The Octonionic Grassmannian II: Exceptional Geometry, Amplituhedra, and Scattering Amplitudes

This is the second paper in a series developing the octonionic amplituhedron program; the first paper \cite{Janik25} establishes foundations including octonionic scattering equations and explicit amplitude formulas. Here we develop the theory of the \emph{octonionic amplituhedron}, a positive geometry in the 27-dimensional Albert algebra h3 (O) that generalizes the amplituhedron of Arkani-Hamed and Trnka to exceptional symmetry. The boundary structure is governed by 27 Pl"ucker-type relations arising from the adjugate $X^\sharp = 0$, whose incidence geometry is isomorphic to that of the 27 lines on a cubic surface. We establish octonionic BCFW recursion, construct BCJ numerators for $E_8$ gauge theory using the Tits bracket, and prove triality relations for amplitudes in 8-dimensional theories. The octonionic Penrose transform is defined via slice regularity, yielding solutions to the 10D Dirac equation with $E_{6(-26)}$ conformal symmetry. We show that the Freudenthal--Tits magic square is a functor from division algebras to Lie algebras, with intertwiners satisfying Yang--Baxter relations. The totally positive Albert algebra is a semialgebraic cell with Bruhat stratification indexed by the $E_6$ root poset. Connections to arithmetic geometry, exceptional theta correspondences, and M-theory compactifications are developed.