  A new efficient approach to the analysis of nonlinear higher-spin equations,
that treats democratically auxiliary spinor variables $Z_A$ and integration
homotopy parameters in the non-linear vertices of the higher-spin theory, is
developed. Being most general, the proposed approach is the same time far
simpler than those available so far. In particular, it is free from the
necessity to use the Schouten identity. Remarkably, the problem of
reconstruction of higher-spin vertices is mapped to certain polyhedra
cohomology in terms of homotopy parameters themselves. The new scheme provides
a powerful tool for the study of higher-order corrections in higher-spin theory
and, in particular, its spin-locality. It is illustrated by the analysis of the
lower order vertices, reproducing not only the results obtained previously by
the shifted homotopy approach but also projectively-compact vertices with the
minimal number of derivatives, that were so far unreachable within that scheme.
