What the paper is about: The authors show that a combination of maximal supersymmetry and a special six-particle consistency condition forces very strong constraints on four-particle scattering in low-energy effective field theories. Working to all orders in the low-energy expansion, they derive closed “master equations” that relate products of four-point amplitudes. These results explain and prove earlier conjectures that the four-point answers take a simple exponential form familiar from string theory, and they clarify when those forms uniquely pick out the classic string amplitudes: the Veneziano amplitude in gauge theory and the Virasoro–Shapiro amplitude in gravity.

What the researchers did: The key move is to study the six-point amplitude in a special collinear limit where two external momenta coincide and several three-particle factorization channels simultaneously go “on shell” (that is, the internal propagators behave as if they carry real particle momenta). In this limit, a scalar-parity projection and a single discrete R-symmetry remove an otherwise unknown regular six-point remainder. What remains are finite contributions determined entirely by products of four-point amplitudes. Equating the two scalar projections produces a closed three-term master equation. The authors derive these equations for maximally supersymmetric Yang–Mills theory (planar, color-ordered) and for maximal supergravity, and then solve them to all orders in the low-energy expansion.

How the method works at a high level: Factorization means that residues of poles in an amplitude are products of lower-point amplitudes. By arranging a limit in which several such poles contribute at once, the unknown six-point pieces drop out after the symmetry projection, leaving algebraic relations among four-point building blocks. Solving those relations forces an exponential structure for the four-point functions. The paper also shows how to reconstruct the full amplitude from its forward limit (the kinematic configuration where the scattering is along the original direction) by interpreting dispersive sum rules as moments of a positive spectral measure.