Octonionic Twistor Theory and Exceptional Intertwiners: A Generalization of Penrose's Program to $E_6$

We develop a generalization of Penrose’s $SU(2,2)$ twistor program to the exceptional Lie
group $E_6$ using the Albert algebra J3 (O) as the fundamental twistor space. The key math-
ematical objects are the octonionic intertwiners: equivariant maps that mediate between
twistor space and spacetime representations. We identify the cubic determinant as the fun-
damental three-vertex intertwiner, construct the symplectic propagator on the Freudenthal
triple system, and address the technical challenges posed by octonionic non-associativity
through slice-regular function theory. This framework provides a natural setting for excep-
tional geometric structures with potential applications to higher-dimensional physics and
scattering amplitudes.