The exact contents of citations.db main_text.text for this paper — one flattened LaTeX string, title through conclusion, appendix excluded, unmodified except for removing email addresses. This is what our citation measures are computed over.
306,116 characters
\documentclass[12pt,english]{article}
\usepackage[T1]{fontenc}
\usepackage[utf8]{inputenc}
\usepackage{geometry}
\geometry{verbose,tmargin=1.2in,bmargin=1.2in,lmargin=1in,rmargin=1in}
\setlength{\parskip}{\medskipamount}
\setlength{\parindent}{0pt}
\usepackage{float}
\usepackage{amsmath}
\usepackage{amsthm}
\usepackage{amssymb}
\usepackage{graphicx}
\usepackage{xcolor}
\usepackage{setspace}
\usepackage{dsfont}
\usepackage{caption}
\usepackage{hyperref}
\usepackage[authoryear,round]{natbib}
\usepackage[multiple]{footmisc}
\onehalfspacing
\allowdisplaybreaks
\usepackage{threeparttable}
\usepackage{xr, refcount}
\makeatletter
[\mathcal{R}^{s}]_{g} & =\big[(\partial_{\beta^{s}\phi'}\mathcal{L})\mathcal{H}^{-1}\big]_{g}
\end{align*}
By definition we have that $\mathcal{H}=-\partial_{\phi\phi'}\mathcal{L}$.
Differentiating gives
\[
\mathcal{H}_{b}=W^{-1}\big(\partial_{\beta\phi\phi'}\mathcal{L}+\sum_{f}(\partial_{\phi\phi'\phi_{f}}\mathcal{L})\big[\mathcal{H}^{-1}(\partial_{\beta\phi}\mathcal{L})\big]_{f}\big)
\]
\begin{align*}
\mathcal{H}_{s_{g}} & =-(\partial_{\beta\phi\phi'}\mathcal{L})(\partial_{s_{g}}B)-\sum_{f}(\partial_{\phi\phi'\phi_{f}}\mathcal{L})(\partial_{s}\Phi)_{fg}\\
& =W^{-1}\big[(\partial_{\beta\phi'}\mathcal{L})\mathcal{H}^{-1}\big]_{g}(\partial_{\beta\phi\phi'}\mathcal{L})\\
& +\sum_{f}(\partial_{\phi\phi'\phi_{f}}\mathcal{L})[\mathcal{H}^{-1}]_{fg}\\
& +W^{-1}\sum_{f}(\partial_{\phi\phi'\phi_{f}}\mathcal{L})\big[(\partial_{\beta\phi'}\mathcal{L})\mathcal{H}^{-1}\big]_{f}\big[(\partial_{\beta\phi'}\mathcal{L})\mathcal{H}^{-1}\big]_{g}
\end{align*}
Differentiating a second time gives
\begin{align*}
\mathcal{H}_{bb} & =-W^{-1}W_{b}\mathcal{H}_{b}+W^{-1}\mathcal{E}_{b}^{1}\\
& -W^{-2}\sum_{f}(\partial_{\beta\phi\phi'\phi_{f}}\mathcal{L})\big[\mathcal{H}^{-1}(\partial_{\beta\phi}\mathcal{L})\big]_{f}\\
& -W^{-2}\sum_{e,f}(\partial_{\phi\phi'\phi_{e}\phi_{f}}\mathcal{L})\big[\mathcal{H}^{-1}(\partial_{\beta\phi}\mathcal{L})\big]_{e}\big[\mathcal{H}^{-1}(\partial_{\beta\phi}\mathcal{L})\big]_{f}\\
& -W^{-1}\sum_{f}(\partial_{\phi\phi'\phi_{f}}\mathcal{L})\big[\mathcal{H}^{-1}\mathcal{H}_{b}\mathcal{H}^{-1}(\partial_{\beta\phi}\mathcal{L})\big]_{f}\\
& -W^{-2}\sum_{f}(\partial_{\phi\phi'\phi_{f}}\mathcal{L})\big[\mathcal{H}^{-1}(\partial_{\beta\beta\phi}\mathcal{L})\big]_{f}\\
& -W^{-2}\sum_{f}(\partial_{\phi\phi'\phi_{f}}\mathcal{L})\big[\mathcal{H}^{-1}\mathcal{E}^{1}\mathcal{H}^{-1}(\partial_{\beta\phi}\mathcal{L})\big]_{f}
\end{align*}
\begin{align*}
\mathcal{H}_{bs_{g}} & =-W^{-2}W_{s_{g}}\mathcal{E}^{1}+W^{-1}\mathcal{E}_{s_{g}}^{1}\\
& -W^{-2}W_{s_{g}}\sum_{f}(\partial_{\phi\phi'\phi_{f}}\mathcal{L})\big[\mathcal{H}^{-1}(\partial_{\beta\phi}\mathcal{L})\big]_{f}\\
& -W^{-2}\sum_{f}(\partial_{\beta\phi\phi'\phi_{f}}\mathcal{L})\big[\mathcal{H}^{-1}(\partial_{\beta\phi}\mathcal{L})\big]_{f}\big[\mathcal{H}^{-1}(\partial_{\beta\phi}\mathcal{L})\big]_{g}\\
& +W^{-1}\sum_{e,f}(\partial_{\phi\phi'\phi_{e}\phi_{f}}\mathcal{L})\big[\mathcal{H}^{-1}(\partial_{\beta\phi}\mathcal{L})\big]_{f}\big[\mathcal{H}^{-1}\big]_{eg}\\
& -W^{-2}\sum_{e,f}(\partial_{\phi\phi'\phi_{e}\phi_{f}}\mathcal{L})\big[\mathcal{H}^{-1}(\partial_{\beta\phi}\mathcal{L})\big]_{e}\big[\mathcal{H}^{-1}(\partial_{\beta\phi}\mathcal{L})\big]_{f}\big[\mathcal{H}^{-1}(\partial_{\beta\phi}\mathcal{L})\big]_{g}\\
& -W^{-1}\sum_{f}(\partial_{\phi\phi'\phi_{f}}\mathcal{L})\big[\mathcal{H}^{-1}\mathcal{H}_{s_{g}}\mathcal{H}^{-1}(\partial_{\beta\phi}\mathcal{L})\big]_{f}\\
& -W^{-2}\sum_{f}(\partial_{\phi\phi'\phi_{f}}\mathcal{L})\big[\mathcal{H}^{-1}(\partial_{\beta\beta\phi}\mathcal{L})\big]_{f}\big[\mathcal{H}^{-1}(\partial_{\beta\phi}\mathcal{L})\big]_{g}\\
& +W^{-1}\sum_{f}(\partial_{\phi\phi'\phi_{f}}\mathcal{L})\big[\mathcal{H}^{-1}\mathcal{E}^{1}\mathcal{H}^{-1}\big]_{fg}\\
& -W^{-2}\sum_{f}(\partial_{\phi\phi'\phi_{f}}\mathcal{L})\big[\mathcal{H}^{-1}\mathcal{E}^{1}\mathcal{H}^{-1}(\partial_{\beta\phi}\mathcal{L})\big]_{f}\big[\mathcal{H}^{-1}(\partial_{\beta\phi}\mathcal{L})\big]_{g}
\end{align*}
\begin{align*}
\mathcal{H}_{s_{g}s_{h}} & =-W^{-2}W_{s_{h}}\big[(\partial_{\beta\phi'}\mathcal{L})\mathcal{H}^{-1}\big]_{g}(\partial_{\beta\phi\phi'}\mathcal{L})\\
& -W^{-2}\big[(\partial_{\beta\beta\phi'}\mathcal{L})\mathcal{H}^{-1}\big]_{g}\big[(\partial_{\beta\phi'}\mathcal{L})\mathcal{H}^{-1}\big]_{h}(\partial_{\beta\phi\phi'}\mathcal{L})\\
& -W^{-1}\big[\mathcal{H}^{-1}(\partial_{\beta\phi\phi'}\mathcal{L})\mathcal{H}^{-1}\big]_{hg}(\partial_{\beta\phi\phi'}\mathcal{L})\\
& -W^{-2}\big[(\partial_{\beta\phi}\mathcal{L})'\mathcal{H}^{-1}\mathcal{E}\mathcal{H}^{-1}\big]_{g}\big[(\partial_{\beta\phi'}\mathcal{L})\mathcal{H}^{-1}\big]_{h}(\partial_{\beta\phi\phi'}\mathcal{L})\\
& -W^{-1}\big[(\partial_{\beta\phi'}\mathcal{L})\mathcal{H}^{-1}\mathcal{H}_{s_{h}}\mathcal{H}^{-1}\big]_{g}(\partial_{\beta\phi\phi'}\mathcal{L})\\
& -W^{-1}\big[(\partial_{\beta\phi'}\mathcal{L})\mathcal{H}^{-1}\big]_{g}\big[(\partial_{\beta\phi'}\mathcal{L})\mathcal{H}^{-1}\big]_{h}(\partial_{\beta\beta\phi\phi'}\mathcal{L})\\
& -W^{-1}\big[(\partial_{\beta\phi'}\mathcal{L})\mathcal{H}^{-1}\big]_{g}\sum_{f}(\partial_{\beta\phi\phi'\phi_{f}}\mathcal{L})\big[\mathcal{H}^{-1}\big]_{fh}\\
& -W^{-2}\big[(\partial_{\beta\phi'}\mathcal{L})\mathcal{H}^{-1}\big]_{g}\sum_{f}(\partial_{\beta\phi\phi'\phi_{f}}\mathcal{L})\big[\mathcal{H}^{-1}(\partial_{\beta\phi}\mathcal{L})\big]_{f}\big[(\partial_{\beta\phi}\mathcal{L})'\mathcal{H}^{-1}\big]_{h}\\
& -W^{-1}\sum_{f}(\partial_{\beta\phi\phi'\phi_{f}}\mathcal{L})[\mathcal{H}^{-1}]_{fg}\big[(\partial_{\beta\phi}\mathcal{L})'\mathcal{H}^{-1}\big]_{h}\\
& -\sum_{e,f}(\partial_{\phi\phi'\phi_{e}\phi_{f}}\mathcal{L})[\mathcal{H}^{-1}]_{fg}[\mathcal{H}^{-1}]_{eh}\\
& -W^{-1}\sum_{e,f}(\partial_{\phi\phi'\phi_{e}\phi_{f}}\mathcal{L})[\mathcal{H}^{-1}]_{fg}\big[(\partial_{\beta\phi}\mathcal{L})'\mathcal{H}^{-1}\big]_{e}\big[(\partial_{\beta\phi}\mathcal{L})'\mathcal{H}^{-1}\big]_{h}\\
& -\sum_{f}(\partial_{\phi\phi'\phi_{f}}\mathcal{L})[\mathcal{H}^{-1}\mathcal{H}_{s_{h}}\mathcal{H}^{-1}]_{fg}\\
& -W^{-2}W_{s_{h}}\sum_{f}(\partial_{\phi\phi'\phi_{f}}\mathcal{L})\big[(\partial_{\beta\phi'}\mathcal{L})\mathcal{H}^{-1}\big]_{f}\big[(\partial_{\beta\phi'}\mathcal{L})\mathcal{H}^{-1}\big]_{g}\\
& -W^{-2}\sum_{f}(\partial_{\beta\phi\phi'\phi_{f}}\mathcal{L})\big[(\partial_{\beta\phi'}\mathcal{L})\mathcal{H}^{-1}\big]_{f}\big[(\partial_{\beta\phi'}\mathcal{L})\mathcal{H}^{-1}\big]_{g}\\
& -W^{-1}\sum_{e,f}(\partial_{\phi\phi'\phi_{e}\phi_{f}}\mathcal{L})[\mathcal{H}^{-1}]_{eh}\big[(\partial_{\beta\phi'}\mathcal{L})\mathcal{H}^{-1}\big]_{f}\big[(\partial_{\beta\phi'}\mathcal{L})\mathcal{H}^{-1}\big]_{g}\\
& -W^{-2}\sum_{e,f}(\partial_{\phi\phi'\phi_{e}\phi_{f}}\mathcal{L})\big[(\partial_{\beta\phi'}\mathcal{L})\mathcal{H}^{-1}\big]_{e}\big[(\partial_{\beta\phi'}\mathcal{L})\mathcal{H}^{-1}\big]_{f}\big[(\partial_{\beta\phi'}\mathcal{L})\mathcal{H}^{-1}\big]_{g}\big[(\partial_{\beta\phi'}\mathcal{L})\mathcal{H}^{-1}\big]_{h}\\
& -W^{-2}\sum_{f}(\partial_{\phi\phi'\phi_{f}}\mathcal{L})\big[(\partial_{\beta\beta\phi'}\mathcal{L})\mathcal{H}^{-1}\big]_{f}\big[(\partial_{\beta\phi'}\mathcal{L})\mathcal{H}^{-1}\big]_{g}\\
& -W^{-1}\sum_{f}(\partial_{\phi\phi'\phi_{f}}\mathcal{L})\big[\mathcal{H}^{-1}(\partial_{\beta\phi\phi'}\mathcal{L})\mathcal{H}^{-1}\big]_{fh}\big[(\partial_{\beta\phi'}\mathcal{L})\mathcal{H}^{-1}\big]_{g}\\
& -W^{-2}\sum_{f}(\partial_{\phi\phi'\phi_{f}}\mathcal{L})\big[(\partial_{\beta\phi'}\mathcal{L})\mathcal{H}^{-1}\mathcal{E}\mathcal{H}^{-1}\big]_{f}\big[(\partial_{\beta\phi'}\mathcal{L})\mathcal{H}^{-1}\big]_{g}\big[(\partial_{\beta\phi'}\mathcal{L})\mathcal{H}^{-1}\big]_{h}\\
& -W^{-1}\sum_{f}(\partial_{\phi\phi'\phi_{f}}\mathcal{L})\big[(\partial_{\beta\phi'}\mathcal{L})\mathcal{H}^{-1}\mathcal{H}_{s_{h}}\mathcal{H}^{-1}\big]_{f}\big[(\partial_{\beta\phi'}\mathcal{L})\mathcal{H}^{-1}\big]_{g}\\
& -W^{-2}\sum_{f}(\partial_{\phi\phi'\phi_{f}}\mathcal{L})\big[(\partial_{\beta\phi'}\mathcal{L})\mathcal{H}^{-1}\big]_{f}\big[(\partial_{\beta\beta\phi'}\mathcal{L})\mathcal{H}^{-1}\big]_{g}\\
& -W^{-1}\sum_{f}(\partial_{\phi\phi'\phi_{f}}\mathcal{L})\big[(\partial_{\beta\phi'}\mathcal{L})\mathcal{H}^{-1}\big]_{f}\big[\mathcal{H}^{-1}(\partial_{\beta\phi\phi'}\mathcal{L})\mathcal{H}^{-1}\big]_{gh}\\
& -W^{-2}\sum_{f}(\partial_{\phi\phi'\phi_{f}}\mathcal{L})\big[(\partial_{\beta\phi'}\mathcal{L})\mathcal{H}^{-1}\big]_{f}\big[(\partial_{\beta\phi'}\mathcal{L})\mathcal{H}^{-1}\mathcal{E}\mathcal{H}^{-1}\big]_{g}\big[(\partial_{\beta\phi'}\mathcal{L})\mathcal{H}^{-1}\big]_{h}\\
& -W^{-1}\sum_{f}(\partial_{\phi\phi'\phi_{f}}\mathcal{L})\big[(\partial_{\beta\phi'}\mathcal{L})\mathcal{H}^{-1}\big]_{f}\big[(\partial_{\beta\phi'}\mathcal{L})\mathcal{H}^{-1}\mathcal{H}_{s_{h}}\mathcal{H}^{-1}\big]_{g}
\end{align*}
The third derivatives are
\begin{align*}
\mathcal{H}_{bbb} & =W^{-2}W_{b}^{2}\mathcal{H}_{b}-W^{-1}W_{bb}\mathcal{H}_{b}-W^{-1}W_{b}\mathcal{H}_{bb}\\
& +W^{-2}W_{b}\mathcal{E}_{b}-W^{-1}\mathcal{E}_{bb}\\
& +2W^{-3}W_{b}\sum_{f}(\partial_{\beta\phi\phi'\phi_{f}}\mathcal{L})\big[\mathcal{H}^{-1}(\partial_{\beta\phi}\mathcal{L})\big]_{f}\\
& -W^{-3}\sum_{f}(\partial_{\beta\beta\phi\phi'\phi_{f}}\mathcal{L})\big[\mathcal{H}^{-1}(\partial_{\beta\phi}\mathcal{L})\big]_{f}\\
& -W^{-3}\sum_{e,f}(\partial_{\beta\phi\phi'\phi_{e}\phi_{f}}\mathcal{L})\big[\mathcal{H}^{-1}(\partial_{\beta\phi}\mathcal{L})\big]_{e}\big[\mathcal{H}^{-1}(\partial_{\beta\phi}\mathcal{L})\big]_{f}\\
& +W^{-2}\sum_{f}(\partial_{\beta\phi\phi'\phi_{f}}\mathcal{L})\big[\mathcal{H}^{-1}\mathcal{H}_{b}\mathcal{H}^{-1}(\partial_{\beta\phi}\mathcal{L})\big]_{f}\\
& -W^{-3}\sum_{f}(\partial_{\beta\phi\phi'\phi_{f}}\mathcal{L})\big[\mathcal{H}^{-1}(\partial_{\beta\beta\phi}\mathcal{L})\big]_{f}\\
& -W^{-3}\sum_{f}(\partial_{\beta\phi\phi'\phi_{f}}\mathcal{L})\big[\mathcal{H}^{-1}\mathcal{E}\mathcal{H}^{-1}(\partial_{\beta\phi}\mathcal{L})\big]_{f}\\
\\
& +2W^{-3}W_{b}\sum_{e,f}(\partial_{\phi\phi'\phi_{e}\phi_{f}}\mathcal{L})\big[\mathcal{H}^{-1}(\partial_{\beta\phi}\mathcal{L})\big]_{e}\big[\mathcal{H}^{-1}(\partial_{\beta\phi}\mathcal{L})\big]_{f}\\
& -W^{-3}\sum_{e,f}(\partial_{\beta\phi\phi'\phi_{e}\phi_{f}}\mathcal{L})\big[\mathcal{H}^{-1}(\partial_{\beta\phi}\mathcal{L})\big]_{e}\big[\mathcal{H}^{-1}(\partial_{\beta\phi}\mathcal{L})\big]_{f}\\
& -W^{-3}\sum_{d,e,f}(\partial_{\phi\phi'\phi_{d}\phi_{e}\phi_{f}}\mathcal{L})\big[\mathcal{H}^{-1}(\partial_{\beta\phi}\mathcal{L})\big]_{d}\big[\mathcal{H}^{-1}(\partial_{\beta\phi}\mathcal{L})\big]_{e}\big[\mathcal{H}^{-1}(\partial_{\beta\phi}\mathcal{L})\big]_{f}\\
& +2W^{-2}\sum_{e,f}(\partial_{\phi\phi'\phi_{e}\phi_{f}}\mathcal{L})\big[\mathcal{H}^{-1}\mathcal{H}_{b}\mathcal{H}^{-1}(\partial_{\beta\phi}\mathcal{L})\big]_{e}\big[\mathcal{H}^{-1}(\partial_{\beta\phi}\mathcal{L})\big]_{f}\\
& -2W^{-3}\sum_{e,f}(\partial_{\phi\phi'\phi_{e}\phi_{f}}\mathcal{L})\big[\mathcal{H}^{-1}(\partial_{\beta\beta\phi}\mathcal{L})\big]_{e}\big[\mathcal{H}^{-1}(\partial_{\beta\phi}\mathcal{L})\big]_{f}\\
& -2W^{-3}\sum_{e,f}(\partial_{\phi\phi'\phi_{e}\phi_{f}}\mathcal{L})\big[\mathcal{H}^{-1}\mathcal{E}\mathcal{H}^{-1}(\partial_{\beta\phi}\mathcal{L})\big]_{e}\big[\mathcal{H}^{-1}(\partial_{\beta\phi}\mathcal{L})\big]_{f}\\
\\
& -W^{-1}W_{b}\sum_{f}(\partial_{\phi\phi'\phi_{f}}\mathcal{L})\big[\mathcal{H}^{-1}\mathcal{H}_{b}\mathcal{H}^{-1}(\partial_{\beta\phi}\mathcal{L})\big]_{f}\\
& +W^{-2}\sum_{f}(\partial_{\beta\phi\phi'\phi_{f}}\mathcal{L})\big[\mathcal{H}^{-1}\mathcal{H}_{b}\mathcal{H}^{-1}(\partial_{\beta\phi}\mathcal{L})\big]_{f}\\
& +W^{-2}\sum_{e,f}(\partial_{\phi\phi'\phi_{e}\phi_{f}}\mathcal{L})\big[\mathcal{H}^{-1}(\partial_{\beta\phi}\mathcal{L})\big]_{e}\big[\mathcal{H}^{-1}\mathcal{H}_{b}\mathcal{H}^{-1}(\partial_{\beta\phi}\mathcal{L})\big]_{f}\\
& -2W^{-1}\sum_{f}(\partial_{\phi\phi'\phi_{f}}\mathcal{L})\big[\mathcal{H}^{-1}\mathcal{H}_{b}\mathcal{H}^{-1}\mathcal{H}_{b}\mathcal{H}^{-1}(\partial_{\beta\phi}\mathcal{L})\big]_{f}\\
& +W^{-1}\sum_{f}(\partial_{\phi\phi'\phi_{f}}\mathcal{L})\big[\mathcal{H}^{-1}\mathcal{H}_{bb}\mathcal{H}^{-1}(\partial_{\beta\phi}\mathcal{L})\big]_{f}\\
& +W^{-2}\sum_{f}(\partial_{\phi\phi'\phi_{f}}\mathcal{L})\big[\mathcal{H}^{-1}\mathcal{H}_{b}\mathcal{H}^{-1}(\partial_{\beta\beta\phi}\mathcal{L})\big]_{f}\\
& +W^{-2}\sum_{f}(\partial_{\phi\phi'\phi_{f}}\mathcal{L})\big[\mathcal{H}^{-1}\mathcal{H}_{b}\mathcal{H}^{-1}\mathcal{E}\mathcal{H}^{-1}(\partial_{\beta\phi}\mathcal{L})\big]_{f}\\
\\
& +2W^{-3}W_{b}\sum_{f}(\partial_{\phi\phi'\phi_{f}}\mathcal{L})\big[\mathcal{H}^{-1}(\partial_{\beta\beta\phi}\mathcal{L})\big]_{f}\\
& -W^{-3}\sum_{f}(\partial_{\beta\phi\phi'\phi_{f}}\mathcal{L})\big[\mathcal{H}^{-1}(\partial_{\beta\beta\phi}\mathcal{L})\big]_{f}\\
& -W^{-3}\sum_{e,f}(\partial_{\phi\phi'\phi_{e}\phi_{f}}\mathcal{L})\big[\mathcal{H}^{-1}(\partial_{\beta\phi}\mathcal{L})\big]_{e}\big[\mathcal{H}^{-1}(\partial_{\beta\beta\phi}\mathcal{L})\big]_{f}\\
& +W^{-2}\sum_{f}(\partial_{\phi\phi'\phi_{f}}\mathcal{L})\big[\mathcal{H}^{-1}\mathcal{H}_{b}\mathcal{H}^{-1}(\partial_{\beta\beta\phi}\mathcal{L})\big]_{f}\\
& -W^{-3}\sum_{f}(\partial_{\phi\phi'\phi_{f}}\mathcal{L})\big[\mathcal{H}^{-1}(\partial_{\beta\beta\beta\phi}\mathcal{L})\big]_{f}\\
& -W^{-3}\sum_{f}(\partial_{\phi\phi'\phi_{f}}\mathcal{L})\big[\mathcal{H}^{-1}\mathcal{E}^{2}\mathcal{H}^{-1}(\partial_{\beta\phi}\mathcal{L})\big]_{f}\\
\\
& +2W^{-3}W_{b}\sum_{f}(\partial_{\phi\phi'\phi_{f}}\mathcal{L})\big[\mathcal{H}^{-1}\mathcal{E}\mathcal{H}^{-1}(\partial_{\beta\phi}\mathcal{L})\big]_{f}\\
& -W^{-3}\sum_{f}(\partial_{\beta\phi\phi'\phi_{f}}\mathcal{L})\big[\mathcal{H}^{-1}\mathcal{E}\mathcal{H}^{-1}(\partial_{\beta\phi}\mathcal{L})\big]_{f}\\
& -W^{-3}\sum_{e,f}(\partial_{\phi\phi'\phi_{e}\phi_{f}}\mathcal{L})\big[\mathcal{H}^{-1}(\partial_{\beta\phi}\mathcal{L})\big]_{e}\big[\mathcal{H}^{-1}\mathcal{E}\mathcal{H}^{-1}(\partial_{\beta\phi}\mathcal{L})\big]_{f}\\
& +W^{-2}\sum_{f}(\partial_{\phi\phi'\phi_{f}}\mathcal{L})\big[\mathcal{H}^{-1}\mathcal{H}_{b}\mathcal{H}^{-1}\mathcal{E}\mathcal{H}^{-1}(\partial_{\beta\phi}\mathcal{L})\big]_{f}\\
& +W^{-2}\sum_{f}(\partial_{\phi\phi'\phi_{f}}\mathcal{L})\big[\mathcal{H}^{-1}\mathcal{E}\mathcal{H}^{-1}\mathcal{H}_{b}\mathcal{H}^{-1}(\partial_{\beta\phi}\mathcal{L})\big]_{f}\\
& -W^{-2}\sum_{f}(\partial_{\phi\phi'\phi_{f}}\mathcal{L})\big[\mathcal{H}^{-1}\mathcal{E}_{b}\mathcal{H}^{-1}(\partial_{\beta\phi}\mathcal{L})\big]_{f}\\
& -W^{-3}\sum_{f}(\partial_{\phi\phi'\phi_{f}}\mathcal{L})\big[\mathcal{H}^{-1}\mathcal{E}\mathcal{H}^{-1}(\partial_{\beta\beta\phi}\mathcal{L})\big]_{f}\\
& -W^{-3}\sum_{f}(\partial_{\phi\phi'\phi_{f}}\mathcal{L})\big[\mathcal{H}^{-1}\mathcal{E}\mathcal{H}^{-1}\mathcal{E}\mathcal{H}^{-1}(\partial_{\beta\phi}\mathcal{L})\big]_{f}
\end{align*}
\begin{align*}
\mathcal{H}_{bbs_{g}} & =-2W^{-3}W_{b}W_{s_{g}}\mathcal{E}^{1}+W^{-2}W_{bs_{g}}\mathcal{E}^{1}+W^{-2}W_{s_{g}}\mathcal{E}_{b}^{1}\\
& +W^{-2}W_{b}\mathcal{E}_{s_{g}}^{1}-W^{-1}\mathcal{E}_{bs_{g}}^{1}\\
& -2W^{-3}W_{b}W_{s_{g}}\sum_{f}(\partial_{\phi\phi'\phi_{f}}\mathcal{L})\big[\mathcal{H}^{-1}(\partial_{\beta\phi}\mathcal{L})\big]_{f}\\
& +W^{-2}W_{bs_{g}}\sum_{f}(\partial_{\phi\phi'\phi_{f}}\mathcal{L})\big[\mathcal{H}^{-1}(\partial_{\beta\phi}\mathcal{L})\big]_{f}\\
& -W^{-3}W_{s_{g}}\sum_{f}(\partial_{\beta\phi\phi'\phi_{f}}\mathcal{L})\big[\mathcal{H}^{-1}(\partial_{\beta\phi}\mathcal{L})\big]_{f}\\
& -W^{-3}W_{s_{g}}\sum_{e,f}(\partial_{\phi\phi'\phi_{e}\phi_{f}}\mathcal{L})\big[\mathcal{H}^{-1}(\partial_{\beta\phi}\mathcal{L})\big]_{e}\big[\mathcal{H}^{-1}(\partial_{\beta\phi}\mathcal{L})\big]_{f}\\
& -W^{-2}W_{s_{g}}\sum_{f}(\partial_{\phi\phi'\phi_{f}}\mathcal{L})\big[\mathcal{H}^{-1}\mathcal{H}_{b}\mathcal{H}^{-1}(\partial_{\beta\phi}\mathcal{L})\big]_{f}\\
& -W^{-3}W_{s_{g}}\sum_{f}(\partial_{\phi\phi'\phi_{f}}\mathcal{L})\big[\mathcal{H}^{-1}(\partial_{\beta\beta\phi}\mathcal{L})\big]_{f}\\
& -W^{-3}W_{s_{g}}\sum_{f}(\partial_{\phi\phi'\phi_{f}}\mathcal{L})\big[\mathcal{H}^{-1}\mathcal{E}^{1}\mathcal{H}^{-1}(\partial_{\beta\phi}\mathcal{L})\big]_{f}\\
& +W^{-3}W_{b}\sum_{f}(\partial_{\beta\phi\phi'\phi_{f}}\mathcal{L})\big[\mathcal{H}^{-1}(\partial_{\beta\phi}\mathcal{L})\big]_{f}\big[\mathcal{H}^{-1}(\partial_{\beta\phi}\mathcal{L})\big]_{g}\\
& +W^{-3}\sum_{f}(\partial_{\beta^{2}\phi\phi'\phi_{f}}\mathcal{L})\big[\mathcal{H}^{-1}(\partial_{\beta\phi}\mathcal{L})\big]_{f}\big[\mathcal{H}^{-1}(\partial_{\beta\phi}\mathcal{L})\big]_{g}\\
& +W^{-3}\sum_{e,f}(\partial_{\beta\phi\phi'\phi_{e}\phi_{f}}\mathcal{L})\big[\mathcal{H}^{-1}(\partial_{\beta\phi}\mathcal{L})\big]_{e}\big[\mathcal{H}^{-1}(\partial_{\beta\phi}\mathcal{L})\big]_{f}\big[\mathcal{H}^{-1}(\partial_{\beta\phi}\mathcal{L})\big]_{g}\\
& +W^{-2}\sum_{f}(\partial_{\beta\phi\phi'\phi_{f}}\mathcal{L})\big[\mathcal{H}^{-1}\mathcal{H}_{b}\mathcal{H}^{-1}(\partial_{\beta\phi}\mathcal{L})\big]_{f}\big[\mathcal{H}^{-1}(\partial_{\beta\phi}\mathcal{L})\big]_{g}\\
& +W^{-3}\sum_{f}(\partial_{\beta\phi\phi'\phi_{f}}\mathcal{L})\big[\mathcal{H}^{-1}(\partial_{\beta\beta\phi}\mathcal{L})\big]_{f}\big[\mathcal{H}^{-1}(\partial_{\beta\phi}\mathcal{L})\big]_{g}\\
& +W^{-3}\sum_{f}(\partial_{\beta\phi\phi'\phi_{f}}\mathcal{L})\big[\mathcal{H}^{-1}(\partial_{\beta\phi\phi'}\mathcal{L})\mathcal{H}^{-1}(\partial_{\beta\phi}\mathcal{L})\big]_{f}\big[\mathcal{H}^{-1}(\partial_{\beta\phi}\mathcal{L})\big]_{g}\\
& +W^{-2}\sum_{f}(\partial_{\beta\phi\phi'\phi_{f}}\mathcal{L})\big[\mathcal{H}^{-1}(\partial_{\beta\phi}\mathcal{L})\big]_{f}\big[\mathcal{H}^{-1}\mathcal{H}_{b}\mathcal{H}^{-1}(\partial_{\beta\phi}\mathcal{L})\big]_{g}\\
& +W^{-3}\sum_{f}(\partial_{\beta\phi\phi'\phi_{f}}\mathcal{L})\big[\mathcal{H}^{-1}(\partial_{\beta\phi}\mathcal{L})\big]_{f}\big[\mathcal{H}^{-1}(\partial_{\beta\beta\phi}\mathcal{L})\big]_{g}\\
& +W^{-3}\sum_{f}(\partial_{\beta\phi\phi'\phi_{f}}\mathcal{L})\big[\mathcal{H}^{-1}(\partial_{\beta\phi}\mathcal{L})\big]_{f}\big[\mathcal{H}^{-1}(\partial_{\beta\phi\phi'}\mathcal{L})\mathcal{H}^{-1}(\partial_{\beta\phi}\mathcal{L})\big]_{g}\\
& -W^{-2}W_{b}\sum_{e,f}(\partial_{\phi\phi'\phi_{e}\phi_{f}}\mathcal{L})\big[\mathcal{H}^{-1}(\partial_{\beta\phi}\mathcal{L})\big]_{f}\big[\mathcal{H}^{-1}\big]_{eg}\\
& -W^{-2}\sum_{e,f}(\partial_{\beta\phi\phi'\phi_{e}\phi_{f}}\mathcal{L})\big[\mathcal{H}^{-1}(\partial_{\beta\phi}\mathcal{L})\big]_{f}\big[\mathcal{H}^{-1}\big]_{eg}\\
& -W^{-2}\sum_{d,e,f}(\partial_{\phi\phi'\phi_{d}\phi_{e}\phi_{f}}\mathcal{L})\big[\mathcal{H}^{-1}(\partial_{\beta\phi}\mathcal{L})\big]_{d}\big[\mathcal{H}^{-1}(\partial_{\beta\phi}\mathcal{L})\big]_{f}\big[\mathcal{H}^{-1}\big]_{eg}\\
& -W^{-1}\sum_{e,f}(\partial_{\phi\phi'\phi_{e}\phi_{f}}\mathcal{L})\big[\mathcal{H}^{-1}\mathcal{H}_{b}\mathcal{H}^{-1}(\partial_{\beta\phi}\mathcal{L})\big]_{f}\big[\mathcal{H}^{-1}\big]_{eg}\\
& -W^{-2}\sum_{e,f}(\partial_{\phi\phi'\phi_{e}\phi_{f}}\mathcal{L})\big[\mathcal{H}^{-1}(\partial_{\beta\beta\phi}\mathcal{L})\big]_{f}\big[\mathcal{H}^{-1}\big]_{eg}\\
& -W^{-2}\sum_{e,f}(\partial_{\phi\phi'\phi_{e}\phi_{f}}\mathcal{L})\big[\mathcal{H}^{-1}(\partial_{\beta\phi\phi'}\mathcal{L})\mathcal{H}^{-1}(\partial_{\beta\phi}\mathcal{L})\big]_{f}\big[\mathcal{H}^{-1}\big]_{eg}\\
& -W^{-1}\sum_{e,f}(\partial_{\phi\phi'\phi_{e}\phi_{f}}\mathcal{L})\big[\mathcal{H}^{-1}(\partial_{\beta\phi}\mathcal{L})\big]_{f}\big[\mathcal{H}^{-1}\mathcal{H}_{b}\mathcal{H}^{-1}\big]_{eg}\\
& +2W^{-3}W_{b}\sum_{e,f}(\partial_{\phi\phi'\phi_{e}\phi_{f}}\mathcal{L})\big[\mathcal{H}^{-1}(\partial_{\beta\phi}\mathcal{L})\big]_{e}\big[\mathcal{H}^{-1}(\partial_{\beta\phi}\mathcal{L})\big]_{f}\big[\mathcal{H}^{-1}(\partial_{\beta\phi}\mathcal{L})\big]_{g}\\
& +W^{-3}\sum_{e,f}(\partial_{\beta\phi\phi'\phi_{e}\phi_{f}}\mathcal{L})\big[\mathcal{H}^{-1}(\partial_{\beta\phi}\mathcal{L})\big]_{e}\big[\mathcal{H}^{-1}(\partial_{\beta\phi}\mathcal{L})\big]_{f}\big[\mathcal{H}^{-1}(\partial_{\beta\phi}\mathcal{L})\big]_{g}\\
& +W^{-3}\sum_{d,e,f}(\partial_{\phi\phi'\phi_{d}\phi_{e}\phi_{f}}\mathcal{L})\big[\mathcal{H}^{-1}(\partial_{\beta\phi}\mathcal{L})\big]_{d}\big[\mathcal{H}^{-1}(\partial_{\beta\phi}\mathcal{L})\big]_{e}\big[\mathcal{H}^{-1}(\partial_{\beta\phi}\mathcal{L})\big]_{f}\big[\mathcal{H}^{-1}(\partial_{\beta\phi}\mathcal{L})\big]_{g}\\
& +2W^{-2}\sum_{e,f}(\partial_{\phi\phi'\phi_{e}\phi_{f}}\mathcal{L})\big[\mathcal{H}^{-1}\mathcal{H}_{b}\mathcal{H}^{-1}(\partial_{\beta\phi}\mathcal{L})\big]_{e}\big[\mathcal{H}^{-1}(\partial_{\beta\phi}\mathcal{L})\big]_{f}\big[\mathcal{H}^{-1}(\partial_{\beta\phi}\mathcal{L})\big]_{g}\\
& +2W^{-3}\sum_{e,f}(\partial_{\phi\phi'\phi_{e}\phi_{f}}\mathcal{L})\big[\mathcal{H}^{-1}(\partial_{\beta\beta\phi}\mathcal{L})\big]_{e}\big[\mathcal{H}^{-1}(\partial_{\beta\phi}\mathcal{L})\big]_{f}\big[\mathcal{H}^{-1}(\partial_{\beta\phi}\mathcal{L})\big]_{g}\\
& +2W^{-3}\sum_{e,f}(\partial_{\phi\phi'\phi_{e}\phi_{f}}\mathcal{L})\big[\mathcal{H}^{-1}(\partial_{\beta\phi\phi'}\mathcal{L})\mathcal{H}^{-1}(\partial_{\beta\phi}\mathcal{L})\big]_{e}\big[\mathcal{H}^{-1}(\partial_{\beta\phi}\mathcal{L})\big]_{f}\big[\mathcal{H}^{-1}(\partial_{\beta\phi}\mathcal{L})\big]_{g}\\
