Pensive billiards, point vortices, and the silver ratio

The paper explicitly proves that, in the ε→0 dipole limit, the interior-to-interior map is a pensive billiard with delay ℓ(p) = L − pL/√(1+p^2) for p = cos θ, using the half-plane fission speeds v± = v(√(1+cos^2 θ) ∓ cos θ), near-boundary constant-speed sliding, and the meeting-distance formula 2L·v+/(v++v−) (yielding L(1 − cos θ/√(1+cos^2 θ))) . The specular (angle-preserving) recombination follows by running the half-plane dynamics backward via the fusion rule, ensuring the same θ on exit , consistent with the paper’s narrative that the dipole “reflects from the boundary at the same angle” . The candidate solution reproduces these same steps and formulas (fission speeds, constant sliding, meeting-distance, and angle invariance), so both are correct and essentially the same argument.