base and fiber Hodge-de Rham operators according to the following proposition, which we prove in Section \[sec:proof-of-operator-decomps\] below.

\[thm:normaloperatordeRham\] Locally over the boundary (see below ), the Hodge-de Rham operator decomposes as $$\begin{split}
      \label{eq:fulldeRham}
      \eth &= \eth_0 + \delta_Z P_1 + P_2 \delta_Z 
      + x^{k -1} E,
    \end{split}$$ where $\eth_0$ acts on sections of ${{}^{\textrm{ice}}\Lambda^{*}}$ decomposed as in by $$\eth_0 = {\left(}\begin{array}{cc}
        x^{-k} \eth^{Z} + \eth^{Y} & - {\partial}_{x} - k x^{-1} (f - \operatorname{\mathbf{N}}) \\
        {\partial}_{x} + k x^{-1} \operatorname{\mathbf{N}}& - x^{-k}\eth^{Z} - \eth^{Y}
      \end{array}
      {\right)},$$ $\eth^{Z}$ depends on the base $Y$ parametrically, acts in only the fiber direction and only on the $\Lambda^{*}(Z)$ factor, and is equal to the Hodge-de Rham operator for the Riemannian manifold ${\textbf{k}}\rvert_{y}$, and where the $P_i$ are polyhomogeneous, bounded endomorphisms on ${{}^{\textrm{ice}}\Lambda^{*}}$, and $E$ is as in -.

To describe the term $E$, we say first it is a differential operator of order one which does not increase the order of blow up of polyhomogeneous distributions; in particular $$\label{eq:errors-in-operator1}
E( x^k \gamma) =
O(x^k),$$ for $\gamma$ a smooth, bounded section of ${{}^{\textrm{ice}}\Lambda^{*}}$. In fact, it is a ${{\mathrm b}}$-differential operators on ${\mathrm{ice}}$-forms with polyhomogeneous coefficients $$\label{eq:errors-in-operator2}
  E  \in {\mathrm{Diff}}^1_{{{\mathrm b}},\operatorname{\mathrm {phg}}}(M; {{}^{\textrm{ice}}\Lambda^{*}}).$$ Letting 