Classify all leading one‑loop saddles with minimal action

Prove that every saddle point of the one‑loop four‑graviton amplitude in type II string theory whose real part of the on‑shell action is minimal is a "half‑period" configuration in which the puncture positions are exactly the half‑periods of the torus, namely {z1,z2,z3,z4} = {0, 1/2, τ/2, (τ+1)/2} and {\bar{z}1,\bar{z}2,\bar{z}3,\bar{z}4} = {0, 1/2, \bar{τ}/2, (\bar{τ}+1)/2}, with the modular parameters satisfying λ(τ) = λ(−\bar{τ}) = −s/t. Establishing this would complete the classification of leading saddles used to derive the high‑energy asymptotics.

Background

In the high‑energy fixed‑angle limit the one‑loop integrand localizes on saddle points. Numerical searches revealed many saddles, but the authors observed that those with the smallest real part of the action appear to have all vertex operator insertions at half‑periods of the torus. Under this ansatz the saddle equations simplify to λ(τ)=λ(−\bar{τ})=−s/t, enabling a tractable analysis of complex Gross–Mende saddles.

A proof that all leading saddles are of this half‑period type would justify the restricted bootstrap for multiplicities and underpin the proposed asymptotic formula.