Acceptance != proven. There were points in history where people accepted the existence of witches too.

So firstly, no scientific theory is ever proven in the strictest sense. The best we can say is that Einstein's theories work better than Newton's in certain situations. And by "work better" I mean they make predictions which more accurately match observations. For most normal situations there is no measurable difference.

You get hung up on the 3 body problem a lot and I don't understand what your issue is.

You're right in that there is no solution. There are no equations which have a variable "t" for time, you plug in the initial positions of the bodies, you put in a value for t and the equations give you the results in terms of where all the bodies will be 't' seconds into the future. I think if anything this is a failure of mathematics, or maybe it's an inherent problem with chaotic systems (chaotic in the mathematical sense). The weather is chaotic too - or our models of it are. That's why weather forecasts are often wrong and there is no such things as a truly long range forecast.

The best we can do is split the n-body problem into a series of 2 body problems. Solve those for a small time increment and then iterate. So yes, it's a bit of a workaround but the key here is

*it does work*. It's got us to the moon, it's put craft on Mars. I don't think any of these models are simply using cycles, they're using Newton's equations to work out from initial positions and velocities what the next positions and velocities will be in small time increments and iterating. But because of this iteration if the initial values are incorrect - in the sense that they are not perfect, which they can't be - then over time those errors will build up.

So yes, this is a weakness of our models, BUT...a model does not have to be perfect to be useful. Someone else said that the value of pi cannot be used perfectly because it's an irrational number. But we know the value well enough to use it to make accurate calculations. If we used the value 3 then that wouldn't be good enough for most purposes, 3.14159265 (which is as much as I know off the top of my head) is good enough for pretty much any purpose.

The headline is while yes, we don't have "a solution" to the n body problem I don't see why you think that is a problem. We have ways of calculating it well enough to send craft into space and land them on other celestial bodies. I'd say that's a triumph of our models, not a failure.