  The Leibniz rule for derivations is invariant under cyclic permutations of
co-multiples within the arguments of derivations. We explore the implications
of this principle: in effect, we construct a class of noncommutative bundles in
which the sheaves of algebras of walks along a tesselated affine manifold form
the base, whereas the fibres are free associative algebras or, at a later
stage, such algebras quotients over the linear relation of equivalence under
cyclic shifts. The calculus of variations is developed on the infinite jet
spaces over such noncommutative bundles.
  In the frames of such field-theoretic extension of the Kontsevich formal
noncommutative symplectic (super)geometry, we prove the main properties of the
Batalin--Vilkovisky Laplacian and Schouten bracket. We show as by-product that
the structures which arise in the classical variational Poisson geometry of
infinite-dimensional integrable systems do actually not refer to the graded
commutativity assumption.
