Stochastic properties of an inverted pendulum on a wheel on a soft surface.

The paper asserts Theorem 3 and computes T0 = dt ∑_{n≥1} n/2^n = 2 dt by positing a 50–50 sign choice near the unstable lines, but it does not justify independence across sampling instants nor the one-step exit/invariance property that makes the residence time geometric; it even references sgn(δ(3)) (β-measurement error) rather than the friction/β̇ sign, suggesting a notation slip. The model’s solution explicitly assumes i.i.d. Rademacher signs and a one-step good/bad dichotomy, then derives the same geometric-law expectation; however, those structural assumptions are not proved from the system dynamics and are only plausible sketches. Thus both reach the same numerical result but lack a fully justified argument under clearly stated hypotheses (paper: sketchy and with a notational mismatch; model: assumes the key properties). See the paper’s statement of the layer near β̇=0 and the 2dt formula, and the relation 2/ρ·k0 A± = ±ν that underlies the attractor lines .