Simple to state. Impossible to solve.

What you are looking at

Three masses, moving under their mutual gravity. Nothing more. And yet after three centuries of work, there is no formula that will tell you where they will be.

Newton solved two bodies completely in 1687. Add a third and the problem becomes something else entirely. In 1890 Poincaré showed why: the three-body problem is not merely hard to solve, it is generically chaotic. Trajectories that begin arbitrarily close together separate exponentially. There is no shortcut to the future. To know where the bodies go, you have to follow them.

The three bodies on the landing page are labelled in our paper. They are latent human intent, machine interpretation, and the world. Their mutual orbit is the trajectory by which lossy intent becomes executable behavior. The trails are what has been observed. The argument of the whole page is in the geometry: you cannot certify where this system is going by looking at a snapshot of it.

The equations we integrate

The Newtonian part is the law everyone knows. Each body is pulled by the others in proportion to their mass and the inverse square of the distance between them:

That alone would be a Newtonian simulation. Ours is relativistic: we integrate the first post-Newtonian correction from the Einstein–Infeld–Hoffmann equations, the same order of correction that accounts for the precession of Mercury's orbit. Each pairwise term picks up a factor in 1/c² that depends on how fast both bodies are moving and on how deep they sit in everyone else's gravitational potential:

Two further terms follow, coupling the velocities and accounting for the fact that each body is itself falling toward every other. The consequence is visible: orbits no longer close. They precess. A path that would have repeated forever instead traces a slowly rotating rosette.

Integration is velocity-Verlet under a fourth-order Yoshida composition. We check it the way you should check anything: by removing the physics and seeing whether it reduces correctly. Setting c to a very large number should make the relativistic terms vanish, and it does, to one part in 10¹⁰.

The masses you cannot see

There are more than three masses in the scene. Two or three others emit no light at all. You can find them anyway, because the starfield behind them is bent: light passing a mass is deflected by an angle

where b is how close the light passes. Every pixel of the background is computed by casting a ray, bending it around each mass, and sampling the sky in the direction it actually came from. Where a hidden mass sits, the stars behind it smear into an arc.

You infer the hidden mass only from the distortion it causes.

That is not decoration. It is the claim of the paper, rendered: readiness is information-limited. A system cannot certify distinctions its trajectory has not extracted from the world. The dark masses are the constraints nobody wrote down, and they are bending the trajectory whether or not anyone has noticed.

The orbits that do close

Chaos is the general case, not the only one. Scattered through the three-body problem are exact periodic solutions: initial conditions so precisely balanced that the three bodies return to exactly where they began, and do it again forever. The first was found by Lagrange in 1772. The most famous, the figure eight, was proved to exist by Chenciner and Montgomery only in 2000, and has all three bodies chasing each other around one shared curve.

The landing page opens on one of these, chosen at random on each visit. They are real solutions, not drawings, and each is verified numerically before it ships: we integrate it for one full period and require it to return to its starting state to within one part in 10⁴.

Figure eight

Chenciner & Montgomery, 2000. All three bodies chase each other around a single closed curve. Period 6.33.

Moth I

Šuvakov & Dmitrašinović, 2013. Three overlapping loops that slowly precess around the centre. Period 14.89.

Moth II

Šuvakov & Dmitrašinović, 2013. Denser than Moth I, and twice as long to return to where it began. Period 28.67.

Watch it happen

Every one of these orbits is unstable. Below, two runs start from almost the same place: the grey trail is the exact periodic orbit, and the coloured one is identical except for a nudge of size ε applied at the very first instant. Same equations, same integrator, same timestep. Nothing else differs.

Drag ε up from zero. What matters is not that the two runs eventually separate, which they must. It is how long they stay indistinguishable first. For that whole stretch, every observation you could make of the coloured run says it is the periodic one.

This is what the paper calls false convergence, in its simplest possible form. The information that determines the outcome entered the system at the first instant. It simply was not observable yet.

Why this is the right picture

A stable three-body orbit is not something you find and keep. It is something you would have to hold.

There is no closed form. There is no configuration you can reach and then stop paying attention to. The only way to know where the system is going is to keep integrating it, step after step, correcting as you go, with the past no guarantee of the next interval.

We think alignment is that kind of object. Not a proof to be discovered and then relied upon, but a trajectory that has to be maintained: intent, interpretation, and world held in a relationship that nothing guarantees will persist on its own. This is why our work is about evidence and permission over time rather than a one-time certificate of safety.

Three bodies. No closed form. The harmony is not something you find. It is something you keep.