Wow.

The closest analogue I have to that is grabbing planet positions and velocities from JPL's HORIZONS system, then doing small time steps holding accelerations constant.

That's how I know the (mathematical) solar system behaves as claimed. Except that Mercury's orbit will eventually become so elliptical and gain so much energy that it careens in and out of the solar system until it flies off to infinity (or people are also right about the limitations of the approximation technique I was using).