The double-logarithmic four-graviton Regge sector as a rank-two twisted period system
We study the double-logarithmic four-graviton Regge sector in $N$-extended supergravity. Its Mellin-space solution is already known in terms of parabolic-cylinder functions. We show that the same answer can be organized as a rank-two twisted period system, meaning that two closely related weighted integrals determine the full Mellin partial wave. These functions satisfy a simple pair of first-order differential equations and a recursion as the number of supersymmetries $N$ changes. This gives a uniform description of the full supergravity family, clarifies the relation between the positive-ray Euler integral and the earlier contour representation, and reproduces the same reduction rule through intersection theory. The reformulation also makes the special cases with four and six supersymmetries particularly transparent and yields a simple Hermite-polynomial construction for the low-even theories.