& +W^{-2}\sum_{e,f}(\partial_{\phi\phi'\phi_{e}\phi_{f}}\mathcal{L})\big[\mathcal{H}^{-1}(\partial_{\beta\phi}\mathcal{L})\big]_{e}\big[\mathcal{H}^{-1}(\partial_{\beta\phi}\mathcal{L})\big]_{f}\big[\mathcal{H}^{-1}\mathcal{H}_{b}\mathcal{H}^{-1}(\partial_{\beta\phi}\mathcal{L})\big]_{g}\\
& +W^{-3}\sum_{e,f}(\partial_{\phi\phi'\phi_{e}\phi_{f}}\mathcal{L})\big[\mathcal{H}^{-1}(\partial_{\beta\phi}\mathcal{L})\big]_{e}\big[\mathcal{H}^{-1}(\partial_{\beta\phi}\mathcal{L})\big]_{f}\big[\mathcal{H}^{-1}(\partial_{\beta\beta\phi}\mathcal{L})\big]_{g}\\
& +W^{-3}\sum_{e,f}(\partial_{\phi\phi'\phi_{e}\phi_{f}}\mathcal{L})\big[\mathcal{H}^{-1}(\partial_{\beta\phi}\mathcal{L})\big]_{e}\big[\mathcal{H}^{-1}(\partial_{\beta\phi}\mathcal{L})\big]_{f}\big[\mathcal{H}^{-1}(\partial_{\beta\phi\phi'}\mathcal{L})\mathcal{H}^{-1}(\partial_{\beta\phi}\mathcal{L})\big]_{g}\\
& -W^{-2}W_{b}\sum_{f}(\partial_{\phi\phi'\phi_{f}}\mathcal{L})\big[\mathcal{H}^{-1}\mathcal{H}_{b}\mathcal{H}^{-1}(\partial_{\beta\phi}\mathcal{L})\big]_{f}\\
& -W^{-2}\sum_{f}(\partial_{\beta\phi\phi'\phi_{f}}\mathcal{L})\big[\mathcal{H}^{-1}\mathcal{H}_{b}\mathcal{H}^{-1}(\partial_{\beta\phi}\mathcal{L})\big]_{f}\\
& -W^{-2}\sum_{e,f}(\partial_{\phi\phi'\phi_{e}\phi_{f}}\mathcal{L})\big[\mathcal{H}^{-1}(\partial_{\beta\phi}\mathcal{L})\big]_{e}\big[\mathcal{H}^{-1}\mathcal{H}_{b}\mathcal{H}^{-1}(\partial_{\beta\phi}\mathcal{L})\big]_{f}\\
& -2W^{-1}\sum_{f}(\partial_{\phi\phi'\phi_{f}}\mathcal{L})\big[\mathcal{H}^{-1}\mathcal{H}_{b}\mathcal{H}^{-1}\mathcal{H}_{b}\mathcal{H}^{-1}(\partial_{\beta\phi}\mathcal{L})\big]_{f}\\
& +W^{-1}\sum_{f}(\partial_{\phi\phi'\phi_{f}}\mathcal{L})\big[\mathcal{H}^{-1}\mathcal{H}_{bb}\mathcal{H}^{-1}(\partial_{\beta\phi}\mathcal{L})\big]_{f}\\
& -W^{-2}\sum_{f}(\partial_{\phi\phi'\phi_{f}}\mathcal{L})\big[\mathcal{H}^{-1}\mathcal{H}_{b}\mathcal{H}^{-1}(\partial_{\beta\beta\phi}\mathcal{L})\big]_{f}\\
& -W^{-2}\sum_{f}(\partial_{\phi\phi'\phi_{f}}\mathcal{L})\big[\mathcal{H}^{-1}\mathcal{H}_{b}\mathcal{H}^{-1}(\partial_{\beta\phi\phi'}\mathcal{L})\mathcal{H}^{-1}(\partial_{\beta\phi}\mathcal{L})\big]_{f}\\
& +W^{-3}W_{b}\sum_{f}(\partial_{\phi\phi'\phi_{f}}\mathcal{L})\big[\mathcal{H}^{-1}(\partial_{\beta\phi}\mathcal{L})\big]_{f}\big[\mathcal{H}^{-1}(\partial_{\beta\phi}\mathcal{L})\big]_{g}\\
& +W^{-3}\sum_{f}(\partial_{\beta\phi\phi'\phi_{f}}\mathcal{L})\big[\mathcal{H}^{-1}(\partial_{\beta\phi}\mathcal{L})\big]_{f}\big[\mathcal{H}^{-1}(\partial_{\beta\phi}\mathcal{L})\big]_{g}\\
& +W^{-3}\sum_{e,f}(\partial_{\phi\phi'\phi_{e}\phi_{f}}\mathcal{L})\big[\mathcal{H}^{-1}(\partial_{\beta\phi}\mathcal{L})\big]_{e}\big[\mathcal{H}^{-1}(\partial_{\beta\phi}\mathcal{L})\big]_{f}\big[\mathcal{H}^{-1}(\partial_{\beta\phi}\mathcal{L})\big]_{g}\\
& +W^{-2}\sum_{f}(\partial_{\phi\phi'\phi_{f}}\mathcal{L})\big[\mathcal{H}^{-1}\mathcal{H}_{b}\mathcal{H}^{-1}(\partial_{\beta\phi}\mathcal{L})\big]_{f}\big[\mathcal{H}^{-1}(\partial_{\beta\phi}\mathcal{L})\big]_{g}\\
& +W^{-3}\sum_{f}(\partial_{\phi\phi'\phi_{f}}\mathcal{L})\big[\mathcal{H}^{-1}(\partial_{\beta\beta\phi}\mathcal{L})\big]_{f}\big[\mathcal{H}^{-1}(\partial_{\beta\phi}\mathcal{L})\big]_{g}\\
& +W^{-3}\sum_{f}(\partial_{\phi\phi'\phi_{f}}\mathcal{L})\big[\mathcal{H}^{-1}(\partial_{\beta\phi\phi'}\mathcal{L})\mathcal{H}^{-1}(\partial_{\beta\phi}\mathcal{L})\big]_{f}\big[\mathcal{H}^{-1}(\partial_{\beta\phi}\mathcal{L})\big]_{g}\\
& +W^{-2}\sum_{f}(\partial_{\phi\phi'\phi_{f}}\mathcal{L})\big[\mathcal{H}^{-1}(\partial_{\beta\phi}\mathcal{L})\big]_{f}\big[\mathcal{H}^{-1}\mathcal{H}_{b}\mathcal{H}^{-1}(\partial_{\beta\phi}\mathcal{L})\big]_{g}\\
& +W^{-3}\sum_{f}(\partial_{\phi\phi'\phi_{f}}\mathcal{L})\big[\mathcal{H}^{-1}(\partial_{\beta\phi}\mathcal{L})\big]_{f}\big[\mathcal{H}^{-1}(\partial_{\beta\beta\phi}\mathcal{L})\big]_{g}\\
& +W^{-3}\sum_{f}(\partial_{\phi\phi'\phi_{f}}\mathcal{L})\big[\mathcal{H}^{-1}(\partial_{\beta\phi}\mathcal{L})\big]_{f}\big[\mathcal{H}^{-1}(\partial_{\beta\phi\phi'}\mathcal{L})\mathcal{H}^{-1}(\partial_{\beta\phi}\mathcal{L})\big]_{g}\\
& -W^{-2}W_{b}\sum_{f}(\partial_{\phi\phi'\phi_{f}}\mathcal{L})\big[\mathcal{H}^{-1}\mathcal{E}^{1}\mathcal{H}^{-1}\big]_{fg}\\
& -W^{-2}\sum_{f}(\partial_{\beta\phi\phi'\phi_{f}}\mathcal{L})\big[\mathcal{H}^{-1}\mathcal{E}^{1}\mathcal{H}^{-1}\big]_{fg}\\
& -W^{-2}\sum_{e,f}(\partial_{\phi\phi'\phi_{e}\phi_{f}}\mathcal{L})\big[\mathcal{H}^{-1}(\partial_{\beta\phi}\mathcal{L})\big]_{e}\big[\mathcal{H}^{-1}\mathcal{E}^{1}\mathcal{H}^{-1}\big]_{fg}\\
& -W^{-1}\sum_{f}(\partial_{\phi\phi'\phi_{f}}\mathcal{L})\big[\mathcal{H}^{-1}\mathcal{H}_{b}\mathcal{H}^{-1}\mathcal{E}^{1}\mathcal{H}^{-1}\big]_{fg}\\
& -W^{-1}\sum_{f}(\partial_{\phi\phi'\phi_{f}}\mathcal{L})\big[\mathcal{H}^{-1}\mathcal{E}^{1}\mathcal{H}^{-1}\mathcal{H}_{b}\mathcal{H}^{-1}\big]_{fg}\\
& +W^{-1}\sum_{f}(\partial_{\phi\phi'\phi_{f}}\mathcal{L})\big[\mathcal{H}^{-1}\mathcal{E}_{b}^{1}\mathcal{H}^{-1}\big]_{fg}\\
& +2W^{-3}W_{b}\sum_{f}(\partial_{\phi\phi'\phi_{f}}\mathcal{L})\big[\mathcal{H}^{-1}\mathcal{E}^{1}\mathcal{H}^{-1}(\partial_{\beta\phi}\mathcal{L})\big]_{f}\big[\mathcal{H}^{-1}(\partial_{\beta\phi}\mathcal{L})\big]_{g}\\
& +W^{-3}\sum_{f}(\partial_{\beta\phi\phi'\phi_{f}}\mathcal{L})\big[\mathcal{H}^{-1}\mathcal{E}^{1}\mathcal{H}^{-1}(\partial_{\beta\phi}\mathcal{L})\big]_{f}\big[\mathcal{H}^{-1}(\partial_{\beta\phi}\mathcal{L})\big]_{g}\\
& +W^{-3}\sum_{e,f}(\partial_{\phi\phi'\phi_{e}\phi_{f}}\mathcal{L})\big[\mathcal{H}^{-1}(\partial_{\beta\phi}\mathcal{L})\big]_{e}\big[\mathcal{H}^{-1}\mathcal{E}^{1}\mathcal{H}^{-1}(\partial_{\beta\phi}\mathcal{L})\big]_{f}\big[\mathcal{H}^{-1}(\partial_{\beta\phi}\mathcal{L})\big]_{g}\\
& +W^{-2}\sum_{f}(\partial_{\phi\phi'\phi_{f}}\mathcal{L})\big[\mathcal{H}^{-1}\mathcal{H}_{b}\mathcal{H}^{-1}\mathcal{E}^{1}\mathcal{H}^{-1}(\partial_{\beta\phi}\mathcal{L})\big]_{f}\big[\mathcal{H}^{-1}(\partial_{\beta\phi}\mathcal{L})\big]_{g}\\
& -W^{-2}\sum_{f}(\partial_{\phi\phi'\phi_{f}}\mathcal{L})\big[\mathcal{H}^{-1}\mathcal{E}_{b}^{1}\mathcal{H}^{-1}(\partial_{\beta\phi}\mathcal{L})\big]_{f}\big[\mathcal{H}^{-1}(\partial_{\beta\phi}\mathcal{L})\big]_{g}\\
& +W^{-2}\sum_{f}(\partial_{\phi\phi'\phi_{f}}\mathcal{L})\big[\mathcal{H}^{-1}\mathcal{E}^{1}\mathcal{H}^{-1}\mathcal{H}_{b}\mathcal{H}^{-1}(\partial_{\beta\phi}\mathcal{L})\big]_{f}\big[\mathcal{H}^{-1}(\partial_{\beta\phi}\mathcal{L})\big]_{g}\\
& +W^{-3}\sum_{f}(\partial_{\phi\phi'\phi_{f}}\mathcal{L})\big[\mathcal{H}^{-1}\mathcal{E}^{1}\mathcal{H}^{-1}(\partial_{\beta\beta\phi}\mathcal{L})\big]_{f}\big[\mathcal{H}^{-1}(\partial_{\beta\phi}\mathcal{L})\big]_{g}\\
& +W^{-3}\sum_{f}(\partial_{\phi\phi'\phi_{f}}\mathcal{L})\big[\mathcal{H}^{-1}\mathcal{E}^{1}\mathcal{H}^{-1}\mathcal{E}^{1}\mathcal{H}^{-1}(\partial_{\beta\phi}\mathcal{L})\big]_{f}\big[\mathcal{H}^{-1}(\partial_{\beta\phi}\mathcal{L})\big]_{g}\\
& +W^{-2}\sum_{f}(\partial_{\phi\phi'\phi_{f}}\mathcal{L})\big[\mathcal{H}^{-1}\mathcal{E}^{1}\mathcal{H}^{-1}(\partial_{\beta\phi}\mathcal{L})\big]_{f}\big[\mathcal{H}^{-1}\mathcal{H}_{b}\mathcal{H}^{-1}(\partial_{\beta\phi}\mathcal{L})\big]_{g}\\
& +W^{-3}\sum_{f}(\partial_{\phi\phi'\phi_{f}}\mathcal{L})\big[\mathcal{H}^{-1}\mathcal{E}^{1}\mathcal{H}^{-1}(\partial_{\beta\phi}\mathcal{L})\big]_{f}\big[\mathcal{H}^{-1}(\partial_{\beta\beta\phi}\mathcal{L})\big]_{g}\\
& +W^{-3}\sum_{f}(\partial_{\phi\phi'\phi_{f}}\mathcal{L})\big[\mathcal{H}^{-1}\mathcal{E}^{1}\mathcal{H}^{-1}(\partial_{\beta\phi}\mathcal{L})\big]_{f}\big[\mathcal{H}^{-1}\mathcal{E}^{1}\mathcal{H}^{-1}(\partial_{\beta\beta\phi}\mathcal{L})\big]_{g}
\end{align*}
\begin{align*}
\mathcal{H}_{bs_{g}s_{h}} & =2W^{-3}W_{b}W_{s_{h}}[\mathcal{R}^{1}]_{g}(\partial_{\beta\phi\phi'}\mathcal{L})-W^{-2}W_{bs_{h}}[\mathcal{R}^{1}]_{g}(\partial_{\beta\phi\phi'}\mathcal{L})\\
& -W^{-2}W_{s_{h}}[\mathcal{R}_{b}^{1}]_{g}(\partial_{\beta\phi\phi'}\mathcal{L})+W^{-3}W_{s_{h}}[\mathcal{R}^{1}]_{g}(\partial_{\beta^{2}\phi\phi'}\mathcal{L})\\
& +W^{-3}W_{s_{h}}[\mathcal{R}^{1}]_{g}\mathcal{P}^{(1,1)}-W^{-2}W_{b}[\mathcal{R}_{s_{h}}^{1}]_{g}(\partial_{\beta\phi\phi'}\mathcal{L})\\
& +W^{-1}[\mathcal{R}_{bs_{h}}^{1}]_{g}(\partial_{\beta\phi\phi'}\mathcal{L})-W^{-2}[\mathcal{R}_{s_{h}}^{1}]_{g}(\partial_{\beta^{2}\phi\phi'}\mathcal{L})-W^{-3}[\mathcal{R}_{s_{h}}^{1}]_{g}\mathcal{P}^{(1,1)}\\
& +2W^{-3}W_{b}[\mathcal{R}^{1}]_{g}(\partial_{\beta^{2}\phi\phi'}\mathcal{L})[\mathcal{R}^{1}]_{h}-W^{-2}[\mathcal{R}_{b}^{1}]_{g}(\partial_{\beta^{2}\phi\phi'}\mathcal{L})[\mathcal{R}^{1}]_{h}\\
& +W^{-3}[\mathcal{R}^{1}]_{g}(\partial_{\beta^{3}\phi\phi'}\mathcal{L})[\mathcal{R}^{1}]_{h}+W^{-3}[\mathcal{R}^{1}]_{g}\mathcal{P}^{(2,1)}[\mathcal{R}^{1}]_{h}\\
& -W^{-2}[\mathcal{R}^{1}]_{g}(\partial_{\beta^{2}\phi\phi'}\mathcal{L})[\mathcal{R}_{b}^{1}]_{h}+W^{-2}W_{b}[\mathcal{R}^{1}]_{g}\mathcal{P}_{(h)}^{1}\\
& -W^{-1}[\mathcal{R}_{b}^{1}]_{g}\mathcal{P}_{(h)}^{1}-W^{-1}[\mathcal{R}^{1}]_{g}\mathcal{P}_{(h),b}^{1}\\
& +2W^{-3}W_{b}[\mathcal{R}^{1}]_{g}\mathcal{P}^{(1,1)}[\mathcal{R}^{1}]_{h}-W^{-2}[\mathcal{R}_{b}^{1}]_{g}\mathcal{P}^{(1,1)}[\mathcal{R}^{1}]_{h}\\
& -W^{-2}[\mathcal{R}^{1}]_{g}\mathcal{P}_{b}^{(1,1)}[\mathcal{R}^{1}]_{h}-W^{-2}[\mathcal{R}^{1}]_{g}\mathcal{P}^{(1,1)}[\mathcal{R}_{b}^{1}]_{h}\\
& +\mathcal{P}_{(g),bs_{h}}^{0}\\
& +2W^{-3}W_{b}W_{s_{h}}\mathcal{P}^{(0,1)}[\mathcal{R}^{1}]_{g}-W^{-2}W_{bs_{h}}\mathcal{P}^{(0,1)}[\mathcal{R}^{1}]_{g}\\
& -W^{-2}W_{s_{h}}\mathcal{P}_{b}^{(0,1)}[\mathcal{R}^{1}]_{g}-W^{-2}W_{s_{h}}\mathcal{P}^{(0,1)}[\mathcal{R}_{b}^{1}]_{g}\\
& -W^{-2}W_{b}\mathcal{P}_{s_{h}}^{(0,1)}[\mathcal{R}^{1}]_{g}+W^{-1}\mathcal{P}_{bs_{h}}^{(0,1)}[\mathcal{R}^{1}]_{g}+W^{-1}\mathcal{P}_{s_{h}}^{(0,1)}[\mathcal{R}_{b}^{1}]_{g}\\
& -W^{-2}W_{b}\mathcal{P}^{(0,1)}[\mathcal{R}_{s_{h}}^{1}]_{g}+W^{-1}\mathcal{P}_{b}^{(0,1)}[\mathcal{R}_{s_{h}}^{1}]_{g}+W^{-1}\mathcal{P}^{(0,1)}[\mathcal{R}_{bs_{h}}^{1}]_{g}
\end{align*}
\begin{align*}
\mathcal{H}_{s_{f}s_{g}s_{h}} & =2W^{-3}W_{s_{f}}W_{s_{h}}[\mathcal{R}^{1}]_{g}(\partial_{\beta\phi\phi'}\mathcal{L})-W^{-2}W_{s_{f}s_{h}}[\mathcal{R}^{1}]_{g}(\partial_{\beta\phi\phi'}\mathcal{L})\\
& -W^{-2}W_{s_{h}}[\mathcal{R}_{s_{f}}^{1}]_{g}(\partial_{\beta\phi\phi'}\mathcal{L})+W^{-3}W_{s_{h}}[\mathcal{R}^{1}]_{g}(\partial_{\beta^{2}\phi\phi'}\mathcal{L})[\mathcal{R}^{1}]_{f}\\
& +W^{-2}W_{s_{h}}[\mathcal{R}^{1}]_{g}\sum_{e}(\partial_{\beta\phi\phi'\phi_{e}}\mathcal{L})[\mathcal{H}^{-1}]_{ef}\\
& +W^{-3}W_{s_{h}}[\mathcal{R}^{1}]_{g}\sum_{e}(\partial_{\beta\phi\phi'\phi_{e}}\mathcal{L})[\mathcal{R}^{1}]_{e}[\mathcal{R}^{1}]_{f}\\
& -W^{-2}W_{s_{f}}[\mathcal{R}_{s_{h}}^{1}]_{g}(\partial_{\beta\phi\phi'}\mathcal{L})+W^{-1}[\mathcal{R}_{s_{f}s_{h}}^{1}]_{g}(\partial_{\beta\phi\phi'}\mathcal{L})\\
& -W^{-2}[\mathcal{R}_{s_{h}}^{1}]_{g}(\partial_{\beta^{2}\phi\phi'}\mathcal{L})[\mathcal{R}^{1}]_{f}-W^{-1}[\mathcal{R}_{s_{h}}^{1}]_{g}\sum_{e}(\partial_{\beta\phi\phi'\phi_{e}}\mathcal{L})[\mathcal{H}^{-1}]_{ef}\\
& -W^{-2}[\mathcal{R}_{s_{h}}^{1}]_{g}\sum_{e}(\partial_{\beta\phi\phi'\phi_{e}}\mathcal{L})[\mathcal{R}^{1}]_{e}[\mathcal{R}^{1}]_{f}\\
& +2W^{-3}W_{s_{f}}[\mathcal{R}^{1}]_{g}(\partial_{\beta^{2}\phi\phi'}\mathcal{L})[\mathcal{R}^{1}]_{h}-W^{-2}[\mathcal{R}_{s_{f}}^{1}]_{g}(\partial_{\beta^{2}\phi\phi'}\mathcal{L})[\mathcal{R}^{1}]_{h}\\
& +W^{-3}[\mathcal{R}^{1}]_{g}(\partial_{\beta^{3}\phi\phi'}\mathcal{L})[\mathcal{R}^{1}]_{f}[\mathcal{R}^{1}]_{h}+W^{-3}[\mathcal{R}^{1}]_{g}(\partial_{\beta^{2}\phi\phi'}\mathcal{L})[\mathcal{R}^{1}]_{h}\\
& -W^{-2}[\mathcal{R}^{1}]_{g}\sum_{e}(\partial_{\beta^{2}\phi\phi'\phi_{e}}\mathcal{L})[\mathcal{R}^{1}]_{e}[\mathcal{R}^{1}]_{f}[\mathcal{R}^{1}]_{h}\\
& +W^{-2}W_{s_{f}}[\mathcal{R}^{1}]_{g}\mathcal{P}_{(h)}^{1}-W^{-1}[\mathcal{R}_{s_{f}}^{1}]_{g}\mathcal{P}_{(h)}^{1}-W^{-1}[\mathcal{R}^{1}]_{g}\mathcal{P}_{(h),s_{f}}^{1}\\
& +2W^{-3}W_{s_{f}}[\mathcal{R}^{1}]_{g}\mathcal{P}^{(1,1)}[\mathcal{R}^{1}]_{h}-W^{-2}[\mathcal{R}_{s_{f}}^{1}]_{g}\mathcal{P}^{(1,1)}[\mathcal{R}^{1}]_{h}\\
& -W^{-2}[\mathcal{R}^{1}]_{g}\mathcal{P}_{s_{f}}^{(1,1)}[\mathcal{R}^{1}]_{h}-W^{-2}[\mathcal{R}^{1}]_{g}\mathcal{P}^{(1,1)}[\mathcal{R}_{s_{f}}^{1}]_{h}\\
& +\mathcal{P}_{(g),s_{f}s_{h}}^{0}\\
& +2W^{-3}W_{s_{f}}W_{s_{h}}\mathcal{P}^{(0,1)}[\mathcal{R}^{1}]_{g}-W^{-2}W_{s_{f}s_{h}}\mathcal{P}^{(0,1)}[\mathcal{R}^{1}]_{g}\\
& -W^{-2}W_{s_{h}}\mathcal{P}_{s_{f}}^{(0,1)}[\mathcal{R}^{1}]_{g}-W^{-2}W_{s_{h}}\mathcal{P}^{(0,1)}[\mathcal{R}_{s_{f}}^{1}]_{g}\\
& -W^{-2}W_{s_{f}}\mathcal{P}_{s_{h}}^{(0,1)}[\mathcal{R}^{1}]_{g}+W^{-1}\mathcal{P}_{s_{f}s_{h}}^{(0,1)}[\mathcal{R}^{1}]_{g}+W^{-1}\mathcal{P}_{s_{h}}^{(0,1)}[\mathcal{R}_{s_{f}}^{1}]_{g}\\
& -W^{-2}W_{s_{f}}\mathcal{P}^{(0,1)}[\mathcal{R}_{s_{h}}^{1}]_{g}+W^{-1}\mathcal{P}_{s_{f}}^{(0,1)}[\mathcal{R}_{s_{h}}^{1}]_{g}+W^{-1}\mathcal{P}^{(0,1)}[\mathcal{R}_{s_{f}s_{h}}^{1}]_{g}
\end{align*}
Fourth derivative:
\begin{align*}
\mathcal{H}_{bs_{f}s_{g}s_{h}} & =-6W^{-4}W_{b}W_{s_{f}}W_{s_{h}}[\mathcal{R}^{1}]_{g}(\partial_{\beta\phi\phi'}\mathcal{L})+2W^{-3}W_{bs_{f}}W_{s_{h}}[\mathcal{R}^{1}]_{g}(\partial_{\beta\phi\phi'}\mathcal{L})\\
& +2W^{-3}W_{s_{f}}W_{bs_{h}}[\mathcal{R}^{1}]_{g}(\partial_{\beta\phi\phi'}\mathcal{L})+2W^{-3}W_{s_{f}}W_{s_{h}}[\mathcal{R}_{b}^{1}]_{g}(\partial_{\beta\phi\phi'}\mathcal{L})\\
& -2W^{-4}W_{s_{f}}W_{s_{h}}[\mathcal{R}^{1}]_{g}(\partial_{\beta^{2}\phi\phi'}\mathcal{L})-2W^{-4}W_{s_{f}}W_{s_{h}}[\mathcal{R}^{1}]_{g}\sum_{e}(\partial_{\beta\phi\phi'\phi_{e}}\mathcal{L})[\mathcal{R}^{1}]_{e}\\
& +2W^{-3}W_{b}W_{s_{f}s_{h}}[\mathcal{R}^{1}]_{g}(\partial_{\beta\phi\phi'}\mathcal{L})-W^{-2}W_{bs_{f}s_{h}}[\mathcal{R}^{1}]_{g}(\partial_{\beta\phi\phi'}\mathcal{L})\\
& -W^{-2}W_{s_{f}s_{h}}[\mathcal{R}_{b}^{1}]_{g}(\partial_{\beta\phi\phi'}\mathcal{L})+W^{-3}W_{s_{f}s_{h}}[\mathcal{R}^{1}]_{g}(\partial_{\beta^{2}\phi\phi'}\mathcal{L})\\
& +W^{-3}W_{s_{f}s_{h}}[\mathcal{R}^{1}]_{g}\sum_{e}(\partial_{\beta\phi\phi'\phi_{e}}\mathcal{L})[\mathcal{R}^{1}]_{e}+2W^{-3}W_{b}W_{s_{h}}[\mathcal{R}_{s_{f}}^{1}]_{g}(\partial_{\beta\phi\phi'}\mathcal{L})\\
& -W^{-2}W_{bs_{h}}[\mathcal{R}_{s_{f}}^{1}]_{g}(\partial_{\beta\phi\phi'}\mathcal{L})-W^{-2}W_{s_{h}}[\mathcal{R}_{bs_{f}}^{1}]_{g}(\partial_{\beta\phi\phi'}\mathcal{L})\\
& +W^{-3}W_{s_{h}}[\mathcal{R}_{s_{f}}^{1}]_{g}(\partial_{\beta^{2}\phi\phi'}\mathcal{L})+W^{-3}W_{s_{h}}[\mathcal{R}_{s_{f}}^{1}]_{g}\sum_{e}(\partial_{\beta\phi\phi'\phi_{e}}\mathcal{L})[\mathcal{R}^{1}]_{e}\\
& -3W^{-4}W_{b}W_{s_{h}}[\mathcal{R}^{1}]_{g}(\partial_{\beta^{2}\phi\phi'}\mathcal{L})[\mathcal{R}^{1}]_{f}+W^{-3}W_{bs_{h}}[\mathcal{R}^{1}]_{g}(\partial_{\beta^{2}\phi\phi'}\mathcal{L})[\mathcal{R}^{1}]_{f}\\
& +W^{-3}W_{s_{h}}[\mathcal{R}_{b}^{1}]_{g}(\partial_{\beta^{2}\phi\phi'}\mathcal{L})[\mathcal{R}^{1}]_{f}-W^{-4}W_{s_{h}}[\mathcal{R}^{1}]_{g}(\partial_{\beta^{3}\phi\phi'}\mathcal{L})[\mathcal{R}^{1}]_{f}\\
& -W^{-4}W_{s_{h}}[\mathcal{R}^{1}]_{g}\sum_{e}(\partial_{\beta^{2}\phi\phi'\phi_{e}}\mathcal{L})[\mathcal{R}^{1}]_{e}[\mathcal{R}^{1}]_{f}\\
& +W^{-3}W_{s_{h}}[\mathcal{R}^{1}]_{g}(\partial_{\beta^{2}\phi\phi'}\mathcal{L})[\mathcal{R}_{b}^{1}]_{f}-2W^{-3}W_{b}W_{s_{h}}[\mathcal{R}^{1}]_{g}\sum_{e}(\partial_{\beta\phi\phi'\phi_{e}}\mathcal{L})[\mathcal{H}^{-1}]_{ef}\\
& +W^{-2}W_{bs_{h}}[\mathcal{R}^{1}]_{g}\sum_{e}(\partial_{\beta\phi\phi'\phi_{e}}\mathcal{L})[\mathcal{H}^{-1}]_{ef}+W^{-2}W_{s_{h}}[\mathcal{R}_{b}^{1}]_{g}\sum_{e}(\partial_{\beta\phi\phi'\phi_{e}}\mathcal{L})[\mathcal{H}^{-1}]_{ef}\\
& -W^{-3}W_{s_{h}}[\mathcal{R}^{1}]_{g}\sum_{e}(\partial_{\beta^{2}\phi\phi'\phi_{e}}\mathcal{L})[\mathcal{H}^{-1}]_{ef}\\
& -W^{-3}W_{s_{h}}[\mathcal{R}^{1}]_{g}\sum_{d,e}(\partial_{\beta\phi\phi'\phi_{d}\phi_{e}}\mathcal{L})[\mathcal{H}^{-1}]_{ef}[\mathcal{R}^{1}]_{d}\\
& -W^{-2}W_{s_{h}}[\mathcal{R}^{1}]_{g}\sum_{e}(\partial_{\beta\phi\phi'\phi_{e}}\mathcal{L})[\mathcal{H}^{-1}\mathcal{H}_{b}\mathcal{H}^{-1}]_{ef}\\
& -3W^{-4}W_{b}W_{s_{h}}[\mathcal{R}^{1}]_{g}\sum_{e}(\partial_{\beta\phi\phi'\phi_{e}}\mathcal{L})[\mathcal{R}^{1}]_{e}[\mathcal{R}^{1}]_{f}\\
& +W^{-3}W_{bs_{h}}[\mathcal{R}^{1}]_{g}\sum_{e}(\partial_{\beta\phi\phi'\phi_{e}}\mathcal{L})[\mathcal{R}^{1}]_{e}[\mathcal{R}^{1}]_{f}\\
& +W^{-3}W_{s_{h}}[\mathcal{R}_{b}^{1}]_{g}\sum_{e}(\partial_{\beta\phi\phi'\phi_{e}}\mathcal{L})[\mathcal{R}^{1}]_{e}[\mathcal{R}^{1}]_{f}\\
& -W^{-4}W_{s_{h}}[\mathcal{R}^{1}]_{g}\sum_{e}(\partial_{\beta^{2}\phi\phi'\phi_{e}}\mathcal{L})[\mathcal{R}^{1}]_{e}[\mathcal{R}^{1}]_{f}\\
& -W^{-4}W_{s_{h}}[\mathcal{R}^{1}]_{g}\sum_{d,e}(\partial_{\beta\phi\phi'\phi_{d}\phi_{e}}\mathcal{L})[\mathcal{R}^{1}]_{d}[\mathcal{R}^{1}]_{e}[\mathcal{R}^{1}]_{f}\\
& +W^{-3}W_{s_{h}}[\mathcal{R}^{1}]_{g}\sum_{e}(\partial_{\beta\phi\phi'\phi_{e}}\mathcal{L})[\mathcal{R}_{b}^{1}]_{e}[\mathcal{R}^{1}]_{f}\\
& +W^{-3}W_{s_{h}}[\mathcal{R}^{1}]_{g}\sum_{e}(\partial_{\beta\phi\phi'\phi_{e}}\mathcal{L})[\mathcal{R}^{1}]_{e}[\mathcal{R}_{b}^{1}]_{f}\\
& +2W^{-3}W_{b}W_{s_{f}}[\mathcal{R}_{s_{h}}^{1}]_{g}(\partial_{\beta\phi\phi'}\mathcal{L})-W^{-2}W_{bs_{f}}[\mathcal{R}_{s_{h}}^{1}]_{g}(\partial_{\beta\phi\phi'}\mathcal{L})\\
& -W^{-2}W_{s_{f}}[\mathcal{R}_{bs_{h}}^{1}]_{g}(\partial_{\beta\phi\phi'}\mathcal{L})+W^{-3}W_{s_{f}}[\mathcal{R}_{s_{h}}^{1}]_{g}(\partial_{\beta^{2}\phi\phi'}\mathcal{L})\\
& +W^{-3}W_{s_{f}}[\mathcal{R}_{s_{h}}^{1}]_{g}\sum_{e}(\partial_{\beta\phi\phi'\phi_{e}}\mathcal{L})[\mathcal{R}^{1}]_{e}-W^{-2}[\mathcal{R}_{s_{f}s_{h}}^{1}]_{g}(\partial_{\beta\phi\phi'}\mathcal{L})\\
& +W^{-1}[\mathcal{R}_{bs_{f}s_{h}}^{1}]_{g}(\partial_{\beta\phi\phi'}\mathcal{L})-W^{-2}[\mathcal{R}_{s_{f}s_{h}}^{1}]_{g}(\partial_{\beta^{2}\phi\phi'}\mathcal{L})\\
& -W^{-2}[\mathcal{R}_{s_{f}s_{h}}^{1}]_{g}\sum_{e}(\partial_{\beta\phi\phi'\phi_{e}}\mathcal{L})[\mathcal{R}^{1}]_{e}+2W^{-3}W_{b}[\mathcal{R}_{s_{h}}^{1}]_{g}(\partial_{\beta^{2}\phi\phi'}\mathcal{L})[\mathcal{R}^{1}]_{f}\\
& -W^{-2}[\mathcal{R}_{bs_{h}}^{1}]_{g}(\partial_{\beta^{2}\phi\phi'}\mathcal{L})[\mathcal{R}^{1}]_{f}+W^{-3}[\mathcal{R}_{s_{h}}^{1}]_{g}(\partial_{\beta^{3}\phi\phi'}\mathcal{L})[\mathcal{R}^{1}]_{f}\\
& +W^{-3}[\mathcal{R}_{s_{h}}^{1}]_{g}\sum_{e}(\partial_{\beta^{2}\phi\phi'\phi_{e}}\mathcal{L})[\mathcal{R}^{1}]_{e}[\mathcal{R}^{1}]_{f}-W^{-2}[\mathcal{R}_{s_{h}}^{1}]_{g}(\partial_{\beta^{2}\phi\phi'}\mathcal{L})[\mathcal{R}_{b}^{1}]_{f}\\
& +W^{-2}W_{b}[\mathcal{R}_{s_{h}}^{1}]_{g}\sum_{e}(\partial_{\beta\phi\phi'\phi_{e}}\mathcal{L})[\mathcal{H}^{-1}]_{ef}-W^{-1}[\mathcal{R}_{bs_{h}}^{1}]_{g}\sum_{e}(\partial_{\beta\phi\phi'\phi_{e}}\mathcal{L})[\mathcal{H}^{-1}]_{ef}\\
& +W^{-2}[\mathcal{R}_{s_{h}}^{1}]_{g}\sum_{e}(\partial_{\beta^{2}\phi\phi'\phi_{e}}\mathcal{L})[\mathcal{H}^{-1}]_{ef}\\
& +W^{-2}[\mathcal{R}_{s_{h}}^{1}]_{g}\sum_{d,e}(\partial_{\beta\phi\phi'\phi_{d}\phi_{e}}\mathcal{L})[\mathcal{H}^{-1}]_{ef}[\mathcal{R}^{1}]_{d}\\
& +W^{-1}[\mathcal{R}_{s_{h}}^{1}]_{g}\sum_{e}(\partial_{\beta\phi\phi'\phi_{e}}\mathcal{L})[\mathcal{H}^{-1}\mathcal{H}_{b}\mathcal{H}^{-1}]_{ef}\\
& +2W^{-3}W_{b}[\mathcal{R}_{s_{h}}^{1}]_{g}\sum_{e}(\partial_{\beta\phi\phi'\phi_{e}}\mathcal{L})[\mathcal{R}^{1}]_{e}[\mathcal{R}^{1}]_{f}\\
& -W^{-2}[\mathcal{R}_{bs_{h}}^{1}]_{g}\sum_{e}(\partial_{\beta\phi\phi'\phi_{e}}\mathcal{L})[\mathcal{R}^{1}]_{e}[\mathcal{R}^{1}]_{f}\\
& +W^{-3}[\mathcal{R}_{s_{h}}^{1}]_{g}\sum_{e}(\partial_{\beta^{2}\phi\phi'\phi_{e}}\mathcal{L})[\mathcal{R}^{1}]_{e}[\mathcal{R}^{1}]_{f}\\
& +W^{-3}[\mathcal{R}_{s_{h}}^{1}]_{g}\sum_{d,e}(\partial_{\beta\phi\phi'\phi_{d}\phi_{e}}\mathcal{L})[\mathcal{R}^{1}]_{d}[\mathcal{R}^{1}]_{e}[\mathcal{R}^{1}]_{f}\\
& -W^{-2}[\mathcal{R}_{s_{h}}^{1}]_{g}\sum_{e}(\partial_{\beta\phi\phi'\phi_{e}}\mathcal{L})[\mathcal{R}_{b}^{1}]_{e}[\mathcal{R}^{1}]_{f}\\
& -W^{-2}[\mathcal{R}_{s_{h}}^{1}]_{g}\sum_{e}(\partial_{\beta\phi\phi'\phi_{e}}\mathcal{L})[\mathcal{R}^{1}]_{e}[\mathcal{R}_{b}^{1}]_{f}\\
& -6W^{-4}W_{b}W_{s_{f}}[\mathcal{R}^{1}]_{g}(\partial_{\beta^{2}\phi\phi'}\mathcal{L})[\mathcal{R}^{1}]_{h}+2W^{-3}W_{bs_{f}}[\mathcal{R}^{1}]_{g}(\partial_{\beta^{2}\phi\phi'}\mathcal{L})[\mathcal{R}^{1}]_{h}\\
& +2W^{-3}W_{s_{f}}[\mathcal{R}_{b}^{1}]_{g}(\partial_{\beta^{2}\phi\phi'}\mathcal{L})[\mathcal{R}^{1}]_{h}-2W^{-4}W_{s_{f}}[\mathcal{R}^{1}]_{g}(\partial_{\beta^{3}\phi\phi'}\mathcal{L})[\mathcal{R}^{1}]_{h}\\
& -2W^{-4}W_{s_{f}}[\mathcal{R}^{1}]_{g}\sum_{e}(\partial_{\beta^{2}\phi\phi'\phi_{e}}\mathcal{L})[\mathcal{R}^{1}]_{e}[\mathcal{R}^{1}]_{h}\\
& +2W^{-3}W_{s_{f}}[\mathcal{R}^{1}]_{g}(\partial_{\beta^{2}\phi\phi'}\mathcal{L})[\mathcal{R}_{b}^{1}]_{h}+2W^{-3}W_{b}[\mathcal{R}_{s_{f}}^{1}]_{g}(\partial_{\beta^{2}\phi\phi'}\mathcal{L})[\mathcal{R}^{1}]_{h}\\
& -W^{-2}[\mathcal{R}_{bs_{f}}^{1}]_{g}(\partial_{\beta^{2}\phi\phi'}\mathcal{L})[\mathcal{R}^{1}]_{h}+W^{-3}[\mathcal{R}_{s_{f}}^{1}]_{g}(\partial_{\beta^{3}\phi\phi'}\mathcal{L})[\mathcal{R}^{1}]_{h}\\
& +W^{-3}[\mathcal{R}_{s_{f}}^{1}]_{g}\sum_{e}(\partial_{\beta^{2}\phi\phi'\phi_{e}}\mathcal{L})[\mathcal{R}^{1}]_{e}[\mathcal{R}^{1}]_{h}\\
& -W^{-2}[\mathcal{R}_{s_{f}}^{1}]_{g}(\partial_{\beta^{2}\phi\phi'}\mathcal{L})[\mathcal{R}_{b}^{1}]_{h}-3W^{-4}W_{b}[\mathcal{R}^{1}]_{g}(\partial_{\beta^{3}\phi\phi'}\mathcal{L})[\mathcal{R}^{1}]_{f}[\mathcal{R}^{1}]_{h}\\
& +W^{-3}[\mathcal{R}_{b}^{1}]_{g}(\partial_{\beta^{3}\phi\phi'}\mathcal{L})[\mathcal{R}^{1}]_{f}[\mathcal{R}^{1}]_{h}-W^{-4}[\mathcal{R}^{1}]_{g}(\partial_{\beta^{4}\phi\phi'}\mathcal{L})[\mathcal{R}^{1}]_{f}[\mathcal{R}^{1}]_{h}\\
& -W^{-4}[\mathcal{R}^{1}]_{g}\sum_{e}(\partial_{\beta^{3}\phi\phi'\phi_{e}}\mathcal{L})[\mathcal{R}^{1}]_{e}[\mathcal{R}^{1}]_{f}[\mathcal{R}^{1}]_{h}\\
