= Tetrahedron counting lemma
A sufficiently regular four-partite three-<uniform hypergraph>, supported on sufficiently regular pair cells of positive density and with positive relative triple densities, has a positive fourth-power number of <three-uniform tetrahedra>. On complete pair supports, telescoping the four edge functions shows that an error of at most $\xi$ in each <three-dimensional box norm> gives a counting error at most $4\xi$. The relative version additionally requires control of the pair-support densities.
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