Algebraic branch points at all loop orders from positive kinematics and wall crossing

There is a remarkable connection between the boundary structure of positive kinematic region and branch points integrated amplitudes in planar $\mathcal{N}=4$ SYM. A long standing question has been precisely how algebraic emerge from this picture. We use wall crossing scattering diagrams to systematically study regions associated with MHV amplitudes. The notion asymptotic chambers diagram naturally explains appearance points. Furthermore, construction also motivates new coordinate system for space that rationalizes relations letters symbol alphabet. As direct application, we conjecture complete list all could appear alphabet 8-point amplitude.