= Counting lemma for octahedrally quasirandom three-uniform hypergraphs
{title2=$|t_F-\prod_ep_e|\leq |E(F)|\alpha^{1/8}$}
Let a fixed simple three-<uniform hypergraph> $F$ have $m$ <hypergraph edges>, with one host <vertex of a hypergraph> class $V_i$ for each <vertex of a hypergraph>. Assume each <hypergraph edge> triple has indicator $h_e$, <subset density> $p_e$, and balanced <three-dimensional box norm> at most $\alpha^{1/8}$. The normalized <labelled hypergraph copy count> is $\mathbb E\prod_{e\in E(F)}h_e$. Telescope this product against $\prod_ep_e$. In each error term, fix all variables outside the three <vertices of a hypergraph> of its balanced <hypergraph edge>. Every other <hypergraph edge> intersects that triple in at most two <vertices of a hypergraph>, so the remaining factors group into three pair <functions> bounded by one. The <pair-factor correlation bound for the three-dimensional box norm> bounds the error by $\alpha^{1/8}$. Summing gives
$$
\left|\#F-\left(\prod_ep_e\right)\left(\prod_i|V_i|\right)\right|\leq m\alpha^{1/8}\prod_i|V_i|.
$$
For the <three-uniform tetrahedron>, $m=4$. The same argument with a common host <set> counts homomorphisms; excluding collisions of distinct abstract <vertices of a hypergraph> changes the count by at most $\binom{|V(F)|}{2}|V|^{|V(F)|-1}$.
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