Defect Spaces and Gram Operators for Tensor-Valued Incidence Maps

We study vector-valued incidence maps obtained from ordinary graph incidence maps by linear observation of the free vertex space. Let $\mathbb{F}$ be a field, $D=(X,E,s,t)$ a finite directed multigraph, $U$ an $\mathbb{F}$-vector space, and $\phi :X\to U$ a vertex labeling with $\mathbb{F}$-linear extension $\widehat{\phi }:{\mathbb{F}}^X\to U$. The vector-valued incidence map ${\mathrm{\partial }}_{\phi }:{\mathbb{F}}^E\to U$, ${\mathrm{\partial }}_{\phi }({\mathbf{1}}_e)=\phi (t(e))-\phi (s(e))$, factors as ${\mathrm{\partial }}_{\phi }=\widehat{\phi }\circ B_D$, where $B_D$ is the classical incidence map of $D$. We prove the formula ${\dim }_{\mathbb{F}}\mathrm{K}\mathrm{e}\mathrm{r}({\mathrm{\partial }}_{\phi })=\mathrm{\mid }E\mathrm{\mid }-\mathrm{\mid }X\mathrm{\mid }+c(D)+{\delta }_{\phi },$ where $c(D)$ is the number of weakly connected components of $D$ and ${\delta }_{\phi }:={\dim }_{\mathbb{F}}(\mathrm{I}\mathrm{m}(B_D)\cap \mathrm{K}\mathrm{e}\mathrm{r}(\widehat{\phi }))$ is the defect invariant. We apply this framework to directed tensor-labeled hypergraphs $\mathcal{H}=(Q_0,Q_1,\beta )$, in which each hyperedge carries a pair of boundary tensors $(A_e,B_e)$ in the tensor algebra $T({\mathbb{F}}^{Q_0})$, and prove that $\delta (\mathcal{H})=0$ over any field for each of the six standard constructions, including symmetric encodings that degenerate in positive characteristic. Over $\mathbb{F}=\mathbb{R}$, the edge Gram operator $\mathcal{L}={\mathrm{\partial }}_{\beta }^{\ast }{\mathrm{\partial }}_{\beta }$ has rank $\mathrm{\mid }{V}_{\mathrm{m}\mathrm{a}\mathrm{c}\mathrm{r}\mathrm{o}}\mathrm{\mid }-c_{\mathrm{m}\mathrm{a}\mathrm{c}\mathrm{r}\mathrm{o}}-\delta (\mathcal{H})$, and its degree-truncated operators form a Loewner-monotone filtration whose rank increments equal the decrements of the defect filtration. We further realize the cycle space of every oriented hypergraph (in the sense of Reff–Rusnak) as $\mathrm{K}\mathrm{e}\mathrm{r}({\mathrm{\partial }}_{\beta })$ within this framework, and exhibit a four-edge inclusion–exclusion example with $\delta (\mathcal{H})=1$.