Vladimir Lefebvre's mathematical model of cognitive reflexion, together with applications; plus thirteen evaluations by relevant authorities. Lefebvre's chapter is an algebraic description of a special hypothetical 'processor', intrinsic to in humans, that models the self and others. The model's 'predictions' of personality constructs, culture conflicts, musicological scales and quantum models are presented, together with experimental confirmations. Most of the critics (Jack Adams-Webber, William Batchelder, Konstantin Bogatyrew, Louis Kauffman, Victorina Lefebvre, L.B. Levitin, Ernest McClain, Anatol Rapoport, Richard Sacksteder, Frederick Turner, Harvey Wheeler and Robert Zajonc) are favorable. Some offer different interpretations and propose modifications.