The underlying simple graph of a pattern $F$ with bipartition $A \uplus B$ has vertex set $A \uplus B$, with $a \in A$ and $b \in B$ adjacent if $F(a, b) > 0$. A pattern $F$ belongs to a class $\mathcal{F}$ of finite simple graphs if its underlying simple graph does. For a graph $G$, let $\gamma(G) = \sum_{v \in V(G)} 2^{\deg_G(v) - 1}$, the number of vertices of a CFI graph of $G$.

Let $\mathbb{K}$ be a field of characteristic zero. Let $\mathcal{F}$ be a class of finite simple graphs that contains $K_1$, is minor-closed, and is closed under weak oddomorphisms. Let $G \notin \mathcal{F}$ be a connected graph, let $F$ be a pattern whose underlying simple graph contains $G$ as a minor, and let $N \geq \gamma(G)$. Then $\hom_{F,N}$ is not a linear combination of homomorphism polynomials $\hom_{F',N}$ of patterns $F' \in \mathcal{F}$.

The proof distinguishes $\hom_{F,N}$ from all $\hom_{F',N}$ with $F' \in \mathcal{F}$ by evaluating at the bi-adjacency matrices of two bipartite graphs with $N$ vertices on either side, obtained from the two CFI graphs of $G$ by replacing the non-edges of $G$ with complete bipartite graphs between the gadgets, passing to the bipartite double cover with a perfect matching added, and padding with isolated vertices.