 Solution

Use octahedral quasirandomness of a three-uniform hypergraph, with its eighth-power normalization. Let $h:X\times Y\times Z\to \{0,1\}$ be the hypergraph edge indicator function, define $p={\mathbb{E}}_{x,y,z}h(x,y,z)$, and put $g=h-p$. All expectations are uniform and independent on the indicated finite sets. The hypergraph is $\alpha$-quasirandom whenThe eight triples form the faces of the tripartite octahedral configuration. Repeated choices within a part are included in this normalized average. For real $g$, the expression is nonnegative, since it equalsThus the condition says that the balanced hypergraph edge indicator function has three-dimensional box norm at most ${\alpha }^{1/8}$. If the norm itself is used as the quasirandomness parameter instead, the eighth-power parameter here is the eighth power of that parameter; stating this convention fixes the quantitative meaning of the counting error.