Determine saddle multiplicities at higher genus

Determine the multiplicities of the higher‑genus (g≥2) complex Gross–Mende‑type saddles, including the tail structure and crossing‑symmetric phase dependence, and derive the corresponding high‑energy asymptotic formula for the genus‑g four‑graviton amplitude.

Background

At genus one, multiplicities were constrained using symmetry, analytic properties, and numerical data, leading to a nearly complete asymptotic formula. For g≥2, the authors expect analogous structures (e.g., elementary tails and phase patterns), but current computational tools are insufficient to fix multiplicities.

A method to compute or bootstrap these multiplicities would extend the precision asymptotics to higher genus.