Why do you think having it fit into a geometric system implies it is correct?
Informally, I don't think that two theorys that vastly differ can give the exact same results, but I haven't got any proof of that.
A more accurate thing to say would be that "it is necessary for a correct theory of the planets to have a geometric model". In the flat earth model the planets follow some path above the earth, so using the shape of the path of the object you should be able to calculate where the object is viewable in the sky.
Now proving that the FET doesn't have a geometric model is hard to do (but I might of found a way to do it, I just need to find a date when three gps satellites form an equilateral triangle), but the fact that no one has found one yet hints that something is up.
Why do you think simplicity implies truth?
Because it's easy to make a false theory that predicts events correctly by adding on ad hoc axioms, and adding them on makes it more complex. Yeah, it's not a rigous way of judging truth, it's more of a hint. Take the Ptolemaic system, it's complexity doesn't disprove it, it just hints that it is wrong, it was finding that Venus has phases and it is full when it looks smaller that disproved it.
Why do you think not having a geometric model shows it is incomplete?
Incomplete means something is missing, in this case it would be the geometric model. If the theory is correct there will be a geometic model, how could there not be.
How do you differentiate a "theory" and an "idea"?
A theory (in the physical sense) accurately describes an element of reality, how it works and how to predict what it will do.
The speed of light being constant is an idea, but special relativity is theory.
So far the FES has described a model of how the earth, planets and stars work. It kind of says how things work, but has got nowhere on the predicting front. So it is more than just an idea, but it's lacking an important element of what I think is a theory. At the end of the day a theory isn't a rigously defined term.
On a seperate issue.
Well, we already know no model can explain every occurring phenomena as it must necessity must contain a Godel Sentence.
I don't think godel's first incompleteness theorem applys to a model of reality. One requiment is that you need to be able to be able to incode all statments about the model within the model itself. If the universe is bounded, you won't be able to incode every statment, there will be a limit.
If you did have an infinate amount of mass to use, then godel's incompleteness theorems would apply, but that's likely because using an infinate amount of mass you can create a turing machine, so you can run a algorithm that checks if a goodstein sequence halts. Basically math existing within the universe would cause the incompleteness.