Dynamics of the no-slip Galton board

Both the paper and the model correctly identify the path-reversing 2-periodic criterion γ = (v/ẋ0)(sin θ cos φ − cos θ sin φ) and give coherent proofs (paper via the β-parameter rotation map; model via a −1 eigenvector condition of the no-slip collision matrix). However, both sources misstate the kinematic “range” relation for the required planar speed v: the paper asserts v = 2√(g d) sin 2θ (its Eq. (3)), while the standard constant-acceleration calculation at equal heights gives d = (v^2/g) sin 2θ and hence v = √(g d / sin 2θ). The model reproduces this incorrect closed form as an “equivalent” expression. Thus, the core no-slip criterion is right, but the stated v(g,d,θ) relation needs correction.