  "Nonconstant cohomologies" are introduced for solutions of set-theoretical
four-simplex equation (FSE). While usual cohomologies lead to solutions of
constant quantum FSE, our "nonconstant cohomologies" lead to solutions of
nonconstant quantum FSE. Computer calculations are presented showing that large
spaces of such cohomologies exist for all Hietarinta's two-color linear
solutions to set-theoretical FSE. After taking a partial trace of the
corresponding quantum operators, combined with one additional trick, this leads
to solutions of tetrahedron equation, including those with non-negative matrix
elements, and not reducible to a permutation, even with cocycle multipliers.
