Disproved, by an elementary counterexample to the statement as printed. The counterexample is the site's, due to Harris, recomputed in the Current assessment: the $3$-uniform hypergraph on $\{1,\dots,9\}$ whose $28$ edges are the $27$ triples with one element in each of $\{1,2,3\}$, $\{4,5,6\}$, $\{7,8,9\}$ and the triple $\{1,2,3\}$ has $3^3+1$ edges on $3\cdot3$ vertices, and no four of its vertices span three edges (the check is in the Current assessment), so the statement fails at $n=3$. The same construction (the complete 3-partite 3-graph on $3n$ vertices plus one triple inside a class) fails it for every $n\ge3$ by the same check, a class needing three vertices to hold the extra triple. The frontmatter standing is derived from the accepted claim page Harris's counterexample, whose acceptance evidence is the site's own and which also carries the links to the Lean checks of the example; a pending claim page, Aristotle's Lean proof published by Alexeev, records an independent Lean proof of the construction for every $n\ge3$. The site's label DISPROVED (LEAN) carries a catalog suffix explained under Formalization: the counterexample has been checked in Lean in external files the corpus has not built.