$1-8/\kappa$ Lower Bound for Inverted-Potential Jacobi Flow

Conservative-flow and fixed-energy optimizers follow the same programme, namely, they intervene after running a conservative flow. When the interventions are coordinated across segments using Chebyshev-root or randomized integration times, the resulting schemes provably accelerate. What no published work provides for such a flow is a worst-case, stopping-rule-independent bound on the value of a single segment, that is, a number that tells one whan the flow contributes and what the coordination contributes. We report this for the case of the inverted-potential shell flow initialized at the negative gradient on strongly convex quadratics. We show that for $\kappa\geq8$ there is a two-dimensional instance and initial point at which no segment reduces the objective gap by $1-8/\kappa$ of its initial value. The reported bound is a property of the unparameterized oriented orbit, rather than of any clock. Compared to Kantorovich's 1948 bound for one line search, the optimally stopped segment is asymptotically no better then and less at every tested $\kappa$.