It's out! The paper of the century---get your copy before they're sold out at the newsstand. (If you're into N-body simulation methods, that is.)
Seriously: I'm very glad to have it out the door. The ball is now in ApJ's court, and I can get back to focusing on my thesis (for which I just submitted a title: "Numerical Relativity from a Gauge Theory Perspective").
Section 6.2.1 uses the OCaml code I posted about earlier for automatic differentiation to show that the integrator algorithm is as symplectic as we claim. This is something I'd like to see more physicists take up---particularly in the cosmological N-body simulations, where preserving the "coldness" of the dark matter phase-space distribution is very important. Everyone uses the leapfrog algorithm (for now---it might turn out that our algorithm is better) because it's symplectic at constant timesteps (and, though we're the first to realize it, with block-power-of-two timesteps---see Section 3, and numerical experiments in Sections 6.1 and 6.2.1), but they never measure whether this is really the case with their algorithms. If they were in the habit of doing this, I'm sure someone before us would have realized that block-power-of-two steps preserve symplecticity while ordinary adaptive steps break it (rather than assuming that any non-constant timestep breaks it).
Anyway, I'm excited to see what the reception of the physics community will be. I've got my fingers crossed....