Observer · Patch · Holography
The three-body
problem,
glued.
Three centuries of "no closed-form solution" assume the wrong meaning of solve. OPH supplies the other one: the underlying principle that makes the problem the shape it is, and turns every hard part into a consequence.
What "solve" means here
Not a formula. A principle.
The weak meaning of solve is to write a closed expression that returns xyz at time t. The strong meaning is to uncover the underlying principle and watch the hard parts fall out as theorems. Newton did the strong version for orbits. Maxwell did it for E&M. OPH does it for the three-body problem and for a dozen problems nobody thought were related.
3-body chaos and Aharonov–Bohm are the same shape.
Berry phase, Foucault, Wilson loops, gauge anomalies, CRDT merge, the 3-body triangle. All of them are loop-holonomy obstructions on a patch net. Different mathematical burdens, same normal form. Solving the shape solves the shape for all of them.
G, 1/r², and the equivalence principle stop being magic.
Newton postulated G. Einstein postulated m_inertial = m_gravitational. OPH derives both from a single √m winding number plus the screen-to-bulk lift. Integrators consume G. OPH produces G.
No energy drift to fight, because none is being created.
RK4 leaks energy and you fight it with smaller dt. OPH has no leak to fight. Exact energy conservation and symplecticity are forced by variational closure of the patch net. They are theorems about the principle, not virtues of a scheme.
Φ_N tells you, per slice, what kind of orbit you are on.
Under Newton alone, 'regular branch or secretly want to collide?' has no local answer. You integrate and wait. OPH gives a per-slice invariant Φ_N and a loop-class label for every trajectory. Choreographies are fixed points of the loop functional, not lucky discoveries.
This is not a faster integrator and it is not a closed formula for chaos. Both would be weak claims and neither is what the problem needs. The problem needs a principle.
Three equal masses, one figure-eight, one closed loop holonomy. Discovered by Chenciner and Montgomery in 2000. Inevitable in OPH terms.
The claim
Local pairwise consistency
does not imply global orbital consistency.
Each pair of bodies feels a simple, exactly-solvable force. The triangle of three pairs does not glue. That gluing failure is what every "unsolvability" theorem about the three-body problem actually measures. OPH names it, quantifies it, and zeros it.
At each cutoff, the patch-net normal form exists and is schedule-independent on the noncollision branch. The refinement limit recovers the Newtonian trajectory, illustrated above for a regular (figure-eight) branch.
For everyone
It is about gluing.
Picture three friends each describing a triangle by saying how to walk between two corners. Locally, every instruction makes sense. Walk all three and you sometimes do not end up where you started. The mismatch is real. Physics calls it a holonomy. OPH says: what is physically real is exactly the data that survives the gluing.
For physicists
It is a normal form.
On a finite patch net, accepted repairs strictly lower a Lyapunov functional. Termination follows on finite state spaces. Under local confluence the normal form is schedule-independent. The 3-body trajectory is the refinement limit of that normal form.Full proofs →
The bigger pattern
The same trick, elsewhere.
The three-body problem is the centerpiece. It is one member of a wider class. Anything where local data is well-defined but loop holonomy is not falls to the same OPH move. The following pages cover the rest.