  We present results from an inductive algebraic approach to the systematic
construction and classification of the `lowest-level' CY3 spaces defined as
zeroes of polynomial loci associated with reflexive polyhedra, derived from
suitable vectors in complex projective spaces. These CY3 spaces may be sorted
into `chains' obtained by combining lower-dimensional projective vectors
classified previously. We analyze all the 4242 (259, 6, 1) two- (three-, four-,
five-) vector chains, which have, respectively, K3 (elliptic, line-segment,
trivial) sections, yielding 174767 (an additional 6189, 1582, 199) distinct
projective vectors that define reflexive polyhedra and thereby CY3 spaces, for
a total of 182737. These CY3 spaces span 10827 (a total of 10882) distinct
pairs of Hodge numbers h_11, h_12. Among these, we list explicitly a total of
212 projective vectors defining three-generation CY3 spaces with K3 sections,
whose characteristics we provide.
