Generic codimension additivity for subdivisions of graphical-model supports

Assume that $\dim(\mathcal{S}_{G,\mathcal{X}})=d$. Let

$$h=\sum_{C\in\mathcal{C}(G)}h_{\mathcal{X}_C,y_C}.$$

Let $\sigma$ be the subdivision of $\mathcal{S}_{G,\mathcal{X}}$ induced by $h$, and let $\sigma_C$ be the subdivision of $\pi_C(\mathcal{S}_{G,\mathcal{X}})$ induced by $h_{\mathcal{X}_C,y_C}$. For $z\in\mathcal{S}_{G,\mathcal{X}}$, let $k_z$ and $l_z$ be the codimensions of the minimal faces of $\sigma$ and $\mathcal{S}_{G,\mathcal{X}}$ containing $z$, respectively. For each $C\in\mathcal{C}(G)$, let $k_{z,C}$ and $l_{z,C}$ be the codimensions of the minimal faces of $\sigma_C$ and $\pi_C(\mathcal{S}_{G,\mathcal{X}})$ containing $z_C$, respectively.

Generic codimension additivity conjecture. If $\mathcal{S}_{G,\mathcal{X}}$ and all tent functions $h_{\mathcal{X}_C,y_C}$ are sufficiently generic, then

$$k_z-l_z=\sum_{C\in\mathcal{C}(G)}k_{z,C}-\sum_{C\in\mathcal{C}(G)}l_{z,C}.$$

The preceding corollary proves only the corresponding inequality for the subdivision codimensions. The generic equality is presented as a conjectural strengthening and concerns the combinatorics of intersections of clique-wise regular subdivisions.