Both M4 and CP2 are highly unique in that they allow twistor structure and in TGD one can overcome the fundamental "googly" problem of the standard twistor program preventing twistorialization in general space-time metric by lifting twistorialization to the level of the imbedding space containg M4 as a Cartesian factor. Also CP2 allows twistor space identifiable as flag manifold SU(3)/U(1)× U(1) as the self-duality of Weyl tensor indeed suggests. This provides an additional "must" in favor of sub-manifold gravity in M4× CP2. Both octonionic interpretation of M8 and triality possible in dimension 8 play a crucial role in the proposed twistorialization of H=M4× CP2. It also turns out that M4× CP2 allows a natural twistorialization respecting Cartesian product: this is far from obvious since it means that one considers space-like geodesics of H with light-like M4 projection as basic objects. p-Adic mass calculations however require tachyonic ground states and in generalized Feynman diagrams fermions propagate as massless particles in M4 sense. Furthermore, light-like H-geodesics lead to non-compact candidates for the twistor space of H. Hence the twistor space would be 12-dimensional manifold CP3× SU(3)/U(1)× U(1).
Generalisation of 2-D conformal invariance extending to infinite-D variant of Yangian symmetry; light-like 3-surfaces as basic objects of TGD Universe and as generalised light-like geodesics; light-likeness condition for momentum generalized to the infinite-dimensional context via super-conformal algebras. These are the facts inspiring the question whether also the "world of classical worlds" (WCW) could allow twistorialization. It turns out that center of mass degrees of freedom (imbedding space) allow natural twistorialization: twistor space for M4× CP2 serves as moduli space for choice of quantization axes in Super Virasoro conditions. Contrary to the original optimistic expectations it turns out that although the analog of incidence relations holds true for Kac-Moody algebra, twistorialization in vibrational degrees of freedom does not look like a good idea since incidence relations force an effective reduction of vibrational degrees of freedom to four. The Grassmannian formalism for scattering amplitudes generalizes practically as such for generalized Feynman diagrams.