Determine the underlying reason for Euclidean-region coefficient relations

Determine whether the relations among the multi-Regge limits of the coefficient functions \(A_{ijkl}\), \(A^{(1)}\), and \(A^{(2)}\) in the three-loop octagon symbol arise from a structural principle rather than being coincidental.

Background

In the appendix, the authors analyze constraints imposed by the vanishing of BDS-normalized amplitudes in Euclidean multi-Regge kinematics. These constraints relate several multi-Regge limits of coefficient functions appearing in the symbol decomposition of the three-loop octagon remainder function.

At three loops, the authors find additional equalities and vanishing conditions among these coefficient functions beyond those already implied at lower loop order. The relations are observed in the explicit symbol data, but the paper does not derive a general explanation for them or establish whether they follow from a deeper property of the amplitude, the Steinmann constraints, symbol integrability, or another structural principle.