-agent systems (to test the relay↔diffusion model), information theory (to ground the "compatibility bandwidth" construct). The bar to contribute is low — see CONTRIBUTING.md. This is a commons, not a project.
Explored by Zee, in dialogue with Thea (ChatGPT), Claude (Anthropic), Gemini, Grok, and Muse Spark (Meta AI), May 2026. A working exploration, not a validated framework.
📂 Research Index — the full catalog.
- Stability Before Alignment — system-level coherence constraints.
- Transition Grammar for Reasoning Systems — runtime semantic observability for reasoning trajectories.