This technical note presents a minimal convex stress test demonstrating that global training time in multi-channel optimization systems is protocol-dependent rather than intrinsic. We consider a strictly convex three-channel quadratic system with a unique global minimum shared across all interface scheduling protocols. Despite identical objective function and minimizer, convergence time varies by up to an order of magnitude depending on interface hierarchy and activation sequence. In certain cyclic hierarchical regimes, global closure fails entirely within finite horizon despite convexity and uniqueness of the solution. We introduce a diagnostic perspective in which temporal metrics (such as time-to-hit and integrated excess loss) are interpreted as closure-dependent functionals rather than intrinsic scalars of the objective landscape. The results suggest that convergence time in multi-interface systems must be evaluated relative to interface hierarchy, even in strictly convex settings.