This is a glue proof: it assumes the four preceding statements of the submission and does nothing but compose them, so the proof network is visible on the archive rather than buried inside a single derivation.

Fix $r$ and $ε$, and set $ε₁ = ε / (3r² + 1)$. Subpolynomial density, applied to the closure of the class under subgraph copies — which is nowhere dense whenever the class is — gives a constant $c$ bounding the edges of every depth-$r$ minor of a subgraph $H$ of a member by $c · k^(1+ε₁)$ in its own vertex count $k$. Since a shallow minor has no more vertices than its host, $k ≤ m$, so that bound rewrites as the per-graph density bound $d = ⌈c · m^ε₁⌉$ for $H$: $k^(1+ε₁) = k · k^ε₁ ≤ k · m^ε₁$. A depth-$r$ topological minor is in particular a depth-$r$ minor, so $d$ bounds the topological density as well, and the admissibility bound yields $adm_{r+1}(H) ≤ 1 + 6(r+1)d³$; admissibility is monotone in the radius, so the same bound holds at radius $r$. The two coloring-number links then give $wcol_r(H) ≤ 1 + r·(scol_r(H) - 1)^r ≤ 1 + r·(adm_r(H) - 1)^(r²) ≤ 1 + r·(6(r+1)d³)^(r²)$ — the arithmetic of Corollary 2.7 of the notes, which has no concept of its own.

What remains is real arithmetic. From $d ≤ (c+1)·m^ε₁$ and $ε₁ · 3r² ≤ ε$ one gets $d^(3r²) ≤ (c+1)^(3r²) · m^ε$, so $wcol_r(H) ≤ 1 + K·m^ε ≤ (K+1)·m^ε$ with $K = r·(6(r+1))^(r²)·(c+1)^(3r²)$, using $m^ε ≥ 1$. The degenerate case $m = 0$ is separate: there $wcol_r(H) = 0$ and $m^ε = 0$.

The statement is Theorem 3.4 of Chapter 2 of the sparsity lecture notes of Pilipczuk and Siebertz (numbering of the 2019/20 edition). The composition reproduces that derivation at the level of the submitted concepts.