Uniform Turán densities of $k$-uniform hypergraphs

For $k\ge 3$, the $(k-2)$-uniform Turán density ${\pi }_{k-2}(F)$ of a $k$-graph $F$ is the supremum of $d$ for which there are arbitrarily large $F$-free $k$-graphs that are uniformly $d$-dense with respect to the $k$-vertex cliques of every $(k-2)$-graph on the same vertex set. We develop a \emph{palette framework} for this density. For every family $\mathcal{F}$ of $k$-graphs, we prove that ${\pi }_{k-2}(\mathcal{F})$ equals the corresponding palette Turán density. We further establish palette classification tools for the existence of $k$-graphs satisfying prescribed palette colorability constraints. Those together allow us to reduce exact density computations to a palette-homomorphism framework without relying on the hypergraph regularity method. As applications, for all $k\ge 3$ and $r\ge 2$, we establish the following values $$\frac{r-1}{r},\frac{(r-1{)}^2}{r^2},\frac{r-1}{2r},\frac{(k-1{)}^k}{k^k},\frac{4(k-2{)}^{k-2}}{k^k},\frac{4(k-2{)}^{k-2}}{3k^k}$$ as $(k-2)$-uniform Turán densities of single $k$-graphs. Finally, for every $k\ge 3$, we show that there exist $k$-graphs $F_1,F_2$ such that $${\pi }_{k-2}(\{F_1,F_2\})<\min \{{\pi }_{k-2}(F_1),{\pi }_{k-2}(F_2)\},$$ which provides the first examples of \emph{non-principal} families for this density.