& +W^{-3}[\mathcal{R}^{1}]_{g}(\partial_{\beta^{3}\phi\phi'}\mathcal{L})[\mathcal{R}_{b}^{1}]_{f}[\mathcal{R}^{1}]_{h}+W^{-3}[\mathcal{R}^{1}]_{g}(\partial_{\beta^{3}\phi\phi'}\mathcal{L})[\mathcal{R}^{1}]_{f}[\mathcal{R}_{b}^{1}]_{h}\\
& -2W^{-3}W_{b}[\mathcal{R}^{1}]_{g}\sum_{e}(\partial_{\beta^{2}\phi\phi'\phi_{e}}\mathcal{L})[\mathcal{H}^{-1}]_{ef}[\mathcal{R}^{1}]_{h}\\
& +W^{-2}[\mathcal{R}_{b}^{1}]_{g}\sum_{e}(\partial_{\beta^{2}\phi\phi'\phi_{e}}\mathcal{L})[\mathcal{H}^{-1}]_{ef}[\mathcal{R}^{1}]_{h}\\
& -W^{-3}[\mathcal{R}^{1}]_{g}\sum_{e}(\partial_{\beta^{3}\phi\phi'\phi_{e}}\mathcal{L})[\mathcal{H}^{-1}]_{ef}[\mathcal{R}^{1}]_{h}\\
& -W^{-3}[\mathcal{R}^{1}]_{g}\sum_{d,e}(\partial_{\beta^{2}\phi\phi'\phi_{d}\phi_{e}}\mathcal{L})[\mathcal{H}^{-1}]_{ef}[\mathcal{R}^{1}]_{d}[\mathcal{R}^{1}]_{h}\\
& -W^{-2}[\mathcal{R}^{1}]_{g}\sum_{e}(\partial_{\beta^{2}\phi\phi'\phi_{e}}\mathcal{L})[\mathcal{H}^{-1}\mathcal{H}_{b}\mathcal{H}^{-1}]_{ef}[\mathcal{R}^{1}]_{h}\\
& +W^{-2}[\mathcal{R}^{1}]_{g}\sum_{e}(\partial_{\beta^{2}\phi\phi'\phi_{e}}\mathcal{L})[\mathcal{H}^{-1}]_{ef}[\mathcal{R}_{b}^{1}]_{h}\\
& -3W^{-4}W_{b}[\mathcal{R}^{1}]_{g}(\partial_{\beta^{2}\phi\phi'}\mathcal{L})[\mathcal{R}^{1}]_{h}+W^{-3}[\mathcal{R}_{b}^{1}]_{g}(\partial_{\beta^{2}\phi\phi'}\mathcal{L})[\mathcal{R}^{1}]_{h}\\
& -W^{-4}[\mathcal{R}^{1}]_{g}(\partial_{\beta^{3}\phi\phi'}\mathcal{L})[\mathcal{R}^{1}]_{h}-W^{-4}[\mathcal{R}^{1}]_{g}\sum_{e}(\partial_{\beta^{2}\phi\phi'\phi_{e}}\mathcal{L})[\mathcal{R}^{1}]_{e}[\mathcal{R}^{1}]_{h}\\
& +W^{-3}[\mathcal{R}^{1}]_{g}(\partial_{\beta^{2}\phi\phi'}\mathcal{L})[\mathcal{R}_{b}^{1}]_{h}\\
& +2W^{-3}W_{b}[\mathcal{R}^{1}]_{g}\sum_{e}(\partial_{\beta^{2}\phi\phi'\phi_{e}}\mathcal{L})[\mathcal{R}^{1}]_{e}[\mathcal{R}^{1}]_{f}[\mathcal{R}^{1}]_{h}\\
& -W^{-2}[\mathcal{R}_{b}^{1}]_{g}\sum_{e}(\partial_{\beta^{2}\phi\phi'\phi_{e}}\mathcal{L})[\mathcal{R}^{1}]_{e}[\mathcal{R}^{1}]_{f}[\mathcal{R}^{1}]_{h}\\
& +W^{-3}[\mathcal{R}^{1}]_{g}\sum_{e}(\partial_{\beta^{3}\phi\phi'\phi_{e}}\mathcal{L})[\mathcal{R}^{1}]_{e}[\mathcal{R}^{1}]_{f}[\mathcal{R}^{1}]_{h}\\
& +W^{-3}[\mathcal{R}^{1}]_{g}\sum_{d,e}(\partial_{\beta^{2}\phi\phi'\phi_{d}\phi_{e}}\mathcal{L})[\mathcal{R}^{1}]_{d}[\mathcal{R}^{1}]_{e}[\mathcal{R}^{1}]_{f}[\mathcal{R}^{1}]_{h}\\
& -W^{-2}[\mathcal{R}^{1}]_{g}\sum_{e}(\partial_{\beta^{2}\phi\phi'\phi_{e}}\mathcal{L})[\mathcal{R}_{b}^{1}]_{e}[\mathcal{R}^{1}]_{f}[\mathcal{R}^{1}]_{h}\\
& -W^{-2}[\mathcal{R}^{1}]_{g}\sum_{e}(\partial_{\beta^{2}\phi\phi'\phi_{e}}\mathcal{L})[\mathcal{R}^{1}]_{e}[\mathcal{R}_{b}^{1}]_{f}[\mathcal{R}^{1}]_{h}\\
& -W^{-2}[\mathcal{R}^{1}]_{g}\sum_{e}(\partial_{\beta^{2}\phi\phi'\phi_{e}}\mathcal{L})[\mathcal{R}^{1}]_{e}[\mathcal{R}^{1}]_{f}[\mathcal{R}_{b}^{1}]_{h}\\
& -2W^{-3}W_{b}W_{s_{f}}[\mathcal{R}^{1}]_{g}\mathcal{P}_{(h)}^{1}+W^{-2}W_{bs_{f}}[\mathcal{R}^{1}]_{g}\mathcal{P}_{(h)}^{1}\\
& +W^{-2}W_{s_{f}}[\mathcal{R}_{b}^{1}]_{g}\mathcal{P}_{(h)}^{1}+W^{-2}W_{s_{f}}[\mathcal{R}^{1}]_{g}\mathcal{P}_{(h),b}^{1}\\
& +W^{-2}W_{b}[\mathcal{R}_{s_{f}}^{1}]_{g}\mathcal{P}_{(h)}^{1}-W^{-1}[\mathcal{R}_{bs_{f}}^{1}]_{g}\mathcal{P}_{(h)}^{1}-W^{-1}[\mathcal{R}_{s_{f}}^{1}]_{g}\mathcal{P}_{(h),b}^{1}\\
& +W^{-2}W_{b}[\mathcal{R}^{1}]_{g}\mathcal{P}_{(h),s_{f}}^{1}-W^{-1}[\mathcal{R}_{b}^{1}]_{g}\mathcal{P}_{(h),s_{f}}^{1}-W^{-1}[\mathcal{R}^{1}]_{g}\mathcal{P}_{(h),bs_{f}}^{1}\\
& -6W^{-4}W_{b}W_{s_{f}}[\mathcal{R}^{1}]_{g}\mathcal{P}^{(1,1)}[\mathcal{R}^{1}]_{h}+2W^{-3}W_{bs_{f}}[\mathcal{R}^{1}]_{g}\mathcal{P}^{(1,1)}[\mathcal{R}^{1}]_{h}\\
& +2W^{-3}W_{s_{f}}[\mathcal{R}_{b}^{1}]_{g}\mathcal{P}^{(1,1)}[\mathcal{R}^{1}]_{h}+2W^{-3}W_{s_{f}}[\mathcal{R}^{1}]_{g}\mathcal{P}_{b}^{(1,1)}[\mathcal{R}^{1}]_{h}\\
& +2W^{-3}W_{s_{f}}[\mathcal{R}^{1}]_{g}\mathcal{P}^{(1,1)}[\mathcal{R}_{b}^{1}]_{h}\\
& +2W^{-3}W_{b}[\mathcal{R}_{s_{f}}^{1}]_{g}\mathcal{P}^{(1,1)}[\mathcal{R}^{1}]_{h}-W^{-2}[\mathcal{R}_{bs_{f}}^{1}]_{g}\mathcal{P}^{(1,1)}[\mathcal{R}^{1}]_{h}\\
& -W^{-2}[\mathcal{R}_{s_{f}}^{1}]_{g}\mathcal{P}_{b}^{(1,1)}[\mathcal{R}^{1}]_{h}-W^{-2}[\mathcal{R}_{s_{f}}^{1}]_{g}\mathcal{P}^{(1,1)}[\mathcal{R}_{b}^{1}]_{h}\\
& +2W^{-3}W_{b}[\mathcal{R}^{1}]_{g}\mathcal{P}_{s_{f}}^{(1,1)}[\mathcal{R}^{1}]_{h}-W^{-2}[\mathcal{R}_{b}^{1}]_{g}\mathcal{P}_{s_{f}}^{(1,1)}[\mathcal{R}^{1}]_{h}\\
& -W^{-2}[\mathcal{R}^{1}]_{g}\mathcal{P}_{bs_{f}}^{(1,1)}[\mathcal{R}^{1}]_{h}-W^{-2}[\mathcal{R}^{1}]_{g}\mathcal{P}_{s_{f}}^{(1,1)}[\mathcal{R}_{b}^{1}]_{h}\\
& +2W^{-3}W_{b}[\mathcal{R}^{1}]_{g}\mathcal{P}^{(1,1)}[\mathcal{R}_{s_{f}}^{1}]_{h}-W^{-2}[\mathcal{R}_{b}^{1}]_{g}\mathcal{P}^{(1,1)}[\mathcal{R}_{s_{f}}^{1}]_{h}\\
& -W^{-2}[\mathcal{R}^{1}]_{g}\mathcal{P}_{b}^{(1,1)}[\mathcal{R}_{s_{f}}^{1}]_{h}-W^{-2}[\mathcal{R}^{1}]_{g}\mathcal{P}^{(1,1)}[\mathcal{R}_{bs_{f}}^{1}]_{h}\\
& +\mathcal{P}_{(g),bs_{f}s_{h}}^{0}\\
& -6W^{-4}W_{b}W_{s_{f}}W_{s_{h}}\mathcal{P}^{(0,1)}[\mathcal{R}^{1}]_{g}+2W^{-3}W_{bs_{f}}W_{s_{h}}\mathcal{P}^{(0,1)}[\mathcal{R}^{1}]_{g}\\
& +2W^{-3}W_{s_{f}}W_{bs_{h}}\mathcal{P}^{(0,1)}[\mathcal{R}^{1}]_{g}+2W^{-3}W_{s_{f}}W_{s_{h}}\mathcal{P}_{b}^{(0,1)}[\mathcal{R}^{1}]_{g}\\
& +2W^{-3}W_{s_{f}}W_{s_{h}}\mathcal{P}^{(0,1)}[\mathcal{R}_{b}^{1}]_{g}\\
& +2W^{-3}W_{b}W_{s_{f}s_{h}}\mathcal{P}^{(0,1)}[\mathcal{R}^{1}]_{g}-W^{-2}W_{bs_{f}s_{h}}\mathcal{P}^{(0,1)}[\mathcal{R}^{1}]_{g}\\
& -W^{-2}W_{s_{f}s_{h}}\mathcal{P}_{b}^{(0,1)}[\mathcal{R}^{1}]_{g}-W^{-2}W_{s_{f}s_{h}}\mathcal{P}^{(0,1)}[\mathcal{R}_{b}^{1}]_{g}\\
& +2W^{-3}W_{b}W_{s_{h}}\mathcal{P}_{s_{f}}^{(0,1)}[\mathcal{R}^{1}]_{g}-W^{-2}W_{bs_{h}}\mathcal{P}_{s_{f}}^{(0,1)}[\mathcal{R}^{1}]_{g}\\
& -W^{-2}W_{s_{h}}\mathcal{P}_{bs_{f}}^{(0,1)}[\mathcal{R}^{1}]_{g}-W^{-2}W_{s_{h}}\mathcal{P}_{s_{f}}^{(0,1)}[\mathcal{R}_{b}^{1}]_{g}\\
& +2W^{-3}W_{b}W_{s_{h}}\mathcal{P}^{(0,1)}[\mathcal{R}_{s_{f}}^{1}]_{g}-W^{-2}W_{bs_{h}}\mathcal{P}^{(0,1)}[\mathcal{R}_{s_{f}}^{1}]_{g}\\
& -W^{-2}W_{s_{h}}\mathcal{P}_{b}^{(0,1)}[\mathcal{R}_{s_{f}}^{1}]_{g}-W^{-2}W_{s_{h}}\mathcal{P}^{(0,1)}[\mathcal{R}_{bs_{f}}^{1}]_{g}\\
& +2W^{-3}W_{b}W_{s_{f}}\mathcal{P}_{s_{h}}^{(0,1)}[\mathcal{R}^{1}]_{g}-W^{-2}W_{bs_{f}}\mathcal{P}_{s_{h}}^{(0,1)}[\mathcal{R}^{1}]_{g}\\
& -W^{-2}W_{s_{f}}\mathcal{P}_{bs_{h}}^{(0,1)}[\mathcal{R}^{1}]_{g}-W^{-2}W_{s_{f}}\mathcal{P}_{s_{h}}^{(0,1)}[\mathcal{R}_{b}^{1}]_{g}\\
& -W^{-2}W_{b}\mathcal{P}_{s_{f}s_{h}}^{(0,1)}[\mathcal{R}^{1}]_{g}+W^{-1}\mathcal{P}_{bs_{f}s_{h}}^{(0,1)}[\mathcal{R}^{1}]_{g}+W^{-1}\mathcal{P}_{s_{f}s_{h}}^{(0,1)}[\mathcal{R}_{b}^{1}]_{g}\\
& -W^{-2}W_{b}\mathcal{P}_{s_{h}}^{(0,1)}[\mathcal{R}_{s_{f}}^{1}]_{g}+W^{-1}\mathcal{P}_{bs_{h}}^{(0,1)}[\mathcal{R}_{s_{f}}^{1}]_{g}+W^{-1}\mathcal{P}_{s_{h}}^{(0,1)}[\mathcal{R}_{bs_{f}}^{1}]_{g}\\
& +2W^{-3}W_{b}W_{s_{f}}\mathcal{P}^{(0,1)}[\mathcal{R}_{s_{h}}^{1}]_{g}-W^{-2}W_{bs_{f}}\mathcal{P}^{(0,1)}[\mathcal{R}_{s_{h}}^{1}]_{g}\\
& -W^{-2}W_{s_{f}}\mathcal{P}_{b}^{(0,1)}[\mathcal{R}_{s_{h}}^{1}]_{g}-W^{-2}W_{s_{f}}\mathcal{P}^{(0,1)}[\mathcal{R}_{bs_{h}}^{1}]_{g}\\
& -W^{-2}W_{b}\mathcal{P}_{s_{f}}^{(0,1)}[\mathcal{R}_{s_{h}}^{1}]_{g}+W^{-1}\mathcal{P}_{bs_{f}}^{(0,1)}[\mathcal{R}_{s_{h}}^{1}]_{g}+W^{-1}\mathcal{P}_{s_{f}}^{(0,1)}[\mathcal{R}_{bs_{h}}^{1}]_{g}\\
& -W^{-2}W_{b}\mathcal{P}^{(0,1)}[\mathcal{R}_{s_{f}s_{h}}^{1}]_{g}+W^{-1}\mathcal{P}_{b}^{(0,1)}[\mathcal{R}_{s_{f}s_{h}}^{1}]_{g}+W^{-1}\mathcal{P}^{(0,1)}[\mathcal{R}_{bs_{f}s_{h}}^{1}]_{g}
\end{align*}
\subsection{Expressions for $\mathcal{P}$ terms}
First derivatives:
\begin{align*}
\mathcal{P}_{b}^{(r,s)} & =-W^{-1}\sum_{f}(\partial_{\beta^{r+1}\phi\phi'\phi_{f}}\mathcal{L})[\mathcal{R}^{s}]_{f}\\
& -W^{-1}\sum_{e,f}(\partial_{\beta^{r}\phi\phi'\phi_{e}\phi_{f}}\mathcal{L})[\mathcal{R}^{s}]_{f}[\mathcal{R}^{1}]_{e}\\
& +\sum_{f}(\partial_{\beta^{r}\phi\phi'\phi_{f}}\mathcal{L})[\mathcal{R}_{b}^{s}]_{f}\\
\mathcal{P}_{(g),b}^{r} & =-W^{-1}\sum_{f}(\partial_{\beta^{r+1}\phi\phi'\phi_{f}}\mathcal{L})\big[\mathcal{H}^{-1}\big]_{fg}\\
& -W^{-1}\sum_{e,f}(\partial_{\beta^{r}\phi\phi'\phi_{e}\phi_{f}}\mathcal{L})[\mathcal{R}^{1}]_{e}\big[\mathcal{H}^{-1}\big]_{fg}\\
& -\sum_{f}(\partial_{\beta^{r}\phi\phi'\phi_{f}}\mathcal{L})\big[\mathcal{H}^{-1}\mathcal{H}_{b}\mathcal{H}^{-1}\big]_{fg}
\end{align*}
\begin{align*}
\mathcal{P}_{s_{h}}^{(a,r)} & =-W^{-1}[\mathcal{R}^{1}]_{h}\sum_{f}(\partial_{\beta^{a+1}\phi\phi'\phi_{f}}\mathcal{L})[\mathcal{R}^{1}]_{f}\\
& -\sum_{e,f}(\partial_{\beta^{a}\phi\phi'\phi_{e}\phi_{f}}\mathcal{L})[\mathcal{R}^{1}]_{f}[\mathcal{H}^{-1}]_{eh}\\
& -W^{-1}\sum_{e,f}(\partial_{\beta^{a}\phi\phi'\phi_{e}\phi_{f}}\mathcal{L})[\mathcal{R}^{1}]_{f}[\mathcal{R}^{1}]_{e}[\mathcal{R}^{1}]_{h}\\
& +\sum_{f}(\partial_{\beta^{a}\phi\phi'\phi_{f}}\mathcal{L})[\mathcal{R}_{s_{h}}^{1}]_{f}
\end{align*}
\begin{align*}
\mathcal{P}_{(g),s_{h}}^{r} & =-W^{-1}[\mathcal{R}^{1}]_{h}\mathcal{P}_{(g)}^{r+1}\\
& -\sum_{e,f}(\partial_{\beta^{r}\phi\phi'\phi_{e}\phi_{f}}\mathcal{L})\big[\mathcal{H}^{-1}\big]_{fg}\big[\mathcal{H}^{-1}\big]_{eh}\\
& -W^{-1}\sum_{e,f}(\partial_{\beta^{r}\phi\phi'\phi_{e}\phi_{f}}\mathcal{L})\big[\mathcal{H}^{-1}\big]_{fg}[\mathcal{R}^{1}]_{e}[\mathcal{R}^{1}]_{h}\\
& -\sum_{f}(\partial_{\beta^{r}\phi\phi'\phi_{f}}\mathcal{L})\big[\mathcal{H}^{-1}\mathcal{H}_{s_{h}}\mathcal{H}^{-1}\big]_{fg}
\end{align*}
Second derivatives:
\begin{align*}
\mathcal{P}_{(g),bs_{h}}^{r} & =W^{-2}W_{b}[\mathcal{R}^{1}]_{h}\mathcal{P}_{(g)}^{r+1}-W^{-1}[\mathcal{R}_{b}^{1}]_{h}\mathcal{P}_{(g)}^{r+1}-W^{-1}[\mathcal{R}^{1}]_{h}\mathcal{P}_{(g),b}^{r+1}\\
& +W^{-1}\sum_{e,f}(\partial_{\beta^{r+1}\phi\phi'\phi_{e}\phi_{f}}\mathcal{L})\big[\mathcal{H}^{-1}\big]_{fg}\big[\mathcal{H}^{-1}\big]_{eh}\\
& +W^{-1}\sum_{d,e,f}(\partial_{\beta^{r}\phi\phi'\phi_{d}\phi_{e}\phi_{f}}\mathcal{L})[\mathcal{R}^{1}]_{d}\big[\mathcal{H}^{-1}\big]_{fg}\big[\mathcal{H}^{-1}\big]_{eh}\\
& +\sum_{e,f}(\partial_{\beta^{r}\phi\phi'\phi_{e}\phi_{f}}\mathcal{L})\big[\mathcal{H}^{-1}\mathcal{H}_{b}\mathcal{H}^{-1}\big]_{fg}\big[\mathcal{H}^{-1}\big]_{eh}\\
& +\sum_{e,f}(\partial_{\beta^{r}\phi\phi'\phi_{e}\phi_{f}}\mathcal{L})\big[\mathcal{H}^{-1}\big]_{fg}\big[\mathcal{H}^{-1}\mathcal{H}_{b}\mathcal{H}^{-1}\big]_{eh}\\
& +W^{-2}W_{b}\sum_{e,f}(\partial_{\beta^{r}\phi\phi'\phi_{e}\phi_{f}}\mathcal{L})\big[\mathcal{H}^{-1}\big]_{fg}[\mathcal{R}^{1}]_{e}[\mathcal{R}^{1}]_{h}\\
& +W^{-2}\sum_{e,f}(\partial_{\beta^{r+1}\phi\phi'\phi_{e}\phi_{f}}\mathcal{L})\big[\mathcal{H}^{-1}\big]_{fg}[\mathcal{R}^{1}]_{e}[\mathcal{R}^{1}]_{h}\\
& +W^{-2}\sum_{d,e,f}(\partial_{\beta^{r}\phi\phi'\phi_{d}\phi_{e}\phi_{f}}\mathcal{L})\big[\mathcal{H}^{-1}\big]_{fg}[\mathcal{R}^{1}]_{d}[\mathcal{R}^{1}]_{e}[\mathcal{R}^{1}]_{h}\\
& +W^{-1}\sum_{e,f}(\partial_{\beta^{r}\phi\phi'\phi_{e}\phi_{f}}\mathcal{L})\big[\mathcal{H}^{-1}\mathcal{H}_{b}\mathcal{H}^{-1}\big]_{fg}[\mathcal{R}^{1}]_{e}[\mathcal{R}^{1}]_{h}\\
& -W^{-1}\sum_{e,f}(\partial_{\beta^{r}\phi\phi'\phi_{e}\phi_{f}}\mathcal{L})\big[\mathcal{H}^{-1}\big]_{fg}[\mathcal{R}_{b}^{1}]_{e}[\mathcal{R}^{1}]_{h}\\
& -W^{-1}\sum_{e,f}(\partial_{\beta^{r}\phi\phi'\phi_{e}\phi_{f}}\mathcal{L})\big[\mathcal{H}^{-1}\big]_{fg}[\mathcal{R}^{1}]_{e}[\mathcal{R}_{b}^{1}]_{h}\\
& +W^{-1}\sum_{f}(\partial_{\beta^{r+1}\phi\phi'\phi_{f}}\mathcal{L})\big[\mathcal{H}^{-1}\mathcal{H}_{s_{h}}\mathcal{H}^{-1}\big]_{fg}\\
& +W^{-1}\sum_{e,f}(\partial_{\beta^{r}\phi\phi'\phi_{e}\phi_{f}}\mathcal{L})[\mathcal{R}^{1}]_{e}\big[\mathcal{H}^{-1}\mathcal{H}_{s_{h}}\mathcal{H}^{-1}\big]_{fg}\\
& +\sum_{f}(\partial_{\beta^{r}\phi\phi'\phi_{f}}\mathcal{L})\big[\mathcal{H}^{-1}\mathcal{H}_{b}\mathcal{H}^{-1}\mathcal{H}_{s_{h}}\mathcal{H}^{-1}\big]_{fg}\\
& +\sum_{f}(\partial_{\beta^{r}\phi\phi'\phi_{f}}\mathcal{L})\big[\mathcal{H}^{-1}\mathcal{H}_{s_{h}}\mathcal{H}^{-1}\mathcal{H}_{b}\mathcal{H}^{-1}\big]_{fg}\\
& -\sum_{f}(\partial_{\beta^{r}\phi\phi'\phi_{f}}\mathcal{L})\big[\mathcal{H}^{-1}\mathcal{H}_{bs_{h}}\mathcal{H}^{-1}\big]_{fg}
\end{align*}
\begin{align*}
\mathcal{P}_{bs_{h}}^{(a,r)} & =W^{-2}W_{b}[\mathcal{R}^{1}]_{h}\sum_{f}(\partial_{\beta^{a+1}\phi\phi'\phi_{f}}\mathcal{L})[\mathcal{R}^{1}]_{f}\\
& -W^{-1}[\mathcal{R}_{b}^{1}]_{h}\sum_{f}(\partial_{\beta^{a+1}\phi\phi'\phi_{f}}\mathcal{L})[\mathcal{R}^{1}]_{f}\\
& -W^{-1}[\mathcal{R}^{1}]_{h}\sum_{f}(\partial_{\beta^{a+1}\phi\phi'\phi_{f}}\mathcal{L})[\mathcal{R}_{b}^{1}]_{f}\\
& +W^{-2}[\mathcal{R}^{1}]_{h}\sum_{f}(\partial_{\beta^{a+2}\phi\phi'\phi_{f}}\mathcal{L})[\mathcal{R}^{1}]_{f}\\
& +W^{-2}[\mathcal{R}^{1}]_{h}\sum_{e,f}(\partial_{\beta^{a+1}\phi\phi'\phi_{e}\phi_{f}}\mathcal{L})[\mathcal{R}^{1}]_{e}[\mathcal{R}^{1}]_{f}\\
& +W^{-1}\sum_{e,f}(\partial_{\beta^{a+1}\phi\phi'\phi_{i}\phi_{f}}\mathcal{L})[\mathcal{R}^{1}]_{f}[\mathcal{H}^{-1}]_{eh}\\
& +W^{-1}\sum_{d,e,f}(\partial_{\beta^{a}\phi\phi'\phi_{d}\phi_{e}\phi_{f}}\mathcal{L})[\mathcal{R}^{1}]_{d}[\mathcal{R}^{1}]_{f}[\mathcal{H}^{-1}]_{eh}\\
& -\sum_{e,f}(\partial_{\beta^{a}\phi\phi'\phi_{i}\phi_{f}}\mathcal{L})[\mathcal{R}_{b}^{1}]_{f}[\mathcal{H}^{-1}]_{eh}\\
& +\sum_{e,f}(\partial_{\beta^{a}\phi\phi'\phi_{i}\phi_{f}}\mathcal{L})[\mathcal{R}^{1}]_{f}[\mathcal{H}^{-1}\mathcal{H}_{b}\mathcal{H}^{-1}]_{eh}\\
& +W^{-2}W_{b}\sum_{e,f}(\partial_{\beta^{a}\phi\phi'\phi_{e}\phi_{f}}\mathcal{L})[\mathcal{R}^{1}]_{f}[\mathcal{R}^{1}]_{e}[\mathcal{R}^{1}]_{h}\\
& +W^{-2}\sum_{e,f}(\partial_{\beta^{a+1}\phi\phi'\phi_{e}\phi_{f}}\mathcal{L})[\mathcal{R}^{1}]_{f}[\mathcal{R}^{1}]_{e}[\mathcal{R}^{1}]_{h}\\
& +W^{-2}\sum_{d,e,f}(\partial_{\beta^{a}\phi\phi'\phi_{d}\phi_{e}\phi_{f}}\mathcal{L})[\mathcal{R}^{1}]_{d}[\mathcal{R}^{1}]_{f}[\mathcal{R}^{1}]_{e}[\mathcal{R}^{1}]_{h}\\
& -W^{-1}\sum_{e,f}(\partial_{\beta^{a}\phi\phi'\phi_{e}\phi_{f}}\mathcal{L})[\mathcal{R}_{b}^{1}]_{f}[\mathcal{R}^{1}]_{e}[\mathcal{R}^{1}]_{h}\\
& -W^{-1}\sum_{e,f}(\partial_{\beta^{a}\phi\phi'\phi_{e}\phi_{f}}\mathcal{L})[\mathcal{R}^{1}]_{f}[\mathcal{R}_{b}^{1}]_{e}[\mathcal{R}^{1}]_{h}\\
& -W^{-1}\sum_{e,f}(\partial_{\beta^{a}\phi\phi'\phi_{e}\phi_{f}}\mathcal{L})[\mathcal{R}^{1}]_{f}[\mathcal{R}^{1}]_{e}[\mathcal{R}_{b}^{1}]_{h}\\
& -W^{-1}\sum_{f}(\partial_{\beta^{a+1}\phi\phi'\phi_{f}}\mathcal{L})[\mathcal{R}_{s_{h}}^{1}]_{f}\\
& -W^{-1}\sum_{e,f}(\partial_{\beta^{a}\phi\phi'\phi_{e}\phi_{f}}\mathcal{L})[\mathcal{R}^{1}]_{e}[\mathcal{R}_{s_{h}}^{1}]_{f}\\
& +\sum_{f}(\partial_{\beta^{a}\phi\phi'\phi_{f}}\mathcal{L})[\mathcal{R}_{bs_{h}}^{1}]_{f}
\end{align*}
\begin{align*}
\mathcal{P}_{s_{g}s_{h}}^{(a,r)} & =W^{-2}W_{s_{g}}[\mathcal{R}^{1}]_{h}\sum_{f}(\partial_{\beta^{a+1}\phi\phi'\phi_{f}}\mathcal{L})[\mathcal{R}^{1}]_{f}\\
& -W^{-1}[\mathcal{R}_{s_{g}}^{1}]_{h}\sum_{f}(\partial_{\beta^{a+1}\phi\phi'\phi_{f}}\mathcal{L})[\mathcal{R}^{1}]_{f}\\
& +W^{-2}[\mathcal{R}^{1}]_{h}\sum_{f}(\partial_{\beta^{a+2}\phi\phi'\phi_{f}}\mathcal{L})[\mathcal{R}^{1}]_{f}[\mathcal{R}^{1}]_{g}\\
& +W^{-1}[\mathcal{R}^{1}]_{h}\sum_{e,f}(\partial_{\beta^{a+1}\phi\phi'\phi_{e}\phi_{f}}\mathcal{L})[\mathcal{R}^{1}]_{f}[\mathcal{H}^{-1}]_{eg}\\
& +W^{-2}[\mathcal{R}^{1}]_{h}\sum_{e,f}(\partial_{\beta^{a+1}\phi\phi'\phi_{e}\phi_{f}}\mathcal{L})[\mathcal{R}^{1}]_{e}[\mathcal{R}^{1}]_{f}[\mathcal{R}^{1}]_{g}\\
& -W^{-1}[\mathcal{R}^{1}]_{h}\sum_{f}(\partial_{\beta^{a+1}\phi\phi'\phi_{f}}\mathcal{L})[\mathcal{R}_{s_{g}}^{1}]_{f}\\
& +W^{-1}\sum_{e,f}(\partial_{\beta^{a+1}\phi\phi'\phi_{i}\phi_{f}}\mathcal{L})[\mathcal{R}^{1}]_{f}[\mathcal{H}^{-1}]_{eh}[\mathcal{R}^{1}]_{g}\\
& +\sum_{d,e,f}(\partial_{\beta^{a}\phi\phi'\phi_{d}\phi_{e}\phi_{f}}\mathcal{L})[\mathcal{R}^{1}]_{f}[\mathcal{H}^{-1}]_{eh}[\mathcal{H}^{-1}]_{dg}\\
& +W^{-1}\sum_{d,e,f}(\partial_{\beta^{a}\phi\phi'\phi_{d}\phi_{e}\phi_{f}}\mathcal{L})[\mathcal{R}^{1}]_{f}[\mathcal{H}^{-1}]_{eh}[\mathcal{R}^{1}]_{d}[\mathcal{R}^{1}]_{g}\\
& -\sum_{e,f}(\partial_{\beta^{a}\phi\phi'\phi_{i}\phi_{f}}\mathcal{L})[\mathcal{R}_{s_{g}}^{1}]_{f}[\mathcal{H}^{-1}]_{eh}\\
& +\sum_{e,f}(\partial_{\beta^{a}\phi\phi'\phi_{i}\phi_{f}}\mathcal{L})[\mathcal{R}^{1}]_{f}[\mathcal{H}^{-1}\mathcal{H}_{s_{g}}\mathcal{H}^{-1}]_{eh}\\
& +W^{-2}W_{s_{g}}\sum_{e,f}(\partial_{\beta^{a}\phi\phi'\phi_{e}\phi_{f}}\mathcal{L})[\mathcal{R}^{1}]_{f}[\mathcal{R}^{1}]_{e}[\mathcal{R}^{1}]_{h}\\
& +W^{-2}\sum_{e,f}(\partial_{\beta^{a+1}\phi\phi'\phi_{e}\phi_{f}}\mathcal{L})[\mathcal{R}^{1}]_{f}[\mathcal{R}^{1}]_{e}[\mathcal{R}^{1}]_{g}[\mathcal{R}^{1}]_{h}\\
& +W^{-1}\sum_{d,e,f}(\partial_{\beta^{a}\phi\phi'\phi_{d}\phi_{e}\phi_{f}}\mathcal{L})[\mathcal{R}^{1}]_{f}[\mathcal{R}^{1}]_{e}[\mathcal{R}^{1}]_{h}[\mathcal{H}^{-1}]_{dg}\\
& +W^{-2}\sum_{d,e,f}(\partial_{\beta^{a}\phi\phi'\phi_{d}\phi_{e}\phi_{f}}\mathcal{L})[\mathcal{R}^{1}]_{f}[\mathcal{R}^{1}]_{d}[\mathcal{R}^{1}]_{e}[\mathcal{R}^{1}]_{g}[\mathcal{R}^{1}]_{h}\\
& -W^{-1}\sum_{e,f}(\partial_{\beta^{a}\phi\phi'\phi_{e}\phi_{f}}\mathcal{L})[\mathcal{R}_{s_{g}}^{1}]_{f}[\mathcal{R}^{1}]_{e}[\mathcal{R}^{1}]_{h}\\
& -W^{-1}\sum_{e,f}(\partial_{\beta^{a}\phi\phi'\phi_{e}\phi_{f}}\mathcal{L})[\mathcal{R}^{1}]_{f}[\mathcal{R}_{s_{g}}^{1}]_{e}[\mathcal{R}^{1}]_{h}\\
& -W^{-1}\sum_{e,f}(\partial_{\beta^{a}\phi\phi'\phi_{e}\phi_{f}}\mathcal{L})[\mathcal{R}^{1}]_{f}[\mathcal{R}^{1}]_{e}[\mathcal{R}_{s_{g}}^{1}]_{h}\\
& -W^{-1}\sum_{f}(\partial_{\beta^{a+1}\phi\phi'\phi_{f}}\mathcal{L})[\mathcal{R}_{s_{h}}^{1}]_{f}[\mathcal{R}^{1}]_{g}\\
& -\sum_{e,f}(\partial_{\beta^{a}\phi\phi'\phi_{e}\phi_{f}}\mathcal{L})[\mathcal{R}_{s_{h}}^{1}]_{f}[\mathcal{H}^{-1}]_{eg}\\
& W^{-1}\sum_{e,f}(\partial_{\beta^{a}\phi\phi'\phi_{e}\phi_{f}}\mathcal{L})[\mathcal{R}_{s_{h}}^{1}]_{f}[\mathcal{R}^{1}]_{e}[\mathcal{R}^{1}]_{g}\\
& +\sum_{f}(\partial_{\beta^{a}\phi\phi'\phi_{f}}\mathcal{L})[\mathcal{R}_{s_{g}s_{h}}^{1}]_{f}
\end{align*}
\begin{align*}
\mathcal{P}_{(g),s_{k}s_{h}}^{r} & =W^{-2}W_{s_{k}}[\mathcal{R}^{1}]_{h}\mathcal{P}_{(g)}^{r+1}-W^{-1}[\mathcal{R}_{s_{k}}^{1}]_{h}\mathcal{P}_{(g)}^{r+1}-W^{-1}[\mathcal{R}^{1}]_{h}\mathcal{P}_{(g),s_{k}}^{r+1}\\
& +W^{-1}\sum_{e,f}(\partial_{\beta^{r+1}\phi\phi'\phi_{e}\phi_{f}}\mathcal{L})\big[\mathcal{H}^{-1}\big]_{fg}\big[\mathcal{H}^{-1}\big]_{eh}[\mathcal{R}^{1}]_{k}\\
& +\sum_{d,e,f}(\partial_{\beta^{r}\phi\phi'\phi_{d}\phi_{e}\phi_{f}}\mathcal{L})\big[\mathcal{H}^{-1}\big]_{fg}\big[\mathcal{H}^{-1}\big]_{eh}\big[\mathcal{H}^{-1}\big]_{dk}\\
& +W^{-1}\sum_{d,e,f}(\partial_{\beta^{r}\phi\phi'\phi_{d}\phi_{e}\phi_{f}}\mathcal{L})\big[\mathcal{H}^{-1}\big]_{fg}\big[\mathcal{H}^{-1}\big]_{eh}[\mathcal{R}^{1}]_{d}[\mathcal{R}^{1}]_{k}\\
& +\sum_{e,f}(\partial_{\beta^{r}\phi\phi'\phi_{e}\phi_{f}}\mathcal{L})\big[\mathcal{H}^{-1}\mathcal{H}_{s_{k}}\mathcal{H}^{-1}\big]_{fg}\big[\mathcal{H}^{-1}\big]_{eh}\\
& +\sum_{e,f}(\partial_{\beta^{r}\phi\phi'\phi_{e}\phi_{f}}\mathcal{L})\big[\mathcal{H}^{-1}\big]_{fg}\big[\mathcal{H}^{-1}\mathcal{H}_{s_{k}}\mathcal{H}^{-1}\big]_{eh}\\
& +W^{-2}W_{s_{k}}\sum_{e,f}(\partial_{\beta^{r}\phi\phi'\phi_{e}\phi_{f}}\mathcal{L})\big[\mathcal{H}^{-1}\big]_{fg}[\mathcal{R}^{1}]_{e}[\mathcal{R}^{1}]_{h}\\
& +W^{-2}\sum_{e,f}(\partial_{\beta^{r+1}\phi\phi'\phi_{e}\phi_{f}}\mathcal{L})\big[\mathcal{H}^{-1}\big]_{fg}[\mathcal{R}^{1}]_{e}[\mathcal{R}^{1}]_{h}[\mathcal{R}^{1}]_{k}\\
& +W^{-1}\sum_{d,e,f}(\partial_{\beta^{r}\phi\phi'\phi_{d}\phi_{e}\phi_{f}}\mathcal{L})\big[\mathcal{H}^{-1}\big]_{fg}\big[\mathcal{H}^{-1}\big]_{dk}[\mathcal{R}^{1}]_{e}[\mathcal{R}^{1}]_{h}\\
& +W^{-2}\sum_{d,e,f}(\partial_{\beta^{r}\phi\phi'\phi_{d}\phi_{e}\phi_{f}}\mathcal{L})\big[\mathcal{H}^{-1}\big]_{fg}[\mathcal{R}^{1}]_{e}[\mathcal{R}^{1}]_{h}[\mathcal{R}^{1}]_{d}[\mathcal{R}^{1}]_{k}\\
& +W^{-1}\sum_{e,f}(\partial_{\beta^{r}\phi\phi'\phi_{e}\phi_{f}}\mathcal{L})\big[\mathcal{H}^{-1}\mathcal{H}_{s_{k}}\mathcal{H}^{-1}\big]_{fg}[\mathcal{R}^{1}]_{e}[\mathcal{R}^{1}]_{h}\\
& -W^{-1}\sum_{e,f}(\partial_{\beta^{r}\phi\phi'\phi_{e}\phi_{f}}\mathcal{L})\big[\mathcal{H}^{-1}\big]_{fg}[\mathcal{R}_{s_{k}}^{1}]_{e}[\mathcal{R}^{1}]_{h}\\
& -W^{-1}\sum_{e,f}(\partial_{\beta^{r}\phi\phi'\phi_{e}\phi_{f}}\mathcal{L})\big[\mathcal{H}^{-1}\big]_{fg}[\mathcal{R}^{1}]_{e}[\mathcal{R}_{s_{k}}^{1}]_{h}\\
& +W^{-1}\sum_{f}(\partial_{\beta^{r+1}\phi\phi'\phi_{f}}\mathcal{L})\big[\mathcal{H}^{-1}\mathcal{H}_{s_{h}}\mathcal{H}^{-1}\big]_{fg}[\mathcal{R}^{1}]_{k}\\
& +\sum_{e,f}(\partial_{\beta^{r}\phi\phi'\phi_{e}\phi_{f}}\mathcal{L})\big[\mathcal{H}^{-1}\mathcal{H}_{s_{h}}\mathcal{H}^{-1}\big]_{fg}\big[\mathcal{H}^{-1}\big]_{ek}\\
& +W^{-1}\sum_{e,f}(\partial_{\beta^{r}\phi\phi'\phi_{e}\phi_{f}}\mathcal{L})\big[\mathcal{H}^{-1}\mathcal{H}_{s_{h}}\mathcal{H}^{-1}\big]_{fg}[\mathcal{R}^{1}]_{e}[\mathcal{R}^{1}]_{k}\\
& +\sum_{f}(\partial_{\beta^{r}\phi\phi'\phi_{f}}\mathcal{L})\big[\mathcal{H}^{-1}\mathcal{H}_{s_{k}}\mathcal{H}^{-1}\mathcal{H}_{s_{h}}\mathcal{H}^{-1}\big]_{fg}\\
& -\sum_{f}(\partial_{\beta^{r}\phi\phi'\phi_{f}}\mathcal{L})\big[\mathcal{H}^{-1}\mathcal{H}_{s_{k}s_{h}}\mathcal{H}^{-1}\big]_{fg}\\
& +\sum_{f}(\partial_{\beta^{r}\phi\phi'\phi_{f}}\mathcal{L})\big[\mathcal{H}^{-1}\mathcal{H}_{s_{h}}\mathcal{H}^{-1}\mathcal{H}_{s_{k}}\mathcal{H}^{-1}\big]_{fg}
\end{align*}
Third derivatives:
\begin{align*}
\mathcal{P}_{bs_{g}s_{h}}^{(a,r)} & =-2W^{-3}W_{b}W_{s_{g}}[\mathcal{R}^{1}]_{h}\sum_{f}(\partial_{\beta^{a+1}\phi\phi'\phi_{f}}\mathcal{L})[\mathcal{R}^{1}]_{f}\\
& +W^{-2}W_{bs_{g}}[\mathcal{R}^{1}]_{h}\sum_{f}(\partial_{\beta^{a+1}\phi\phi'\phi_{f}}\mathcal{L})[\mathcal{R}^{1}]_{f}\\
& +W^{-2}W_{s_{g}}[\mathcal{R}_{b}^{1}]_{h}\sum_{f}(\partial_{\beta^{a+1}\phi\phi'\phi_{f}}\mathcal{L})[\mathcal{R}^{1}]_{f}\\
& -W^{-3}W_{s_{g}}[\mathcal{R}^{1}]_{h}\sum_{f}(\partial_{\beta^{a+2}\phi\phi'\phi_{f}}\mathcal{L})[\mathcal{R}^{1}]_{f}\\
