Kontsevich graphs act on Nambu--Poisson brackets, IV. When the invisible becomes crucial

Kontsevich's graphs allow encoding multi-vectors whose coefficients are differential-polynomial in the coefficients of a given Poisson bracket on an affine real manifold. Encoding formulas by directed graphs adapts to the class of Nambu-determinant Poisson brackets, yet the graph topology becomes dimension-specific. To inspect whether a given Kontsevich graph cocycle $γ$ acts (non)trivially -- in the second Poisson cohomology -- on the space of Nambu brackets, taking a vector field solution $\smash{\vec{X}^γ_d}$ from dimension $d$ does not work in $d+1$. For $2 \leqslant d \leqslant 4$, the action of tetrahedron $γ_3$ on Nambu brackets is known to be a Poisson coboundary, $\dot{P} = [[ P,\smash{\vec{X}^{γ_3}_d} (P)]]$. We explore which minimal (sub)sets of graphs, encoding (non)vanishing objects over $\mathbb{R}^d_{\text{aff}}$, generate the topological data that suffice for a solution $\smash{\vec{X}^{γ_3}_{d+1}}$ to appear. We detect that there can be no solution in higher dimension without invisible graphs that vanish as formulas in $d=3$, but whose descendants do not all vanish over $d=4$.