Switching Dynamics of Shallow Arches

Three of the four regimes match and are correct: the paper’s time‑independent overdamped (its Eq. 22) and undamped (Eq. 26) formulas, and the overdamped ramp result via the first Airy zero 2.3381 (Eqs. 39–40), all agree with the candidate solution and with standard derivations . In the undamped ramp case, the paper asserts τ∞ = τc + 3.22 (K v)^{-1/5} (Eqs. 47–48) after an approximation that reduces Painlevé I to an elliptic-integrable surrogate; this constant conflicts with the rigorous tritronquée constant s0 ≈ 3.409 in the canonical u'' = u^2 + s normalization. The candidate solution correctly identifies the Painlevé I/tritronquée mechanism but incorrectly claims the paper’s 3.22 is a mere normalization variant of 3.409. Hence, the paper’s derivation is incomplete for the undamped ramp, and the model’s reconciliation there is also incomplete .