& -W^{-3}W_{s_{g}}[\mathcal{R}^{1}]_{h}\sum_{e,f}(\partial_{\beta^{a+1}\phi\phi'\phi_{e}\phi_{f}}\mathcal{L})[\mathcal{R}^{1}]_{e}[\mathcal{R}^{1}]_{f}\\
& +W^{-2}W_{s_{g}}[\mathcal{R}^{1}]_{h}\sum_{f}(\partial_{\beta^{a+1}\phi\phi'\phi_{f}}\mathcal{L})[\mathcal{R}_{b}^{1}]_{f}\\
& +W^{-2}W_{b}[\mathcal{R}_{s_{g}}^{1}]_{h}\sum_{f}(\partial_{\beta^{a+1}\phi\phi'\phi_{f}}\mathcal{L})[\mathcal{R}^{1}]_{f}\\
& -W^{-1}[\mathcal{R}_{bs_{g}}^{1}]_{h}\sum_{f}(\partial_{\beta^{a+1}\phi\phi'\phi_{f}}\mathcal{L})[\mathcal{R}^{1}]_{f}\\
& +W^{-2}[\mathcal{R}_{s_{g}}^{1}]_{h}\sum_{f}(\partial_{\beta^{a+2}\phi\phi'\phi_{f}}\mathcal{L})[\mathcal{R}^{1}]_{f}\\
& +W^{-2}[\mathcal{R}_{s_{g}}^{1}]_{h}\sum_{e,f}(\partial_{\beta^{a+1}\phi\phi'\phi_{e}\phi_{f}}\mathcal{L})[\mathcal{R}^{1}]_{f}\\
& -W^{-1}[\mathcal{R}_{s_{g}}^{1}]_{h}\sum_{f}(\partial_{\beta^{a+1}\phi\phi'\phi_{f}}\mathcal{L})[\mathcal{R}_{b}^{1}]_{f}\\
& -2W^{-3}W_{b}[\mathcal{R}^{1}]_{h}\sum_{f}(\partial_{\beta^{a+2}\phi\phi'\phi_{f}}\mathcal{L})[\mathcal{R}^{1}]_{f}[\mathcal{R}^{1}]_{g}\\
& +W^{-2}[\mathcal{R}_{b}^{1}]_{h}\sum_{f}(\partial_{\beta^{a+2}\phi\phi'\phi_{f}}\mathcal{L})[\mathcal{R}^{1}]_{f}[\mathcal{R}^{1}]_{g}\\
& -W^{-3}[\mathcal{R}^{1}]_{h}\sum_{f}(\partial_{\beta^{a+3}\phi\phi'\phi_{f}}\mathcal{L})[\mathcal{R}^{1}]_{f}[\mathcal{R}^{1}]_{g}\\
& -W^{-3}[\mathcal{R}^{1}]_{h}\sum_{e,f}(\partial_{\beta^{a+2}\phi\phi'\phi_{e}\phi_{f}}\mathcal{L})[\mathcal{R}^{1}]_{f}[\mathcal{R}^{1}]_{g}\\
& +W^{-2}[\mathcal{R}^{1}]_{h}\sum_{f}(\partial_{\beta^{a+2}\phi\phi'\phi_{f}}\mathcal{L})[\mathcal{R}_{b}^{1}]_{f}[\mathcal{R}^{1}]_{g}\\
& +W^{-2}[\mathcal{R}^{1}]_{h}\sum_{f}(\partial_{\beta^{a+2}\phi\phi'\phi_{f}}\mathcal{L})[\mathcal{R}^{1}]_{f}[\mathcal{R}_{b}^{1}]_{g}\\
& -W^{-2}W_{b}[\mathcal{R}^{1}]_{h}\sum_{e,f}(\partial_{\beta^{a+1}\phi\phi'\phi_{e}\phi_{f}}\mathcal{L})[\mathcal{R}^{1}]_{f}[\mathcal{H}^{-1}]_{eg}\\
& +W^{-1}[\mathcal{R}_{b}^{1}]_{h}\sum_{e,f}(\partial_{\beta^{a+1}\phi\phi'\phi_{e}\phi_{f}}\mathcal{L})[\mathcal{R}^{1}]_{f}[\mathcal{H}^{-1}]_{eg}\\
& -W^{-2}[\mathcal{R}^{1}]_{h}\sum_{e,f}(\partial_{\beta^{a+2}\phi\phi'\phi_{e}\phi_{f}}\mathcal{L})[\mathcal{R}^{1}]_{f}[\mathcal{H}^{-1}]_{eg}\\
& -W^{-2}[\mathcal{R}^{1}]_{h}\sum_{d,e,f}(\partial_{\beta^{a+1}\phi\phi'\phi_{d}\phi_{e}\phi_{f}}\mathcal{L})[\mathcal{R}^{1}]_{d}[\mathcal{R}^{1}]_{f}[\mathcal{H}^{-1}]_{eg}\\
& +W^{-1}[\mathcal{R}^{1}]_{h}\sum_{e,f}(\partial_{\beta^{a+1}\phi\phi'\phi_{e}\phi_{f}}\mathcal{L})[\mathcal{R}_{b}^{1}]_{f}[\mathcal{H}^{-1}]_{eg}\\
& -W^{-1}[\mathcal{R}^{1}]_{h}\sum_{e,f}(\partial_{\beta^{a+1}\phi\phi'\phi_{e}\phi_{f}}\mathcal{L})[\mathcal{R}^{1}]_{f}[\mathcal{H}^{-1}\mathcal{H}_{b}\mathcal{H}^{-1}]_{eg}\\
& -2W^{-3}W_{b}[\mathcal{R}^{1}]_{h}\sum_{e,f}(\partial_{\beta^{a+1}\phi\phi'\phi_{e}\phi_{f}}\mathcal{L})[\mathcal{R}^{1}]_{e}[\mathcal{R}^{1}]_{f}[\mathcal{R}^{1}]_{g}\\
& +W^{-2}[\mathcal{R}_{b}^{1}]_{h}\sum_{e,f}(\partial_{\beta^{a+1}\phi\phi'\phi_{e}\phi_{f}}\mathcal{L})[\mathcal{R}^{1}]_{e}[\mathcal{R}^{1}]_{f}[\mathcal{R}^{1}]_{g}\\
& -W^{-3}[\mathcal{R}^{1}]_{h}\sum_{e,f}(\partial_{\beta^{a+2}\phi\phi'\phi_{e}\phi_{f}}\mathcal{L})[\mathcal{R}^{1}]_{e}[\mathcal{R}^{1}]_{f}[\mathcal{R}^{1}]_{g}\\
& -W^{-3}[\mathcal{R}^{1}]_{h}\sum_{d,e,f}(\partial_{\beta^{a+1}\phi\phi'\phi_{d}\phi_{e}\phi_{f}}\mathcal{L})[\mathcal{R}^{1}]_{d}[\mathcal{R}^{1}]_{e}[\mathcal{R}^{1}]_{f}[\mathcal{R}^{1}]_{g}\\
& +W^{-2}[\mathcal{R}^{1}]_{h}\sum_{e,f}(\partial_{\beta^{a+1}\phi\phi'\phi_{e}\phi_{f}}\mathcal{L})[\mathcal{R}_{b}^{1}]_{e}[\mathcal{R}^{1}]_{f}[\mathcal{R}^{1}]_{g}\\
& +W^{-2}[\mathcal{R}^{1}]_{h}\sum_{e,f}(\partial_{\beta^{a+1}\phi\phi'\phi_{e}\phi_{f}}\mathcal{L})[\mathcal{R}^{1}]_{e}[\mathcal{R}_{b}^{1}]_{f}[\mathcal{R}^{1}]_{g}\\
& +W^{-2}[\mathcal{R}^{1}]_{h}\sum_{e,f}(\partial_{\beta^{a+1}\phi\phi'\phi_{e}\phi_{f}}\mathcal{L})[\mathcal{R}^{1}]_{e}[\mathcal{R}^{1}]_{f}[\mathcal{R}_{b}^{1}]_{g}\\
& +W^{-2}W_{b}[\mathcal{R}^{1}]_{h}\sum_{f}(\partial_{\beta^{a+1}\phi\phi'\phi_{f}}\mathcal{L})[\mathcal{R}_{s_{g}}^{1}]_{f}\\
& -W^{-1}[\mathcal{R}_{b}^{1}]_{h}\sum_{f}(\partial_{\beta^{a+1}\phi\phi'\phi_{f}}\mathcal{L})[\mathcal{R}_{s_{g}}^{1}]_{f}\\
& +W^{-2}[\mathcal{R}^{1}]_{h}\sum_{f}(\partial_{\beta^{a+2}\phi\phi'\phi_{f}}\mathcal{L})[\mathcal{R}_{s_{g}}^{1}]_{f}\\
& +W^{-2}[\mathcal{R}^{1}]_{h}\sum_{e,f}(\partial_{\beta^{a+1}\phi\phi'\phi_{e}\phi_{f}}\mathcal{L})[\mathcal{R}_{s_{g}}^{1}]_{f}\\
& -W^{-1}[\mathcal{R}^{1}]_{h}\sum_{f}(\partial_{\beta^{a+1}\phi\phi'\phi_{f}}\mathcal{L})[\mathcal{R}_{bs_{g}}^{1}]_{f}\\
& -W^{-2}W_{b}\sum_{e,f}(\partial_{\beta^{a+1}\phi\phi'\phi_{e}\phi_{f}}\mathcal{L})[\mathcal{R}^{1}]_{f}[\mathcal{H}^{-1}]_{eh}[\mathcal{R}^{1}]_{g}\\
& -W^{-2}\sum_{e,f}(\partial_{\beta^{a+2}\phi\phi'\phi_{e}\phi_{f}}\mathcal{L})[\mathcal{R}^{1}]_{f}[\mathcal{H}^{-1}]_{eh}[\mathcal{R}^{1}]_{g}\\
& -W^{-2}\sum_{d,e,f}(\partial_{\beta^{a+1}\phi\phi'\phi_{d}\phi_{e}\phi_{f}}\mathcal{L})[\mathcal{R}^{1}]_{d}[\mathcal{R}^{1}]_{f}[\mathcal{H}^{-1}]_{eh}[\mathcal{R}^{1}]_{g}\\
& +W^{-1}\sum_{e,f}(\partial_{\beta^{a+1}\phi\phi'\phi_{e}\phi_{f}}\mathcal{L})[\mathcal{R}_{b}^{1}]_{f}[\mathcal{H}^{-1}]_{eh}[\mathcal{R}^{1}]_{g}\\
& -W^{-1}\sum_{e,f}(\partial_{\beta^{a+1}\phi\phi'\phi_{e}\phi_{f}}\mathcal{L})[\mathcal{R}^{1}]_{f}[\mathcal{H}^{-1}\mathcal{H}_{b}\mathcal{H}^{-1}]_{eh}[\mathcal{R}^{1}]_{g}\\
& +W^{-1}\sum_{e,f}(\partial_{\beta^{a+1}\phi\phi'\phi_{e}\phi_{f}}\mathcal{L})[\mathcal{R}^{1}]_{f}[\mathcal{H}^{-1}]_{eh}[\mathcal{R}_{b}^{1}]_{g}\\
& -W^{-1}\sum_{d,e,f}(\partial_{\beta^{a+1}\phi\phi'\phi_{d}\phi_{e}\phi_{f}}\mathcal{L})[\mathcal{R}^{1}]_{f}[\mathcal{H}^{-1}]_{eh}[\mathcal{H}^{-1}]_{dg}\\
& -W^{-1}\sum_{c,d,e,f}(\partial_{\beta^{a}\phi\phi'\phi_{c}\phi_{d}\phi_{e}\phi_{f}}\mathcal{L})[\mathcal{R}^{1}]_{c}[\mathcal{R}^{1}]_{f}[\mathcal{H}^{-1}]_{eh}[\mathcal{H}^{-1}]_{dg}\\
& +\sum_{d,e,f}(\partial_{\beta^{a}\phi\phi'\phi_{d}\phi_{e}\phi_{f}}\mathcal{L})[\mathcal{R}_{b}^{1}]_{f}[\mathcal{H}^{-1}]_{eh}[\mathcal{H}^{-1}]_{dg}\\
& -\sum_{d,e,f}(\partial_{\beta^{a}\phi\phi'\phi_{d}\phi_{e}\phi_{f}}\mathcal{L})[\mathcal{R}^{1}]_{f}[\mathcal{H}^{-1}\mathcal{H}_{b}\mathcal{H}^{-1}]_{eh}[\mathcal{H}^{-1}]_{dg}\\
& -\sum_{d,e,f}(\partial_{\beta^{a}\phi\phi'\phi_{d}\phi_{e}\phi_{f}}\mathcal{L})[\mathcal{R}^{1}]_{f}[\mathcal{H}^{-1}]_{eh}[\mathcal{H}^{-1}\mathcal{H}_{b}\mathcal{H}^{-1}]_{dg}\\
& -W^{-2}W_{b}\sum_{d,e,f}(\partial_{\beta^{a}\phi\phi'\phi_{d}\phi_{e}\phi_{f}}\mathcal{L})[\mathcal{R}^{1}]_{f}[\mathcal{H}^{-1}]_{eh}[\mathcal{R}^{1}]_{d}[\mathcal{R}^{1}]_{g}\\
& -W^{-2}\sum_{d,e,f}(\partial_{\beta^{a+1}\phi\phi'\phi_{d}\phi_{e}\phi_{f}}\mathcal{L})[\mathcal{R}^{1}]_{f}[\mathcal{H}^{-1}]_{eh}[\mathcal{R}^{1}]_{d}[\mathcal{R}^{1}]_{g}\\
& -W^{-2}\sum_{c,d,e,f}(\partial_{\beta^{a}\phi\phi'\phi_{c}\phi_{d}\phi_{e}\phi_{f}}\mathcal{L})[\mathcal{R}^{1}]_{c}[\mathcal{R}^{1}]_{f}[\mathcal{H}^{-1}]_{eh}[\mathcal{R}^{1}]_{d}[\mathcal{R}^{1}]_{g}\\
& +W^{-1}\sum_{d,e,f}(\partial_{\beta^{a}\phi\phi'\phi_{d}\phi_{e}\phi_{f}}\mathcal{L})[\mathcal{R}_{b}^{1}]_{f}[\mathcal{H}^{-1}]_{eh}[\mathcal{R}^{1}]_{d}[\mathcal{R}^{1}]_{g}\\
& -W^{-1}\sum_{d,e,f}(\partial_{\beta^{a}\phi\phi'\phi_{d}\phi_{e}\phi_{f}}\mathcal{L})[\mathcal{R}^{1}]_{f}[\mathcal{H}^{-1}\mathcal{H}_{b}\mathcal{H}^{-1}]_{eh}[\mathcal{R}^{1}]_{d}[\mathcal{R}^{1}]_{g}\\
& +W^{-1}\sum_{d,e,f}(\partial_{\beta^{a}\phi\phi'\phi_{d}\phi_{e}\phi_{f}}\mathcal{L})[\mathcal{R}^{1}]_{f}[\mathcal{H}^{-1}]_{eh}[\mathcal{R}_{b}^{1}]_{d}[\mathcal{R}^{1}]_{g}\\
& +W^{-1}\sum_{d,e,f}(\partial_{\beta^{a}\phi\phi'\phi_{d}\phi_{e}\phi_{f}}\mathcal{L})[\mathcal{R}^{1}]_{f}[\mathcal{H}^{-1}]_{eh}[\mathcal{R}^{1}]_{d}[\mathcal{R}_{b}^{1}]_{g}\\
& +W^{-1}\sum_{e,f}(\partial_{\beta^{a+1}\phi\phi'\phi_{e}\phi_{f}}\mathcal{L})[\mathcal{R}_{s_{g}}^{1}]_{f}[\mathcal{H}^{-1}]_{eh}\\
& +W^{-1}\sum_{d,e,f}(\partial_{\beta^{a}\phi\phi'\phi_{d}\phi_{e}\phi_{f}}\mathcal{L})[\mathcal{R}^{1}]_{d}[\mathcal{R}_{s_{g}}^{1}]_{f}[\mathcal{H}^{-1}]_{eh}\\
& -\sum_{e,f}(\partial_{\beta^{a}\phi\phi'\phi_{e}\phi_{f}}\mathcal{L})[\mathcal{R}_{bs_{g}}^{1}]_{f}[\mathcal{H}^{-1}]_{eh}\\
& +\sum_{e,f}(\partial_{\beta^{a}\phi\phi'\phi_{e}\phi_{f}}\mathcal{L})[\mathcal{R}_{s_{g}}^{1}]_{f}[\mathcal{H}^{-1}\mathcal{H}_{b}\mathcal{H}^{-1}]_{eh}\\
& -W^{-1}\sum_{e,f}(\partial_{\beta^{a+1}\phi\phi'\phi_{e}\phi_{f}}\mathcal{L})[\mathcal{R}^{1}]_{f}[\mathcal{H}^{-1}\mathcal{H}_{s_{g}}\mathcal{H}^{-1}]_{eh}\\
& -W^{-1}\sum_{d,e,f}(\partial_{\beta^{a}\phi\phi'\phi_{d}\phi_{e}\phi_{f}}\mathcal{L})[\mathcal{R}^{1}]_{d}[\mathcal{R}^{1}]_{f}[\mathcal{H}^{-1}\mathcal{H}_{s_{g}}\mathcal{H}^{-1}]_{eh}\\
& +\sum_{e,f}(\partial_{\beta^{a}\phi\phi'\phi_{e}\phi_{f}}\mathcal{L})[\mathcal{R}_{b}^{1}]_{f}[\mathcal{H}^{-1}\mathcal{H}_{s_{g}}\mathcal{H}^{-1}]_{eh}\\
& -\sum_{e,f}(\partial_{\beta^{a}\phi\phi'\phi_{e}\phi_{f}}\mathcal{L})[\mathcal{R}^{1}]_{f}[\mathcal{H}^{-1}\mathcal{H}_{b}\mathcal{H}^{-1}\mathcal{H}_{s_{g}}\mathcal{H}^{-1}]_{eh}\\
& +\sum_{e,f}(\partial_{\beta^{a}\phi\phi'\phi_{e}\phi_{f}}\mathcal{L})[\mathcal{R}^{1}]_{f}[\mathcal{H}^{-1}\mathcal{H}_{bs_{g}}\mathcal{H}^{-1}]_{eh}\\
& -\sum_{e,f}(\partial_{\beta^{a}\phi\phi'\phi_{e}\phi_{f}}\mathcal{L})[\mathcal{R}^{1}]_{f}[\mathcal{H}^{-1}\mathcal{H}_{s_{g}}\mathcal{H}^{-1}\mathcal{H}_{b}\mathcal{H}^{-1}]_{eh}\\
& -2W^{-3}W_{b}W_{s_{g}}\sum_{e,f}(\partial_{\beta^{a}\phi\phi'\phi_{e}\phi_{f}}\mathcal{L})[\mathcal{R}^{1}]_{f}[\mathcal{R}^{1}]_{e}[\mathcal{R}^{1}]_{h}\\
& +W^{-2}W_{bs_{g}}\sum_{e,f}(\partial_{\beta^{a}\phi\phi'\phi_{e}\phi_{f}}\mathcal{L})[\mathcal{R}^{1}]_{f}[\mathcal{R}^{1}]_{e}[\mathcal{R}^{1}]_{h}\\
& -W^{-3}W_{s_{g}}\sum_{e,f}(\partial_{\beta^{a+1}\phi\phi'\phi_{e}\phi_{f}}\mathcal{L})[\mathcal{R}^{1}]_{f}[\mathcal{R}^{1}]_{e}[\mathcal{R}^{1}]_{h}\\
& -W^{-3}W_{s_{g}}\sum_{d,e,f}(\partial_{\beta^{a}\phi\phi'\phi_{d}\phi_{e}\phi_{f}}\mathcal{L})[\mathcal{R}^{1}]_{d}[\mathcal{R}^{1}]_{f}[\mathcal{R}^{1}]_{e}[\mathcal{R}^{1}]_{h}\\
& +W^{-2}W_{s_{g}}\sum_{e,f}(\partial_{\beta^{a}\phi\phi'\phi_{e}\phi_{f}}\mathcal{L})[\mathcal{R}_{b}^{1}]_{f}[\mathcal{R}^{1}]_{e}[\mathcal{R}^{1}]_{h}\\
& +W^{-2}W_{s_{g}}\sum_{e,f}(\partial_{\beta^{a}\phi\phi'\phi_{e}\phi_{f}}\mathcal{L})[\mathcal{R}^{1}]_{f}[\mathcal{R}_{b}^{1}]_{e}[\mathcal{R}^{1}]_{h}\\
& +W^{-2}W_{s_{g}}\sum_{e,f}(\partial_{\beta^{a}\phi\phi'\phi_{e}\phi_{f}}\mathcal{L})[\mathcal{R}^{1}]_{f}[\mathcal{R}^{1}]_{e}[\mathcal{R}_{b}^{1}]_{h}\\
& -2W^{-3}W_{b}\sum_{e,f}(\partial_{\beta^{a+1}\phi\phi'\phi_{e}\phi_{f}}\mathcal{L})[\mathcal{R}^{1}]_{f}[\mathcal{R}^{1}]_{e}[\mathcal{R}^{1}]_{g}[\mathcal{R}^{1}]_{h}\\
& -W^{-3}\sum_{e,f}(\partial_{\beta^{a+2}\phi\phi'\phi_{e}\phi_{f}}\mathcal{L})[\mathcal{R}^{1}]_{f}[\mathcal{R}^{1}]_{e}[\mathcal{R}^{1}]_{g}[\mathcal{R}^{1}]_{h}\\
& -W^{-3}\sum_{d,e,f}(\partial_{\beta^{a+1}\phi\phi'\phi_{d}\phi_{e}\phi_{f}}\mathcal{L})[\mathcal{R}^{1}]_{d}[\mathcal{R}^{1}]_{f}[\mathcal{R}^{1}]_{e}[\mathcal{R}^{1}]_{g}[\mathcal{R}^{1}]_{h}\\
& +W^{-2}\sum_{e,f}(\partial_{\beta^{a+1}\phi\phi'\phi_{e}\phi_{f}}\mathcal{L})[\mathcal{R}_{b}^{1}]_{f}[\mathcal{R}^{1}]_{e}[\mathcal{R}^{1}]_{g}[\mathcal{R}^{1}]_{h}\\
& +W^{-2}\sum_{e,f}(\partial_{\beta^{a+1}\phi\phi'\phi_{e}\phi_{f}}\mathcal{L})[\mathcal{R}^{1}]_{f}[\mathcal{R}_{b}^{1}]_{e}[\mathcal{R}^{1}]_{g}[\mathcal{R}^{1}]_{h}\\
& +W^{-2}\sum_{e,f}(\partial_{\beta^{a+1}\phi\phi'\phi_{e}\phi_{f}}\mathcal{L})[\mathcal{R}^{1}]_{f}[\mathcal{R}^{1}]_{e}[\mathcal{R}_{b}^{1}]_{g}[\mathcal{R}^{1}]_{h}\\
& +W^{-2}\sum_{e,f}(\partial_{\beta^{a+1}\phi\phi'\phi_{e}\phi_{f}}\mathcal{L})[\mathcal{R}^{1}]_{f}[\mathcal{R}^{1}]_{e}[\mathcal{R}^{1}]_{g}[\mathcal{R}_{b}^{1}]_{h}\\
& -W^{-2}W_{b}\sum_{d,e,f}(\partial_{\beta^{a}\phi\phi'\phi_{d}\phi_{e}\phi_{f}}\mathcal{L})[\mathcal{R}^{1}]_{f}[\mathcal{R}^{1}]_{e}[\mathcal{R}^{1}]_{h}[\mathcal{H}^{-1}]_{dg}\\
& -W^{-2}\sum_{d,e,f}(\partial_{\beta^{a+1}\phi\phi'\phi_{d}\phi_{e}\phi_{f}}\mathcal{L})[\mathcal{R}^{1}]_{f}[\mathcal{R}^{1}]_{e}[\mathcal{R}^{1}]_{h}[\mathcal{H}^{-1}]_{dg}\\
& -W^{-2}\sum_{c,d,e,f}(\partial_{\beta^{a}\phi\phi'\phi_{c}\phi_{d}\phi_{e}\phi_{f}}\mathcal{L})[\mathcal{R}^{1}]_{c}[\mathcal{R}^{1}]_{f}[\mathcal{R}^{1}]_{e}[\mathcal{R}^{1}]_{h}[\mathcal{H}^{-1}]_{dg}\\
& +W^{-1}\sum_{d,e,f}(\partial_{\beta^{a}\phi\phi'\phi_{d}\phi_{e}\phi_{f}}\mathcal{L})[\mathcal{R}_{b}^{1}]_{f}[\mathcal{R}^{1}]_{e}[\mathcal{R}^{1}]_{h}[\mathcal{H}^{-1}]_{dg}\\
& +W^{-1}\sum_{d,e,f}(\partial_{\beta^{a}\phi\phi'\phi_{d}\phi_{e}\phi_{f}}\mathcal{L})[\mathcal{R}^{1}]_{f}[\mathcal{R}_{b}^{1}]_{e}[\mathcal{R}^{1}]_{h}[\mathcal{H}^{-1}]_{dg}\\
& +W^{-1}\sum_{d,e,f}(\partial_{\beta^{a}\phi\phi'\phi_{d}\phi_{e}\phi_{f}}\mathcal{L})[\mathcal{R}^{1}]_{f}[\mathcal{R}^{1}]_{e}[\mathcal{R}_{b}^{1}]_{h}[\mathcal{H}^{-1}]_{dg}\\
& -W^{-1}\sum_{d,e,f}(\partial_{\beta^{a}\phi\phi'\phi_{d}\phi_{e}\phi_{f}}\mathcal{L})[\mathcal{R}^{1}]_{f}[\mathcal{R}^{1}]_{e}[\mathcal{R}^{1}]_{h}[\mathcal{H}^{-1}\mathcal{H}_{b}\mathcal{H}^{-1}]_{dg}\\
& -2W^{-3}W_{b}\sum_{d,e,f}(\partial_{\beta^{a}\phi\phi'\phi_{d}\phi_{e}\phi_{f}}\mathcal{L})[\mathcal{R}^{1}]_{f}[\mathcal{R}^{1}]_{d}[\mathcal{R}^{1}]_{e}[\mathcal{R}^{1}]_{g}[\mathcal{R}^{1}]_{h}\\
& -W^{-3}\sum_{d,e,f}(\partial_{\beta^{a+1}\phi\phi'\phi_{d}\phi_{e}\phi_{f}}\mathcal{L})[\mathcal{R}^{1}]_{f}[\mathcal{R}^{1}]_{d}[\mathcal{R}^{1}]_{e}[\mathcal{R}^{1}]_{g}[\mathcal{R}^{1}]_{h}\\
& -W^{-3}\sum_{c,d,e,f}(\partial_{\beta^{a}\phi\phi'\phi_{c}\phi_{d}\phi_{e}\phi_{f}}\mathcal{L})[\mathcal{R}^{1}]_{c}[\mathcal{R}^{1}]_{f}[\mathcal{R}^{1}]_{d}[\mathcal{R}^{1}]_{e}[\mathcal{R}^{1}]_{g}[\mathcal{R}^{1}]_{h}\\
& +W^{-2}\sum_{d,e,f}(\partial_{\beta^{a}\phi\phi'\phi_{d}\phi_{e}\phi_{f}}\mathcal{L})[\mathcal{R}_{b}^{1}]_{f}[\mathcal{R}^{1}]_{d}[\mathcal{R}^{1}]_{e}[\mathcal{R}^{1}]_{g}[\mathcal{R}^{1}]_{h}\\
& +W^{-2}\sum_{d,e,f}(\partial_{\beta^{a}\phi\phi'\phi_{d}\phi_{e}\phi_{f}}\mathcal{L})[\mathcal{R}^{1}]_{f}[\mathcal{R}_{b}^{1}]_{d}[\mathcal{R}^{1}]_{e}[\mathcal{R}^{1}]_{g}[\mathcal{R}^{1}]_{h}\\
& +W^{-2}\sum_{d,e,f}(\partial_{\beta^{a}\phi\phi'\phi_{d}\phi_{e}\phi_{f}}\mathcal{L})[\mathcal{R}^{1}]_{f}[\mathcal{R}^{1}]_{d}[\mathcal{R}_{b}^{1}]_{e}[\mathcal{R}^{1}]_{g}[\mathcal{R}^{1}]_{h}\\
& +W^{-2}\sum_{d,e,f}(\partial_{\beta^{a}\phi\phi'\phi_{d}\phi_{e}\phi_{f}}\mathcal{L})[\mathcal{R}^{1}]_{f}[\mathcal{R}^{1}]_{d}[\mathcal{R}^{1}]_{e}[\mathcal{R}_{b}^{1}]_{g}[\mathcal{R}^{1}]_{h}\\
& +W^{-2}\sum_{d,e,f}(\partial_{\beta^{a}\phi\phi'\phi_{d}\phi_{e}\phi_{f}}\mathcal{L})[\mathcal{R}^{1}]_{f}[\mathcal{R}^{1}]_{d}[\mathcal{R}^{1}]_{e}[\mathcal{R}^{1}]_{g}[\mathcal{R}_{b}^{1}]_{h}\\
& +W^{-2}W_{b}\sum_{e,f}(\partial_{\beta^{a}\phi\phi'\phi_{e}\phi_{f}}\mathcal{L})[\mathcal{R}_{s_{g}}^{1}]_{f}[\mathcal{R}^{1}]_{e}[\mathcal{R}^{1}]_{h}\\
& +W^{-2}\sum_{e,f}(\partial_{\beta^{a+1}\phi\phi'\phi_{e}\phi_{f}}\mathcal{L})[\mathcal{R}_{s_{g}}^{1}]_{f}[\mathcal{R}^{1}]_{e}[\mathcal{R}^{1}]_{h}\\
& +W^{-2}\sum_{d,e,f}(\partial_{\beta^{a}\phi\phi'\phi_{d}\phi_{e}\phi_{f}}\mathcal{L})[\mathcal{R}^{1}]_{d}[\mathcal{R}_{s_{g}}^{1}]_{f}[\mathcal{R}^{1}]_{e}[\mathcal{R}^{1}]_{h}\\
& -W^{-1}\sum_{e,f}(\partial_{\beta^{a}\phi\phi'\phi_{e}\phi_{f}}\mathcal{L})[\mathcal{R}_{bs_{g}}^{1}]_{f}[\mathcal{R}^{1}]_{e}[\mathcal{R}^{1}]_{h}\\
& -W^{-1}\sum_{e,f}(\partial_{\beta^{a}\phi\phi'\phi_{e}\phi_{f}}\mathcal{L})[\mathcal{R}_{s_{g}}^{1}]_{f}[\mathcal{R}_{b}^{1}]_{e}[\mathcal{R}^{1}]_{h}\\
& -W^{-1}\sum_{e,f}(\partial_{\beta^{a}\phi\phi'\phi_{e}\phi_{f}}\mathcal{L})[\mathcal{R}_{s_{g}}^{1}]_{f}[\mathcal{R}^{1}]_{e}[\mathcal{R}_{b}^{1}]_{h}\\
& +W^{-2}W_{b}\sum_{e,f}(\partial_{\beta^{a}\phi\phi'\phi_{e}\phi_{f}}\mathcal{L})[\mathcal{R}^{1}]_{f}[\mathcal{R}_{s_{g}}^{1}]_{e}[\mathcal{R}^{1}]_{h}\\
& +W^{-2}\sum_{e,f}(\partial_{\beta^{a+1}\phi\phi'\phi_{e}\phi_{f}}\mathcal{L})[\mathcal{R}^{1}]_{f}[\mathcal{R}_{s_{g}}^{1}]_{e}[\mathcal{R}^{1}]_{h}\\
& +W^{-2}\sum_{d,e,f}(\partial_{\beta^{a}\phi\phi'\phi_{d}\phi_{e}\phi_{f}}\mathcal{L})[\mathcal{R}^{1}]_{d}[\mathcal{R}^{1}]_{f}[\mathcal{R}_{s_{g}}^{1}]_{e}[\mathcal{R}^{1}]_{h}\\
& -W^{-1}\sum_{e,f}(\partial_{\beta^{a}\phi\phi'\phi_{e}\phi_{f}}\mathcal{L})[\mathcal{R}_{b}^{1}]_{f}[\mathcal{R}_{s_{g}}^{1}]_{e}[\mathcal{R}^{1}]_{h}\\
& -W^{-1}\sum_{e,f}(\partial_{\beta^{a}\phi\phi'\phi_{e}\phi_{f}}\mathcal{L})[\mathcal{R}^{1}]_{f}[\mathcal{R}_{bs_{g}}^{1}]_{e}[\mathcal{R}^{1}]_{h}\\
& -W^{-1}\sum_{e,f}(\partial_{\beta^{a}\phi\phi'\phi_{e}\phi_{f}}\mathcal{L})[\mathcal{R}^{1}]_{f}[\mathcal{R}_{s_{g}}^{1}]_{e}[\mathcal{R}_{b}^{1}]_{h}\\
& +W^{-2}W_{b}\sum_{e,f}(\partial_{\beta^{a}\phi\phi'\phi_{e}\phi_{f}}\mathcal{L})[\mathcal{R}^{1}]_{f}[\mathcal{R}^{1}]_{e}[\mathcal{R}_{s_{g}}^{1}]_{h}\\
& +W^{-2}\sum_{e,f}(\partial_{\beta^{a+1}\phi\phi'\phi_{e}\phi_{f}}\mathcal{L})[\mathcal{R}^{1}]_{f}[\mathcal{R}^{1}]_{e}[\mathcal{R}_{s_{g}}^{1}]_{h}\\
& +W^{-2}\sum_{d,e,f}(\partial_{\beta^{a}\phi\phi'\phi_{d}\phi_{e}\phi_{f}}\mathcal{L})[\mathcal{R}^{1}]_{d}[\mathcal{R}^{1}]_{f}[\mathcal{R}^{1}]_{e}[\mathcal{R}_{s_{g}}^{1}]_{h}\\
& -W^{-1}\sum_{e,f}(\partial_{\beta^{a}\phi\phi'\phi_{e}\phi_{f}}\mathcal{L})[\mathcal{R}_{b}^{1}]_{f}[\mathcal{R}^{1}]_{e}[\mathcal{R}_{s_{g}}^{1}]_{h}\\
& -W^{-1}\sum_{e,f}(\partial_{\beta^{a}\phi\phi'\phi_{e}\phi_{f}}\mathcal{L})[\mathcal{R}^{1}]_{f}[\mathcal{R}_{b}^{1}]_{e}[\mathcal{R}_{s_{g}}^{1}]_{h}\\
& -W^{-1}\sum_{e,f}(\partial_{\beta^{a}\phi\phi'\phi_{e}\phi_{f}}\mathcal{L})[\mathcal{R}^{1}]_{f}[\mathcal{R}^{1}]_{e}[\mathcal{R}_{bs_{g}}^{1}]_{h}\\
& +W^{-2}W_{b}\sum_{f}(\partial_{\beta^{a+1}\phi\phi'\phi_{f}}\mathcal{L})[\mathcal{R}_{s_{h}}^{1}]_{f}[\mathcal{R}^{1}]_{g}\\
& +W^{-2}\sum_{f}(\partial_{\beta^{a+2}\phi\phi'\phi_{f}}\mathcal{L})[\mathcal{R}_{s_{h}}^{1}]_{f}[\mathcal{R}^{1}]_{g}\\
& +W^{-2}\sum_{e,f}(\partial_{\beta^{a+1}\phi\phi'\phi_{e}\phi_{f}}\mathcal{L})[\mathcal{R}^{1}]_{e}[\mathcal{R}_{s_{h}}^{1}]_{f}[\mathcal{R}^{1}]_{g}\\
& -W^{-1}\sum_{f}(\partial_{\beta^{a+1}\phi\phi'\phi_{f}}\mathcal{L})[\mathcal{R}_{bs_{h}}^{1}]_{f}[\mathcal{R}^{1}]_{g}\\
& -W^{-1}\sum_{f}(\partial_{\beta^{a+1}\phi\phi'\phi_{f}}\mathcal{L})[\mathcal{R}_{s_{h}}^{1}]_{f}[\mathcal{R}_{b}^{1}]_{g}\\
& +W^{-1}\sum_{e,f}(\partial_{\beta^{a+1}\phi\phi'\phi_{e}\phi_{f}}\mathcal{L})[\mathcal{R}_{s_{h}}^{1}]_{f}[\mathcal{H}^{-1}]_{eg}\\
& +W^{-1}\sum_{d,e,f}(\partial_{\beta^{a}\phi\phi'\phi_{d}\phi_{e}\phi_{f}}\mathcal{L})[\mathcal{R}^{1}]_{d}[\mathcal{R}_{s_{h}}^{1}]_{f}[\mathcal{H}^{-1}]_{eg}\\
& -\sum_{e,f}(\partial_{\beta^{a}\phi\phi'\phi_{e}\phi_{f}}\mathcal{L})[\mathcal{R}_{bs_{h}}^{1}]_{f}[\mathcal{H}^{-1}]_{eg}\\
& +\sum_{e,f}(\partial_{\beta^{a}\phi\phi'\phi_{e}\phi_{f}}\mathcal{L})[\mathcal{R}_{s_{h}}^{1}]_{f}[\mathcal{H}^{-1}\mathcal{H}_{b}\mathcal{H}^{-1}]_{eg}\\
& -W^{-2}W_{b}\sum_{e,f}(\partial_{\beta^{a}\phi\phi'\phi_{e}\phi_{f}}\mathcal{L})[\mathcal{R}_{s_{h}}^{1}]_{f}[\mathcal{R}^{1}]_{e}[\mathcal{R}^{1}]_{g}\\
& -W^{-2}\sum_{e,f}(\partial_{\beta^{a+1}\phi\phi'\phi_{e}\phi_{f}}\mathcal{L})[\mathcal{R}_{s_{h}}^{1}]_{f}[\mathcal{R}^{1}]_{e}[\mathcal{R}^{1}]_{g}\\
& -W^{-2}\sum_{d,e,f}(\partial_{\beta^{a}\phi\phi'\phi_{d}\phi_{e}\phi_{f}}\mathcal{L})[\mathcal{R}^{1}]_{d}[\mathcal{R}_{s_{h}}^{1}]_{f}[\mathcal{R}^{1}]_{e}[\mathcal{R}^{1}]_{g}\\
& +W^{-1}\sum_{e,f}(\partial_{\beta^{a}\phi\phi'\phi_{e}\phi_{f}}\mathcal{L})[\mathcal{R}_{bs_{h}}^{1}]_{f}[\mathcal{R}^{1}]_{e}[\mathcal{R}^{1}]_{g}\\
& +W^{-1}\sum_{e,f}(\partial_{\beta^{a}\phi\phi'\phi_{e}\phi_{f}}\mathcal{L})[\mathcal{R}_{s_{h}}^{1}]_{f}[\mathcal{R}_{b}^{1}]_{e}[\mathcal{R}^{1}]_{g}\\
& +W^{-1}\sum_{e,f}(\partial_{\beta^{a}\phi\phi'\phi_{e}\phi_{f}}\mathcal{L})[\mathcal{R}_{s_{h}}^{1}]_{f}[\mathcal{R}^{1}]_{e}[\mathcal{R}_{b}^{1}]_{g}\\
& -W^{-1}\sum_{f}(\partial_{\beta^{a+1}\phi\phi'\phi_{f}}\mathcal{L})[\mathcal{R}_{s_{g}s_{h}}^{1}]_{f}\\
& -W^{-1}\sum_{e,f}(\partial_{\beta^{a}\phi\phi'\phi_{e}\phi_{f}}\mathcal{L})[\mathcal{R}^{1}]_{e}[\mathcal{R}_{s_{g}s_{h}}^{1}]_{f}\\
& +\sum_{f}(\partial_{\beta^{a}\phi\phi'\phi_{f}}\mathcal{L})[\mathcal{R}_{bs_{g}s_{h}}^{1}]_{f}
\end{align*}
\begin{align*}
\mathcal{P}_{(g),bs_{k}s_{h}}^{r} & =-2W^{-3}W_{b}W_{s_{k}}[\mathcal{R}^{1}]_{h}\mathcal{P}_{(g)}^{r+1}+W^{-2}W_{bs_{k}}[\mathcal{R}^{1}]_{h}\mathcal{P}_{(g)}^{r+1}\\
& +W^{-2}W_{s_{k}}[\mathcal{R}_{b}^{1}]_{h}\mathcal{P}_{(g)}^{r+1}+W^{-2}W_{s_{k}}[\mathcal{R}^{1}]_{h}\mathcal{P}_{(g),b}^{r+1}\\
& +W^{-2}W_{b}[\mathcal{R}_{s_{k}}^{1}]_{h}\mathcal{P}_{(g)}^{r+1}-W^{-1}[\mathcal{R}_{bs_{k}}^{1}]_{h}\mathcal{P}_{(g)}^{r+1}-W^{-1}[\mathcal{R}_{s_{k}}^{1}]_{h}\mathcal{P}_{(g),b}^{r+1}\\
& +W^{-2}W_{b}[\mathcal{R}^{1}]_{h}\mathcal{P}_{(g),s_{k}}^{r+1}-W^{-1}[\mathcal{R}_{b}^{1}]_{h}\mathcal{P}_{(g),s_{k}}^{r+1}-W^{-1}[\mathcal{R}^{1}]_{h}\mathcal{P}_{(g),bs_{k}}^{r+1}\\
& -W^{-2}W_{b}\sum_{e,f}(\partial_{\beta^{r+1}\phi\phi'\phi_{e}\phi_{f}}\mathcal{L})\big[\mathcal{H}^{-1}\big]_{fg}\big[\mathcal{H}^{-1}\big]_{eh}[\mathcal{R}^{1}]_{k}\\
& -W^{-2}\sum_{e,f}(\partial_{\beta^{r+2}\phi\phi'\phi_{e}\phi_{f}}\mathcal{L})\big[\mathcal{H}^{-1}\big]_{fg}\big[\mathcal{H}^{-1}\big]_{eh}[\mathcal{R}^{1}]_{k}\\
& -W^{-2}\sum_{d,e,f}(\partial_{\beta^{r+1}\phi\phi'\phi_{d}\phi_{e}\phi_{f}}\mathcal{L})\big[\mathcal{H}^{-1}\big]_{fg}\big[\mathcal{H}^{-1}\big]_{eh}[\mathcal{R}^{1}]_{d}[\mathcal{R}^{1}]_{k}\\
& -W^{-1}\sum_{e,f}(\partial_{\beta^{r+1}\phi\phi'\phi_{e}\phi_{f}}\mathcal{L})\big[\mathcal{H}^{-1}\mathcal{H}_{b}\mathcal{H}^{-1}\big]_{fg}\big[\mathcal{H}^{-1}\big]_{eh}[\mathcal{R}^{1}]_{k}\\
& -W^{-1}\sum_{e,f}(\partial_{\beta^{r+1}\phi\phi'\phi_{e}\phi_{f}}\mathcal{L})\big[\mathcal{H}^{-1}\big]_{fg}\big[\mathcal{H}^{-1}\mathcal{H}_{b}\mathcal{H}^{-1}\big]_{eh}[\mathcal{R}^{1}]_{k}\\
