ON THE OPEN TODA CHAIN WITH EXTERNAL FORCING

The paper proves for c>0 that endpoint interactions decay super-Gaussianly, the inner core exhibits exponential relaxation to ballistic motion, and that wave operators W# and Ŵ# exist; composing them yields an intertwining map W transporting integrals, from which Liouville integrability, a Lax pair, and action–angle variables follow. These are established with explicit bounds, including e^{q1−q2}, e^{q_{N−1}−qN} = O(e^{−γ t^2}) and pn(t) = pn,∞ + O(e^{−μ t}), qn(t) = qn,∞ + t pn,∞ + O(e^{−μ t}) for 2 ≤ n ≤ N−1, and with detailed construction of wave operators and the intertwining relation Û_t ∘ W = W ∘ U_t . The candidate solution mirrors this structure: (A) the same long-time asymptotics with super-Gaussian boundary decoupling and exponential core convergence, (B) existence of W#, Ŵ# and the intertwining map W, and (C) transporting N integrals and a Lax pair/action–angle variables from the decoupled model. Minor differences are cosmetic: the paper proves exponential rates via a tailored S(t) argument in Flaschka variables, while the model sketches a variation-of-constants viewpoint and slightly overstates rate control by spectral gaps; the paper also notes technical reasons for not constructing W directly as lim_{t→∞} Û_{−t}∘U_t. Overall, both are correct and closely aligned in method and conclusions .