Distortion of countably branching diamonds from midpoint and tree energies

We derive quantitative distortion bounds for countably branching diamond graphs from midpoint estimates and direct tree energies. In the dual of a finite-height segment forest, every distortion-D embedding of the depth-k diamond satisfies $D^2\ge 1+k/4$. The same bound holds in the infinite-height coordinate predual. We also obtain power-type distortion bounds for path-cost and recursive tree norms.