& +W^{-1}\sum_{e,f}(\partial_{\beta^{r+1}\phi\phi'\phi_{e}\phi_{f}}\mathcal{L})\big[\mathcal{H}^{-1}\big]_{fg}\big[\mathcal{H}^{-1}\big]_{eh}[\mathcal{R}_{b}^{1}]_{k}\\
& -W^{-1}\sum_{d,e,f}(\partial_{\beta^{r+1}\phi\phi'\phi_{d}\phi_{e}\phi_{f}}\mathcal{L})\big[\mathcal{H}^{-1}\big]_{fg}\big[\mathcal{H}^{-1}\big]_{eh}\big[\mathcal{H}^{-1}\big]_{dk}\\
& -W^{-1}\sum_{c,d,e,f}(\partial_{\beta^{r}\phi\phi'\phi_{c}\phi_{d}\phi_{e}\phi_{f}}\mathcal{L})[\mathcal{R}^{1}]_{c}\big[\mathcal{H}^{-1}\big]_{fg}\big[\mathcal{H}^{-1}\big]_{eh}\big[\mathcal{H}^{-1}\big]_{dk}\\
& -\sum_{d,e,f}(\partial_{\beta^{r}\phi\phi'\phi_{d}\phi_{e}\phi_{f}}\mathcal{L})\big[\mathcal{H}^{-1}\mathcal{H}_{b}\mathcal{H}^{-1}\big]_{fg}\big[\mathcal{H}^{-1}\big]_{eh}\big[\mathcal{H}^{-1}\big]_{dk}\\
& -\sum_{d,e,f}(\partial_{\beta^{r}\phi\phi'\phi_{d}\phi_{e}\phi_{f}}\mathcal{L})\big[\mathcal{H}^{-1}\big]_{fg}\big[\mathcal{H}^{-1}\mathcal{H}_{b}\mathcal{H}^{-1}\big]_{eh}\big[\mathcal{H}^{-1}\big]_{dk}\\
& -\sum_{d,e,f}(\partial_{\beta^{r}\phi\phi'\phi_{d}\phi_{e}\phi_{f}}\mathcal{L})\big[\mathcal{H}^{-1}\big]_{fg}\big[\mathcal{H}^{-1}\big]_{eh}\big[\mathcal{H}^{-1}\mathcal{H}_{b}\mathcal{H}^{-1}\big]_{dk}\\
& -W^{-2}W_{b}\sum_{d,e,f}(\partial_{\beta^{r}\phi\phi'\phi_{d}\phi_{e}\phi_{f}}\mathcal{L})\big[\mathcal{H}^{-1}\big]_{fg}\big[\mathcal{H}^{-1}\big]_{eh}[\mathcal{R}^{1}]_{d}[\mathcal{R}^{1}]_{k}\\
& -W^{-2}\sum_{d,e,f}(\partial_{\beta^{r+1}\phi\phi'\phi_{d}\phi_{e}\phi_{f}}\mathcal{L})\big[\mathcal{H}^{-1}\big]_{fg}\big[\mathcal{H}^{-1}\big]_{eh}[\mathcal{R}^{1}]_{d}[\mathcal{R}^{1}]_{k}\\
& -W^{-2}\sum_{c,d,e,f}(\partial_{\beta^{r}\phi\phi'\phi_{c}\phi_{d}\phi_{e}\phi_{f}}\mathcal{L})\big[\mathcal{H}^{-1}\big]_{fg}\big[\mathcal{H}^{-1}\big]_{eh}[\mathcal{R}^{1}]_{c}[\mathcal{R}^{1}]_{d}[\mathcal{R}^{1}]_{k}\\
& -W^{-1}\sum_{d,e,f}(\partial_{\beta^{r}\phi\phi'\phi_{d}\phi_{e}\phi_{f}}\mathcal{L})\big[\mathcal{H}^{-1}\mathcal{H}_{b}\mathcal{H}^{-1}\big]_{fg}\big[\mathcal{H}^{-1}\big]_{eh}[\mathcal{R}^{1}]_{d}[\mathcal{R}^{1}]_{k}\\
& -W^{-1}\sum_{d,e,f}(\partial_{\beta^{r}\phi\phi'\phi_{d}\phi_{e}\phi_{f}}\mathcal{L})\big[\mathcal{H}^{-1}\big]_{fg}\big[\mathcal{H}^{-1}\mathcal{H}_{b}\mathcal{H}^{-1}\big]_{eh}[\mathcal{R}^{1}]_{d}[\mathcal{R}^{1}]_{k}\\
& +W^{-1}\sum_{d,e,f}(\partial_{\beta^{r}\phi\phi'\phi_{d}\phi_{e}\phi_{f}}\mathcal{L})\big[\mathcal{H}^{-1}\big]_{fg}\big[\mathcal{H}^{-1}\big]_{eh}[\mathcal{R}_{b}^{1}]_{d}[\mathcal{R}^{1}]_{k}\\
& +W^{-1}\sum_{d,e,f}(\partial_{\beta^{r}\phi\phi'\phi_{d}\phi_{e}\phi_{f}}\mathcal{L})\big[\mathcal{H}^{-1}\big]_{fg}\big[\mathcal{H}^{-1}\big]_{eh}[\mathcal{R}^{1}]_{d}[\mathcal{R}_{b}^{1}]_{k}\\
& -W^{-1}\sum_{e,f}(\partial_{\beta^{r+1}\phi\phi'\phi_{e}\phi_{f}}\mathcal{L})\big[\mathcal{H}^{-1}\mathcal{H}_{s_{k}}\mathcal{H}^{-1}\big]_{fg}\big[\mathcal{H}^{-1}\big]_{eh}\\
& -W^{-1}\sum_{d,e,f}(\partial_{\beta^{r}\phi\phi'\phi_{d}\phi_{e}\phi_{f}}\mathcal{L})\big[\mathcal{H}^{-1}\mathcal{H}_{s_{k}}\mathcal{H}^{-1}\big]_{fg}\big[\mathcal{H}^{-1}\big]_{eh}[\mathcal{R}^{1}]_{d}\\
& -\sum_{e,f}(\partial_{\beta^{r}\phi\phi'\phi_{e}\phi_{f}}\mathcal{L})\big[\mathcal{H}^{-1}\mathcal{H}_{b}\mathcal{H}^{-1}\mathcal{H}_{s_{k}}\mathcal{H}^{-1}\big]_{fg}\big[\mathcal{H}^{-1}\big]_{eh}\\
& +\sum_{e,f}(\partial_{\beta^{r}\phi\phi'\phi_{e}\phi_{f}}\mathcal{L})\big[\mathcal{H}^{-1}\mathcal{H}_{bs_{k}}\mathcal{H}^{-1}\big]_{fg}\big[\mathcal{H}^{-1}\big]_{eh}\\
& -\sum_{e,f}(\partial_{\beta^{r}\phi\phi'\phi_{e}\phi_{f}}\mathcal{L})\big[\mathcal{H}^{-1}\mathcal{H}_{s_{k}}\mathcal{H}^{-1}\mathcal{H}_{b}\mathcal{H}^{-1}\big]_{fg}\big[\mathcal{H}^{-1}\big]_{eh}\\
& -\sum_{e,f}(\partial_{\beta^{r}\phi\phi'\phi_{e}\phi_{f}}\mathcal{L})\big[\mathcal{H}^{-1}\mathcal{H}_{s_{k}}\mathcal{H}^{-1}\big]_{fg}\big[\mathcal{H}^{-1}\mathcal{H}_{b}\mathcal{H}^{-1}\big]_{eh}\\
& -W^{-1}\sum_{e,f}(\partial_{\beta^{r+1}\phi\phi'\phi_{e}\phi_{f}}\mathcal{L})\big[\mathcal{H}^{-1}\big]_{fg}\big[\mathcal{H}^{-1}\mathcal{H}_{s_{k}}\mathcal{H}^{-1}\big]_{eh}\\
& -W^{-1}\sum_{d,e,f}(\partial_{\beta^{r}\phi\phi'\phi_{d}\phi_{e}\phi_{f}}\mathcal{L})\big[\mathcal{H}^{-1}\big]_{fg}\big[\mathcal{H}^{-1}\mathcal{H}_{s_{k}}\mathcal{H}^{-1}\big]_{eh}[\mathcal{R}^{1}]_{d}\\
& -\sum_{e,f}(\partial_{\beta^{r}\phi\phi'\phi_{e}\phi_{f}}\mathcal{L})\big[\mathcal{H}^{-1}\mathcal{H}_{b}\mathcal{H}^{-1}\big]_{fg}\big[\mathcal{H}^{-1}\mathcal{H}_{s_{k}}\mathcal{H}^{-1}\big]_{eh}\\
& -\sum_{e,f}(\partial_{\beta^{r}\phi\phi'\phi_{e}\phi_{f}}\mathcal{L})\big[\mathcal{H}^{-1}\big]_{fg}\big[\mathcal{H}^{-1}\mathcal{H}_{b}\mathcal{H}^{-1}\mathcal{H}_{s_{k}}\mathcal{H}^{-1}\big]_{eh}\\
& +\sum_{e,f}(\partial_{\beta^{r}\phi\phi'\phi_{e}\phi_{f}}\mathcal{L})\big[\mathcal{H}^{-1}\big]_{fg}\big[\mathcal{H}^{-1}\mathcal{H}_{bs_{k}}\mathcal{H}^{-1}\big]_{eh}\\
& -\sum_{e,f}(\partial_{\beta^{r}\phi\phi'\phi_{e}\phi_{f}}\mathcal{L})\big[\mathcal{H}^{-1}\big]_{fg}\big[\mathcal{H}^{-1}\mathcal{H}_{s_{k}}\mathcal{H}^{-1}\mathcal{H}_{b}\mathcal{H}^{-1}\big]_{eh}\\
& -2W^{-3}W_{b}W_{s_{k}}\sum_{e,f}(\partial_{\beta^{r}\phi\phi'\phi_{e}\phi_{f}}\mathcal{L})\big[\mathcal{H}^{-1}\big]_{fg}[\mathcal{R}^{1}]_{e}[\mathcal{R}^{1}]_{h}\\
& +W^{-2}W_{bs_{k}}\sum_{e,f}(\partial_{\beta^{r}\phi\phi'\phi_{e}\phi_{f}}\mathcal{L})\big[\mathcal{H}^{-1}\big]_{fg}[\mathcal{R}^{1}]_{e}[\mathcal{R}^{1}]_{h}\\
& -W^{-3}W_{s_{k}}\sum_{e,f}(\partial_{\beta^{r+1}\phi\phi'\phi_{e}\phi_{f}}\mathcal{L})\big[\mathcal{H}^{-1}\big]_{fg}[\mathcal{R}^{1}]_{e}[\mathcal{R}^{1}]_{h}\\
& -W^{-3}W_{s_{k}}\sum_{d,e,f}(\partial_{\beta^{r}\phi\phi'\phi_{d}\phi_{e}\phi_{f}}\mathcal{L})\big[\mathcal{H}^{-1}\big]_{fg}[\mathcal{R}^{1}]_{d}[\mathcal{R}^{1}]_{e}[\mathcal{R}^{1}]_{h}\\
& -W^{-2}W_{s_{k}}\sum_{e,f}(\partial_{\beta^{r}\phi\phi'\phi_{e}\phi_{f}}\mathcal{L})\big[\mathcal{H}^{-1}\mathcal{H}_{b}\mathcal{H}^{-1}\big]_{fg}[\mathcal{R}^{1}]_{e}[\mathcal{R}^{1}]_{h}\\
& +W^{-2}W_{s_{k}}\sum_{e,f}(\partial_{\beta^{r}\phi\phi'\phi_{e}\phi_{f}}\mathcal{L})\big[\mathcal{H}^{-1}\big]_{fg}[\mathcal{R}_{b}^{1}]_{e}[\mathcal{R}^{1}]_{h}\\
& +W^{-2}W_{s_{k}}\sum_{e,f}(\partial_{\beta^{r}\phi\phi'\phi_{e}\phi_{f}}\mathcal{L})\big[\mathcal{H}^{-1}\big]_{fg}[\mathcal{R}^{1}]_{e}[\mathcal{R}_{b}^{1}]_{h}\\
& -2W^{-3}W_{b}\sum_{e,f}(\partial_{\beta^{r+1}\phi\phi'\phi_{e}\phi_{f}}\mathcal{L})\big[\mathcal{H}^{-1}\big]_{fg}[\mathcal{R}^{1}]_{e}[\mathcal{R}^{1}]_{h}[\mathcal{R}^{1}]_{k}\\
& -W^{-3}\sum_{e,f}(\partial_{\beta^{r+2}\phi\phi'\phi_{e}\phi_{f}}\mathcal{L})\big[\mathcal{H}^{-1}\big]_{fg}[\mathcal{R}^{1}]_{e}[\mathcal{R}^{1}]_{h}[\mathcal{R}^{1}]_{k}\\
& -W^{-3}\sum_{d,e,f}(\partial_{\beta^{r+1}\phi\phi'\phi_{d}\phi_{e}\phi_{f}}\mathcal{L})\big[\mathcal{H}^{-1}\big]_{fg}[\mathcal{R}^{1}]_{d}[\mathcal{R}^{1}]_{e}[\mathcal{R}^{1}]_{h}[\mathcal{R}^{1}]_{k}\\
& -W^{-2}\sum_{e,f}(\partial_{\beta^{r+1}\phi\phi'\phi_{e}\phi_{f}}\mathcal{L})\big[\mathcal{H}^{-1}\mathcal{H}_{b}\mathcal{H}^{-1}\big]_{fg}[\mathcal{R}^{1}]_{e}[\mathcal{R}^{1}]_{h}[\mathcal{R}^{1}]_{k}\\
& +W^{-2}\sum_{e,f}(\partial_{\beta^{r+1}\phi\phi'\phi_{e}\phi_{f}}\mathcal{L})\big[\mathcal{H}^{-1}\big]_{fg}[\mathcal{R}_{b}^{1}]_{e}[\mathcal{R}^{1}]_{h}[\mathcal{R}^{1}]_{k}\\
& +W^{-2}\sum_{e,f}(\partial_{\beta^{r+1}\phi\phi'\phi_{e}\phi_{f}}\mathcal{L})\big[\mathcal{H}^{-1}\big]_{fg}[\mathcal{R}^{1}]_{e}[\mathcal{R}_{b}^{1}]_{h}[\mathcal{R}^{1}]_{k}\\
& +W^{-2}\sum_{e,f}(\partial_{\beta^{r+1}\phi\phi'\phi_{e}\phi_{f}}\mathcal{L})\big[\mathcal{H}^{-1}\big]_{fg}[\mathcal{R}^{1}]_{e}[\mathcal{R}^{1}]_{h}[\mathcal{R}_{b}^{1}]_{k}\\
& -W^{-2}W_{b}\sum_{d,e,f}(\partial_{\beta^{r}\phi\phi'\phi_{d}\phi_{e}\phi_{f}}\mathcal{L})\big[\mathcal{H}^{-1}\big]_{fg}\big[\mathcal{H}^{-1}\big]_{dk}[\mathcal{R}^{1}]_{e}[\mathcal{R}^{1}]_{h}\\
& -W^{-2}\sum_{d,e,f}(\partial_{\beta^{r+1}\phi\phi'\phi_{d}\phi_{e}\phi_{f}}\mathcal{L})\big[\mathcal{H}^{-1}\big]_{fg}\big[\mathcal{H}^{-1}\big]_{dk}[\mathcal{R}^{1}]_{e}[\mathcal{R}^{1}]_{h}\\
& -W^{-2}\sum_{c,d,e,f}(\partial_{\beta^{r}\phi\phi'\phi_{c}\phi_{d}\phi_{e}\phi_{f}}\mathcal{L})\big[\mathcal{H}^{-1}\big]_{fg}\big[\mathcal{H}^{-1}\big]_{dk}[\mathcal{R}^{1}]_{c}[\mathcal{R}^{1}]_{e}[\mathcal{R}^{1}]_{h}\\
& -W^{-1}\sum_{d,e,f}(\partial_{\beta^{r}\phi\phi'\phi_{d}\phi_{e}\phi_{f}}\mathcal{L})\big[\mathcal{H}^{-1}\mathcal{H}_{b}\mathcal{H}^{-1}\big]_{fg}\big[\mathcal{H}^{-1}\big]_{dk}[\mathcal{R}^{1}]_{e}[\mathcal{R}^{1}]_{h}\\
& -W^{-1}\sum_{d,e,f}(\partial_{\beta^{r}\phi\phi'\phi_{d}\phi_{e}\phi_{f}}\mathcal{L})\big[\mathcal{H}^{-1}\big]_{fg}\big[\mathcal{H}^{-1}\mathcal{H}_{b}\mathcal{H}^{-1}\big]_{dk}[\mathcal{R}^{1}]_{e}[\mathcal{R}^{1}]_{h}\\
& +W^{-1}\sum_{d,e,f}(\partial_{\beta^{r}\phi\phi'\phi_{d}\phi_{e}\phi_{f}}\mathcal{L})\big[\mathcal{H}^{-1}\big]_{fg}\big[\mathcal{H}^{-1}\big]_{dk}[\mathcal{R}_{b}^{1}]_{e}[\mathcal{R}^{1}]_{h}\\
& +W^{-1}\sum_{d,e,f}(\partial_{\beta^{r}\phi\phi'\phi_{d}\phi_{e}\phi_{f}}\mathcal{L})\big[\mathcal{H}^{-1}\big]_{fg}\big[\mathcal{H}^{-1}\big]_{dk}[\mathcal{R}^{1}]_{e}[\mathcal{R}_{b}^{1}]_{h}\\
& -2W^{-3}W_{b}\sum_{d,e,f}(\partial_{\beta^{r}\phi\phi'\phi_{d}\phi_{e}\phi_{f}}\mathcal{L})\big[\mathcal{H}^{-1}\big]_{fg}[\mathcal{R}^{1}]_{e}[\mathcal{R}^{1}]_{h}[\mathcal{R}^{1}]_{d}[\mathcal{R}^{1}]_{k}\\
& -W^{-2}\sum_{d,e,f}(\partial_{\beta^{r+1}\phi\phi'\phi_{d}\phi_{e}\phi_{f}}\mathcal{L})\big[\mathcal{H}^{-1}\big]_{fg}[\mathcal{R}^{1}]_{e}[\mathcal{R}^{1}]_{h}[\mathcal{R}^{1}]_{d}[\mathcal{R}^{1}]_{k}\\
& -W^{-2}\sum_{c,d,e,f}(\partial_{\beta^{r}\phi\phi'\phi_{c}\phi_{d}\phi_{e}\phi_{f}}\mathcal{L})\big[\mathcal{H}^{-1}\big]_{fg}[\mathcal{R}^{1}]_{c}[\mathcal{R}^{1}]_{e}[\mathcal{R}^{1}]_{h}[\mathcal{R}^{1}]_{d}[\mathcal{R}^{1}]_{k}\\
& -W^{-2}\sum_{d,e,f}(\partial_{\beta^{r}\phi\phi'\phi_{d}\phi_{e}\phi_{f}}\mathcal{L})\big[\mathcal{H}^{-1}\mathcal{H}_{b}\mathcal{H}^{-1}\big]_{fg}[\mathcal{R}^{1}]_{e}[\mathcal{R}^{1}]_{h}[\mathcal{R}^{1}]_{d}[\mathcal{R}^{1}]_{k}\\
& +W^{-2}\sum_{d,e,f}(\partial_{\beta^{r}\phi\phi'\phi_{d}\phi_{e}\phi_{f}}\mathcal{L})\big[\mathcal{H}^{-1}\big]_{fg}[\mathcal{R}_{b}^{1}]_{e}[\mathcal{R}^{1}]_{h}[\mathcal{R}^{1}]_{d}[\mathcal{R}^{1}]_{k}\\
& +W^{-2}\sum_{d,e,f}(\partial_{\beta^{r}\phi\phi'\phi_{d}\phi_{e}\phi_{f}}\mathcal{L})\big[\mathcal{H}^{-1}\big]_{fg}[\mathcal{R}^{1}]_{e}[\mathcal{R}_{b}^{1}]_{h}[\mathcal{R}^{1}]_{d}[\mathcal{R}^{1}]_{k}\\
& +W^{-2}\sum_{d,e,f}(\partial_{\beta^{r}\phi\phi'\phi_{d}\phi_{e}\phi_{f}}\mathcal{L})\big[\mathcal{H}^{-1}\big]_{fg}[\mathcal{R}^{1}]_{e}[\mathcal{R}^{1}]_{h}[\mathcal{R}_{b}^{1}]_{d}[\mathcal{R}^{1}]_{k}\\
& +W^{-2}\sum_{d,e,f}(\partial_{\beta^{r}\phi\phi'\phi_{d}\phi_{e}\phi_{f}}\mathcal{L})\big[\mathcal{H}^{-1}\big]_{fg}[\mathcal{R}^{1}]_{e}[\mathcal{R}^{1}]_{h}[\mathcal{R}^{1}]_{d}[\mathcal{R}_{b}^{1}]_{k}\\
& -W^{-2}W_{b}\sum_{e,f}(\partial_{\beta^{r}\phi\phi'\phi_{e}\phi_{f}}\mathcal{L})\big[\mathcal{H}^{-1}\mathcal{H}_{s_{k}}\mathcal{H}^{-1}\big]_{fg}[\mathcal{R}^{1}]_{e}[\mathcal{R}^{1}]_{h}\\
& -W^{-2}\sum_{e,f}(\partial_{\beta^{r+1}\phi\phi'\phi_{e}\phi_{f}}\mathcal{L})\big[\mathcal{H}^{-1}\mathcal{H}_{s_{k}}\mathcal{H}^{-1}\big]_{fg}[\mathcal{R}^{1}]_{e}[\mathcal{R}^{1}]_{h}\\
& -W^{-2}\sum_{d,e,f}(\partial_{\beta^{r}\phi\phi'\phi_{d}\phi_{e}\phi_{f}}\mathcal{L})\big[\mathcal{H}^{-1}\mathcal{H}_{s_{k}}\mathcal{H}^{-1}\big]_{fg}[\mathcal{R}^{1}]_{d}[\mathcal{R}^{1}]_{e}[\mathcal{R}^{1}]_{h}\\
& -W^{-1}\sum_{e,f}(\partial_{\beta^{r}\phi\phi'\phi_{e}\phi_{f}}\mathcal{L})\big[\mathcal{H}^{-1}\mathcal{H}_{b}\mathcal{H}^{-1}\mathcal{H}_{s_{k}}\mathcal{H}^{-1}\big]_{fg}[\mathcal{R}^{1}]_{e}[\mathcal{R}^{1}]_{h}\\
& +W^{-1}\sum_{e,f}(\partial_{\beta^{r}\phi\phi'\phi_{e}\phi_{f}}\mathcal{L})\big[\mathcal{H}^{-1}\mathcal{H}_{bs_{k}}\mathcal{H}^{-1}\big]_{fg}[\mathcal{R}^{1}]_{e}[\mathcal{R}^{1}]_{h}\\
& -W^{-1}\sum_{e,f}(\partial_{\beta^{r}\phi\phi'\phi_{e}\phi_{f}}\mathcal{L})\big[\mathcal{H}^{-1}\mathcal{H}_{s_{k}}\mathcal{H}^{-1}\mathcal{H}_{b}\mathcal{H}^{-1}\big]_{fg}[\mathcal{R}^{1}]_{e}[\mathcal{R}^{1}]_{h}\\
& +W^{-1}\sum_{e,f}(\partial_{\beta^{r}\phi\phi'\phi_{e}\phi_{f}}\mathcal{L})\big[\mathcal{H}^{-1}\mathcal{H}_{s_{k}}\mathcal{H}^{-1}\big]_{fg}[\mathcal{R}_{b}^{1}]_{e}[\mathcal{R}^{1}]_{h}\\
& +W^{-1}\sum_{e,f}(\partial_{\beta^{r}\phi\phi'\phi_{e}\phi_{f}}\mathcal{L})\big[\mathcal{H}^{-1}\mathcal{H}_{s_{k}}\mathcal{H}^{-1}\big]_{fg}[\mathcal{R}^{1}]_{e}[\mathcal{R}_{b}^{1}]_{h}\\
& +W^{-2}W_{b}\sum_{e,f}(\partial_{\beta^{r}\phi\phi'\phi_{e}\phi_{f}}\mathcal{L})\big[\mathcal{H}^{-1}\big]_{fg}[\mathcal{R}_{s_{k}}^{1}]_{e}[\mathcal{R}^{1}]_{h}\\
& +W^{-2}\sum_{e,f}(\partial_{\beta^{r+1}\phi\phi'\phi_{e}\phi_{f}}\mathcal{L})\big[\mathcal{H}^{-1}\big]_{fg}[\mathcal{R}_{s_{k}}^{1}]_{e}[\mathcal{R}^{1}]_{h}\\
& +W^{-2}\sum_{d,e,f}(\partial_{\beta^{r}\phi\phi'\phi_{d}\phi_{e}\phi_{f}}\mathcal{L})\big[\mathcal{H}^{-1}\big]_{fg}[\mathcal{R}_{s_{k}}^{1}]_{e}[\mathcal{R}^{1}]_{d}[\mathcal{R}^{1}]_{h}\\
& +W^{-1}\sum_{e,f}(\partial_{\beta^{r}\phi\phi'\phi_{e}\phi_{f}}\mathcal{L})\big[\mathcal{H}^{-1}\mathcal{H}_{b}\mathcal{H}^{-1}\big]_{fg}[\mathcal{R}_{s_{k}}^{1}]_{e}[\mathcal{R}^{1}]_{h}\\
& -W^{-1}\sum_{e,f}(\partial_{\beta^{r}\phi\phi'\phi_{e}\phi_{f}}\mathcal{L})\big[\mathcal{H}^{-1}\big]_{fg}[\mathcal{R}_{bs_{k}}^{1}]_{e}[\mathcal{R}^{1}]_{h}\\
& -W^{-1}\sum_{e,f}(\partial_{\beta^{r}\phi\phi'\phi_{e}\phi_{f}}\mathcal{L})\big[\mathcal{H}^{-1}\big]_{fg}[\mathcal{R}_{s_{k}}^{1}]_{e}[\mathcal{R}_{b}^{1}]_{h}\\
& +W^{-2}W_{b}\sum_{e,f}(\partial_{\beta^{r}\phi\phi'\phi_{e}\phi_{f}}\mathcal{L})\big[\mathcal{H}^{-1}\big]_{fg}[\mathcal{R}^{1}]_{e}[\mathcal{R}_{s_{k}}^{1}]_{h}\\
& +W^{-2}\sum_{e,f}(\partial_{\beta^{r+1}\phi\phi'\phi_{e}\phi_{f}}\mathcal{L})\big[\mathcal{H}^{-1}\big]_{fg}[\mathcal{R}^{1}]_{e}[\mathcal{R}_{s_{k}}^{1}]_{h}\\
& +W^{-2}\sum_{d,e,f}(\partial_{\beta^{r}\phi\phi'\phi_{d}\phi_{e}\phi_{f}}\mathcal{L})\big[\mathcal{H}^{-1}\big]_{fg}[\mathcal{R}^{1}]_{d}[\mathcal{R}^{1}]_{e}[\mathcal{R}_{s_{k}}^{1}]_{h}\\
& +W^{-1}\sum_{e,f}(\partial_{\beta^{r}\phi\phi'\phi_{e}\phi_{f}}\mathcal{L})\big[\mathcal{H}^{-1}\mathcal{H}_{b}\mathcal{H}^{-1}\big]_{fg}[\mathcal{R}^{1}]_{e}[\mathcal{R}_{s_{k}}^{1}]_{h}\\
& -W^{-1}\sum_{e,f}(\partial_{\beta^{r}\phi\phi'\phi_{e}\phi_{f}}\mathcal{L})\big[\mathcal{H}^{-1}\big]_{fg}[\mathcal{R}_{b}^{1}]_{e}[\mathcal{R}_{s_{k}}^{1}]_{h}\\
& -W^{-1}\sum_{e,f}(\partial_{\beta^{r}\phi\phi'\phi_{e}\phi_{f}}\mathcal{L})\big[\mathcal{H}^{-1}\big]_{fg}[\mathcal{R}^{1}]_{e}[\mathcal{R}_{bs_{k}}^{1}]_{h}\\
& -W^{-2}W_{b}\sum_{f}(\partial_{\beta^{r+1}\phi\phi'\phi_{f}}\mathcal{L})\big[\mathcal{H}^{-1}\mathcal{H}_{s_{h}}\mathcal{H}^{-1}\big]_{fg}[\mathcal{R}^{1}]_{k}\\
& -W^{-2}\sum_{f}(\partial_{\beta^{r+2}\phi\phi'\phi_{f}}\mathcal{L})\big[\mathcal{H}^{-1}\mathcal{H}_{s_{h}}\mathcal{H}^{-1}\big]_{fg}[\mathcal{R}^{1}]_{k}\\
& -W^{-2}\sum_{e,f}(\partial_{\beta^{r+1}\phi\phi'\phi_{e}\phi_{f}}\mathcal{L})\big[\mathcal{H}^{-1}\mathcal{H}_{s_{h}}\mathcal{H}^{-1}\big]_{fg}[\mathcal{R}^{1}]_{e}[\mathcal{R}^{1}]_{k}\\
& -W^{-1}\sum_{f}(\partial_{\beta^{r+1}\phi\phi'\phi_{f}}\mathcal{L})\big[\mathcal{H}^{-1}\mathcal{H}_{b}\mathcal{H}^{-1}\mathcal{H}_{s_{h}}\mathcal{H}^{-1}\big]_{fg}[\mathcal{R}^{1}]_{k}\\
& +W^{-1}\sum_{f}(\partial_{\beta^{r+1}\phi\phi'\phi_{f}}\mathcal{L})\big[\mathcal{H}^{-1}\mathcal{H}_{bs_{h}}\mathcal{H}^{-1}\big]_{fg}[\mathcal{R}^{1}]_{k}\\
& -W^{-1}\sum_{f}(\partial_{\beta^{r+1}\phi\phi'\phi_{f}}\mathcal{L})\big[\mathcal{H}^{-1}\mathcal{H}_{s_{h}}\mathcal{H}^{-1}\mathcal{H}_{b}\mathcal{H}^{-1}\big]_{fg}[\mathcal{R}^{1}]_{k}\\
& +W^{-1}\sum_{f}(\partial_{\beta^{r+1}\phi\phi'\phi_{f}}\mathcal{L})\big[\mathcal{H}^{-1}\mathcal{H}_{s_{h}}\mathcal{H}^{-1}\big]_{fg}[\mathcal{R}_{b}^{1}]_{k}\\
& -W^{-1}\sum_{e,f}(\partial_{\beta^{r+1}\phi\phi'\phi_{e}\phi_{f}}\mathcal{L})\big[\mathcal{H}^{-1}\mathcal{H}_{s_{h}}\mathcal{H}^{-1}\big]_{fg}\big[\mathcal{H}^{-1}\big]_{ek}\\
& -W^{-1}\sum_{d,e,f}(\partial_{\beta^{r}\phi\phi'\phi_{d}\phi_{e}\phi_{f}}\mathcal{L})\big[\mathcal{H}^{-1}\mathcal{H}_{s_{h}}\mathcal{H}^{-1}\big]_{fg}\big[\mathcal{H}^{-1}\big]_{ek}[\mathcal{R}^{1}]_{d}\\
& -\sum_{e,f}(\partial_{\beta^{r}\phi\phi'\phi_{e}\phi_{f}}\mathcal{L})\big[\mathcal{H}^{-1}\mathcal{H}_{b}\mathcal{H}^{-1}\mathcal{H}_{s_{h}}\mathcal{H}^{-1}\big]_{fg}\big[\mathcal{H}^{-1}\big]_{ek}\\
& +\sum_{e,f}(\partial_{\beta^{r}\phi\phi'\phi_{e}\phi_{f}}\mathcal{L})\big[\mathcal{H}^{-1}\mathcal{H}_{bs_{h}}\mathcal{H}^{-1}\big]_{fg}\big[\mathcal{H}^{-1}\big]_{ek}\\
& -\sum_{e,f}(\partial_{\beta^{r}\phi\phi'\phi_{e}\phi_{f}}\mathcal{L})\big[\mathcal{H}^{-1}\mathcal{H}_{s_{h}}\mathcal{H}^{-1}\mathcal{H}_{b}\mathcal{H}^{-1}\big]_{fg}\big[\mathcal{H}^{-1}\big]_{ek}\\
& -\sum_{e,f}(\partial_{\beta^{r}\phi\phi'\phi_{e}\phi_{f}}\mathcal{L})\big[\mathcal{H}^{-1}\mathcal{H}_{s_{h}}\mathcal{H}^{-1}\big]_{fg}\big[\mathcal{H}^{-1}\mathcal{H}_{b}\mathcal{H}^{-1}\big]_{ek}\\
& -W^{-2}W_{b}\sum_{e,f}(\partial_{\beta^{r}\phi\phi'\phi_{e}\phi_{f}}\mathcal{L})\big[\mathcal{H}^{-1}\mathcal{H}_{s_{h}}\mathcal{H}^{-1}\big]_{fg}[\mathcal{R}^{1}]_{e}[\mathcal{R}^{1}]_{k}\\
& -W^{-2}\sum_{e,f}(\partial_{\beta^{r+1}\phi\phi'\phi_{e}\phi_{f}}\mathcal{L})\big[\mathcal{H}^{-1}\mathcal{H}_{s_{h}}\mathcal{H}^{-1}\big]_{fg}[\mathcal{R}^{1}]_{e}[\mathcal{R}^{1}]_{k}\\
& -W^{-2}\sum_{d,e,f}(\partial_{\beta^{r}\phi\phi'\phi_{d}\phi_{e}\phi_{f}}\mathcal{L})\big[\mathcal{H}^{-1}\mathcal{H}_{s_{h}}\mathcal{H}^{-1}\big]_{fg}[\mathcal{R}^{1}]_{d}[\mathcal{R}^{1}]_{e}[\mathcal{R}^{1}]_{k}\\
& -W^{-1}\sum_{e,f}(\partial_{\beta^{r}\phi\phi'\phi_{e}\phi_{f}}\mathcal{L})\big[\mathcal{H}^{-1}\mathcal{H}_{b}\mathcal{H}^{-1}\mathcal{H}_{s_{h}}\mathcal{H}^{-1}\big]_{fg}[\mathcal{R}^{1}]_{e}[\mathcal{R}^{1}]_{k}\\
& +W^{-1}\sum_{e,f}(\partial_{\beta^{r}\phi\phi'\phi_{e}\phi_{f}}\mathcal{L})\big[\mathcal{H}^{-1}\mathcal{H}_{bs_{h}}\mathcal{H}^{-1}\big]_{fg}[\mathcal{R}^{1}]_{e}[\mathcal{R}^{1}]_{k}\\
& -W^{-1}\sum_{e,f}(\partial_{\beta^{r}\phi\phi'\phi_{e}\phi_{f}}\mathcal{L})\big[\mathcal{H}^{-1}\mathcal{H}_{s_{h}}\mathcal{H}^{-1}\mathcal{H}_{b}\mathcal{H}^{-1}\big]_{fg}[\mathcal{R}^{1}]_{e}[\mathcal{R}^{1}]_{k}\\
& +W^{-1}\sum_{e,f}(\partial_{\beta^{r}\phi\phi'\phi_{e}\phi_{f}}\mathcal{L})\big[\mathcal{H}^{-1}\mathcal{H}_{s_{h}}\mathcal{H}^{-1}\big]_{fg}[\mathcal{R}_{b}^{1}]_{e}[\mathcal{R}^{1}]_{k}\\
& +W^{-1}\sum_{e,f}(\partial_{\beta^{r}\phi\phi'\phi_{e}\phi_{f}}\mathcal{L})\big[\mathcal{H}^{-1}\mathcal{H}_{s_{h}}\mathcal{H}^{-1}\big]_{fg}[\mathcal{R}^{1}]_{e}[\mathcal{R}_{b}^{1}]_{k}\\
& -W^{-1}\sum_{f}(\partial_{\beta^{r+1}\phi\phi'\phi_{f}}\mathcal{L})\big[\mathcal{H}^{-1}\mathcal{H}_{s_{k}}\mathcal{H}^{-1}\mathcal{H}_{s_{h}}\mathcal{H}^{-1}\big]_{fg}\\
& -W^{-1}\sum_{e,f}(\partial_{\beta^{r}\phi\phi'\phi_{e}\phi_{f}}\mathcal{L})\big[\mathcal{H}^{-1}\mathcal{H}_{s_{k}}\mathcal{H}^{-1}\mathcal{H}_{s_{h}}\mathcal{H}^{-1}\big]_{fg}[\mathcal{R}^{1}]_{e}\\
& -\sum_{f}(\partial_{\beta^{r}\phi\phi'\phi_{f}}\mathcal{L})\big[\mathcal{H}^{-1}\mathcal{H}_{b}\mathcal{H}^{-1}\mathcal{H}_{s_{k}}\mathcal{H}^{-1}\mathcal{H}_{s_{h}}\mathcal{H}^{-1}\big]_{fg}\\
& +\sum_{f}(\partial_{\beta^{r}\phi\phi'\phi_{f}}\mathcal{L})\big[\mathcal{H}^{-1}\mathcal{H}_{bs_{k}}\mathcal{H}^{-1}\mathcal{H}_{s_{h}}\mathcal{H}^{-1}\big]_{fg}\\
& -\sum_{f}(\partial_{\beta^{r}\phi\phi'\phi_{f}}\mathcal{L})\big[\mathcal{H}^{-1}\mathcal{H}_{s_{k}}\mathcal{H}^{-1}\mathcal{H}_{b}\mathcal{H}^{-1}\mathcal{H}_{s_{h}}\mathcal{H}^{-1}\big]_{fg}\\
& +\sum_{f}(\partial_{\beta^{r}\phi\phi'\phi_{f}}\mathcal{L})\big[\mathcal{H}^{-1}\mathcal{H}_{s_{k}}\mathcal{H}^{-1}\mathcal{H}_{bs_{h}}\mathcal{H}^{-1}\big]_{fg}\\
& -\sum_{f}(\partial_{\beta^{r}\phi\phi'\phi_{f}}\mathcal{L})\big[\mathcal{H}^{-1}\mathcal{H}_{s_{k}}\mathcal{H}^{-1}\mathcal{H}_{s_{h}}\mathcal{H}^{-1}\mathcal{H}_{b}\mathcal{H}^{-1}\big]_{fg}\\
& +W^{-1}\sum_{f}(\partial_{\beta^{r+1}\phi\phi'\phi_{f}}\mathcal{L})\big[\mathcal{H}^{-1}\mathcal{H}_{s_{k}s_{h}}\mathcal{H}^{-1}\big]_{fg}\\
& +W^{-1}\sum_{e,f}(\partial_{\beta^{r}\phi\phi'\phi_{e}\phi_{f}}\mathcal{L})\big[\mathcal{H}^{-1}\mathcal{H}_{s_{k}s_{h}}\mathcal{H}^{-1}\big]_{fg}[\mathcal{R}^{1}]_{e}\\
& +\sum_{f}(\partial_{\beta^{r}\phi\phi'\phi_{f}}\mathcal{L})\big[\mathcal{H}^{-1}\mathcal{H}_{b}\mathcal{H}^{-1}\mathcal{H}_{s_{k}s_{h}}\mathcal{H}^{-1}\big]_{fg}\\
& -\sum_{f}(\partial_{\beta^{r}\phi\phi'\phi_{f}}\mathcal{L})\big[\mathcal{H}^{-1}\mathcal{H}_{bs_{k}s_{h}}\mathcal{H}^{-1}\big]_{fg}\\
& +\sum_{f}(\partial_{\beta^{r}\phi\phi'\phi_{f}}\mathcal{L})\big[\mathcal{H}^{-1}\mathcal{H}_{s_{k}s_{h}}\mathcal{H}^{-1}\mathcal{H}_{b}\mathcal{H}^{-1}\big]_{fg}\\
& -W^{-1}\sum_{f}(\partial_{\beta^{r+1}\phi\phi'\phi_{f}}\mathcal{L})\big[\mathcal{H}^{-1}\mathcal{H}_{s_{k}}\mathcal{H}^{-1}\mathcal{H}_{s_{h}}\mathcal{H}^{-1}\big]_{fg}\\
& -W^{-1}\sum_{e,f}(\partial_{\beta^{r}\phi\phi'\phi_{e}\phi_{f}}\mathcal{L})\big[\mathcal{H}^{-1}\mathcal{H}_{s_{h}}\mathcal{H}^{-1}\mathcal{H}_{s_{k}}\mathcal{H}^{-1}\big]_{fg}[\mathcal{R}^{1}]_{e}\\
& -\sum_{f}(\partial_{\beta^{r}\phi\phi'\phi_{f}}\mathcal{L})\big[\mathcal{H}^{-1}\mathcal{H}_{b}\mathcal{H}^{-1}\mathcal{H}_{s_{h}}\mathcal{H}^{-1}\mathcal{H}_{s_{k}}\mathcal{H}^{-1}\big]_{fg}\\
& +\sum_{f}(\partial_{\beta^{r}\phi\phi'\phi_{f}}\mathcal{L})\big[\mathcal{H}^{-1}\mathcal{H}_{bs_{h}}\mathcal{H}^{-1}\mathcal{H}_{s_{k}}\mathcal{H}^{-1}\big]_{fg}\\
& -\sum_{f}(\partial_{\beta^{r}\phi\phi'\phi_{f}}\mathcal{L})\big[\mathcal{H}^{-1}\mathcal{H}_{s_{h}}\mathcal{H}^{-1}\mathcal{H}_{b}\mathcal{H}^{-1}\mathcal{H}_{s_{k}}\mathcal{H}^{-1}\big]_{fg}\\
& +\sum_{f}(\partial_{\beta^{r}\phi\phi'\phi_{f}}\mathcal{L})\big[\mathcal{H}^{-1}\mathcal{H}_{s_{h}}\mathcal{H}^{-1}\mathcal{H}_{bs_{k}}\mathcal{H}^{-1}\big]_{fg}\\
