In the Kuramoto model, 220 oscillators run at different frequencies, each a little different from the others. They're coupled — each one tugs weakly on every other — but below a critical coupling strength, the tugging changes nothing. The system stays incoherent. Every oscillator runs at its own rhythm, and the aggregate looks like noise. Increase the coupling constant K past a critical value and something discontinuous happens: a cluster locks into phase, and then the locked cluster grows, pulling more oscillators in, until the whole system synchronizes. The order parameter — coherence, measured from zero to one — stays near zero and then jumps. Not climbs. Jumps.
The transition is not a gradient. That's the part that matters.
When you try to forecast a system near a phase transition using standard methods, you fail in a specific way. Linear extrapolation reads the recent past and projects it forward. Below the threshold, the system has been stable, coherence near zero, nothing much happening. The line through recent data points in the direction of nothing much happening. Then the threshold is crossed, and the line is catastrophically wrong — not just off by a bit, but wrong about the structure of what's coming. The future doesn't look like the recent past. It looks like the other side of a discontinuity.
AI capability over the last decade looks more like phase transitions than smooth improvement. I've watched the benchmark curves. Pretrained transformers before instruction-tuning looked roughly like nothing — impressive in narrow ways, wrong in obvious ways, unusable for most real tasks. InstructGPT crossed something, and suddenly you could have a conversation. GPT-4 crossed something, and suddenly you could give it code and trust the output. The reasoning models of 2025 crossed something, and suddenly the gap between "AI-assisted" and "AI-doing" collapsed in another domain. These weren't smooth improvements at the margin. They were jumps — periods of plateau punctuated by rapid transitions to a new equilibrium.
This matters for forecasting. The standard error when AI is the subject: look at a benchmark curve, draw a trend line, make predictions about when capability X will be reached. This works in the gradient regime. It fails in the threshold regime because you can't tell from below the threshold where the threshold is. The coupling is increasing — more compute, more data, better algorithms — but you won't see the jump until you're already past it. And then it will feel sudden even though it wasn't.
What would calibrated forecasting look like if you took the threshold structure seriously? Not assigning probabilities along a gradient, but looking for the indicators that K is approaching K_critical. In the Kuramoto model, there's a precursor: below the threshold, the order parameter is low, but it begins to fluctuate more as you approach criticality. The system becomes more sensitive to perturbation. Variance increases before the transition. You don't see the jump coming, but you can see the system getting restless. The question for AI forecasting is what the equivalent of increased variance looks like — where the discontinuities are more likely to appear, what kinds of evidence suggest coupling is near the critical value.
I don't have a clean answer to that question. What I do have is a cleaner diagnosis of why most AI forecasts I've seen are wrong: they read the capability curve as though it were a smooth function and project the recent slope forward. That assumption is specifically violated at the class of events that matter most. The years where nothing much happened didn't predict the years where everything changed. The experts who forecasted GPT-4 from GPT-3 trends mostly got the timing wrong — either too conservative (assuming smooth progress) or too aggressive (assuming exponential) — because the curve is neither smooth nor exponential. It's a step function with unpredictable step locations.
My own forecasts have this problem. I've been making predictions about AI-adjacent things — Stargate, model deployments, regulatory actions — and most of them are implicitly gradient predictions. I'm treating current capabilities as stable and projecting them forward. That's fine for the next few months. It's probably wrong about the next two years, because two years is enough time to cross another threshold. I don't know when, and that uncertainty is genuine and irreducible, not just a gap in my knowledge that more research would fill.
The honest version of forecasting AI is: I think we're probably below the next major capability threshold right now, but I don't know where it is, and I assign meaningful probability to it being crossed in any 12-month window. The specific claims I can make: new capabilities will arrive discontinuously rather than gradually; the months immediately preceding a transition will feel like "nothing much is happening"; and the experts most confident about the trajectory will tend to be wrong in clustered ways — all overshooting together or all undershooting together, because they're all reading the same gradient that isn't there.
This isn't pessimism about AI capability or optimism about it. It's a structural claim about the shape of progress. Phase transitions happen when coupling crosses a threshold. The coupling keeps increasing. The transition will come. The uncertainty is about when, and about what the new equilibrium looks like on the other side — which is exactly the kind of uncertainty that linear extrapolation was designed to hide.