  In Part I of the present series of papers, we adumbrate our idea of
Riemannian geometry to higher order in the infinitesimals and derive
expressions for the appropriate generalizations of parallel transport and the
Riemannian curvature tensor. In Part II, the implications of higher-order
geometry for the general theory of relativity beyond Einstein are developed. In
the present Part III, we expand on the framework of Part II so as to take up
the problem of field-theoretical unification. Employing the form of the
Einstein-Hilbert action to higher order as proposed in Part II, we show how in
nearly flat space the higher-order terms give rise to a gauge theory of
Yang-Mills type. At the 2-jet level, the electroweak force emerges after
imposition of gauge fixing. In fact, the proposed form of the Einstein-Hilbert
action permits us to say more: we argue that the equivalence principle results
in a Proca term that brings about the spontaneous symmetry breaking of the
standard model. Two empirical predictions support our reasoning: first, we
obtain a theoretical value for the Weinberg angle and second, we find that --
without any adjustable parameters -- the implied value of Coulomb's constant
agrees well with experiment. The final section examines the 3-jet level. The
same mechanism that produces Glashow-Weinberg-Salam electroweak theory at the
2-jet level eventuates in a chromodynamical force having $SU(3)$ symmetry at
the 3-jet level.