& -\sum_{f}(\partial_{\beta^{r}\phi\phi'\phi_{f}}\mathcal{L})\big[\mathcal{H}^{-1}\mathcal{H}_{s_{h}}\mathcal{H}^{-1}\mathcal{H}_{s_{k}}\mathcal{H}^{-1}\mathcal{H}_{b}\mathcal{H}^{-1}\big]_{fg}
\end{align*}
\subsection{Expressions for $\mathcal{R}$ terms}
First derivatives:
\begin{align*}
\mathcal{R}_{s_{h}}^{a} & =-W^{-1}[\mathcal{R}^{1}]_{h}\mathcal{R}^{a+1}\\
& -[\mathcal{H}^{-1}]_{h,\cdot}(\partial_{\beta^{a}\phi\phi'}\mathcal{L})\mathcal{H}^{-1}\\
& -W^{-1}[\mathcal{R}^{1}]_{h}\mathcal{R}^{1}(\partial_{\beta^{a}\phi\phi'}\mathcal{L})\mathcal{H}^{-1}\\
& -\mathcal{R}^{a}\mathcal{H}_{s_{h}}\mathcal{H}^{-1}
\end{align*}
\begin{align*}
\mathcal{R}_{b}^{a} & =-W^{-1}\mathcal{R}^{a+1}\\
& -W^{-1}\mathcal{R}_{}^{1}{}'(\partial_{\beta^{a}\phi\phi'}\mathcal{L})\mathcal{H}^{-1}\\
& -\mathcal{R}^{a+1}{}'\mathcal{H}_{b}\mathcal{H}^{-1}
\end{align*}
Second derivatives:
\begin{align*}
\mathcal{R}_{bs_{h}}^{a} & =W^{-2}W_{b}[\mathcal{R}^{1}]_{h}\mathcal{R}^{a+1}-W^{-1}[\mathcal{R}_{b}^{1}]_{h}\mathcal{R}^{a+1}\\
& +W^{-2}[\mathcal{R}^{1}]_{h}\mathcal{R}_{b}^{a+1}\\
& +[\mathcal{H}^{-1}\mathcal{H}_{b}\mathcal{H}^{-1}]_{h,\cdot}(\partial_{\beta^{a}\phi\phi'}\mathcal{L})\mathcal{H}^{-1}\\
& +W^{-1}[\mathcal{H}^{-1}]_{h,\cdot}(\partial_{\beta^{a+1}\phi\phi'}\mathcal{L})\mathcal{H}^{-1}+W^{-1}[\mathcal{H}^{-1}]_{h,\cdot}\mathcal{P}^{a+1,1}\mathcal{H}^{-1}\\
& +[\mathcal{H}^{-1}]_{h,\cdot}(\partial_{\beta^{a}\phi\phi'}\mathcal{L})\mathcal{H}^{-1}\mathcal{H}_{b}\mathcal{H}^{-1}\\
& +W^{-2}W_{b}[\mathcal{R}^{1}]_{h}\mathcal{R}^{1}(\partial_{\beta^{a}\phi\phi'}\mathcal{L})\mathcal{H}^{-1}-W^{-1}[\mathcal{R}_{b}^{1}]_{h}\mathcal{R}^{1}(\partial_{\beta^{a}\phi\phi'}\mathcal{L})\mathcal{H}^{-1}\\
& -W^{-1}[\mathcal{R}^{1}]_{h}\mathcal{R}_{b}^{1}(\partial_{\beta^{a}\phi\phi'}\mathcal{L})\mathcal{H}^{-1}\\
& +W^{-2}[\mathcal{R}^{1}]_{h}\mathcal{R}^{1}(\partial_{\beta^{a+1}\phi\phi'}\mathcal{L})\mathcal{H}^{-1}+W^{-2}[\mathcal{R}^{1}]_{h}\mathcal{R}^{1}\mathcal{P}^{a+1,1}\mathcal{H}^{-1}\\
& +W^{-1}[\mathcal{R}^{1}]_{h}\mathcal{R}^{1}(\partial_{\beta^{a}\phi\phi'}\mathcal{L})\mathcal{H}^{-1}\mathcal{H}_{b}\mathcal{H}^{-1}\\
& -\mathcal{R}_{b}^{a}\mathcal{H}_{s_{h}}\mathcal{H}^{-1}-\mathcal{R}^{a}\mathcal{H}_{bs_{h}}\mathcal{H}^{-1}\\
& +\mathcal{R}^{a}\mathcal{H}_{s_{h}}\mathcal{H}^{-1}\mathcal{H}_{b}\mathcal{H}^{-1}
\end{align*}
\begin{align*}
\mathcal{R}_{s_{g}s_{h}}^{a} & =W^{-2}W_{s_{g}}[\mathcal{R}^{1}]_{h}\mathcal{R}^{a+1}\\
& -W^{-1}[\mathcal{R}_{s_{g}}^{1}]_{h}\mathcal{R}^{a+1}\\
& -W^{-1}[\mathcal{R}^{1}]_{h}\mathcal{R}_{s_{g}}^{a+1}\\
& +[\mathcal{H}^{-1}\mathcal{H}_{s_{g}}^{-1}\mathcal{H}^{-1}]_{h,\cdot}(\partial_{\beta^{a}\phi\phi'}\mathcal{L})\mathcal{H}^{-1}\\
& +W^{-1}[\mathcal{H}^{-1}]_{h,\cdot}(\partial_{\beta^{a+1}\phi\phi'}\mathcal{L})\mathcal{H}^{-1}[\mathcal{R}^{1}]_{g}\\
& +[\mathcal{H}^{-1}]_{h,\cdot}\sum_{e}(\partial_{\beta^{a}\phi\phi'\phi_{e}}\mathcal{L})\mathcal{H}^{-1}[\mathcal{H}^{-1}]_{eg}\\
& +W^{-1}[\mathcal{H}^{-1}]_{h,\cdot}\sum_{e}(\partial_{\beta^{a}\phi\phi'\phi_{e}}\mathcal{L})\mathcal{H}^{-1}[\mathcal{R}^{1}]_{e}[\mathcal{R}^{1}]_{g}\\
& +[\mathcal{H}^{-1}]_{h,\cdot}(\partial_{\beta^{a}\phi\phi'}\mathcal{L})\mathcal{H}^{-1}\mathcal{H}_{s_{g}}^{-1}\mathcal{H}^{-1}\\
& +W^{-2}W_{s_{g}}[\mathcal{R}^{1}]_{h}\mathcal{R}^{1}(\partial_{\beta^{a}\phi\phi'}\mathcal{L})\mathcal{H}^{-1}\\
& -W^{-1}[\mathcal{R}_{s_{g}}^{1}]_{h}\mathcal{R}^{1}(\partial_{\beta^{a}\phi\phi'}\mathcal{L})\mathcal{H}^{-1}\\
& -W^{-1}[\mathcal{R}^{1}]_{h}\mathcal{R}_{s_{g}}^{1}(\partial_{\beta^{a}\phi\phi'}\mathcal{L})\mathcal{H}^{-1}\\
& +W^{-2}[\mathcal{R}^{1}]_{h}\mathcal{R}^{1}(\partial_{\beta^{a+1}\phi\phi'}\mathcal{L})\mathcal{H}^{-1}[\mathcal{R}^{1}]_{g}\\
& +W^{-1}[\mathcal{R}^{1}]_{h}\mathcal{R}^{1}\sum_{e}(\partial_{\beta^{a}\phi\phi'\phi_{e}}\mathcal{L})\mathcal{H}^{-1}[\mathcal{H}^{-1}]_{eg}\\
& +W^{-2}[\mathcal{R}^{1}]_{h}\mathcal{R}^{1}\sum_{e}(\partial_{\beta^{a}\phi\phi'\phi_{e}}\mathcal{L})\mathcal{H}^{-1}[\mathcal{R}^{1}]_{e}[\mathcal{R}^{1}]_{g}\\
& +W^{-1}[\mathcal{R}^{1}]_{h}\mathcal{R}^{1}(\partial_{\beta^{a}\phi\phi'}\mathcal{L})\mathcal{H}^{-1}\mathcal{H}_{s_{g}}^ {}\mathcal{H}^{-1}\\
& -\mathcal{R}_{s_{g}}^{a}\mathcal{H}_{s_{h}}\mathcal{H}^{-1}\\
& -\mathcal{R}^{a}\mathcal{H}_{s_{g}s_{h}}\mathcal{H}^{-1}\\
& +\mathcal{R}^{a}\mathcal{H}_{s_{h}}\mathcal{H}^{-1}\mathcal{H}_{s_{g}}^{-1}\mathcal{H}^{-1}
\end{align*}
Third derivatives:
\begin{align*}
\mathcal{R}_{bs_{g}s_{h}}^{a} & =-2W^{-3}W_{b}W_{s_{g}}[\mathcal{R}^{1}]_{h}\mathcal{R}^{a+1}+W^{-2}W_{bs_{g}}[\mathcal{R}^{1}]_{h}\mathcal{R}^{a+1}\\
& +W^{-2}W_{s_{g}}[\mathcal{R}_{b}^{1}]_{h}\mathcal{R}^{a+1}+W^{-2}W_{s_{g}}[\mathcal{R}^{1}]_{h}\mathcal{R}_{b}^{a+1}\\
& +W^{-2}W_{b}[\mathcal{R}_{s_{g}}^{1}]_{h}\mathcal{R}^{a+1}-W^{-1}[\mathcal{R}_{bs_{g}}^{1}]_{h}\mathcal{R}^{a+1}\\
& -W^{-1}[\mathcal{R}_{s_{g}}^{1}]_{h}\mathcal{R}_{b}^{a+1}+W^{-2}W_{b}[\mathcal{R}^{1}]_{h}\mathcal{R}_{s_{g}}^{a+1}\\
& -W^{-1}[\mathcal{R}_{b}^{1}]_{h}\mathcal{R}_{s_{g}}^{a+1}-W^{-1}[\mathcal{R}^{1}]_{h}\mathcal{R}_{bs_{g}}^{a+1}\\
& -[\mathcal{H}^{-1}\mathcal{H}_{b}^{-1}\mathcal{H}^{-1}\mathcal{H}_{s_{g}}^{-1}\mathcal{H}^{-1}]_{h,\cdot}(\partial_{\beta^{a}\phi\phi'}\mathcal{L})\mathcal{H}^{-1}\\
& +[\mathcal{H}^{-1}\mathcal{H}_{bs_{g}}^{-1}\mathcal{H}^{-1}]_{h,\cdot}(\partial_{\beta^{a}\phi\phi'}\mathcal{L})\mathcal{H}^{-1}\\
& -[\mathcal{H}^{-1}\mathcal{H}_{s_{g}}^{-1}\mathcal{H}^{-1}\mathcal{H}_{b}^{-1}\mathcal{H}^{-1}]_{h,\cdot}(\partial_{\beta^{a}\phi\phi'}\mathcal{L})\mathcal{H}^{-1}\\
& -W^{-1}[\mathcal{H}^{-1}\mathcal{H}_{s_{g}}^{-1}\mathcal{H}^{-1}]_{h,\cdot}(\partial_{\beta^{a+1}\phi\phi'}\mathcal{L})\mathcal{H}^{-1}\\
& -W^{-1}[\mathcal{H}^{-1}\mathcal{H}_{s_{g}}^{-1}\mathcal{H}^{-1}]_{h,\cdot}\sum_{f}(\partial_{\beta^{a}\phi\phi'\phi_{f}}\mathcal{L})\mathcal{H}^{-1}[\mathcal{R}^{1}]_{f}\\
& -[\mathcal{H}^{-1}\mathcal{H}_{s_{g}}^{-1}\mathcal{H}^{-1}]_{h,\cdot}(\partial_{\beta^{a}\phi\phi'}\mathcal{L})\mathcal{H}^{-1}\mathcal{H}_{b}^{-1}\mathcal{H}^{-1}\\
& -W^{-2}W_{b}[\mathcal{H}^{-1}]_{h,\cdot}(\partial_{\beta^{a+1}\phi\phi'}\mathcal{L})\mathcal{H}^{-1}[\mathcal{R}^{1}]_{g}\\
& -W^{-1}[\mathcal{H}^{-1}\mathcal{H}_{b}^{-1}\mathcal{H}^{-1}]_{h,\cdot}(\partial_{\beta^{a+1}\phi\phi'}\mathcal{L})\mathcal{H}^{-1}[\mathcal{R}^{1}]_{g}\\
& -W^{-2}[\mathcal{H}^{-1}]_{h,\cdot}(\partial_{\beta^{a+2}\phi\phi'}\mathcal{L})\mathcal{H}^{-1}[\mathcal{R}^{1}]_{g}\\
& -W^{-2}[\mathcal{H}^{-1}]_{h,\cdot}\sum_{f}(\partial_{\beta^{a+1}\phi\phi'\phi_{f}}\mathcal{L})\mathcal{H}^{-1}[\mathcal{R}^{1}]_{f}[\mathcal{R}^{1}]_{g}\\
& -W^{-1}[\mathcal{H}^{-1}]_{h,\cdot}(\partial_{\beta^{a+1}\phi\phi'}\mathcal{L})\mathcal{H}^{-1}\mathcal{H}_{b}^{-1}\mathcal{H}^{-1}[\mathcal{R}^{1}]_{g}\\
& +W^{-1}[\mathcal{H}^{-1}]_{h,\cdot}(\partial_{\beta^{a+1}\phi\phi'}\mathcal{L})\mathcal{H}^{-1}[\mathcal{R}_{b}^{1}]_{g}\\
& -[\mathcal{H}^{-1}\mathcal{H}_{b}^{-1}\mathcal{H}^{-1}]_{h,\cdot}\sum_{e}(\partial_{\beta^{a}\phi\phi'\phi_{e}}\mathcal{L})\mathcal{H}^{-1}[\mathcal{H}^{-1}]_{eg}\\
& -W^{-1}[\mathcal{H}^{-1}]_{h,\cdot}\sum_{e}(\partial_{\beta^{a+1}\phi\phi'\phi_{e}}\mathcal{L})\mathcal{H}^{-1}[\mathcal{H}^{-1}]_{eg}\\
& -W^{-1}[\mathcal{H}^{-1}]_{h,\cdot}\sum_{e,f}(\partial_{\beta^{a}\phi\phi'\phi_{e}\phi_{f}}\mathcal{L})\mathcal{H}^{-1}[\mathcal{H}^{-1}]_{eg}[\mathcal{R}^{1}]_{f}\\
& -[\mathcal{H}^{-1}]_{h,\cdot}\sum_{e}(\partial_{\beta^{a}\phi\phi'\phi_{e}}\mathcal{L})\mathcal{H}^{-1}\mathcal{H}_{b}^{-1}\mathcal{H}^{-1}[\mathcal{H}^{-1}]_{eg}\\
& -[\mathcal{H}^{-1}]_{h,\cdot}\sum_{e}(\partial_{\beta^{a}\phi\phi'\phi_{e}}\mathcal{L})\mathcal{H}^{-1}[\mathcal{H}^{-1}\mathcal{H}_{b}^{-1}\mathcal{H}^{-1}]_{eg}\\
& -W^{-2}W_{b}[\mathcal{H}^{-1}]_{h,\cdot}\sum_{e}(\partial_{\beta^{a}\phi\phi'\phi_{e}}\mathcal{L})\mathcal{H}^{-1}[\mathcal{R}^{1}]_{e}[\mathcal{R}^{1}]_{g}\\
& -W^{-1}[\mathcal{H}^{-1}\mathcal{H}_{b}^{-1}\mathcal{H}^{-1}]_{h,\cdot}\sum_{e}(\partial_{\beta^{a}\phi\phi'\phi_{e}}\mathcal{L})\mathcal{H}^{-1}[\mathcal{R}^{1}]_{e}[\mathcal{R}^{1}]_{g}\\
& -W^{-2}[\mathcal{H}^{-1}]_{h,\cdot}\sum_{e}(\partial_{\beta^{a+1}\phi\phi'\phi_{e}}\mathcal{L})\mathcal{H}^{-1}[\mathcal{R}^{1}]_{e}[\mathcal{R}^{1}]_{g}\\
& -W^{-2}[\mathcal{H}^{-1}]_{h,\cdot}\sum_{e,f}(\partial_{\beta^{a}\phi\phi'\phi_{e}\phi_{f}}\mathcal{L})\mathcal{H}^{-1}[\mathcal{R}^{1}]_{e}[\mathcal{R}^{1}]_{f}[\mathcal{R}^{1}]_{g}\\
& -W^{-1}[\mathcal{H}^{-1}]_{h,\cdot}\sum_{e}(\partial_{\beta^{a}\phi\phi'\phi_{e}}\mathcal{L})\mathcal{H}^{-1}\mathcal{H}_{b}^{-1}\mathcal{H}^{-1}[\mathcal{R}^{1}]_{e}[\mathcal{R}^{1}]_{g}\\
& +W^{-1}[\mathcal{H}^{-1}]_{h,\cdot}\sum_{e}(\partial_{\beta^{a}\phi\phi'\phi_{e}}\mathcal{L})\mathcal{H}^{-1}[\mathcal{R}_{b}^{1}]_{e}[\mathcal{R}^{1}]_{g}\\
& +W^{-1}[\mathcal{H}^{-1}]_{h,\cdot}\sum_{e}(\partial_{\beta^{a}\phi\phi'\phi_{e}}\mathcal{L})\mathcal{H}^{-1}[\mathcal{R}^{1}]_{e}[\mathcal{R}_{b}^{1}]_{g}\\
& -[\mathcal{H}^{-1}\mathcal{H}_{b}^{-1}\mathcal{H}^{-1}]_{h,\cdot}(\partial_{\beta^{a}\phi\phi'}\mathcal{L})\mathcal{H}^{-1}\mathcal{H}_{s_{g}}^{-1}\mathcal{H}^{-1}\\
& -W^{-1}[\mathcal{H}^{-1}]_{h,\cdot}(\partial_{\beta^{a+1}\phi\phi'}\mathcal{L})\mathcal{H}^{-1}\mathcal{H}_{s_{g}}^ {}\mathcal{H}^{-1}\\
& -W^{-1}[\mathcal{H}^{-1}]_{h,\cdot}\sum_{f}(\partial_{\beta^{a}\phi\phi'\phi_{f}}\mathcal{L})\mathcal{H}^{-1}\mathcal{H}_{s_{g}}^ {}\mathcal{H}^{-1}[\mathcal{R}^{1}]_{f}\\
& -[\mathcal{H}^{-1}]_{h,\cdot}(\partial_{\beta^{a}\phi\phi'}\mathcal{L})\mathcal{H}^{-1}\mathcal{H}_{b}^{-1}\mathcal{H}^{-1}\mathcal{H}_{s_{g}}^ {}\mathcal{H}^{-1}\\
& +[\mathcal{H}^{-1}]_{h,\cdot}(\partial_{\beta^{a}\phi\phi'}\mathcal{L})\mathcal{H}^{-1}\mathcal{H}_{bs_{g}}^ {}\mathcal{H}^{-1}\\
& -[\mathcal{H}^{-1}]_{h,\cdot}(\partial_{\beta^{a}\phi\phi'}\mathcal{L})\mathcal{H}^{-1}\mathcal{H}_{s_{g}}^ {}\mathcal{H}^{-1}\mathcal{H}_{b}^{-1}\mathcal{H}^{-1}\\
& -2W^{-3}W_{b}W_{s_{g}}[\mathcal{R}^{1}]_{h}\mathcal{R}^{1}(\partial_{\beta^{a}\phi\phi'}\mathcal{L})\mathcal{H}^{-1}\\
& +W^{-2}W_{bs_{g}}[\mathcal{R}^{1}]_{h}\mathcal{R}^{1}(\partial_{\beta^{a}\phi\phi'}\mathcal{L})\mathcal{H}^{-1}\\
& +W^{-2}W_{s_{g}}[\mathcal{R}_{b}^{1}]_{h}\mathcal{R}^{1}(\partial_{\beta^{a}\phi\phi'}\mathcal{L})\mathcal{H}^{-1}\\
& +W^{-2}W_{s_{g}}[\mathcal{R}^{1}]_{h}\mathcal{R}_{b}^{1}(\partial_{\beta^{a}\phi\phi'}\mathcal{L})\mathcal{H}^{-1}\\
& -W^{-3}W_{s_{g}}[\mathcal{R}^{1}]_{h}\mathcal{R}^{1}(\partial_{\beta^{a+1}\phi\phi'}\mathcal{L})\mathcal{H}^{-1}\\
& -W^{-3}W_{s_{g}}[\mathcal{R}^{1}]_{h}\mathcal{R}^{1}\sum_{f}(\partial_{\beta^{a}\phi\phi'\phi_{f}}\mathcal{L})\mathcal{H}^{-1}[\mathcal{R}^{1}]_{f}\\
& -W^{-2}W_{s_{g}}[\mathcal{R}^{1}]_{h}\mathcal{R}^{1}(\partial_{\beta^{a}\phi\phi'}\mathcal{L})\mathcal{H}^{-1}\mathcal{H}_{b}^{-1}\mathcal{H}^{-1}\\
& +W^{-2}W_{b}[\mathcal{R}_{s_{g}}^{1}]_{h}\mathcal{R}^{1}(\partial_{\beta^{a}\phi\phi'}\mathcal{L})\mathcal{H}^{-1}\\
& -W^{-1}[\mathcal{R}_{bs_{g}}^{1}]_{h}\mathcal{R}^{1}(\partial_{\beta^{a}\phi\phi'}\mathcal{L})\mathcal{H}^{-1}\\
& -W^{-1}[\mathcal{R}_{s_{g}}^{1}]_{h}\mathcal{R}_{b}^{1}(\partial_{\beta^{a}\phi\phi'}\mathcal{L})\mathcal{H}^{-1}\\
& +W^{-2}[\mathcal{R}_{s_{g}}^{1}]_{h}\mathcal{R}^{1}(\partial_{\beta^{a+1}\phi\phi'}\mathcal{L})\mathcal{H}^{-1}\\
& +W^{-2}[\mathcal{R}_{s_{g}}^{1}]_{h}\mathcal{R}^{1}\sum_{f}(\partial_{\beta^{a}\phi\phi'\phi_{f}}\mathcal{L})\mathcal{H}^{-1}[\mathcal{R}^{1}]_{f}\\
& +W^{-1}[\mathcal{R}_{s_{g}}^{1}]_{h}\mathcal{R}^{1}(\partial_{\beta^{a}\phi\phi'}\mathcal{L})\mathcal{H}^{-1}\mathcal{H}_{b}^{-1}\mathcal{H}^{-1}\\
& +W^{-2}W_{b}[\mathcal{R}^{1}]_{h}\mathcal{R}_{s_{g}}^{1}(\partial_{\beta^{a}\phi\phi'}\mathcal{L})\mathcal{H}^{-1}\\
& -W^{-1}[\mathcal{R}_{b}^{1}]_{h}\mathcal{R}_{s_{g}}^{1}(\partial_{\beta^{a}\phi\phi'}\mathcal{L})\mathcal{H}^{-1}\\
& -W^{-1}[\mathcal{R}^{1}]_{h}\mathcal{R}_{bs_{g}}^{1}(\partial_{\beta^{a}\phi\phi'}\mathcal{L})\mathcal{H}^{-1}\\
& +W^{-2}[\mathcal{R}^{1}]_{h}\mathcal{R}_{s_{g}}^{1}(\partial_{\beta^{a+1}\phi\phi'}\mathcal{L})\mathcal{H}^{-1}\\
& +W^{-2}[\mathcal{R}^{1}]_{h}\mathcal{R}_{s_{g}}^{1}\sum_{f}(\partial_{\beta^{a}\phi\phi'\phi_{f}}\mathcal{L})\mathcal{H}^{-1}[\mathcal{R}^{1}]_{f}\\
& +W^{-1}[\mathcal{R}^{1}]_{h}\mathcal{R}_{s_{g}}^{1}(\partial_{\beta^{a}\phi\phi'}\mathcal{L})\mathcal{H}^{-1}\mathcal{H}_{b}^{-1}\mathcal{H}^{-1}\\
& -2W^{-3}W_{b}[\mathcal{R}^{1}]_{h}\mathcal{R}^{1}(\partial_{\beta^{a+1}\phi\phi'}\mathcal{L})\mathcal{H}^{-1}[\mathcal{R}^{1}]_{g}\\
& +W^{-2}[\mathcal{R}_{b}^{1}]_{h}\mathcal{R}^{1}(\partial_{\beta^{a+1}\phi\phi'}\mathcal{L})\mathcal{H}^{-1}[\mathcal{R}^{1}]_{g}\\
& +W^{-2}[\mathcal{R}^{1}]_{h}\mathcal{R}_{b}^{1}(\partial_{\beta^{a+1}\phi\phi'}\mathcal{L})\mathcal{H}^{-1}[\mathcal{R}^{1}]_{g}\\
& -W^{-3}[\mathcal{R}^{1}]_{h}\mathcal{R}^{1}(\partial_{\beta^{a+2}\phi\phi'}\mathcal{L})\mathcal{H}^{-1}[\mathcal{R}^{1}]_{g}\\
& -W^{-3}[\mathcal{R}^{1}]_{h}\mathcal{R}^{1}\sum_{f}(\partial_{\beta^{a+1}\phi\phi'\phi_{f}}\mathcal{L})\mathcal{H}^{-1}[\mathcal{R}^{1}]_{f}[\mathcal{R}^{1}]_{g}\\
& -W^{-2}[\mathcal{R}^{1}]_{h}\mathcal{R}^{1}(\partial_{\beta^{a+1}\phi\phi'}\mathcal{L})\mathcal{H}^{-1}\mathcal{H}_{b}^{-1}\mathcal{H}^{-1}[\mathcal{R}^{1}]_{g}\\
& +W^{-2}[\mathcal{R}^{1}]_{h}\mathcal{R}^{1}(\partial_{\beta^{a+1}\phi\phi'}\mathcal{L})\mathcal{H}^{-1}[\mathcal{R}_{b}^{1}]_{g}\\
& -W_{b}^{-2}[\mathcal{R}^{1}]_{h}\mathcal{R}^{1}\sum_{e}(\partial_{\beta^{a}\phi\phi'\phi_{e}}\mathcal{L})\mathcal{H}^{-1}[\mathcal{H}^{-1}]_{eg}\\
& +W^{-1}[\mathcal{R}_{b}^{1}]_{h}\mathcal{R}^{1}\sum_{e}(\partial_{\beta^{a}\phi\phi'\phi_{e}}\mathcal{L})\mathcal{H}^{-1}[\mathcal{H}^{-1}]_{eg}\\
& +W^{-1}[\mathcal{R}^{1}]_{h}\mathcal{R}_{b}^{1}\sum_{e}(\partial_{\beta^{a}\phi\phi'\phi_{e}}\mathcal{L})\mathcal{H}^{-1}[\mathcal{H}^{-1}]_{eg}\\
& -W^{-2}[\mathcal{R}^{1}]_{h}\mathcal{R}^{1}\sum_{e}(\partial_{\beta^{a+1}\phi\phi'\phi_{e}}\mathcal{L})\mathcal{H}^{-1}[\mathcal{H}^{-1}]_{eg}\\
& -W^{-2}[\mathcal{R}^{1}]_{h}\mathcal{R}^{1}\sum_{e,f}(\partial_{\beta^{a}\phi\phi'\phi_{e}\phi_{f}}\mathcal{L})\mathcal{H}^{-1}[\mathcal{H}^{-1}]_{eg}[\mathcal{R}^{1}]_{f}\\
& -W^{-1}[\mathcal{R}^{1}]_{h}\mathcal{R}^{1}\sum_{e}(\partial_{\beta^{a}\phi\phi'\phi_{e}}\mathcal{L})\mathcal{H}^{-1}\mathcal{H}_{b}^{-1}\mathcal{H}^{-1}[\mathcal{H}^{-1}]_{eg}\\
& -W^{-1}[\mathcal{R}^{1}]_{h}\mathcal{R}^{1}\sum_{e}(\partial_{\beta^{a}\phi\phi'\phi_{e}}\mathcal{L})\mathcal{H}^{-1}[\mathcal{H}^{-1}\mathcal{H}_{b}^{-1}\mathcal{H}^{-1}]_{eg}\\
& -2W^{-3}W_{b}[\mathcal{R}^{1}]_{h}\mathcal{R}^{1}\sum_{e}(\partial_{\beta^{a}\phi\phi'\phi_{e}}\mathcal{L})\mathcal{H}^{-1}[\mathcal{R}^{1}]_{e}[\mathcal{R}^{1}]_{g}\\
& +W^{-2}[\mathcal{R}_{b}^{1}]_{h}\mathcal{R}^{1}\sum_{e}(\partial_{\beta^{a}\phi\phi'\phi_{e}}\mathcal{L})\mathcal{H}^{-1}[\mathcal{R}^{1}]_{e}[\mathcal{R}^{1}]_{g}\\
& +W^{-2}[\mathcal{R}^{1}]_{h}\mathcal{R}_{b}^{1}\sum_{e}(\partial_{\beta^{a}\phi\phi'\phi_{e}}\mathcal{L})\mathcal{H}^{-1}[\mathcal{R}^{1}]_{e}[\mathcal{R}^{1}]_{g}\\
& -W^{-3}[\mathcal{R}^{1}]_{h}\mathcal{R}^{1}\sum_{e}(\partial_{\beta^{a+1}\phi\phi'\phi_{e}}\mathcal{L})\mathcal{H}^{-1}[\mathcal{R}^{1}]_{e}[\mathcal{R}^{1}]_{g}\\
& -W^{-3}[\mathcal{R}^{1}]_{h}\mathcal{R}^{1}\sum_{e,f}(\partial_{\beta^{a}\phi\phi'\phi_{e}\phi_{f}}\mathcal{L})\mathcal{H}^{-1}[\mathcal{R}^{1}]_{e}[\mathcal{R}^{1}]_{f}[\mathcal{R}^{1}]_{g}\\
& -W^{-2}[\mathcal{R}^{1}]_{h}\mathcal{R}^{1}\sum_{e}(\partial_{\beta^{a}\phi\phi'\phi_{e}}\mathcal{L})\mathcal{H}^{-1}\mathcal{H}_{b}^{-1}\mathcal{H}^{-1}[\mathcal{R}^{1}]_{e}[\mathcal{R}^{1}]_{g}\\
& +W^{-2}[\mathcal{R}^{1}]_{h}\mathcal{R}^{1}\sum_{e}(\partial_{\beta^{a}\phi\phi'\phi_{e}}\mathcal{L})\mathcal{H}^{-1}[\mathcal{R}_{b}^{1}]_{e}[\mathcal{R}^{1}]_{g}\\
& +W^{-2}[\mathcal{R}^{1}]_{h}\mathcal{R}^{1}\sum_{e}(\partial_{\beta^{a}\phi\phi'\phi_{e}}\mathcal{L})\mathcal{H}^{-1}[\mathcal{R}^{1}]_{e}[\mathcal{R}_{b}^{1}]_{g}\\
& -W^{-2}W_{b}[\mathcal{R}^{1}]_{h}\mathcal{R}^{1}(\partial_{\beta^{a}\phi\phi'}\mathcal{L})\mathcal{H}^{-1}\mathcal{H}_{s_{g}}^{-1}\mathcal{H}^{-1}\\
& +W^{-1}[\mathcal{R}_{b}^{1}]_{h}\mathcal{R}^{1}(\partial_{\beta^{a}\phi\phi'}\mathcal{L})\mathcal{H}^{-1}\mathcal{H}_{s_{g}}^{-1}\mathcal{H}^{-1}\\
& +W^{-1}[\mathcal{R}^{1}]_{h}\mathcal{R}_{b}^{1}(\partial_{\beta^{a}\phi\phi'}\mathcal{L})\mathcal{H}^{-1}\mathcal{H}_{s_{g}}^{-1}\mathcal{H}^{-1}\\
& -W^{-2}[\mathcal{R}^{1}]_{h}\mathcal{R}^{1}(\partial_{\beta^{a+1}\phi\phi'}\mathcal{L})\mathcal{H}^{-1}\mathcal{H}_{s_{g}}^{-1}\mathcal{H}^{-1}\\
& -W^{-2}[\mathcal{R}^{1}]_{h}\mathcal{R}^{1}\sum_{f}(\partial_{\beta^{a}\phi\phi'\phi_{f}}\mathcal{L})\mathcal{H}^{-1}\mathcal{H}_{s_{g}}^{-1}\mathcal{H}^{-1}[\mathcal{R}^{1}]_{f}\\
& -W^{-1}[\mathcal{R}^{1}]_{h}\mathcal{R}^{1}(\partial_{\beta^{a}\phi\phi'}\mathcal{L})\mathcal{H}^{-1}\mathcal{H}_{b}^{-1}\mathcal{H}^{-1}\mathcal{H}_{s_{g}}^{-1}\mathcal{H}^{-1}\\
& +W^{-1}[\mathcal{R}^{1}]_{h}\mathcal{R}^{1}(\partial_{\beta^{a}\phi\phi'}\mathcal{L})\mathcal{H}^{-1}\mathcal{H}_{bs_{g}}^{-1}\mathcal{H}^{-1}\\
& -W^{-1}[\mathcal{R}^{1}]_{h}\mathcal{R}^{1}(\partial_{\beta^{a}\phi\phi'}\mathcal{L})\mathcal{H}^{-1}\mathcal{H}_{s_{g}}^{-1}\mathcal{H}^{-1}\mathcal{H}_{b}^{-1}\mathcal{H}^{-1}\\
& -\mathcal{R}_{bs_{g}}^{a}\mathcal{H}_{s_{h}}\mathcal{H}^{-1}-\mathcal{R}_{s_{g}}^{a}\mathcal{H}_{bs_{h}}\mathcal{H}^{-1}+\mathcal{R}_{s_{g}}^{a}\mathcal{H}_{s_{h}}\mathcal{H}^{-1}\mathcal{H}_{b}^{-1}\mathcal{H}^{-1}\\
& -\mathcal{R}_{b}^{a}\mathcal{H}_{s_{g}s_{h}}\mathcal{H}^{-1}-\mathcal{R}^{a}\mathcal{H}_{bs_{g}s_{h}}\mathcal{H}^{-1}+\mathcal{R}^{a}\mathcal{H}_{s_{g}s_{h}}\mathcal{H}^{-1}\mathcal{H}_{b}^{-1}\mathcal{H}^{-1}\\
& +\mathcal{R}_{b}^{a}\mathcal{H}_{s_{h}}\mathcal{H}^{-1}\mathcal{H}_{s_{g}}^{-1}\mathcal{H}^{-1}+\mathcal{R}^{a}\mathcal{H}_{bs_{h}}\mathcal{H}^{-1}\mathcal{H}_{s_{g}}^{-1}\mathcal{H}^{-1}\\
& -\mathcal{R}^{a}\mathcal{H}_{s_{h}}\mathcal{H}^{-1}\mathcal{H}_{b}^{-1}\mathcal{H}^{-1}\mathcal{H}_{s_{g}}^{-1}\mathcal{H}^{-1}+\mathcal{R}^{a}\mathcal{H}_{s_{h}}\mathcal{H}^{-1}\mathcal{H}_{bs_{g}}^{-1}\mathcal{H}^{-1}\\
& -\mathcal{R}^{a}\mathcal{H}_{s_{h}}\mathcal{H}^{-1}\mathcal{H}_{s_{g}}^{-1}\mathcal{H}^{-1}\mathcal{H}_{b}^{-1}\mathcal{H}^{-1}
\end{align*}
\subsection{Expressions for $\mathcal{F}$ terms}
Define $\mathcal{E}^{s}=\partial_{\beta^{s}\phi\phi'}\mathcal{L}$,
$\mathcal{F}^{s,t}=(\partial_{\beta^{s}\phi'}\mathcal{L})\mathcal{H}^{-1}(\partial_{\beta^{t}\phi}\mathcal{L})$,
and
\begin{align*}
\mathcal{F}^{s,(r),t} & =(\partial_{\beta^{s}\phi'}\mathcal{L})\mathcal{H}^{-1}\mathcal{E}^{r}\mathcal{H}^{-1}(\partial_{\beta^{t}\phi}\mathcal{L})\\
\mathcal{F}^{s,(r_{1},r_{2}),t} & =(\partial_{\beta^{s}\phi'}\mathcal{L})\mathcal{H}^{-1}\mathcal{E}^{r_{1}}\mathcal{H}^{-1}\mathcal{E}^{r_{2}}\mathcal{H}^{-1}(\partial_{\beta^{t}\phi}\mathcal{L})
\end{align*}
Then, differentiating gives
(1)
\begin{align*}
\mathcal{F}_{b}^{s,t} & =-W^{-1}\big((\partial_{\beta^{s+1}\phi'}\mathcal{L})+(\partial_{\beta\phi'}\mathcal{L})\mathcal{H}^{-1}\mathcal{E}^{s}\big)\mathcal{H}^{-1}(\partial_{\beta^{t}\phi}\mathcal{L})\\
& \quad-W^{-1}(\partial_{\beta^{s}\phi'}\mathcal{L})\mathcal{H}^{-1}\big((\partial_{\beta^{t+1}\phi}\mathcal{L})+\mathcal{E}^{t}\mathcal{H}^{-1}(\partial_{\beta\phi}\mathcal{L})\big)\\
& \quad-(\partial_{\beta^{s}\phi'}\mathcal{L})\mathcal{H}^{-1}\mathcal{H}_{b}\mathcal{H}^{-1}(\partial_{\beta^{t}\phi}\mathcal{L})\\
& =-W^{-1}\Big(\mathcal{F}^{s+1,t}+\mathcal{F}^{1,(s),t}+\mathcal{F}^{s,t+1}+\mathcal{F}^{s,(t),1}\Big)\\
& \quad-(\partial_{\beta^{s}\phi'}\mathcal{L})\mathcal{H}^{-1}\mathcal{H}_{b}\mathcal{H}^{-1}(\partial_{\beta^{t}\phi}\mathcal{L})
\end{align*}
(2)
\begin{align*}
\mathcal{F}_{s}^{s,t} & =-W^{-1}\big(\mathcal{H}^{-1}(\partial_{\beta\phi}\mathcal{L})(\partial_{\beta^{s+1}\phi'}\mathcal{L})+\mathcal{G}(\partial_{\beta^{s}\phi\phi'}\mathcal{L})\big)\mathcal{H}^{-1}(\partial_{\beta^{t}\phi}\mathcal{L})\\
& \quad-\mathcal{H}^{-1}(\partial_{\beta^{s}\phi\phi'}\mathcal{L})\mathcal{H}^{-1}(\partial_{\beta^{t}\phi}\mathcal{L})\\
& \quad-W^{-1}(\partial_{\beta^{s}\phi'}\mathcal{L})\mathcal{H}^{-1}\big((\partial_{\beta^{t+1}\phi}\mathcal{L})(\partial_{\beta\phi'}\mathcal{L})\mathcal{H}^{-1}+(\partial_{\beta^{t}\phi\phi'}\mathcal{L})\mathcal{G}\big)\\
& \quad-(\partial_{\beta^{s}\phi'}\mathcal{L})\mathcal{H}^{-1}(\partial_{\beta^{t}\phi\phi'}\mathcal{L})\mathcal{H}^{-1}\\
& \quad-(\partial_{\beta^{s}\phi'}\mathcal{L})\mathcal{H}^{-1}\mathcal{H}_{s}\mathcal{H}^{-1}(\partial_{\beta^{t}\phi}\mathcal{L})\\
& =-W^{-1}\big(\mathcal{G}^{1,s+1}+\mathcal{G}\mathcal{E}^{s}\mathcal{H}^{-1}\big)(\partial_{\beta^{t}\phi}\mathcal{L})-\mathcal{H}^{-1}\mathcal{E}^{s}\mathcal{H}^{-1}(\partial_{\beta^{t}\phi}\mathcal{L})\\
& \quad-W^{-1}(\partial_{\beta^{s}\phi}\mathcal{L})\big(\mathcal{G}^{t+1,1}+\mathcal{H}^{-1}\mathcal{E}^{t}\mathcal{G}\big)-(\partial_{\beta^{s}\phi'}\mathcal{L})\mathcal{H}^{-1}\mathcal{E}^{t}\mathcal{H}^{-1}\\
& \quad-(\partial_{\beta^{s}\phi'}\mathcal{L})\mathcal{H}^{-1}\mathcal{H}_{s}\mathcal{H}^{-1}(\partial_{\beta^{t}\phi}\mathcal{L})
\end{align*}
Differentiating a second time gives
(1)
\begin{align*}
\mathcal{F}_{bb}^{s,t} & =W^{-1}W_{b}\mathcal{F}_{b}^{s,t}\\
& -W^{-1}\Big(\mathcal{F}_{b}^{s+1,t}+\mathcal{F}_{b}^{1,(s),t}+\mathcal{F}_{b}^{s,t+1}+\mathcal{F}_{b}^{s,(t),1}\Big)\\
& +W^{-1}(\partial_{\beta^{s+1}\phi'}\mathcal{L})\mathcal{H}^{-1}\mathcal{H}_{b}\mathcal{H}^{-1}(\partial_{\beta^{t}\phi}\mathcal{L})\\
& +W^{-1}(\partial_{\beta\phi'}\mathcal{L})\mathcal{H}^{-1}\mathcal{E}^{s}\mathcal{H}^{-1}\mathcal{H}_{b}\mathcal{H}^{-1}(\partial_{\beta^{t}\phi}\mathcal{L})\\
& -2(\partial_{\beta^{s}\phi'}\mathcal{L})\mathcal{H}^{-1}\mathcal{H}_{b}\mathcal{H}^{-1}\mathcal{H}_{b}\mathcal{H}^{-1}(\partial_{\beta^{t}\phi}\mathcal{L})\\
