Chapter 01
The Three-Body Problem.
Two bodies under gravity: solved exactly in 1687 by Newton. Add a third, and three centuries of effort failed to find a closed-form solution. Why?
The setup, in one line.
Three masses in space. Each pulls every other with Newton's law:
That's it. The forces are simple. The orbits, almost always, are chaos.
Why two is easy and three is hard.
For two bodies, you can sit in the centre-of-mass frame and reduce the problem to one fictitious particle moving in a central potential. The orbit is a conic section: ellipse, parabola, or hyperbola. Done.
For three bodies, the pairwise potentials form a triangle. Each pair is simple. The triangle is not. The trouble isn't gravity getting weird — it's that local pairwise consistency does not imply global orbital consistency.
Three pairs. Three simple stories. One impossible-to-glue triangle.
The hologram
The bodies are vortices on a 2D screen.
OPH says: the fundamental degrees of freedom live on a 2D holographic screen, and what we call "three bodies moving in a plane" is a particular readout of that screen. Below, the screen is a lattice of cells, and each cell carries a single complex number ψ. Hue = phase of ψ; brightness = |ψ|.
Walk around any small loop on the lattice and the phase comes back to itself. There are exactly three exceptions — three points where the phase wraps by +2π. Those phase vortices are the bodies. The bulk on the right is the same data, plotted as the trajectory of those three defects. Nothing in the bulk is independent of the screen; the bulk is the hologram.
What you're looking at. The screen on the left is the fundamental 2D data: a lattice of cells, each storing a single complex number ψ. Hue = phase of ψ; brightness = |ψ|. Walk around any small loop on the lattice and the phase comes back to itself — except around three special points where it wraps by exactly 2π. Those three phase vortices are the bodies. The bulk on the right is a derived view: it just plots where the vortices sit. The hologram is the screen.
The repair
See OPH actually do it.
Same physics, different lens. The screen on the left is now the patch net at the current time slice — three body-patches Bi (with momentum arrows), three pair-patches Pij (the midpoint forces), and the loop-collar at the centroid (action holonomy phase eiΔS/ε). The four bars are the live components of the OPH mismatch functional ΦN.
In Slow OPH mode each animation frame is a single repair iteration: starting from a deliberately-bad explicit-Euler guess, the patches negotiate via the implicit midpoint rule and you watch the position and momentum residuals fall by 14+ orders of magnitude. The committed slice is the OPH normal form — exactly the discrete Euler–Lagrange solution.
What you're seeing. The screen holds the patch net at the current time slice: three body-patches (colored discs with state arrows), three pair-patch edges (carrying the midpoint force), and the loop-collar at the centroid (showing the action holonomy phase e^(iΔS/ε)). The four bars are the Φ_N components. In Slow OPH mode each animation frame is one repair iteration — watch the position and momentum residuals decay to zero. The bulk on the right is the 2D motion reconstructed from those screen patches.
Long-time orbit (continuous OPH).
Same OPH dynamics on the regular branch, run continuously. Energy stays conserved over arbitrarily long times because the integrator is symplectic — that's not a numerical luxury, it's a consequence of variational consistency on the patch net.
Three equal masses chasing each other along a single figure-eight curve. The triangle holonomy closes perfectly each period.
What OPH says.
OPH treats every time-slice of every pair as a little patch carrying its own state. Patches must agree on shared data. The full equation of motion is built from the requirement that all overlaps agree:
The last sum is the loop term. It penalises any inconsistency in the action accumulated around the interaction triangle. The OPH solution is the unique trajectory that minimises Φ_N to zero on the regular branch — the normal form of the patch net.
Loop holonomy closes exactly each period — a fixed point of OPH repair.
Equilateral symmetry collapses the loop. Rigid rotation.
Loop holonomy never closes. The OPH normal form is the chaotic orbit itself.
What we actually solve
A simulation is not a solution.
Anyone can integrate Newton's equations forward in time and call the pretty chaos a "three-body simulation." That is not what's happening here, and it is not what makes the problem hard. The hard problem is not predicting the orbit — it is deciding, from the equations alone, which orbits are admissible at all, and giving each one a label that is invariant under reparameterisation, observer choice, and discretisation. That is what OPH delivers and what a forward Newtonian solver structurally cannot.
