  Motivated by potential applications to holography on space-times of positive
curvature, and by the successful twistor description of scattering amplitudes,
we propose a new dual matrix formulation of N = 4 gauge theory on S(4). The
matrix model is defined by taking the low energy limit of a holomorphic
Chern-Simons theory on CP(3|4), in the presence of a large instanton flux. The
theory comes with a choice of S(4) radius L and a parameter N controlling the
overall size of the matrices. The flat space variant of the 4D effective theory
arises by taking the large N scaling limit of the matrix model, with l_pl^2 ~
L^2 / N held fixed. Its massless spectrum contains both spin one and spin two
excitations, which we identify with gluons and gravitons. As shown in the
companion paper, the matrix model correlation functions of both these
excitations correctly reproduce the corresponding MHV scattering amplitudes. We
present evidence that the scaling limit defines a gravitational theory with a
finite Planck length. In particular we find that in the l_pl -> 0 limit, the
matrix model makes contact with the CSW rules for amplitudes of pure gauge
theory, which are uncontaminated by conformal supergravity. We also propose a
UV completion for the system by embedding the matrix model in the physical
superstring.