& -(\partial_{\beta^{s}\phi'}\mathcal{L})\mathcal{H}^{-1}\mathcal{H}_{bb}\mathcal{H}^{-1}(\partial_{\beta^{t}\phi}\mathcal{L})\\
& +W^{-1}(\partial_{\beta^{s}\phi'}\mathcal{L})\mathcal{H}^{-1}\mathcal{H}_{b}\mathcal{H}^{-1}(\partial_{\beta^{t+1}\phi}\mathcal{L})\\
& +W^{-1}(\partial_{\beta^{s}\phi'}\mathcal{L})\mathcal{H}^{-1}\mathcal{H}_{b}\mathcal{H}^{-1}\mathcal{E}^{t}\mathcal{H}^{-1}(\partial_{\beta\phi}\mathcal{L})
\end{align*}
(2)
\begin{align*}
\mathcal{F}_{bs}^{s,t} & =-W^{-1}\big(\mathcal{G}^{1,s+1}+\mathcal{G}\mathcal{E}^{s}\mathcal{H}^{-1}\big)(\partial_{\beta^{t}\phi}\mathcal{L})\\
& -\mathcal{H}^{-1}\mathcal{E}^{s}\mathcal{H}^{-1}(\partial_{\beta^{t}\phi}\mathcal{L})\\
& \quad-W^{-1}(\partial_{\beta^{s}\phi}\mathcal{L})\big(\mathcal{G}^{t+1,1}+\mathcal{H}^{-1}\mathcal{E}^{t}\mathcal{G}\big)\\
& -(\partial_{\beta^{s}\phi'}\mathcal{L})\mathcal{H}^{-1}\mathcal{E}^{t}\mathcal{H}^{-1}\\
& \quad-(\partial_{\beta^{s}\phi'}\mathcal{L})\mathcal{H}^{-1}\mathcal{H}_{s}\mathcal{H}^{-1}(\partial_{\beta^{t}\phi}\mathcal{L})\\
\end{align*}
(3)
\begin{align*}\mathcal{F}_{s_{f}s_{g}} & =W^{-2}W_{s_{f}}(\mathcal{F}^{2,1}+\mathcal{F}^{1,2}+2\mathcal{F}^{1,(1),1})[\mathcal{R}^{1}]_{g}\\
& -W^{-1}(\mathcal{F}_{s_{f}}^{2,1}+\mathcal{F}_{s_{f}}^{1,2}+2\mathcal{F}_{s_{f}}^{1,(1),1})[\mathcal{R}^{1}]_{g}\\
& +W^{-1}(\mathcal{F}^{2,1}+\mathcal{F}^{1,2}+2\mathcal{F}^{1,(1),1})\big[\mathcal{H}^{-1}\mathcal{H}_{s_{f}}\mathcal{R}^{1}\big]_{g}\\
& +W^{-2}(\mathcal{F}^{2,1}+\mathcal{F}^{1,2}+2\mathcal{F}^{1,(1),1})[\mathcal{R}^{2}]_{g}[\mathcal{R}^{1}]_{f}\\
& +W^{-2}(\mathcal{F}^{2,1}+\mathcal{F}^{1,2}+2\mathcal{F}^{1,(1),1})\big[\mathcal{H}^{-1}\mathcal{E}\mathcal{R}^{1}\big]_{g}[\mathcal{R}^{1}]_{f}\\
& +W^{-1}(\mathcal{F}^{2,1}+\mathcal{F}^{1,2}+2\mathcal{F}^{1,(1),1})\big[\mathcal{H}^{-1}\mathcal{E}\mathcal{H}^{-1}\big]_{fg}\\
& +2[\mathcal{H}^{-1}\mathcal{H}_{s_{f}}\mathcal{H}^{-1}]_{\cdot,g}\mathcal{E}\mathcal{R}^{1}-2[\mathcal{H}^{-1}]_{\cdot,g}\mathcal{E}_{s_{f}}\mathcal{R}^{1}\\
& +2[\mathcal{H}^{-1}]_{\cdot,g}\mathcal{E}\mathcal{H}^{-1}\mathcal{H}_{s_{f}}\mathcal{R}^{1}+2W^{-1}[\mathcal{H}^{-1}]_{\cdot,g}\mathcal{E}\mathcal{R}^{2}[\mathcal{R}^{1}]_{f}\\
& +2W^{-1}[\mathcal{H}^{-1}]_{\cdot,g}\mathcal{E}\mathcal{H}^{-1}\mathcal{E}\mathcal{R}^{1}[\mathcal{R}^{1}]_{f}+2[\mathcal{H}^{-1}\mathcal{E}\mathcal{H}^{-1}\mathcal{E}\mathcal{H}^{-1}]_{fg}\\
& +2W^{-1}(\mathcal{R}^{2})'\mathcal{H}_{s_{g}}\mathcal{R}^{1}[\mathcal{R}^{1}]_{f}+2W^{-1}(\mathcal{R}^{1})'\mathcal{E}\mathcal{H}^{-1}\mathcal{H}_{s_{g}}\mathcal{R}^{1}[\mathcal{R}^{1}]_{f}\\
& +2[\mathcal{H}^{-1}]_{\cdot,g}\mathcal{E}\mathcal{H}^{-1}\mathcal{H}_{s_{g}}\mathcal{R}^{1}+2(\mathcal{R}^{1})'\mathcal{H}_{s_{f}}\mathcal{H}^{-1}\mathcal{H}_{s_{g}}\mathcal{R}^{1}\\
& -(\mathcal{R}^{1})'\mathcal{H}_{s_{f}s_{g}}\mathcal{R}^{1}
\end{align*}
Third derivative:
\begin{align*}\mathcal{F}_{bs_{f}s_{g}} & =-2W^{-3}W_{b}W_{s_{f}}(\mathcal{F}^{2,1}+\mathcal{F}^{1,2}+2\mathcal{F}^{1,(1),1})[\mathcal{R}^{1}]_{g}\\
& +W^{-2}W_{bs_{f}}(\mathcal{F}^{2,1}+\mathcal{F}^{1,2}+2\mathcal{F}^{1,(1),1})[\mathcal{R}^{1}]_{g}\\
& +W^{-2}W_{s_{f}}(\mathcal{F}_{b}^{2,1}+\mathcal{F}_{b}^{1,2}+2\mathcal{F}_{b}^{1,(1),1})[\mathcal{R}^{1}]_{g}\\
& +W^{-2}W_{s_{f}}(\mathcal{F}^{2,1}+\mathcal{F}^{1,2}+2\mathcal{F}^{1,(1),1})[\mathcal{R}_{b}^{1}]_{g}\\
& +W^{-2}W_{b}(\mathcal{F}_{s_{f}}^{2,1}+\mathcal{F}_{s_{f}}^{1,2}+2\mathcal{F}_{s_{f}}^{1,(1),1})[\mathcal{R}^{1}]_{g}\\
& -W^{-1}(\mathcal{F}_{bs_{f}}^{2,1}+\mathcal{F}_{bs_{f}}^{1,2}+2\mathcal{F}_{bs_{f}}^{1,(1),1})[\mathcal{R}^{1}]_{g}\\
& -W^{-1}(\mathcal{F}_{s_{f}}^{2,1}+\mathcal{F}_{s_{f}}^{1,2}+2\mathcal{F}_{s_{f}}^{1,(1),1})[\mathcal{R}_{b}^{1}]_{g}\\
& -W^{-2}W_{b}(\mathcal{F}^{2,1}+\mathcal{F}^{1,2}+2\mathcal{F}^{1,(1),1})\big[\mathcal{H}^{-1}\mathcal{H}_{s_{f}}\mathcal{R}^{1}\big]_{g}\\
& +W^{-1}(\mathcal{F}_{b}^{2,1}+\mathcal{F}_{b}^{1,2}+2\mathcal{F}_{b}^{1,(1),1})\big[\mathcal{H}^{-1}\mathcal{H}_{s_{f}}\mathcal{R}^{1}\big]_{g}\\
& -W^{-1}(\mathcal{F}^{2,1}+\mathcal{F}^{1,2}+2\mathcal{F}^{1,(1),1})\big[\mathcal{H}^{-1}\mathcal{H}_{b}\mathcal{H}^{-1}\mathcal{H}_{s_{f}}\mathcal{R}^{1}\big]_{g}\\
& +W^{-1}(\mathcal{F}^{2,1}+\mathcal{F}^{1,2}+2\mathcal{F}^{1,(1),1})\big[\mathcal{H}^{-1}\mathcal{H}_{bs_{f}}\mathcal{R}^{1}\big]_{g}\\
& +W^{-1}(\mathcal{F}^{2,1}+\mathcal{F}^{1,2}+2\mathcal{F}^{1,(1),1})\big[\mathcal{H}^{-1}\mathcal{H}_{s_{f}}\mathcal{R}_{b}^{1}\big]_{g}\\
& -2W^{-3}W_{b}(\mathcal{F}^{2,1}+\mathcal{F}^{1,2}+2\mathcal{F}^{1,(1),1})[\mathcal{R}^{2}]_{g}[\mathcal{R}^{1}]_{f}\\
& +W^{-2}(\mathcal{F}_{b}^{2,1}+\mathcal{F}_{b}^{1,2}+2\mathcal{F}_{b}^{1,(1),1})[\mathcal{R}^{2}]_{g}[\mathcal{R}^{1}]_{f}\\
& +W^{-2}(\mathcal{F}^{2,1}+\mathcal{F}^{1,2}+2\mathcal{F}^{1,(1),1})[\mathcal{R}_{b}^{2}]_{g}[\mathcal{R}^{1}]_{f}\\
& +W^{-2}(\mathcal{F}^{2,1}+\mathcal{F}^{1,2}+2\mathcal{F}^{1,(1),1})[\mathcal{R}^{2}]_{g}[\mathcal{R}_{b}^{1}]_{f}\\
& -2W^{-3}W_{b}(\mathcal{F}^{2,1}+\mathcal{F}^{1,2}+2\mathcal{F}^{1,(1),1})\big[\mathcal{H}^{-1}\mathcal{E}\mathcal{R}^{1}\big]_{g}[\mathcal{R}^{1}]_{f}\\
& +W^{-2}(\mathcal{F}_{b}^{2,1}+\mathcal{F}_{b}^{1,2}+2\mathcal{F}_{b}^{1,(1),1})\big[\mathcal{H}^{-1}\mathcal{E}\mathcal{R}^{1}\big]_{g}[\mathcal{R}^{1}]_{f}\\
& -W^{-2}(\mathcal{F}^{2,1}+\mathcal{F}^{1,2}+2\mathcal{F}^{1,(1),1})\big[\mathcal{H}^{-1}\mathcal{H}_{b}\mathcal{H}^{-1}\mathcal{E}\mathcal{R}^{1}\big]_{g}[\mathcal{R}^{1}]_{f}\\
& +W^{-2}(\mathcal{F}^{2,1}+\mathcal{F}^{1,2}+2\mathcal{F}^{1,(1),1})\big[\mathcal{H}^{-1}\mathcal{E}_{b}\mathcal{R}^{1}\big]_{g}[\mathcal{R}^{1}]_{f}\\
& +W^{-2}(\mathcal{F}^{2,1}+\mathcal{F}^{1,2}+2\mathcal{F}^{1,(1),1})\big[\mathcal{H}^{-1}\mathcal{E}\mathcal{R}_{b}^{1}\big]_{g}[\mathcal{R}^{1}]_{f}\\
& +W^{-2}(\mathcal{F}^{2,1}+\mathcal{F}^{1,2}+2\mathcal{F}^{1,(1),1})\big[\mathcal{H}^{-1}\mathcal{E}\mathcal{R}^{1}\big]_{g}[\mathcal{R}_{b}^{1}]_{f}\\
& -W^{-2}W_{b}(\mathcal{F}^{2,1}+\mathcal{F}^{1,2}+2\mathcal{F}^{1,(1),1})\big[\mathcal{H}^{-1}\mathcal{E}\mathcal{H}^{-1}\big]_{fg}\\
& +W^{-1}(\mathcal{F}_{b}^{2,1}+\mathcal{F}_{b}^{1,2}+2\mathcal{F}_{b}^{1,(1),1})\big[\mathcal{H}^{-1}\mathcal{E}\mathcal{H}^{-1}\big]_{fg}\\
& -W^{-1}(\mathcal{F}^{2,1}+\mathcal{F}^{1,2}+2\mathcal{F}_{}^{1,(1),1})\big[\mathcal{H}^{-1}\mathcal{H}_{b}\mathcal{H}^{-1}\mathcal{E}\mathcal{H}^{-1}\big]_{fg}\\
& +W^{-1}(\mathcal{F}^{2,1}+\mathcal{F}^{1,2}+2\mathcal{F}^{1,(1),1})\big[\mathcal{H}^{-1}\mathcal{E}_{b}\mathcal{H}^{-1}\big]_{fg}\\
& -W^{-1}(\mathcal{F}^{2,1}+\mathcal{F}^{1,2}+2\mathcal{F}^{1,(1),1})\big[\mathcal{H}^{-1}\mathcal{E}\mathcal{H}^{-1}\mathcal{H}_{b}\mathcal{H}^{-1}\big]_{fg}\\
& -2[\mathcal{H}^{-1}\mathcal{H}_{b}\mathcal{H}^{-1}\mathcal{H}_{s_{f}}\mathcal{H}^{-1}]_{\cdot,g}\mathcal{E}\mathcal{R}^{1}\\
& +2[\mathcal{H}^{-1}\mathcal{H}_{bs_{f}}\mathcal{H}^{-1}]_{\cdot,g}\mathcal{E}\mathcal{R}^{1}\\
& -2[\mathcal{H}^{-1}\mathcal{H}_{s_{f}}\mathcal{H}^{-1}\mathcal{H}_{b}\mathcal{H}^{-1}]_{\cdot,g}\mathcal{E}\mathcal{R}^{1}\\
& +2[\mathcal{H}^{-1}\mathcal{H}_{s_{f}}\mathcal{H}^{-1}]_{\cdot,g}(\mathcal{E}_{b}\mathcal{R}^{1}+\mathcal{E}\mathcal{R}_{b}^{1})\\
& +2[\mathcal{H}^{-1}\mathcal{H}_{b}\mathcal{H}^{-1}]_{\cdot,g}\mathcal{E}_{s_{f}}\mathcal{R}^{1}\\
& -2[\mathcal{H}^{-1}]_{\cdot,g}(\mathcal{E}_{bs_{f}}\mathcal{R}^{1}+\mathcal{E}_{s_{f}}\mathcal{R}_{b}^{1})\\
& -2[\mathcal{H}^{-1}\mathcal{H}_{b}\mathcal{H}^{-1}]_{\cdot,g}\mathcal{E}\mathcal{H}^{-1}\mathcal{H}_{s_{f}}\mathcal{R}^{1}\\
& +2[\mathcal{H}^{-1}]_{\cdot,g}\mathcal{E}_{b}\mathcal{H}^{-1}\mathcal{H}_{s_{f}}\mathcal{R}^{1}\\
& -2[\mathcal{H}^{-1}]_{\cdot,g}\mathcal{E}\mathcal{H}^{-1}\mathcal{H}_{b}\mathcal{H}^{-1}\mathcal{H}_{s_{f}}\mathcal{R}^{1}\\
& +2[\mathcal{H}^{-1}]_{\cdot,g}\mathcal{E}\mathcal{H}^{-1}(\mathcal{H}_{bs_{f}}\mathcal{R}^{1}+\mathcal{H}_{s_{f}}\mathcal{R}_{b}^{1})\\
& -2W^{-2}W_{b}[\mathcal{H}^{-1}]_{\cdot,g}\mathcal{E}\mathcal{R}^{2}[\mathcal{R}^{1}]_{f}\\
& -2W^{-1}[\mathcal{H}^{-1}\mathcal{H}_{b}\mathcal{H}^{-1}]_{\cdot,g}\mathcal{E}\mathcal{R}^{2}[\mathcal{R}^{1}]_{f}\\
& +2W^{-1}[\mathcal{H}^{-1}]_{\cdot,g}(\mathcal{E}_{b}\mathcal{R}^{2}[\mathcal{R}^{1}]_{f}+\mathcal{E}\mathcal{R}_{b}^{2}[\mathcal{R}^{1}]_{f}+\mathcal{E}\mathcal{R}^{2}[\mathcal{R}_{b}^{1}]_{f})\\
& -2W^{-2}W_{b}[\mathcal{H}^{-1}]_{\cdot,g}\mathcal{E}\mathcal{H}^{-1}\mathcal{E}\mathcal{R}^{1}[\mathcal{R}^{1}]_{f}\\
& +2W^{-1}[\mathcal{H}^{-1}\mathcal{H}_{b}\mathcal{H}^{-1}]_{\cdot,g}\mathcal{E}\mathcal{H}^{-1}\mathcal{E}\mathcal{R}^{1}[\mathcal{R}^{1}]_{f}\\
& +2W^{-1}[\mathcal{H}^{-1}]_{\cdot,g}\mathcal{E}_{b}\mathcal{H}^{-1}\mathcal{E}\mathcal{R}^{1}[\mathcal{R}^{1}]_{f}\\
& +2W^{-1}[\mathcal{H}^{-1}]_{\cdot,g}\mathcal{E}\mathcal{H}^{-1}\mathcal{H}_{b}\mathcal{H}^{-1}\mathcal{E}\mathcal{R}^{1}[\mathcal{R}^{1}]_{f}\\
& +2W^{-1}[\mathcal{H}^{-1}]_{\cdot,g}\mathcal{E}\mathcal{H}^{-1}(\mathcal{E}_{b}\mathcal{R}^{1}[\mathcal{R}^{1}]_{f}+\mathcal{E}\mathcal{R}_{b}^{1}[\mathcal{R}^{1}]_{f}+\mathcal{E}\mathcal{R}^{1}[\mathcal{R}_{b}^{1}]_{f})\\
& -2[\mathcal{H}^{-1}\mathcal{H}_{b}\mathcal{H}^{-1}\mathcal{E}\mathcal{H}^{-1}\mathcal{E}\mathcal{H}^{-1}]_{fg}-2[\mathcal{H}^{-1}\mathcal{E}\mathcal{H}^{-1}\mathcal{H}_{b}\mathcal{H}^{-1}\mathcal{E}\mathcal{H}^{-1}]_{fg}\\
& -2[\mathcal{H}^{-1}\mathcal{E}\mathcal{H}^{-1}\mathcal{E}\mathcal{H}^{-1}\mathcal{H}_{b}\mathcal{H}^{-1}]_{fg}+2[\mathcal{H}^{-1}\mathcal{E}_{b}\mathcal{H}^{-1}\mathcal{E}\mathcal{H}^{-1}]_{fg}\\
& +2[\mathcal{H}^{-1}\mathcal{E}\mathcal{H}^{-1}\mathcal{E}_{b}\mathcal{H}^{-1}]_{fg}-2W^{-2}W_{b}(\mathcal{R}^{2})'\mathcal{H}_{s_{g}}\mathcal{R}^{1}[\mathcal{R}^{1}]_{f}\\
& +2W^{-1}(\mathcal{R}_{b}^{2})'\mathcal{H}_{s_{g}}\mathcal{R}^{1}[\mathcal{R}^{1}]_{f}+2W^{-1}(\mathcal{R}^{2})'\mathcal{H}_{bs_{g}}\mathcal{R}^{1}[\mathcal{R}^{1}]_{f}\\
& +2W^{-1}(\mathcal{R}^{2})'\mathcal{H}_{s_{g}}\mathcal{R}_{b}^{1}[\mathcal{R}^{1}]_{f}+2W^{-1}(\mathcal{R}^{2})'\mathcal{H}_{s_{g}}\mathcal{R}^{1}[\mathcal{R}_{b}^{1}]_{f}\\
& -2W^{-2}W_{b}(\mathcal{R}^{1})'\mathcal{E}\mathcal{H}^{-1}\mathcal{H}_{s_{g}}\mathcal{R}^{1}[\mathcal{R}^{1}]_{f}\\
& +2W^{-1}(\mathcal{R}_{b}^{1})'\mathcal{E}\mathcal{H}^{-1}\mathcal{H}_{s_{g}}\mathcal{R}^{1}[\mathcal{R}^{1}]_{f}\\
& +2W^{-1}(\mathcal{R}^{1})'\mathcal{E}_{b}\mathcal{H}^{-1}\mathcal{H}_{s_{g}}\mathcal{R}^{1}[\mathcal{R}^{1}]_{f}\\
& -2W^{-1}(\mathcal{R}^{1})'\mathcal{E}\mathcal{H}^{-1}\mathcal{H}_{b}\mathcal{H}^{-1}\mathcal{H}_{s_{g}}\mathcal{R}^{1}[\mathcal{R}^{1}]_{f}\\
& +2W^{-1}(\mathcal{R}^{1})'\mathcal{E}\mathcal{H}^{-1}\mathcal{H}_{bs_{g}}\mathcal{R}^{1}[\mathcal{R}^{1}]_{f}\\
& +2W^{-1}(\mathcal{R}^{1})'\mathcal{E}\mathcal{H}^{-1}\mathcal{H}_{s_{g}}\mathcal{R}_{b}^{1}[\mathcal{R}^{1}]_{f}\\
& +2W^{-1}(\mathcal{R}^{1})'\mathcal{E}\mathcal{H}^{-1}\mathcal{H}_{s_{g}}\mathcal{R}^{1}[\mathcal{R}_{b}^{1}]_{f}\\
& -2[\mathcal{H}^{-1}\mathcal{H}_{b}\mathcal{H}^{-1}]_{\cdot,g}\mathcal{E}\mathcal{H}^{-1}\mathcal{H}_{s_{g}}\mathcal{R}^{1}\\
& +2[\mathcal{H}^{-1}]_{\cdot,g}\mathcal{E}_{b}\mathcal{H}^{-1}\mathcal{H}_{s_{g}}\mathcal{R}^{1}-2[\mathcal{H}^{-1}]_{\cdot,g}\mathcal{E}\mathcal{H}^{-1}\mathcal{H}_{b}\mathcal{H}^{-1}\mathcal{H}_{s_{g}}\mathcal{R}^{1}\\
& +2[\mathcal{H}^{-1}]_{\cdot,g}\mathcal{E}\mathcal{H}^{-1}\mathcal{H}_{bs_{g}}\mathcal{R}^{1}+2[\mathcal{H}^{-1}]_{\cdot,g}\mathcal{E}\mathcal{H}^{-1}\mathcal{H}_{s_{g}}\mathcal{R}_{b}^{1}\\
& +2(\mathcal{R}_{b}^{1})'\mathcal{H}_{s_{f}}\mathcal{H}^{-1}\mathcal{H}_{s_{g}}\mathcal{R}^{1}+2(\mathcal{R}^{1})'\mathcal{H}_{bs_{f}}\mathcal{H}^{-1}\mathcal{H}_{s_{g}}\mathcal{R}^{1}\\
& -2(\mathcal{R}^{1})'\mathcal{H}_{s_{f}}\mathcal{H}^{-1}\mathcal{H}_{b}\mathcal{H}^{-1}\mathcal{H}_{s_{g}}\mathcal{R}^{1}+2(\mathcal{R}^{1})'\mathcal{H}_{s_{f}}\mathcal{H}^{-1}\mathcal{H}_{bs_{g}}\mathcal{R}^{1}\\
& +2(\mathcal{R}^{1})'\mathcal{H}_{s_{f}}\mathcal{H}^{-1}\mathcal{H}_{s_{g}}\mathcal{R}_{b}^{1}-(\mathcal{R}_{b}^{1})'\mathcal{H}_{s_{f}s_{g}}\mathcal{R}^{1}\\
& -(\mathcal{R}^{1})'\mathcal{H}_{bs_{f}s_{g}}\mathcal{R}^{1}-(\mathcal{R}^{1})'\mathcal{H}_{s_{f}s_{g}}\mathcal{R}_{b}^{1}
\end{align*}
\subsection{Expressions for $\mathcal{G}$ terms}
Differentiating $\mathcal{G}=\mathcal{H}^{-1}(\partial_{\beta\phi}\mathcal{L})(\partial_{\beta\phi'}\mathcal{L})\mathcal{H}^{-1}$
with respect to $b$ and $s$ gives
\begin{align*}
\mathcal{G}_{b} & =\mathcal{H}^{-1}\mathcal{H}_{b}\mathcal{G}+\mathcal{G}\mathcal{H}_{b}\mathcal{H}^{-1}\\
& +W^{-1}\mathcal{G}^{2,1}+W^{-1}\mathcal{H}^{-1}\mathcal{E}\mathcal{G}\\
& +W^{-1}\mathcal{G}^{1,2}+W^{-1}\mathcal{G}\mathcal{E}\mathcal{H}^{-1}
\end{align*}
\begin{align*}
\mathcal{G}_{s_{g}} & =-\mathcal{H}^{-1}\mathcal{H}_{s_{g}}\mathcal{G}-\mathcal{G}\mathcal{H}_{s_{g}}\mathcal{H}^{-1}\\
& -W^{-1}(\mathcal{G}^{2,1}+\mathcal{G}^{1,2})\big[\mathcal{H}^{-1}(\partial_{\beta\phi}\mathcal{L})\big]_{g}\\
& +\mathcal{H}^{-1}\mathcal{E}^{1}\big[\mathcal{H}^{-1}\big]_{g}(\partial_{\beta^{t}\phi'}\mathcal{L})\mathcal{H}^{-1}\\
& +\mathcal{H}^{-1}(\partial_{\beta^{s}\phi}\mathcal{L})\big[\mathcal{H}^{-1}\big]_{g}'\mathcal{E}^{1}\mathcal{H}^{-1}\\
& -W^{-1}\mathcal{H}^{-1}\mathcal{E}^{1}\mathcal{G}\big[\mathcal{H}^{-1}(\partial_{\beta\phi}\mathcal{L})\big]_{g}\\
& -W^{-1}\mathcal{G}\mathcal{E}^{1}\mathcal{H}^{-1}\big[\mathcal{H}^{-1}(\partial_{\beta\phi}\mathcal{L})\big]_{g}
\end{align*}
Differentiating a second time gives
\begin{align*}
\mathcal{G}_{bb} & =-\mathcal{H}^{-1}\mathcal{H}_{b}\mathcal{H}^{-1}\mathcal{H}_{b}\mathcal{G}+\mathcal{H}^{-1}\mathcal{H}_{bb}\mathcal{G}+\mathcal{H}^{-1}\mathcal{H}_{b}\mathcal{G}_{b}\\
& +\mathcal{G}_{b}\mathcal{H}_{b}\mathcal{H}^{-1}+\mathcal{G}\mathcal{H}_{bb}\mathcal{H}^{-1}-\mathcal{G}\mathcal{H}_{b}\mathcal{H}^{-1}\mathcal{H}_{b}\mathcal{H}^{-1}\\
& -W^{-2}W_{b}\mathcal{G}^{2,1}+W^{-1}\mathcal{G}_{b}^{2,1}-W^{-2}W_{b}\mathcal{G}^{1,2}+W^{-1}\mathcal{G}_{b}^{1,2}\\
& -W^{-2}W_{b}\mathcal{H}^{-1}\mathcal{E}\mathcal{G}-W^{-2}W_{b}\mathcal{G}\mathcal{E}\mathcal{H}^{-1}\\
& -W^{-1}\mathcal{H}^{-1}\mathcal{H}_{b}\mathcal{H}^{-1}\mathcal{E}\mathcal{G}-W^{-1}\mathcal{G}\mathcal{E}\mathcal{H}^{-1}\mathcal{H}_{b}\mathcal{H}^{-1}\\
& +W^{-1}\mathcal{H}^{-1}\mathcal{E}_{b}\mathcal{G}\\
& +W^{-1}\mathcal{G}\mathcal{E}_{b}\mathcal{H}^{-1}\\
& +W^{-1}\mathcal{H}^{-1}\mathcal{E}\mathcal{G}_{b}+W^{-1}\mathcal{G}_{b}\mathcal{E}\mathcal{H}^{-1}
\end{align*}
\begin{align*}
\mathcal{G}_{bs_{g}} & =\mathcal{H}^{-1}\mathcal{H}_{s_{g}}\mathcal{H}^{-1}\mathcal{H}_{b}\mathcal{G}+\mathcal{G}\mathcal{H}_{b}\mathcal{H}^{-1}\mathcal{H}_{s_{g}}\mathcal{H}^{-1}\\
& -\mathcal{H}^{-1}\mathcal{H}_{bs_{g}}\mathcal{G}-\mathcal{G}\mathcal{H}_{b_{s_{g}}}\mathcal{H}^{-1}\\
& -\mathcal{H}^{-1}\mathcal{H}_{b}\mathcal{G}_{s_{g}}-\mathcal{G}_{s_{g}}\mathcal{H}_{b}\mathcal{H}^{-1}\\
& +W^{-2}W_{s_{g}}\mathcal{G}^{2,1}-W^{-1}\mathcal{G}_{s_{g}}^{2,1}\\
& +W^{-2}W_{s_{g}}\mathcal{G}^{1,2}-W^{-1}\mathcal{G}_{s_{g}}^{1,2}\\
& +W^{-2}W_{s_{g}}(\mathcal{H}^{-1}\mathcal{E}\mathcal{G}+\mathcal{G}\mathcal{E}\mathcal{H}^{-1})\\
& +W^{-1}\mathcal{H}^{-1}\mathcal{H}_{s_{g}}\mathcal{H}^{-1}\mathcal{E}\mathcal{G}+W^{-1}\mathcal{G}\mathcal{E}\mathcal{H}^{-1}\mathcal{H}_{s_{g}}\mathcal{H}^{-1}\\
& -W^{-1}\mathcal{H}^{-1}\mathcal{E}_{s_{g}}\mathcal{G}-W^{-1}\mathcal{H}^{-1}\mathcal{E}\mathcal{G}_{s_{g}}\\
& -W^{-1}\mathcal{G}\mathcal{E}_{s_{g}}\mathcal{H}^{-1}-W^{-1}\mathcal{G}_{s_{g}}\mathcal{E}\mathcal{H}^{-1}
\end{align*}
\begin{align*}
\mathcal{G}_{s_{f}s_{g}} & =\mathcal{H}^{-1}\mathcal{H}_{s_{f}}\mathcal{H}^{-1}\mathcal{H}_{s_{g}}\mathcal{G}+\mathcal{G}\mathcal{H}_{s_{g}}\mathcal{H}^{-1}\mathcal{H}_{s_{f}}\mathcal{H}^{-1}\\
& -\mathcal{H}^{-1}\mathcal{H}_{s_{f}s_{g}}\mathcal{G}-\mathcal{G}\mathcal{H}_{s_{f}s_{g}}\mathcal{H}^{-1}\\
& -\mathcal{H}^{-1}\mathcal{H}_{s_{g}}\mathcal{G}_{s_{f}}-\mathcal{G}_{s_{f}}\mathcal{H}_{s_{g}}\mathcal{H}^{-1}\\
& +W^{-2}W_{s_{f}}(\mathcal{G}^{2,1}+\mathcal{G}^{1,2})\big[\mathcal{H}^{-1}(\partial_{\beta\phi}\mathcal{L})\big]_{g}\\
& -W^{-1}(\mathcal{G}_{s_{f}}^{2,1}+\mathcal{G}_{s_{f}}^{1,2})\big[\mathcal{H}^{-1}(\partial_{\beta\phi}\mathcal{L})\big]_{g}\\
& +W^{-1}(\mathcal{G}^{2,1}+\mathcal{G}^{1,2})\big[\mathcal{H}^{-1}\mathcal{H}_{s_{f}}\mathcal{H}^{-1}(\partial_{\beta\phi}\mathcal{L})\big]_{g}\\
& +W^{-2}(\mathcal{G}^{2,1}+\mathcal{G}^{1,2})\big[\mathcal{H}^{-1}(\partial_{\beta\beta\phi}\mathcal{L})\big]_{g}\big[\mathcal{H}^{-1}(\partial_{\beta\phi}\mathcal{L})\big]_{f}\\
& +W^{-1}(\mathcal{G}^{2,1}+\mathcal{G}^{1,2})\big[\mathcal{H}^{-1}\mathcal{E}\mathcal{H}^{-1}\big]_{gf}\\
& +W^{-2}(\mathcal{G}^{2,1}+\mathcal{G}^{1,2})\big[\mathcal{H}^{-1}\mathcal{E}\mathcal{H}^{-1}(\partial_{\beta\phi}\mathcal{L})\big]_{g}\big[\mathcal{H}^{-1}(\partial_{\beta\phi}\mathcal{L})\big]_{f}\\
& -\mathcal{H}^{-1}\mathcal{H}_{s_{f}}\mathcal{H}^{-1}\mathcal{E}\big[\mathcal{H}^{-1}\big]_{g,\cdot}(\partial_{\beta\phi'}\mathcal{L})\mathcal{H}^{-1}\\
& +\mathcal{H}^{-1}\mathcal{E}_{s_{f}}\big[\mathcal{H}^{-1}\big]_{g,\cdot}(\partial_{\beta\phi'}\mathcal{L})\mathcal{H}^{-1}\\
& -\mathcal{H}^{-1}\mathcal{E}\big[\mathcal{H}^{-1}\mathcal{H}_{s_{f}}\mathcal{H}^{-1}\big]_{g,\cdot}(\partial_{\beta\phi'}\mathcal{L})\mathcal{H}^{-1}\\
& -W^{-1}\mathcal{H}^{-1}\mathcal{E}^{1}\big[\mathcal{H}^{-1}\big]_{g}(\partial_{\beta\beta\phi'}\mathcal{L})\mathcal{H}^{-1}\big[\mathcal{H}^{-1}(\partial_{\beta\phi}\mathcal{L})\big]_{f}\\
& -\mathcal{H}^{-1}\mathcal{E}^{1}\big[\mathcal{H}^{-1}\big]_{g}\big[\mathcal{H}^{-1}\mathcal{E}\mathcal{H}^{-1}\big]_{\cdot,f}\\
& -W^{-1}\mathcal{H}^{-1}\mathcal{E}^{1}\big[\mathcal{H}^{-1}\big]_{g}(\partial_{\beta\phi'}\mathcal{L})\mathcal{H}^{-1}\mathcal{E}\mathcal{H}^{-1}\big[\mathcal{H}^{-1}(\partial_{\beta\phi}\mathcal{L})\big]_{f}\\
& -\mathcal{H}^{-1}\mathcal{E}^{1}\big[\mathcal{H}^{-1}\big]_{g}(\partial_{\beta\phi'}\mathcal{L})\mathcal{H}^{-1}\mathcal{H}_{s_{f}}\mathcal{H}^{-1}\\
& -\mathcal{H}^{-1}\mathcal{H}_{s_{f}}\mathcal{H}^{-1}(\partial_{\beta\phi}\mathcal{L})\big[\mathcal{H}^{-1}\big]_{g}'\mathcal{E}\mathcal{H}^{-1}\\
& -W^{-1}\mathcal{H}^{-1}(\partial_{\beta\beta\phi}\mathcal{L})\big[\mathcal{H}^{-1}\big]_{g}'\mathcal{E}\mathcal{H}^{-1}\big[\mathcal{H}^{-1}(\partial_{\beta\phi}\mathcal{L})\big]_{f}\\
& -\big[\mathcal{H}^{-1}\mathcal{E}\mathcal{H}^{-1}\big]_{f,\cdot}\big[\mathcal{H}^{-1}\big]_{g,\cdot}'\mathcal{E}\mathcal{H}^{-1}\\
& -W^{-1}\mathcal{H}^{-1}\mathcal{E}\mathcal{H}^{-1}(\partial_{\beta\phi}\mathcal{L})\big[\mathcal{H}^{-1}\big]_{g}'\mathcal{E}\mathcal{H}^{-1}\big[\mathcal{H}^{-1}(\partial_{\beta\phi}\mathcal{L})\big]_{f}\\
& -\mathcal{H}^{-1}(\partial_{\beta^{s}\phi}\mathcal{L})\big[\mathcal{H}^{-1}\mathcal{H}_{s_{f}}\mathcal{H}^{-1}\big]_{g}'\mathcal{E}\mathcal{H}^{-1}\\
& +\mathcal{H}^{-1}(\partial_{\beta^{s}\phi}\mathcal{L})\big[\mathcal{H}^{-1}\big]_{g}'\mathcal{E}_{s_{f}}\mathcal{H}^{-1}\\
& -\mathcal{H}^{-1}(\partial_{\beta^{s}\phi}\mathcal{L})\big[\mathcal{H}^{-1}\big]_{g}'\mathcal{E}^{1}\mathcal{H}^{-1}\mathcal{H}_{s_{f}}\mathcal{H}^{-1}\\
& +W^{-2}W_{s_{g}}\mathcal{H}^{-1}\mathcal{E}\mathcal{G}\big[\mathcal{H}^{-1}(\partial_{\beta\phi}\mathcal{L})\big]_{g}\\
& +W^{-1}\mathcal{H}^{-1}\mathcal{H}_{s_{f}}\mathcal{H}^{-1}\mathcal{E}\mathcal{G}\big[\mathcal{H}^{-1}(\partial_{\beta\phi}\mathcal{L})\big]_{g}\\
& -W^{-1}\mathcal{H}^{-1}\mathcal{E}_{s_{f}}\mathcal{G}\big[\mathcal{H}^{-1}(\partial_{\beta\phi}\mathcal{L})\big]_{g}\\
& -W^{-1}\mathcal{H}^{-1}\mathcal{E}\mathcal{G}_{s_{f}}\big[\mathcal{H}^{-1}(\partial_{\beta\phi}\mathcal{L})\big]_{g}\\
& +W^{-1}\mathcal{H}^{-1}\mathcal{E}\mathcal{G}\big[\mathcal{H}^{-1}\mathcal{H}_{s_{f}}\mathcal{H}^{-1}(\partial_{\beta\phi}\mathcal{L})\big]_{g}\\
& +W^{-2}\mathcal{H}^{-1}\mathcal{E}\mathcal{G}\big[\mathcal{H}^{-1}(\partial_{\beta\beta\phi}\mathcal{L})\big]_{g}\big[\mathcal{H}^{-1}(\partial_{\beta\phi}\mathcal{L})\big]_{f}\\
& +W^{-1}\mathcal{H}^{-1}\mathcal{E}\mathcal{G}\big[\mathcal{H}^{-1}(\partial_{\beta\phi\phi'}\mathcal{L})\mathcal{H}^{-1}\big]_{g}\\
& +W^{-2}\mathcal{H}^{-1}\mathcal{E}\mathcal{G}\big[\mathcal{H}^{-1}\mathcal{E}\mathcal{H}^{-1}(\partial_{\beta\phi}\mathcal{L})\big]_{g}\big[\mathcal{H}^{-1}(\partial_{\beta\phi}\mathcal{L})\big]_{f}\\
& +W^{-2}W_{s_{f}}\mathcal{G}\mathcal{E}\mathcal{H}^{-1}\big[\mathcal{H}^{-1}(\partial_{\beta\phi}\mathcal{L})\big]_{g}\\
& -W^{-1}\mathcal{G}_{s_{f}}\mathcal{E}\mathcal{H}^{-1}\big[\mathcal{H}^{-1}(\partial_{\beta\phi}\mathcal{L})\big]_{g}\\
& -W^{-1}\mathcal{G}\mathcal{E}_{s_{f}}\mathcal{H}^{-1}\big[\mathcal{H}^{-1}(\partial_{\beta\phi}\mathcal{L})\big]_{g}\\
& +W^{-1}\mathcal{G}\mathcal{E}\mathcal{H}^{-1}\mathcal{H}_{s_{f}}\mathcal{H}^{-1}\big[\mathcal{H}^{-1}(\partial_{\beta\phi}\mathcal{L})\big]_{g}\\
& +W^{-1}\mathcal{G}\mathcal{E}\mathcal{H}^{-1}\big[\mathcal{H}^{-1}\mathcal{H}_{s_{f}}\mathcal{H}^{-1}(\partial_{\beta\phi}\mathcal{L})\big]_{g}\\
& +W^{-2}\mathcal{G}\mathcal{E}\mathcal{H}^{-1}\big[\mathcal{H}^{-1}(\partial_{\beta\beta\phi}\mathcal{L})\big]_{g}\big[\mathcal{H}^{-1}(\partial_{\beta\phi}\mathcal{L})\big]_{f}\\
& +W^{-1}\mathcal{G}\mathcal{E}\mathcal{H}^{-1}\big[\mathcal{H}^{-1}(\partial_{\beta\phi\phi'}\mathcal{L})\mathcal{H}^{-1}\big]_{g}\\
& +W^{-2}\mathcal{G}\mathcal{E}\mathcal{H}^{-1}\big[\mathcal{H}^{-1}\mathcal{E}\mathcal{H}^{-1}(\partial_{\beta\phi}\mathcal{L})\big]_{g}\big[\mathcal{H}^{-1}(\partial_{\beta\phi}\mathcal{L})\big]_{f}
\end{align*}
Finally, the third derivatives are
\begin{align*}
\mathcal{G}_{bbb} & =2\mathcal{H}^{-1}\mathcal{H}_{b}\mathcal{H}^{-1}\mathcal{H}_{b}\mathcal{H}^{-1}\mathcal{H}_{b}\mathcal{G}-2\mathcal{H}^{-1}\mathcal{H}_{b}\mathcal{H}^{-1}\mathcal{H}_{b}\mathcal{G}_{b}\\
& -\mathcal{H}^{-1}\mathcal{H}_{bb}\mathcal{H}^{-1}\mathcal{H}_{b}\mathcal{G}-2\mathcal{H}^{-1}\mathcal{H}_{b}\mathcal{H}^{-1}\mathcal{H}_{bb}\mathcal{G}\\
