  The geometric approach [1312.1262] to iterated variations of local
functionals -- e.g., of the (master-)action functional -- resulted in an
extension of the deformation quantisation technique to the set-up of Poisson
models of field theory [IHES/M/15/13]. It also allowed of a rigorous proof
([1312.1262],[1210.0726]) for the main inter-relations between the
Batalin-Vilkovisky (BV) Laplacian $\Delta$ and variational Schouten bracket.
The ad hoc use of these relations had been a known analytic difficulty in the
BV-formalism for quantisation of gauge systems; now achieved, the proof does
actually not require the assumption of graded-commutativity [1210.0726].
Explained in our previous work, geometry's self-regularisation is rendered by
Gel'fand's calculus of singular linear integral operators supported on the
diagonal.
  We now illustrate that analytic technique by inspecting the validity
mechanism [1312.4140] for the graded Jacobi identity which the variational
Schouten bracket does satisfy (whence $\Delta^2=0$, i.e., the BV-Laplacian is a
differential acting in the algebra of local functionals). By using one tuple of
three variational multi-vectors twice, we contrast the new logic of iterated
variations -- when the right-hand side of Jacobi's identity vanishes altogether
-- with the old method: interlacing its steps and stops, it could produce some
non-zero representative of the trivial class in the top-degree horizontal
cohomology. But we then show at once by an elementary counterexample why, in
the frames of the old approach that did not rely on Gel'fand's calculus, the
BV-Laplacian failed to be a graded derivation of the variational Schouten
bracket.
  Keywords: Variational multi-vectors, Schouten bracket, Jacobi identity,
Batalin-Vilkovisky Laplacian, symbolic computations.
  PACS: 02.40.-k, 11.10.-z; also 02.30.Ik, 02.30.Jr, 11.15.-q, 11.30.-j
