The Butterfly and the Fork
Two ways a prediction can fail. In the first, you never know the start well enough. In the second, knowing it perfectly still isn’t enough. Weather models are famous for the first. A paper released in October 2026 claims the equation for a gas can do the second.
1 · The butterfly: tiny differences grow
In 1963 Edward Lorenz boiled air rising and falling in convection down to three numbers that feed into each other. Below, his model runs twice. The two runs start apart by the distance you choose. Press Run and watch the first number over time.
Lorenz’s equations with his 1963 values (σ = 10, ρ = 28, β = 8/3), solved step by step on this page. Starting 0.1 apart, the runs part at about time 14; starting a trillionth apart, at about time 44. Making the start a hundred billion times more precise buys only about thirty more time units. Set it to exactly zero and they never part, however long you wait. That is chaos: hard to predict, but still one future for each exact start.
2 · The fork: one exact start, many futures
A textbook example, not the gas equationSome equations do not pick. The simplest known is one line: the rate of growth equals the square root of the amount. Start at exactly zero. Staying at zero obeys the rule. So does waiting any length of time and then taking off. Choose when the curve leaves.
The equation is dx/dt = √x, starting at x = 0. Every curve here satisfies it exactly at every moment: at zero, the growth rate is √0 = 0; after take-off at time c, the curve is (t − c)²/4, whose growth rate (t − c)/2 equals its square root. Nothing in the equation decides which one happens. Something outside it would have to.
3 · What the Boltzmann papers claim
Ludwig Boltzmann’s 1872 equation describes a gas as particles that fly and collide. Two papers in OpenAI’s October 2026 release claim that, for hard spheres, it can behave more like the fork than the butterfly: one starting state, two different futures. That happens even with the conservation laws and the entropy law imposed.
“Their common datum has bounded velocities but unbounded amplitudes concentrated at successively smaller spatial scales.”
In plain terms, the start is not ordinary air. It packs gas into narrow jets in thinner and thinner rings, piling up at a point. That is how it slips past every theorem that guarantees one future, all of which assume a gentler start.
| Paper | Claim | Machine-checked |
|---|---|---|
| September 23, 2026 · 65 pages | Two solutions; mass conserved locally, momentum in total; energy and entropy as inequalities | Yes, in Lean |
| October 5, 2026 · 78 pages | Two solutions with exact local conservation of mass, momentum and energy, and the entropy law | No — “outside” the formalization |
Neither paper has been reviewed by specialists, and no public comment from one was found as of October 14, 2026. Three mathematicians had conjectured this kind of result in print on August 13, 2026. This page does not claim the proof is correct.
The long road to the question
- 1872Boltzmann publishes his equation and the H-theorem: entropy only moves one way.
- 1876Loschmidt objects that reversible collisions cannot produce a one-way arrow. The debate shapes statistical physics.
- 1963Lorenz finds that his convection model never repeats, and that tiny differences grow.
- 1989DiPerna and Lions prove solutions to Boltzmann’s equation exist for all time, in a weak sense. Whether they are unique is left open.
- 2002Villani’s review: “many fundamental questions are still unsolved: in particular uniqueness.”
- 2009–2022The equations of fluids are shown to allow many weak solutions, even with an energy condition imposed.
- 2024Deng, Hani and Ma derive Boltzmann’s equation from colliding spheres for as long as its solutions exist.
- Aug 13, 2026Gismondi, Golding and Novack conjecture non-uniqueness for a dense class of starting states.
- Oct 6, 2026OpenAI releases two papers claiming it. Only the weaker one is machine-checked.