This submission states Theorem 2 of Dreier, Mählmann, McCarty, Pilipczuk, Toruńczyk, Neighborhood Complexity and Radius-1 Merge-Width in Monadically Dependent Graph Classes (2026): every monadically dependent class of finite graphs has almost linear neighborhood complexity — for every $\varepsilon > 0$there is a$c$such that every member$G$ and every nonempty vertex subset $A$ satisfy $|\{N(v) \cap A : v \in V(G)\}| \le c\,|A|^{1+\varepsilon}$.

The concept surface has eleven review units. Six definitions: graph classes on the canonical vertex types; non-copying first-order transductions between classes of arbitrary relational structures (a domain formula and one formula per target relation symbol, over a monadic color expansion); graph transductions as the special case over the language of graphs; monadic dependence (the class does not transduce all graphs); neighborhood complexity, with the shared bound predicate $|A|^{1+o(1)}$; and nowhere denseness via shallow-minor models. Five theorems: weakly sparse monadically dependent classes are nowhere dense; nowhere dense classes have subpolynomial weak coloring numbers, uniformly over subgraphs of members; nowhere dense classes have almost linear neighborhood complexity; the headline theorem; and nowhere dense classes are monadically dependent (Adler–Adler).

The proof package discharges the headline theorem via the paper's VC-dimension sparsification argument; the nowhere dense counting statement from the coloring-number theorem, via radius-1 trace counting along a weak coloring order and localization; the weakly sparse theorem via Mählmann's Ramsey-theoretic extraction of induced subdivided bicliques (thesis, Lemma 13.8) together with a star-crossing transduction of all graphs; and the Adler–Adler direction via uniform quasi-wideness and a semantic locality argument — the deletion specialization of the flip-breakability route, with hereditarily finite rank-bounded local types of decorated balls and a ball-swap back-and-forth system in place of Gaifman's theorem. A transduction of all graphs would shatter arbitrarily large sets; quasi-wide scattering, a local-type pigeonhole, and the swap lemma refute this. With that, all five theorem concepts are proved, and every one reports only Lean's standard logical axioms — the surface carries the full classical equivalence that on weakly sparse classes, monadic dependence and nowhere denseness coincide.