I don’t think I follow the last paragraph.
This \epsilon_j + \delta_j + \zeta_j diagram, is that considering this expression at different values of j? I’m confused how it can be evaluated at S(\gamma_j), or really how it forms a diagram at all.
Also generally, to prove a diagram is everywhere ≥ than another, you need to show both that all points of the first lie ≥ the second, but also that all points of the second lie ≤ the first.