\wedge x^{k \operatorname{\mathbf{N}}}\Lambda^{*}(Z),$$ where $\operatorname{\mathbf{N}}$ is a multiplication operator on $\Lambda^{*}(Z)$ which takes a form $\alpha$ of degree $q$ to $q  \alpha$. Note that with respect to this decomposition, if $\operatorname{\mathbf{N}}$ is the operator acting on $\Lambda^{\operatorname{\mathbf{N}}}(Y)$ by multipliction by $\operatorname{\mathbf{N}}$, the exterior derivative $d$ acts as $$\label{eq:exterior-derivative}
d=\begin{pmatrix}d^Y+ (-1)^{q_Y} x^{-k}d^Z&0\\
\partial_x+kx^{-1}\operatorname{\mathbf{N}}&-d^Y-(-1)^{q_Y} x^{-k}d^Z\end{pmatrix}.$$

Moreover, we will use the space of vector fields which are locally $C^\infty(M)$ linear combinations of the vector fields $$\label{eq:ice-fields}
  {\partial}_x, \quad {\partial}_{y_{i}}, \quad x^{-k} {\partial}_{z_{\alpha}}.$$ These vector fields are local sections of a bundle ${{}^{ice} T}(M)$ which is dual to ${{}^{ice}T^{*}}(M) = \Lambda^1_{ice}(M)$. We denote sections of ${{}^{ice} T}(M)$ by $\mathcal{V}_{ice}$.

We consider metrics $g$ on $M$ which are sections of ${\operatorname{Sym}}^{0,2}({{}^{ice}T^{*}}(M))$, and which locally over the boundary take the form $$\label{eq:cuspedgemetric}
  g =  
{\left(}\begin{array}{ccc}
  dx & dy^i & x^{k} dz^\alpha
\end{array}
{\right)}{\left(}{\left(}\begin{array}{ccc}
1 & 0 & 0 \\
0 & (h_{ij}) & 0 \\
0 & 0 & ({\textbf{k}}_{\alpha \beta})
\end{array}
{\right)}+ O(x^k, g_0)
{\right)}{\left(}\begin{array}{c}
  dx \\ dy^i \\ x^{k} dz^\alpha
\end{array}
{\right)},$$ with $h_{ij}$ and and ${\textbf{k}}_{\alpha \beta}$ independent of $x$. Here $O(x^k, g_0)$ refers to a $O(x^k)$ norm bound with respect to an exact incomplete cusp edge metric $g_0$ as in below, and 