Rolling systems and their billiard limits

The paper’s main theorem states that, as the rolling ball’s radius r→0, the edge-crossing map preserves ΠxSΠx, flips the sheet-normal n(x), and acts on (ū, W) by the 2×2 block [ [cos(πη), sin(πη)], [sin(πη), −cos(πη)] ], together with the parameter identification γr/√(1+γr^2) = (1/π) arccos((1−γb^2)/(1+γb^2)) (Theorem 1) . The paper derives this by reducing the motion near the edge to a constant-coefficient planar system (Example 7/9) and proving convergence via a time-rescaling and homothety argument in Section 6, yielding the same rotation by angle πη across the rounded meridian and the same sign flip for n(x) (Equation (6), Theorem 19) . The candidate solution reproduces this structure: it isolates the 2D normal plane, writes the reduced ODE Z′ = ηJZ in the edge angle θ, integrates to obtain R(πη), then accounts for the sheet-normal flip to produce the identical 2×2 block; it also matches the no-slip billiard block (Equation (3)) and recovers the same parameter relation . The differences are stylistic (θ-parameterization versus the paper’s τ-rescaling), not substantive; both proofs hinge on the same harmonic-oscillator/rotation mechanism on the meridian and the bounded-curvature error control in tangential directions.