Fix the remaining multiplicity parameter in the final asymptotic formula

Determine the value of the single undetermined multiplicity parameter α in the conjectured high‑energy asymptotic expansion (equation (4.28) in the paper) for the one‑loop four‑graviton amplitude A(s,θ), which sums contributions from elementary complex Gross–Mende saddles labeled by Γ(2). Establishing α would complete the closed‑form expression for the asymptotics.

Background

The authors formulate a saddle‑sum expression for the high‑energy asymptotics in terms of contributions A_γ(s,θ) from complex Gross–Mende saddles with multiplicities M_γ(s,θ). Using symmetry and numerical data they fix almost all multiplicities, arriving at a compact crossing‑symmetric formula that depends on a single constant α.

This parameter does not affect the imaginary part or values at integer s, making it difficult to fix from available data; determining it would finalize the precise asymptotics.