Establish the asymptotic quartic repulsion for HTT amplitudes

Prove quartic peak repulsion for the tree-level three-point amplitude involving one highly excited string and two tachyons in the simultaneous limit of string excitation level N tending to infinity and peak spacing s tending to zero.

Background

The paper analytically derives a quartic small-spacing law for sufficiently smooth amplitude ensembles and numerically observes an exponent close to four for highly excited string amplitudes at large but finite excitation level N. However, the numerical calculations use finite-N contact windows and do not establish the true joint asymptotic limit. A rigorous result would require controlling the distribution of the relevant amplitude derivatives after conditioning the occupation numbers on a partition of N, thereby making the order of the limits N → ∞ and s → 0 uniform.