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There is a remarkable connection between the boundary structure of thepositive kinematic region and branch points of integrated amplitudes in planar$\mathcal{N}=4$ SYM. A long standing question has been precisely how algebraicbranch points emerge from this picture. We use wall crossing and scatteringdiagrams to systematically study the boundary structure of the positivekinematic regions associated with MHV amplitudes. The notion of asymptoticchambers in the scattering diagram naturally explains the appearance ofalgebraic branch points. Furthermore, the scattering diagram construction alsomotivates a new coordinate system for kinematic space that rationalizes therelations between algebraic letters in the symbol alphabet. As a directapplication, we conjecture a complete list of all algebraic letters that couldappear in the symbol alphabet of the 8-point MHV amplitude.

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