2 bodies.
That is all.
There seems to be a belief creeping into the thread that all numerical solutions of the n-body problem are based on, or are somehow limited to 2-body calculations. That is simply not the case at all. Hamiltonian splitting is one such option, but there are countless others using a wide variety of methods. There are even guides out there on building your own n-body simulator, if you're interested. Here's one:
https://medium.com/swlh/create-your-own-n-body-simulation-with-python-f417234885e9 - note that the code involves building a set of matrices with the gravitational force of each n-body acting on each of the other bodies. The simulation works in a step-wise manner - the smaller the step, the longer it takes to process, but the more accurate it is. Whilst you can't necessarily know how accurate the end result is, you can check the energy states of the system - the more constant, the better the simulation.
Here's another study:
https://lup.lub.lu.se/luur/download?func=downloadFile&recordOId=4780668&fileOId=4780676 - note they back test the models against 2 and 3 body problems with known solutions, demonstrating the accuracy of the different methods.
This stuff isn't new. The planet Neptune was actually discovered because people noticed that the known planets weren't conforming to the models at the time, meaning that something else must be in the system causing the error. So they looked at the right piece of sky at the right time and...voila...a new planet. That's pretty awesome, considering it was mid-19th century.
Watching something for several years will give you a good idea of what is going to happen. Insurance companies make a killing off of this principle. Past practice is the best indicator of future performance.
You do not need gravity to make this true which is a good thing since it doesn't exist.
You can run these models yourself and verify the accuracy if you wish (although you need major computing power to get decent results). If you are suggesting that all the cutting edge solar-system and indeed galaxy/ universal modals are actually just fudges based on observations then, once again, we have a FET proponent making an appeal to a massive conspiracy where just one dissenting voice could achieve global superstar status if they blew the cover off...but nobody ever does. Could it not just be that, actually, the models are pretty good and the solar system is actually the way we think it is, with a round earth in orbit around the sun and gravity operating as theorised and measured?
Your skepticism is a positive trait, but unless you have an alternative predictive model that outperforms what we already have and aligns with our observed world...why believe in it?
Your arguments that the mathematical difficulty of solving n-body is indicative of gravity not existing is just ludicrous - the same problems exist in countless of other scientific problems. Do you doubt aerodynamics for the same reason?