The bottom of this potential energy is the region satisfying sin x -0.25 ≤ y ≤ sin x + 0.25 (22) which is colored white in Figure . Under this potential energy, we optimized a path with fixed endpoints at (x, y) = (0, 0) and (2π, 0) using the direct MaxFlux method. The initial path was a straight line connecting the two endpoints (dashed line). The obtained optimal path (blue curve) closely approximates the shortest path passing through the flat bottom of the potential energy (orange curve). Although there is a slight deviation between the optimal and shortest paths due to the finite number of basis functions used to expand the path, it is important to note that the optimal path passes through the interior of the bottom. This is because, when β is large, minimizing the objective function in eq 18 prioritizes reducing the exponential term over shortening the path.