Everything up to here on this instrument's ladder has been a lattice — a grid of things sitting next to other things. This one has no grid at all. It has two thousand oscillators: little clocks, each running at its own natural speed, none of them wired to any particular neighbour. They are coupled only through the crowd. Each one feels a pull toward wherever the average of all of them happens to be pointing at that instant, and the strength of that pull is a single number you can turn up or down.
At zero pull, nothing happens. The clocks drift apart, spread evenly around the circle, and the crowd's average sits at essentially nothing — the way a thousand people walking in random directions have no collective direction. Turn the pull up and for a long while still nothing happens; the fast clocks and the slow clocks simply refuse. And then, at a particular value of the coupling, a clump nucleates. A group near the middle of the speed range locks together, and once locked they pull harder, and the clump grows. The crowd acquires a direction it did not have a moment before.
The remarkable part is that theory says exactly where this happens. For the spread of speeds used here the critical coupling is precisely twice the width of that spread — a closed-form answer with nothing fitted. So this rung is a calibration: point the instrument at a number somebody derived on paper, and see how close it lands.
It landed at 1.0007 against an exact 1.0000. But the stronger claim is not the tipping point at all — it is the curve. Above the transition, theory also gives a closed form for how synchronized the crowd should be at every coupling, and the measured coherence matches that formula to fifteen parts in a hundred thousand across seven couplings the run never fitted anything to. A pipeline that manufactured a plausible-looking transition would still have to reproduce seven numbers it was never shown. Below, the same crowd, running live, with the coupling under your hand.
left · the crowd — ember dots are
phase-locked, slate dots are the fast and slow tails still drifting; the arrow is the
crowd's own direction, its length the coherence r. right · r against K,
with the un-fitted closed form r = √(1 − Kc/K) drawn as a line
and your measurements dropped onto it. The points landing on a line nobody fitted is
the whole argument.
N — · settled r — · locked fraction — · simulated time 0
Two details in that panel are load-bearing rather than decorative. The natural speeds
are not drawn from a random generator: they are placed at deterministic quantiles of the
Lorentzian, on the grid (i + ½)/N. That makes the set exactly antisymmetric,
so the crowd has no spurious drift, and it adds no sampling noise on top of the very
finite-size effect being measured. And every one of RK4's four stages recomputes the mean
field from that stage's phases — reusing the stage-start centroid would silently
demote the integrator to Euler.
The published numbers are from the 2026-08-02 promoted run — human-reviewed the same day as the machine pass. An independent rerun on 2026-08-14 returned 0.9955, a 0.5% error, with the closed form matched to 2.0 × 10⁻⁴; both sit well inside the ±0.10 band, and both are worth having on the page rather than only the flattering one.
The finite-size systematic was measured before the tolerance was declared, not after: the estimate approaches 2γ from above as the crowd grows — 1.0504 / 1.0256 / 1.0083 / 1.0007 at N = 250 / 500 / 1000 / 2000 — and a second seed at N = 2000 gave 0.9910, meaning run-to-run scatter has already overtaken the finite-size shift by that population. What happens to that same crowd when you park it exactly at the critical coupling and change only its size is K02; what happens when you try to measure how it responds to a push, and the instrument refuses to answer, is K03.