Equivalences of 5-dimensional CR-manifolds V: Six initial frames and coframes; Explicitness obstacles

Abstract: Local CR-generic submanifolds of C^N are in one-to-one correspondence with their respective graphing functions, but it is well known that (despite their importance) the Cartan-Hachtroudi-Chern-Moser invariants and coframes for Levi nondegenerate hypersurfaces M in C^n+1 have been fully computed in CR dimension n >= 2 only in special cases which show off a tremendous collapse of computational complexity in comparison to the general case. One of the goals of this Part V is to set up systematic initial data that are essentially explicit in terms of the concerned graphing functions, for the six already studied general classes I, II, III-1, III-2, IV-1, IV-2. Incredibly, for Class III-1 CR-generic submanifolds M^5 in C^4 that are the geometry-preserving deformations of one of the natural models of Beloshapka, even the initial frame and coframe are not absorbable by an individual personal computer, for some of the concerned coefficient-functions incorporate nearly 100 000 000 of monomials in 165 jet variables, not to mention that the exploration of biholomorphic equivalences yet requires to differentiate such functions at least four times. As will appear later on, deep (archaic) mathematical links are extant between the effective Cartan theory and the famous hyperbolicity conjecture of Kobayashi.