= 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 \b[$\alpha$-quasirandom] when
$$
\boxed{\|g\|_{\square^3}^8=\mathbb E_{x_0,x_1,y_0,y_1,z_0,z_1}\prod_{i,j,k\in\{0,1\}}g(x_i,y_j,z_k)\leq\alpha.}
$$
The 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 equals
$$
\mathbb E_{x_0,x_1,y_0,y_1}\left(\mathbb E_z\prod_{i,j\in\{0,1\}}g(x_i,y_j,z)\right)^2.
$$
Thus 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.
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