Tree-derived ideals: Fubini iterations, limit amalgamations, and Katetov obstructions

We develop a machinery for deriving ideals on a countable set from a partition of $\omega$ indexed by $\omega^{<\omega}$. A derivative operator on trace trees, parametrized by an auxiliary ideal $\mathcal J$, yields a strict transfinite hierarchy $\mathcal H^{\mathcal J}_\alpha$ of proper ideals, tall from level one onward and independent of the chosen partition. Its finite levels are exactly the Fubini powers, $\mathcal H_n\cong\mathrm{Fin}^{\otimes(n+1)}$, while $\mathcal H_\omega=\bigcup_n\mathcal H_n$ amalgamates all finite powers. We establish presentation independence, local homogeneity, Fubini recursion, and $\Pi^1_1$-completeness of the full hierarchy. Let $\mathcal F_\omega$ be Kwela's canonical inductive limit and $\mathcal F'_\omega$ his independent-partitions limit. We prove $\mathcal F_\omega\not\leq_K\mathcal H_\omega$, although $\mathcal F'_\omega\sqsubseteq\mathcal H_\omega$ and every finite coherent fragment of a putative reduction is realizable over $\mathcal H_\omega$. The proof introduces essential depth, an invariant monotone along Katetov reductions of $\mathrm{Fin}\otimes\mathrm{Fin}$, and gives the sharp non-extension bound $N(m)=m+2$. Thus $\mathcal H_\omega$ contains no isomorphic copy of $\mathcal F_\omega$, and $\mathcal F_\omega\not\leq_K\mathcal F'_\omega$. Applications include chromatic ideals $\mathcal G_k\in\mathcal H_2\setminus\mathcal H_1$ whose inclusion order records divisibility and whose Katetov order records arithmetic. We also prove the orthogonality of Cantor--Bendixson and derivative ranks. Finally, $\mathcal P(\omega)/\mathcal H$ is a $\sigma$-closed reduced power $\mathbb B\cong\mathbb B^\omega/\mathrm{Fin}$, contains $\mathcal P(\omega)/\mathrm{Fin}$ regularly, and under CH is forcing-equivalent to $(\mathcal P(\omega)/\mathrm{Fin})^+$.