An earlier run of this lab's coherence work noticed something appealing. Sweep the coupling of a crowd of oscillators and the fluctuations of their synchronization do not peak at the ordered end or the disordered end — they peak somewhere in the middle, at partial order. That run fitted a tidy closed form to the shape of that peak and located its maximum at a coherence of two-fifths. It looked like a finding.
This rung went to check it, and the checking is the result. Two things came back.
The first is a calibration, and it passes. Rather than hunting for a peak at all, park the coupling at exactly the critical value theory pins — no argmax to locate, no fitted family to inherit assumptions from — and simply ask how synchronized crowds of different sizes get. The answer shrinks as a power of the crowd size: r ~ N−0.401 ± 0.017 across populations from 250 to 4000, against a published exponent of 0.39 ± 0.02 for exactly this configuration. Agreement at 0.42 σ. That is this instrument checking itself against the literature, and it is the honest headline.
The second is a refutation, and it lands on this lab's own earlier work. Because the coherence at the critical point falls with population size, the peak's position inherits that scaling — so it cannot be a fixed number, and any single-population reading of it, the earlier two-fifths included, is partly measuring the population size. Pin the earlier proposed form's exponents where that run put them and it scores a negative R² at every population size — worse than a flat horizontal line drawn through the data's own mean. What survives unchanged is the qualitative fact: the fluctuation peak really is interior, standing 2.2× to 8.9× above both ends of the swept range, at every size tried.
left · the ladder as objects — every crowd is at the identical coupling; the only difference between them is how many oscillators are in the jar. right · top: ⟨r⟩ against N on log axes with the published slope band; bottom: the running average still falling with measurement time, which is the whole reason the early reading is wrong.
rungs 3 / 5 · simulated time 0 · window full · fitted slope — · published 0.39 ± 0.02
The counterintuitive result is the whole show: the smallest crowd looks the most synchronized. Every panel sits at the same coupling, and the largest one is the least ordered. A viewer expecting "more oscillators, more consensus" gets the opposite, which is exactly the finite-size effect this rung is about.
Give the panel about a minute before you read its slope. The average has to fall
before it means anything, and the page says so rather than hiding it: until roughly
t = 1500 the browser's points are drawn faded and the slope is labelled
still settling. Left running, it walks down to about −0.40 and lands inside the
published band — which is the same convergence the receipt paid 3797 seconds for.
Press read too early and you can manufacture the defect yourself. It shortens
the measurement window to the original t_measure = 200. The biggest panel
immediately reports a visibly inflated, still-falling number: on the traced run the
average was still dropping through 0.070 → 0.049 → 0.036 → 0.028 → 0.027 across
t ∈ [100, 3000), so reading the short window measures a transient
and inflates coherence by roughly two and a half times. The bottom-right panel shows that
fall happening live; the dashed line is where the original window stopped looking.
The measured coherences behind that exponent are 0.08423 / 0.06439 / 0.04875 / 0.03689 / 0.02772 at N = 250 / 500 / 1000 / 2000 / 4000, twelve initial conditions per rung, 3797 seconds of wall time. They are drawn on the plot above in moss, alongside whatever your browser has managed so far in ember.
This milestone claims no novelty, and says so. An adversarial pass against the literature found that essentially every element of it is published already — this model, this frequency distribution, this sampling rule, this estimator, over a wider range of sizes and far longer runs. It was re-scoped from a finding to what it honestly is: a calibration against a published exponent, plus the retirement of a closed form only this lab ever proposed. The calibration gate also excludes the wrong universality class: random frequency draws would give an exponent near 0.20, and this run sits 2.4 bands away from that.
The crowd itself, and the closed form its transition is calibrated against, is K01. The attempt to measure how that crowd responds to an external push — which this instrument refused to complete — is K03.