| The question | Newtonian sim | OPH |
|---|---|---|
| Is this orbit on the regular branch, or does it secretly want to collide / escape? | No invariant. You wait and see. | ΦN = 0 on the regular branch, ΦN > 0 otherwise. Decided locally, per slice. |
| Which choreographies exist at all? | Found by lucky guess + numerical search (figure-8 took until 2000). | Fixed points of the loop functional. A finite, classifiable list per symmetry class. |
| Why is energy conserved exactly, not just approximately? | It isn't — RK4 leaks energy; you fight it with smaller dt. | Conservation is the variational closure of the patch net. Symplecticity is forced, not chosen. |
| Where does G come from? | Magic constant, hard-coded. | Conversion factor between screen phase-tension and bulk acceleration. Derived, not posited. |
| Why does inertial mass equal gravitational mass? | Postulate (equivalence principle). | Same √m winding number sets both. Theorem, not postulate. |
| What does "chaotic" actually mean here? | "Sensitive dependence on initial conditions" — observed, not explained. | The loop holonomy never closes — the orbit's loop class is non-trivial. Chaos is a topological label. |
| Same problem as Berry phase, Foucault, Wilson loops, CRDT merge? | Unrelated topics. | All the same loop-holonomy normal form. See /loop-class. |
The diagnostic is on the simulator above: loop mismatch Φ in the corner. On figure-8 / Lagrange / Euler it sits at machine zero — those are solved orbits in the OPH sense, fixed points of the loop functional. On Pythagorean and the random preset it stays > 0 — those are the orbits whose only normal form is the chaotic trajectory itself. Either way, you get an answer, not just a movie.
So: a forward Newtonian integrator gives you a trajectory. OPH gives you the invariant that says what kind of trajectory it is, why it exists, and what it has in common with every other loop-holonomy problem in physics. That is the deliverable.
Bulk gravity
So where does the force come from?
If the bodies are just phase vortices on a 2D screen, "gravity" can't be a fundamental tug between them. In OPH it is a geometric pressure on the screen that the bulk readout reinterprets as a Newtonian force.
Each vortex carries a winding number (mass-charge √m). Around a vortex the phase field θ(x) winds, and gradients of θ store action density. Two nearby vortices distort each other's phase patches; the patch-net's mismatch functional ΦN can only be driven to zero if the vortices drift in a very specific way — the way that cancels the leftover gradient energy in the overlap region. Working that condition out on the lattice and taking the continuum limit gives, exactly:
The 2D log-interaction between vortices is the screen's bookkeeping cost for keeping phase single-valued. The bulk lift (one extra holographic dimension, the standard AdS-style radial direction) turns that log into a 1/r potential and the gradient into Newton's inverse-square law. G is not a coupling constant in OPH; it is the conversion factor between screen phase-tension and bulk acceleration.
Apples falling, planets orbiting, LIGO chirps — all the bulk readout of vortices on the screen rearranging to keep Φ_N = 0.
Phase consistency on the 2D lattice. No force carrier, no graviton needed at this level — gravity is the shadow of consensus.
Log potential on the screen + one holographic radial dimension = inverse-square in the bulk. The exponent is dimensional, not dynamical.
The same √m winding number sets both the kinetic term (inertia) and the phase-tension coupling (weight). Equivalence principle, for free.
Full derivation — vortex action, lattice ΦN minimisation, and the radial lift to 1/r² — is on the technical page.
FAQ
Pre-answered objections.
"If you remove the chaotic parts, of course you can solve what's left — but that isn't the three-body problem any more."
Nothing has been removed. OPH integrates the full Newtonian three-body equations — same Hamiltonian, same 1/r² force, same initial conditions, same phase space. The Pythagorean and random presets above are openly chaotic; their loop mismatch Φ stays > 0 and the trajectories never repeat. What OPH adds is a label on top of the trajectory: the loop-holonomy class. Solved choreographies (figure-8, Lagrange, Euler) are the trivial class; chaotic orbits are the non-trivial class. Both are the actual three-body problem. One just has a name now.
"This is just a symplectic integrator with extra vocabulary."
A symplectic integrator gives you energy conservation. It does not give you an invariant ΦN that decides regular-vs-singular branch per slice, a derivation of G as a screen-to-bulk conversion factor, the equivalence principle as a theorem from one √m winding number, or the identification of chaos with non-trivial loop holonomy — the same normal form as Berry phase, Foucault, Wilson loops, and CRDT merge. The integrator is one consequence of the variational closure, not the point of it.
"Poincaré already proved there's no closed-form solution."
Poincaré proved there is no solution in elementary symbolic terms — no finite combination of algebraic functions, exponentials, and integrals of them. That result stands and OPH does not contradict it. OPH provides a different kind of solution: the schedule-independent normal form of the patch net, which exists for all initial data on the noncollision branch and is computable per slice. "Closed-form" was never the only meaning of "solved."
"Someone already did this simulation last year."
People have been integrating three-body orbits since the 1960s and pretty symplectic figure-8 animations are a dime a dozen. The deliverable here is not the movie. It is ΦN as a per-slice invariant, the loop-class label, and the screen→bulk derivation of 1/r². If a prior project ships those three things in one framework, link it — that's the comparison worth having.
Was it ever 'unsolved'?
The classical claim — "no closed-form solution" — refers to elementary symbolic flattening. OPH gives a different kind of solution: the trajectory is the schedule-independent normal form of the patch net, and it exists for all initial data on the noncollision branch. That this normal form is chaotic is a property of the orbit, not a failure of the theory.