Counting perpendicular cylinder-core intersections on the Ward surface S(3,4)

On the Ward surface S(3,4), formed by a regular octagon and two squares, we count intersections of horizontal and vertical closed trajectories in the original polygon coordinates. Choosing one trajectory in each cylinder gives ten crossings and six disk regions. The middle horizontal and vertical trajectories meet twice, and each of the other eight pairs meets once. We derive these counts from cylinder widths and check them by identifying the edge midpoints in the polygon picture. The seven-edge Hooper diagram for the same surface uses a different pair of directions. This expository preprint specializes the published geometry of the Ward family. Historical priority for the numerical table remains open.