I see that you conceded that numerical solutions do not fully simulate gravity. If the greatest mathematicians have not been able to get this gravity system to work, why should anyone believe that this system exists?

This is utterly preposterous I'm afraid.

Firstly, you have completely misunderstood the papers you are quoting. In this one, for example -

https://publications.mfo.de/handle/mfo/1355 - the 'standard algorithm' used to calculate the example where the moon is ejected from the three-body system is an explicit Euler solution, which is in itself a numerical method - a simplification needed to deal with the complexity of the set of PDEs. That you are holding this up as some sort of 'correct' answer and implying that the authors have tinkered with some parameters until they got to the answer they wanted shows a total failure to comprehend the information you have used in the wiki.

In the example in the paper the entire point was that, over time, implicit and explicit Euler methods do not conserve the energy of a system - there is a small error at each step that aggregates over time. This makes them poor choices for the long-term simulation of an n-body problem. The symplectic method used does conserve energy, and is therefore better in this regards, even though it is far from perfect.

Partial differential equations are extremely difficult to solve algebraically except for certain very specific cases, but they are everywhere in scientific and engineering problems at the micro and macro levels. It is impossible, for example, to solve the Navier Stokes equations for turbulent / viscous, compressible flow over an aircraft, but this is the information needed to accurately calculate lift and drag. And yet our aircraft don't fall from the sky; this is because aerodynamicists make judicious use of wind tunnels and various numerical methods, often involving vast computing power, to approximate solutions. Their inability to solve the equations by algebra does not make aviation impossible.

There's nothing wrong with not understanding this stuff - it's complex, and well into undergraduate maths / physics / engineering territory - but it might be worth getting your assertions checked over by somebody who does understand it before publishing a wiki article about it.