  This is the first in a series of papers presenting a new understanding of
scattering amplitudes based on fundamentally combinatorial ideas in the
kinematic space of the scattering data. We study the simplest theory of colored
scalar particles with cubic interactions, at all loop orders and to all orders
in the topological 't Hooft expansion. We find a novel formula for
loop-integrated amplitudes, with no trace of the conventional sum over Feynman
diagrams, but instead determined by a beautifully simple counting problem
attached to any order of the topological expansion. These results represent a
significant step forward in the decade-long quest to formulate the fundamental
physics of the real world in a radically new language, where the rules of
spacetime and quantum mechanics, as reflected in the principles of locality and
unitarity, are seen to emerge from deeper mathematical structures.
