Show that higher‑genus dominant saddles are exactly branched covers of the sphere

Establish that, for any genus g≥2, the dominant saddle geometries in the high‑energy fixed‑angle limit of the closed‑string four‑point amplitude are precisely the (g+1)‑fold branched covers of the sphere branched at the four vertex insertions, and prove that no other saddle geometries with equal or smaller real part of the action exist.

Background

Motivated by the one‑loop analysis, the authors hypothesize that at genus g the leading saddles are obtained by lifting the tree‑level solution to a (g+1)‑fold branched cover. This would imply a universal rescaling of the on‑shell action and predictable phase structure.

However, a general proof excluding other candidate saddles at higher genus is not known, and establishing such a classification would generalize the one‑loop picture.