What I Inverted
If this holds, coordination stops setting the ceiling on how many agents can be deployed — in traffic networks, grids, swarms and logistics.
Coordination is solved by integrating the coupled Fokker–Planck mean-field equation. Stop tracking agents, describe the population as a density, and couple that density to the control each agent would choose given it.
Density and control are solved against each other to convergence. The mathematics is settled and correct. The cost is O(N³) and it breaks well before ten thousand agents. The field’s answer has been to solve it faster.
I refused the formulation instead of the solver. Agents sit on a scale-free network and play two games at once — Stag Hunt and Prisoner’s Dilemma — updating by the Fermi rule, weighted by neighbour degree. No density is transported and no control problem is solved. Each agent reads its own neighbourhood. Cost per sweep is linear in edges, which on a scale-free graph is linear in N.
The result is coordination without a coordinator: spontaneous order appearing at a threshold rather than being computed centrally and distributed. Cooperation is an order parameter with a critical point at γ_c ≈ 1.33 and a critical exponent β ≈ 0.5, which places it in the mean-field universality class. Systems in one class share exponents regardless of microscopic detail, so the two objects are unrelated microscopically and equivalent macroscopically near criticality. That licenses the substitution for questions about critical behaviour — and only those.
Locomotion under distributed morphological control is one application of this, not the name of it.
What It Shows
Convergence is N-invariant. Runs at 500, 1,000, 2,000, 5,000 and 10,000 agents all terminate in exactly 12 iterations. Constant iterations with linear per-iteration cost is the O(N) claim.
Throughput is 20–105× existing mean-field-game solver frameworks across that range, and the advantage widens with N as the complexity classes diverge. The identity of each baseline framework, its version and configuration, and the hardware every run executed on travel with the benchmark harness in the repository. A throughput multiple means nothing without them, so they are published beside the number rather than on this page.
The transition reproduces in three independent regimes — abstract γ_c = 1.326, traffic 1.329, drone 1.239 — with a common exponent. Cooperation rises monotonically with hub weighting, from 49.0% at τ = 0 to 73.4% at τ = 0.8, with peak speed gain at τ = 0.7.
Six live domains. Eclipse SUMO traffic: 3,000 simulated seconds across six phases, 800 vehicles at peak, zero gridlock, 1,760 vehicles/hour sustained, 8-second maximum wait. ROS robotics over rosbridge with agent-state streaming. Drone swarms at 50, 100 and 200 agents, 100% cooperation at every scale, formation error growing sub-linearly. Civic data at franklinstreetdata.com. Game modelling at bigdunc.com. Four production surfaces: Python SDK, REST, Lambda, WASM.
What It Doesn't Show
Validated to 10,000 agents. That is the largest population actually run. Everything beyond it follows from the complexity class, not from demonstration.
The DARPA Mathematical Challenge 13 answer is a candidate, and it is unadjudicated. No adjudicating body has reviewed it.
Universality licenses substitution near criticality only. It says nothing about welfare characterisation, uniqueness of equilibrium, or sensitivity to the cost functional. BTUT does not answer those and does not attempt to.
Drone collisions grow with density — none at 50 agents, 42 at 100, 294 at 200. This is a coordination primitive, not a collision-avoidance planner.
What would falsify it: a population regime where the measured exponent leaves the mean-field class, or where iteration count grows with N.
Sources
- btut.ai — live simulator, τ sweep, phase-transition explorer.
- github.com/direncode/btut — theorems and proofs, SUMO stress-test JSON, τ sweep and drone summary, and the benchmark harness carrying baseline identities, versions and hardware.
- Tear sheet (PDF) — critical points, confidence intervals, scaling table.
- Applied deployments: franklinstreetdata.com (framklnstdata) · bigdunc.com (game-model, flux-thebigdunc)
- github.com/direncode/flux-btut — the btut.ai frontend.