Billiard-Ball Time Loop (2D): A Self-Consistent Paradox

This 2D companion builds the classic Polchinski billiard-ball time-travel paradox as a real elastic-collision problem and solves it numerically, in the same top-down table-plane coordinates the underlying physics already lives in. A ball heading for one mouth of a wormhole is struck — just before it arrives — by a second ball that is really its own future self, already returned from an earlier trip through the wormhole. Instead of picking a trajectory by hand, the solver runs a damped fixed-point iteration on the struck-ball's velocity until the loop closes: the velocity the ball carries into the wormhole, after being deflected by the collision, must be exactly the velocity that produced the deflection in the first place. Adjust the impact geometry (δ), the wormhole's relative twist (ψ), the launch speed and the solver's own damping factor α to watch the fixed point converge differently each time; a dedicated velocity-space panel plots v₁, v₂ and the residual history directly, and the table view can be dragged and zoomed to inspect the collision geometry up close.