& +\mathcal{H}^{-1}\mathcal{H}_{bbb}\mathcal{G}+2\mathcal{H}^{-1}\mathcal{H}_{bb}\mathcal{G}_{b}\\
& +\mathcal{H}^{-1}\mathcal{H}_{b}\mathcal{G}_{bb}\\
& +\mathcal{G}_{bb}\mathcal{H}_{b}\mathcal{H}^{-1}+2\mathcal{G}_{b}\mathcal{H}_{bb}\mathcal{H}^{-1}-2\mathcal{G}_{b}\mathcal{H}_{b}\mathcal{H}^{-1}\mathcal{H}_{b}\mathcal{H}^{-1}\\
& +\mathcal{G}\mathcal{H}_{bbb}\mathcal{H}^{-1}-2\mathcal{G}\mathcal{H}_{bb}\mathcal{H}^{-1}\mathcal{H}_{b}\mathcal{H}^{-1}-\mathcal{G}\mathcal{H}_{b}\mathcal{H}^{-1}\mathcal{H}_{bb}\mathcal{H}^{-1}\\
& +2\mathcal{G}\mathcal{H}_{b}\mathcal{H}^{-1}\mathcal{H}_{b}\mathcal{H}^{-1}\mathcal{H}_{b}\mathcal{H}^{-1}\\
& +2W^{-3}W_{b}^{2}(\mathcal{G}^{2,1}+\mathcal{G}^{1,2})-W^{-2}W_{bb}(\mathcal{G}^{2,1}+\mathcal{G}^{1,2})-W^{-2}W_{b}(\mathcal{G}_{b}^{2,1}+\mathcal{G}_{b}^{1,2})\\
& -W^{-2}W_{b}(\mathcal{G}_{b}^{2,1}+\mathcal{G}_{b}^{1,2})+W^{-1}(\mathcal{G}_{bb}^{2,1}+\mathcal{G}_{bb}^{1,2})\\
& +2W^{-3}W_{b}^{2}\mathcal{H}^{-1}\mathcal{E}\mathcal{G}-W^{-2}W_{bb}\mathcal{H}^{-1}\mathcal{E}\mathcal{G}+W^{-2}W_{b}\mathcal{H}^{-1}\mathcal{H}_{b}\mathcal{H}^{-1}\mathcal{E}\mathcal{G}\\
& -W^{-2}W_{b}\mathcal{H}^{-1}\mathcal{E}_{b}\mathcal{G}-W^{-2}W_{b}\mathcal{H}^{-1}\mathcal{E}\mathcal{G}_{b}\\
& +2W^{-3}W_{b}^{2}\mathcal{G}\mathcal{E}\mathcal{H}^{-1}-W^{-2}W_{bb}\mathcal{G}\mathcal{E}\mathcal{H}^{-1}-W^{-2}W_{b}\mathcal{G}_{b}\mathcal{E}\mathcal{H}^{-1}\\
& -W^{-2}W_{b}\mathcal{G}\mathcal{E}_{b}\mathcal{H}^{-1}+W^{-2}W_{b}\mathcal{G}\mathcal{E}\mathcal{H}^{-1}\mathcal{H}_{b}\mathcal{H}^{-1}\\
& +W^{-2}W_{b}\mathcal{H}^{-1}\mathcal{H}_{b}\mathcal{H}^{-1}\mathcal{E}\mathcal{G}+2W^{-1}\mathcal{H}^{-1}\mathcal{H}_{b}\mathcal{H}^{-1}\mathcal{H}_{b}\mathcal{H}^{-1}\mathcal{E}\mathcal{G}\\
& -W^{-1}\mathcal{H}^{-1}\mathcal{H}_{bb}\mathcal{H}^{-1}\mathcal{E}\mathcal{G}-W^{-1}\mathcal{H}^{-1}\mathcal{H}_{b}\mathcal{H}^{-1}\mathcal{E}_{b}\mathcal{G}-W^{-1}\mathcal{H}^{-1}\mathcal{H}_{b}\mathcal{H}^{-1}\mathcal{E}\mathcal{G}_{b}\\
& +W^{-2}W_{b}\mathcal{G}\mathcal{E}\mathcal{H}^{-1}\mathcal{H}_{b}\mathcal{H}^{-1}-W^{-1}\mathcal{G}_{b}\mathcal{E}\mathcal{H}^{-1}\mathcal{H}_{b}\mathcal{H}^{-1}-W^{-1}\mathcal{G}\mathcal{E}_{b}\mathcal{H}^{-1}\mathcal{H}_{b}\mathcal{H}^{-1}\\
& -W^{-1}\mathcal{G}\mathcal{E}\mathcal{H}^{-1}\mathcal{H}_{bb}\mathcal{H}^{-1}+W^{-1}\mathcal{G}\mathcal{E}\mathcal{H}^{-1}\mathcal{H}_{b}\mathcal{H}^{-1}\mathcal{H}_{b}\mathcal{H}^{-1}\\
& -W^{-2}W_{b}\mathcal{H}^{-1}(\mathcal{E}_{b}\mathcal{G}+\mathcal{E}\mathcal{G}_{b})-W^{-1}\mathcal{H}^{-1}\mathcal{H}_{b}\mathcal{H}^{-1}(\mathcal{E}_{b}\mathcal{G}+\mathcal{E}\mathcal{G}_{b})\\
& +W^{-1}\mathcal{H}^{-1}(\mathcal{E}_{bb}\mathcal{G}+\mathcal{E}\mathcal{G}_{bb})\\
& -W^{-2}W_{b}(\mathcal{E}_{b}\mathcal{G}+\mathcal{E}\mathcal{G}_{b})\mathcal{H}^{-1}+W^{-1}(\mathcal{E}_{bb}\mathcal{G}+\mathcal{E}\mathcal{G}_{bb})\mathcal{H}^{-1}\\
& -W^{-1}(\mathcal{E}_{b}\mathcal{G}+\mathcal{E}\mathcal{G}_{b})\mathcal{H}^{-1}\mathcal{H}_{b}\mathcal{H}^{-1}
\end{align*}
\subsection{Expressions for $W$ terms}
We begin by expressing the first three derivatives of $W$ in terms
of $\mathcal{F}$. To do this, first note that by the definition of
$\mathcal{F}$
\begin{align*}
W & =\partial_{\beta\beta}\mathcal{L}+(\partial_{\beta\phi'}\mathcal{L})\mathcal{H}^{-1}(\partial_{\beta\phi}\mathcal{L})\\
& =(\partial_{\beta\beta}\mathcal{L})+\mathcal{F}
\end{align*}
Differentiating with respect to $b$ and $s$ gives
\begin{align*}
W_{b} & =-W^{-1}(\partial_{\beta\beta\beta}\mathcal{L})-W^{-1}\mathcal{F}^{2,1}+\mathcal{F}_{b}
\end{align*}
\begin{align*}
W_{s} & =-W^{-1}(\partial_{\beta\beta\beta}\mathcal{L})\mathcal{H}^{-1}(\partial_{\beta\phi}\mathcal{L})-\Big[\mathcal{H}^{-1}+W^{-1}\mathcal{G}\big](\partial_{\beta\beta\phi}\mathcal{L})+\mathcal{F}_{s}
\end{align*}
Differentiating a second time gives
(1)
\begin{align*}
W_{bb} & =W^{-2}W_{b}(\partial_{\beta\beta\beta}\mathcal{L})+W^{-2}(\partial_{\beta^{4}}\mathcal{L})+W^{-2}\mathcal{F}^{3,1}\\
& +W^{-2}W_{b}\mathcal{F}^{2,1}-W^{-1}\mathcal{F}_{b}^{2,1}\\
& +\mathcal{F}_{bb}
\end{align*}
(2)
\begin{align*}
W_{bs} & =W^{-2}W_{b}(\partial_{\beta\beta\beta}\mathcal{L})\mathcal{H}^{-1}(\partial_{\beta\phi}\mathcal{L})+W^{-2}\big[(\partial_{\beta^{4}}\mathcal{L})+\mathcal{F}^{3,1}\big]\mathcal{H}^{-1}(\partial_{\beta\phi}\mathcal{L})\\
& +W^{-1}(\partial_{\beta\beta\beta}\mathcal{L})\mathcal{H}^{-1}\mathcal{H}_{b}\mathcal{H}^{-1}(\partial_{\beta\phi}\mathcal{L})+W^{-2}(\partial_{\beta\beta\beta}\mathcal{L})\mathcal{H}^{-1}\big[(\partial_{\beta\beta\phi}\mathcal{L})+(\partial_{\beta\phi\phi'}\mathcal{L})\mathcal{H}^{-1}(\partial_{\beta\phi}\mathcal{L})\big]\\
& +W^{-1}\big[\mathcal{H}^{-1}+W^{-1}\mathcal{G}\big]\big[(\partial_{\beta\beta\beta\phi}\mathcal{L})+(\partial_{\beta\beta\phi\phi'}\mathcal{L})\mathcal{H}^{-1}(\partial_{\beta\phi}\mathcal{L})\big]\\
& +\Big[\mathcal{H}^{-1}\mathcal{H}_{b}\mathcal{H}^{-1}+W^{-2}W_{b}\mathcal{G}-W^{-1}\mathcal{G}_{b}\big](\partial_{\beta\beta\phi}\mathcal{L})\\
& +\mathcal{F}_{sb}
\end{align*}
(3)
\begin{align*}W_{ss_{g}} & =W^{-2}W_{s_{g}}(\partial_{\beta\beta\beta}\mathcal{L})\mathcal{R}^{1}+W^{-2}(\partial_{\beta^{4}}\mathcal{L})[\mathcal{R}^{1}]_{g}\mathcal{R}^{1}\\
& +W^{-1}(\partial_{\beta^{3}\phi'}\mathcal{L})[\mathcal{H}^{-1}]_{\cdot,g}\mathcal{R}^{1}+W^{-2}(\partial_{\beta^{3}\phi'}\mathcal{L})\mathcal{R}^{1}[\mathcal{R}^{1}]_{g}\mathcal{R}^{1}\\
& -W^{-1}(\partial_{\beta\beta\beta}\mathcal{L})\mathcal{H}^{-1}\mathcal{R}_{b}^{1}+\mathcal{H}^{-1}\mathcal{H}_{s_{g}}\mathcal{R}^{2}\\
& +W^{-2}W_{s_{g}}\mathcal{G}(\partial_{\beta\beta\phi}\mathcal{L})-W^{-1}\mathcal{G}_{b}(\partial_{\beta\beta\phi}\mathcal{L})\\
& +W^{-1}\Big[\mathcal{H}^{-1}+W^{-1}\mathcal{G}\big](\partial_{\beta^{3}\phi}\mathcal{L})[\mathcal{R}^{1}]_{g}\\
& +\Big[\mathcal{H}^{-1}+W^{-1}\mathcal{G}\big](\partial_{\beta\beta\phi\phi'}\mathcal{L})[\mathcal{H}^{-1}]_{\cdot,g}\\
& +W^{-1}\Big[\mathcal{H}^{-1}+W^{-1}\mathcal{G}\big](\partial_{\beta\beta\phi\phi'}\mathcal{L})\mathcal{R}^{1}[\mathcal{R}^{1}]_{g}\\
& +\mathcal{F}_{ss_{g}}
\end{align*}
Third derivative:
\begin{align*}W_{bss_{g}} & =-2W^{-3}W_{b}W_{s_{g}}(\partial_{\beta\beta\beta}\mathcal{L})\mathcal{R}^{1}+W^{-2}W_{bs_{g}}(\partial_{\beta\beta\beta}\mathcal{L})\mathcal{R}^{1}\\
& -W^{-3}W_{s_{g}}(\partial_{\beta^{4}}\mathcal{L})\mathcal{R}^{1}-W^{-3}W_{s_{g}}\mathcal{R}^{1}(\partial_{\beta^{3}\phi'}\mathcal{L})\mathcal{R}^{1}\\
& +W^{-2}W_{s_{g}}(\partial_{\beta\beta\beta}\mathcal{L})\mathcal{R}_{b}^{1}\\
& -2W^{-3}W_{b}(\partial_{\beta^{4}}\mathcal{L})[\mathcal{R}^{1}]_{g}\mathcal{R}^{1}\\
& -W^{-3}(\partial_{\beta^{5}}\mathcal{L})[\mathcal{R}^{1}]_{g}\mathcal{R}^{1}-W^{-3}\mathcal{R}^{1}(\partial_{\beta^{4}\phi'}\mathcal{L})\mathcal{R}^{1}[\mathcal{R}^{1}]_{g}\\
& +W^{-2}(\partial_{\beta^{4}}\mathcal{L})[\mathcal{R}_{b}^{1}]_{g}\mathcal{R}^{1}+W^{-2}(\partial_{\beta^{4}}\mathcal{L})[\mathcal{R}_{b}^{1}]_{g}\mathcal{R}^{1}\\
& -W^{-2}W_{b}(\partial_{\beta^{3}\phi'}\mathcal{L})[\mathcal{H}^{-1}]_{\cdot,g}\mathcal{R}^{1}\\
& -W^{-2}(\partial_{\beta^{4}\phi'}\mathcal{L})[\mathcal{H}^{-1}]_{\cdot,g}\mathcal{R}^{1}-W^{-2}(\mathcal{R}^{1})'(\partial_{\beta^{3}\phi\phi'}\mathcal{L})[\mathcal{H}^{-1}]_{\cdot,g}\mathcal{R}^{1}\\
& -W^{-1}(\partial_{\beta^{3}\phi'}\mathcal{L})[\mathcal{H}^{-1}\mathcal{H}_{b}\mathcal{H}^{-1}]_{\cdot,g}\mathcal{R}^{1}+W^{-1}(\partial_{\beta^{3}\phi'}\mathcal{L})[\mathcal{H}^{-1}]_{\cdot,g}\mathcal{R}_{b}^{1}\\
& -2W^{-3}W_{b}(\partial_{\beta^{3}\phi'}\mathcal{L})\mathcal{R}^{1}[\mathcal{R}^{1}]_{g}\mathcal{R}^{1}\\
& -W^{-3}(\partial_{\beta^{4}\phi'}\mathcal{L})\mathcal{R}^{1}[\mathcal{R}^{1}]_{g}\mathcal{R}^{1}-W^{-3}(\mathcal{R}^{1})'(\partial_{\beta^{3}\phi\phi'}\mathcal{L})\mathcal{R}^{1}[\mathcal{R}^{1}]_{g}\mathcal{R}^{1}\\
& +W^{-2}(\partial_{\beta^{3}\phi'}\mathcal{L})\mathcal{R}_{b}^{1}[\mathcal{R}^{1}]_{g}\mathcal{R}^{1}+W^{-2}(\partial_{\beta^{3}\phi'}\mathcal{L})\mathcal{R}^{1}[\mathcal{R}_{b}^{1}]_{g}\mathcal{R}^{1}\\
& +W^{-2}(\partial_{\beta^{3}\phi'}\mathcal{L})\mathcal{R}^{1}[\mathcal{R}^{1}]_{g}\mathcal{R}_{b}^{1}\\
& +W^{-2}W_{b}(\partial_{\beta\beta\beta}\mathcal{L})\mathcal{H}^{-1}\mathcal{R}_{b}^{1}\\
& +W^{-2}(\partial_{\beta^{4}}\mathcal{L})\mathcal{H}^{-1}\mathcal{R}_{b}^{1}+W^{-2}(\partial_{\beta^{3}\phi'}\mathcal{L})\mathcal{R}^{1}\mathcal{H}^{-1}\mathcal{R}_{b}^{1}\\
& +W^{-1}(\partial_{\beta\beta\beta}\mathcal{L})\mathcal{H}^{-1}\mathcal{H}_{b}\mathcal{H}^{-1}\mathcal{R}_{b}^{1}-W^{-1}(\partial_{\beta\beta\beta}\mathcal{L})\mathcal{H}^{-1}\mathcal{R}_{bb}^{1}\\
& -\mathcal{H}^{-1}\mathcal{H}_{b}\mathcal{H}^{-1}\mathcal{H}_{s_{g}}\mathcal{R}^{2}+\mathcal{H}^{-1}\mathcal{H}_{bs_{g}}\mathcal{R}^{2}\\
& -\mathcal{H}^{-1}\mathcal{H}_{s_{g}}\mathcal{R}_{b}^{2}\\
& -2W^{-3}W_{b}W_{s_{g}}\mathcal{G}(\partial_{\beta\beta\phi}\mathcal{L})+W^{-2}W_{bs_{g}}\mathcal{G}(\partial_{\beta\beta\phi}\mathcal{L})\\
& +W^{-2}W_{s_{g}}\mathcal{G}_{b}(\partial_{\beta\beta\phi}\mathcal{L})\\
& -W^{-3}W_{s_{g}}\mathcal{G}(\partial_{\beta^{3}\phi}\mathcal{L})-W^{-3}W_{s_{g}}\mathcal{G}(\partial_{\beta\beta\phi\phi'}\mathcal{L})\mathcal{R}^{1}\\
& +W^{-2}W_{b}\mathcal{G}_{b}(\partial_{\beta\beta\phi}\mathcal{L})-W^{-1}\mathcal{G}_{bb}(\partial_{\beta\beta\phi}\mathcal{L})\\
& +W^{-2}\mathcal{G}_{b}(\partial_{\beta^{3}\phi}\mathcal{L})+W^{-2}\mathcal{G}_{b}(\partial_{\beta\beta\phi\phi'}\mathcal{L})\mathcal{R}^{1}\\
& -W^{-2}W_{b}\Big[\mathcal{H}^{-1}+W^{-1}\mathcal{G}\big](\partial_{\beta^{3}\phi}\mathcal{L})[\mathcal{R}^{1}]_{g}\\
& -W^{-1}\Big[\mathcal{H}^{-1}\mathcal{H}_{b}\mathcal{H}^{-1}+W^{-2}W_{b}\mathcal{G}-W^{-1}\mathcal{G}_{b}\big](\partial_{\beta^{3}\phi}\mathcal{L})[\mathcal{R}^{1}]_{g}\\
& -W^{-2}\Big[\mathcal{H}^{-1}+W^{-1}\mathcal{G}\big](\partial_{\beta^{4}\phi}\mathcal{L})[\mathcal{R}^{1}]_{g}\\
& -W^{-2}\Big[\mathcal{H}^{-1}+W^{-1}\mathcal{G}\big](\partial_{\beta^{3}\phi\phi'}\mathcal{L})\mathcal{R}^{1}[\mathcal{R}^{1}]_{g}\\
& +W^{-1}\Big[\mathcal{H}^{-1}+W^{-1}\mathcal{G}\big](\partial_{\beta^{3}\phi}\mathcal{L})[\mathcal{R}_{b}^{1}]_{g}\\
& -\Big[\mathcal{H}^{-1}\mathcal{H}_{b}\mathcal{H}^{-1}+W^{-2}W_{b}\mathcal{G}-W^{-1}\mathcal{G}_{b}\big](\partial_{\beta\beta\phi\phi'}\mathcal{L})[\mathcal{H}^{-1}]_{\cdot,g}\\
& -W^{-1}\Big[\mathcal{H}^{-1}+W^{-1}\mathcal{G}\big](\partial_{\beta^{3}\phi\phi'}\mathcal{L})[\mathcal{H}^{-1}]_{\cdot,g}\\
& -W^{-1}\Big[\mathcal{H}^{-1}+W^{-1}\mathcal{G}\big]\sum_{f}(\partial_{\beta^{2}\phi\phi'\phi_{f}}\mathcal{L})[\mathcal{H}^{-1}]_{\cdot,g}[\mathcal{R}^{1}]_{f}\\
& -\Big[\mathcal{H}^{-1}+W^{-1}\mathcal{G}\big](\partial_{\beta\beta\phi\phi'}\mathcal{L})[\mathcal{H}^{-1}\mathcal{H}_{b}\mathcal{H}^{-1}]_{\cdot,g}\\
& -W^{-2}W_{b}\Big[\mathcal{H}^{-1}+W^{-1}\mathcal{G}\big](\partial_{\beta\beta\phi\phi'}\mathcal{L})\mathcal{R}^{1}[\mathcal{R}^{1}]_{g}\\
& -W^{-1}\Big[\mathcal{H}^{-1}\mathcal{H}_{b}\mathcal{H}^{-1}+W^{-2}W_{b}\mathcal{G}-W^{-1}\mathcal{G}_{b}\big](\partial_{\beta\beta\phi\phi'}\mathcal{L})\mathcal{R}^{1}[\mathcal{R}^{1}]_{g}\\
& -W^{-2}\Big[\mathcal{H}^{-1}+W^{-1}\mathcal{G}\big](\partial_{\beta^{3}\phi\phi'}\mathcal{L})\mathcal{R}^{1}[\mathcal{R}^{1}]_{g}\\
& -W^{-2}\Big[\mathcal{H}^{-1}+W^{-1}\mathcal{G}\big]\sum_{f}(\partial_{\beta^{2}\phi\phi'\phi_{f}}\mathcal{L})[\mathcal{R}^{1}]_{f}\mathcal{R}^{1}[\mathcal{R}^{1}]_{g}\\
& +W^{-1}\Big[\mathcal{H}^{-1}+W^{-1}\mathcal{G}\big](\partial_{\beta\beta\phi\phi'}\mathcal{L})\mathcal{R}_{b}^{1}[\mathcal{R}^{1}]_{g}\\
& +W^{-1}\Big[\mathcal{H}^{-1}+W^{-1}\mathcal{G}\big](\partial_{\beta\beta\phi\phi'}\mathcal{L})\mathcal{R}^{1}[\mathcal{R}_{b}^{1}]_{g}\\
& +\mathcal{F}_{bss_{g}}
\end{align*}
\subsection{\label{sec:beta_expansion} Expansion for $\widehat{\beta}$}
In Section \ref{sec:full_expansion} we provide bounds for each of the above terms that
by Assumptions 1 and 2 can be shown to hold uniformly over a neighborhood
of the truth. Using these results, we will be able to show that
\begin{align*}
\partial_{b}\mathcal{L}^{*}(0,0) & =\partial_{b}\mathcal{L}^{*}-(\partial_{bb}\mathcal{L}^{*})\mathcal{S}_{\beta}-(\partial_{bs'}\mathcal{L}^{*})\mathcal{S}\\
& +\frac{1}{2}(\partial_{bbb}\mathcal{L}^{*})\mathcal{S}_{\beta}^{2}+(\partial_{bbs'}\mathcal{L}^{*})\mathcal{S}\mathcal{S}_{\beta}+\frac{1}{2}\mathcal{S}'(\partial_{bss'}\mathcal{L}^{*})\mathcal{S}\\
& -\frac{1}{2}\mathcal{S}'(\partial_{bbss'}\mathcal{L}_{(1)}^{*})\mathcal{S}\mathcal{S}_{\beta}-\frac{1}{6}\sum_{g}\mathcal{S}'(\partial_{bss's_{g}}\mathcal{L}_{(1)}^{*})\mathcal{S}\mathcal{S}_{g}\\
& +\frac{1}{4}\mathcal{S}'(\partial_{bbbss'}\mathcal{L}_{(1)}^{*}(\bar{b},\bar{s}))\mathcal{S}\mathcal{S}_{\beta}^{2}+\frac{1}{24}\sum_{f,g}\mathcal{S}'(\partial_{bss's_{f}s_{g}}\mathcal{L}_{(2)}^{*})\mathcal{S}\mathcal{S}_{f}\mathcal{S}_{g}\\
& +o_{p}(N^{-2}) \\
&=\partial_{b}\mathcal{L}^{*}-(\partial_{bb}\mathcal{L}^{*})\mathcal{S}_{\beta}-(\partial_{bs'}\mathcal{L}^{*})\mathcal{S}+\frac{1}{2}\mathcal{S}'(\partial_{bss'}\mathcal{L}^{*})\mathcal{S}+o_{p}(1)
\end{align*}
where $(\bar{b},\bar{s})$ are intermediate values between $(0,0)$
and $(\mathcal{S}_{\beta},\mathcal{S})$, which
under Assumptions 1 and 2 must lie in $\mathcal{B}(r_\beta,\beta_0)\times\mathcal{B}_q(r_\phi,\phi_0)$ for sufficiently large $N$ (see Corollary B.3 in FW16).
Hence, we have the first-order expansion
\begin{align*}
\partial_{b}\mathcal{L}^{*}(0,0) & =W_{N}^{-1}\mathcal{S}_{\beta}+W_{N}^{-1}(\partial_{\beta\phi'}\mathcal{L})\mathcal{H}^{-1}\mathcal{S}+N\frac{1}{2}\mathcal{S}'\mathcal{H}^{-1}\mathcal{H}_{b}\mathcal{H}^{-1}\mathcal{S}+o_{p}(1)\\
NW_{N}(\widehat{\beta}-\beta) & =(\partial_{\beta}\mathcal{L})+(\partial_{\beta\phi'}\mathcal{L})\mathcal{H}^{-1}\mathcal{S}+\frac{1}{2}\mathcal{S}'\mathcal{H}^{-1}(\partial_{\beta\phi\phi'}\mathcal{L})\mathcal{H}^{-1}\mathcal{S}\\
& \quad+\frac{1}{2}\mathcal{S}'\mathcal{H}^{-1}\sum_{f}(\partial_{\phi\phi'\phi_{f}}\mathcal{L})\big[\mathcal{H}^{-1}(\partial_{\beta\phi}\mathcal{L})\big]_{f}\mathcal{H}^{-1}\mathcal{S}+o_{p}(1)
\end{align*}
\section{\label{sec:basic_lemmas}Lemmas for bounding individual components}
In order to provide bounds on the components in the asymptotic expansion above, we first demonstrate some basic results that are used frequently. We begin with a result from FW16 (Lemma S.6 in that paper). Through the proofs we apply the theorem with $q=16$ which is justified by Assumption 2.
\begin{lem}
\label{lem:S6} Let Assumptions 1 and
2 hold for the dyadic network M-estimator described in the main paper.
Let $\mathcal{B}(r_{\beta},\beta_{0})$ an\textup{d} $\mathcal{B}_{q}(r_{\phi},\phi_{0})$
be neighborhoods of the true parameter values, with $r_{\beta}=o(N^{-1/q-2\epsilon})$
and $r_{\phi}=o(N^{-2\epsilon})$, and let $4<q\leq16$ and $\epsilon=\frac{1}{2q}$.
The following conditions hold.
(i) for $s\leq3$
\begin{align*}
\frac{1}{N-1}\sum_{i}\sum_{j\ne i}\partial_{\beta}\ell_{ij} & =O_{p}(1),\quad\frac{1}{N-1}\sum_{i}\sum_{j\ne i}\partial_{\beta^{3}}\tilde{\ell}_{ij}=O_{p}(1)\\
\sup_{\beta\in\mathcal{B}(r_{\beta},\beta_{0})}\sup_{\phi\in\mathcal{B}_{q}(r_{\phi},\phi_{0})} & \frac{1}{N(N-1)}\sum_{i}\sum_{j\ne i}\partial_{\beta^{s}}\ell_{ij} = O_p(1)
\end{align*}
(ii) for $s\leq3$
\[
\sup_{\beta\in\mathcal{B}(r_{\beta},\beta_{0})}\sup_{\phi\in\mathcal{B}_{q}(r_{\phi},\phi_{0})}\frac{1}{N}\sum_{i}\lvert\frac{1}{N-1}\sum_{j\ne i}\partial_{\beta^{s}\pi}\ell_{ij}\rvert^{q}=O_{p}(1)
\]
(iii) for $s\leq3$, $t\leq5$ and $s+t\leq6$
\[
\sup_{\beta\in\mathcal{B}(r_{\beta},\beta_{0})}\sup_{\phi\in\mathcal{B}_{q}(r_{\phi},\phi_{0})}\max_{i}\frac{1}{N-1}\sum_{j\ne i}\lvert\partial_{\beta^{s}\pi^{t}}\ell_{ij}\rvert^{q}=O_{p}(N^{2\epsilon})
\]
(iv)
\[
\frac{1}{N}\sum_{i}\lvert\frac{1}{\sqrt{N-1}}\sum_{j\ne i}\partial_{\pi}\tilde{\ell}_{ij}\rvert^{q}=O_{p}(1),\quad\frac{1}{N}\sum_{i}\lvert\frac{1}{\sqrt{N-1}}\sum_{j\ne i}\partial_{\beta\pi}\tilde{\ell}_{ij}\rvert^{2}=O_{p}(1)
\]
(v) for $s\leq3$, $t\leq5$ and $s+t\leq6$
\begin{align*}
\max_{i}\bar{E}\big[\partial_{\beta^{s}\pi^{t}}\tilde{\ell}_{ij}^{q}\big] & \leq C,\quad\max_{i}\bar{E}\Big[\Big(\frac{1}{\sqrt{N-1}}\sum_{j\ne i}\partial_{\beta^{s}\pi^{t}}\tilde{\ell}_{ij}\Big)^{q}\Big]\leq C
\end{align*}
\end{lem}
\subsection{Bounds for $\bar{\mathcal{H}}^{-1}$ and $\overline{W}_{N}^{-1}$}
Here we provide important results related to $\bar{\mathcal{H}}^{-1}$ and $\overline{W}_{N}^{-1}$. The objective function for the estimator is given by
\[
\mathcal{L}_{N}(\beta,\phi_{N})=\frac{1}{N-1}\sum_{i}\sum_{j\ne i}\ell(Z_{ij},\beta,\alpha_{i},\gamma_{j})-\frac{1}{N}b(v_{N}'\phi_{N})^{2}/2
\]
As in FW16 and Dzemski 2019, we may write
\[
\bar{\mathcal{H}}=\begin{bmatrix}\bar{\mathcal{H}}_{\alpha\alpha}^{*} & \bar{\mathcal{H}}_{\alpha\gamma}^{*}\\
\bar{\mathcal{H}}_{\gamma\alpha}^{*} & \bar{\mathcal{H}}_{\gamma\gamma}^{*}
\end{bmatrix}+\frac{1}{N}bv_{N}v_{N}'
\]
where $\bar{\mathcal{H}}_{\alpha\alpha}^{*}$ and $\bar{\mathcal{H}}_{\gamma\gamma}^{*}$
are the diagonal matrices with elements
\begin{align*}
\big(\bar{\mathcal{H}}_{\alpha\alpha}^{*}\big)_{ii} & =-\frac{1}{N-1}\sum_{j\ne i}\bar{E}[\partial_{\pi^{2}}\ell_{ij}]\\
\big(\bar{\mathcal{H}}_{\gamma\gamma}^{*}\big)_{ii} & =-\frac{1}{N-1}\sum_{j\ne i}\bar{E}[\partial_{\pi^{2}}\ell_{ji}]
\end{align*}
and $\bar{\mathcal{H}}_{\alpha\gamma}^{*}=(\bar{\mathcal{H}}_{\gamma\alpha}^{*})'$
has off-diagonal entries $\big(\bar{\mathcal{H}}_{\alpha\gamma}^{*}\big)_{ij}=-\bar{E}[\partial_{\pi^{2}}\ell_{ij}]/(N-1)$
and zeroes in diagonal entries. Following Lemma A.1 in D19 and Lemma
D.1 in FW16, we may prove the following approximation result.
\begin{lem}
\label{lem:H_approx}Under Assumptions 1 and 2
\begin{align*}
\lVert\bar{\mathcal{H}}^{-1}-D^{-1}\rVert_{max} & =O_{p}(N^{-1})\\
\lVert\bar{\mathcal{H}}^{-1}\rVert_{q} & =O_{p}(1)
\end{align*}
where $D=diag\big(\bar{\mathcal{H}}_{\alpha\alpha}^{*},\bar{\mathcal{H}}_{\gamma\gamma}^{*}\big)$.
\end{lem}
The proof of this Lemma is identical to the versions in D19 and FW16.
We next bound some deviations between the sample average values and
conditional expectations of the Hessian matrix.
\begin{lem}
\label{lem:H_tilde}Let Assumptions 1 and 2 hold. Then
\begin{align*}
\lVert\mathcal{H}-\bar{\mathcal{H}}\rVert & =O_{p}(N^{-\frac{1}{2}+2\epsilon})\\
\lVert\mathcal{H}_{b}-\bar{\mathcal{H}}_{b}\rVert & =O_{p}(N^{-\frac{3}{2}+6\epsilon})
\end{align*}
\end{lem}
\begin{proof}
The proof of the first result follows the equivalent result in FW16.
We have
\[
\lVert\mathcal{H}-\bar{\mathcal{H}}\rVert\leq\lVert\partial_{\alpha\alpha}\mathcal{L}-\partial_{\alpha\alpha}\bar{\mathcal{L}}\rVert+2\lVert\partial_{\alpha\gamma}\mathcal{L}-\partial_{\alpha\gamma}\bar{\mathcal{L}}\rVert+\lVert\partial_{\gamma\gamma}\mathcal{L}-\partial_{\gamma\gamma}\bar{\mathcal{L}}\rVert
\]
To bound the first term, note that from Lemma \ref{lem:S6}
\begin{align*}
\bar{E}\lVert\partial_{\alpha\alpha}\mathcal{L}-\partial_{\alpha\alpha}\bar{\mathcal{L}}\rVert^{q} & =\bar{E}\Big[\max_{i}\Big(\frac{1}{N-1}\sum_{j\ne i}\partial_{\pi^{2}}\tilde{\ell}_{ij}\Big)^{q}\Big]\\
& =O_{p}(N^{1-q/2})
\end{align*}
and hence $\lVert\partial_{\alpha\alpha}\mathcal{L}-\partial_{\alpha\alpha}\bar{\mathcal{L}}\rVert=O_{p}(N^{-\frac{1}{2}+\frac{1}{q}})$
and similarly for $\lVert\partial_{\gamma\gamma}\mathcal{L}-\partial_{\gamma\gamma}\bar{\mathcal{L}}\rVert$.
The bound $\lVert\partial_{\alpha\gamma}\mathcal{L}-\partial_{\alpha\gamma}\bar{\mathcal{L}}\rVert=O_{p}(N^{-\frac{1}{2}+\frac{1}{q}})$
follows from Lemma \ref{lem:S6}. This gives $\lVert\mathcal{H}-\bar{\mathcal{H}}\rVert=O_{p}(N^{-\frac{1}{2}+\frac{1}{q}})=O_{p}(N^{-\frac{1}{2}+2\epsilon})$.
For the second result, recall that by definition we have
\[
W\mathcal{H}_{b}=\partial_{\beta\phi\phi'}\mathcal{L}+\sum_{f}(\partial_{\phi\phi'\phi_{f}}\mathcal{L})\big[\mathcal{H}^{-1}(\partial_{\beta\phi}\mathcal{L})\big]_{f}
\]
The bound $\lVert\partial_{\beta\phi\phi'}\tilde{\mathcal{L}}\rVert=O_{p}(N^{-\frac{1}{2}+2\epsilon})$
follows identically to the first result. For the second part
\begin{align*}
\lVert & \sum_{f}(\partial_{\phi\phi'\phi_{f}}\mathcal{L})\big[\mathcal{H}^{-1}(\partial_{\beta\phi}\mathcal{L})\big]_{f}-\sum_{f}(\partial_{\phi\phi'\phi_{f}}\bar{\mathcal{L}})\big[\bar{\mathcal{H}}^{-1}(\partial_{\beta\phi}\mathcal{\bar{L}})\big]_{f}\rVert\\
\leq & \rVert\sum_{f}(\partial_{\phi\phi'\phi_{f}}\tilde{\mathcal{L}})\big[\mathcal{H}^{-1}(\partial_{\beta\phi}\mathcal{L})\big]_{f}\rVert+\rVert\sum_{f}(\partial_{\phi\phi'\phi_{f}}\bar{\mathcal{L}})\big[(\mathcal{H}^{-1}-\bar{\mathcal{H}}^{-1})(\partial_{\beta\phi}\mathcal{L})\big]_{f}\rVert\\
& +\rVert\sum_{f}(\partial_{\phi\phi'\phi_{f}}\bar{\mathcal{L}})\big[\mathcal{\bar{H}}^{-1}(\partial_{\beta\phi}\tilde{\mathcal{L}})\big]_{f}\rVert
\end{align*}
For the first term, we can write
\begin{align*}
\rVert\sum_{f}(\partial_{\phi\phi'\phi_{f}}\tilde{\mathcal{L}})\big[\mathcal{H}^{-1}(\partial_{\beta\phi}\mathcal{L})\big]_{f}\rVert & \leq\rVert\sum_{f}(\partial_{\phi\phi'\phi_{f}}\tilde{\mathcal{L}})\big[\bar{\mathcal{H}}^{-1}(\partial_{\beta\phi}\mathcal{L})\big]_{f}\rVert+\rVert\partial_{\phi\phi\phi}\tilde{\mathcal{L}}\rVert\lVert\mathcal{H}^{-1}-\bar{\mathcal{H}}^{-1}\rVert\lVert\partial_{\beta\phi}\mathcal{L}\rVert\\
& =\rVert\sum_{f}(\partial_{\phi\phi'\phi_{f}}\tilde{\mathcal{L}})\big[\bar{\mathcal{H}}^{-1}(\partial_{\beta\phi}\mathcal{L})\big]_{f}\rVert+O_{p}(N^{-\frac{1}{2}+4\epsilon})
\end{align*}
where the bound on $\rVert\partial_{\phi\phi\phi}\tilde{\mathcal{L}}\rVert$
follows from the same reasoning as the bound on $\rVert\partial_{\phi\phi}\tilde{\mathcal{L}}\rVert$,
applying the result on tensor norms in Lemma S.5 of FW16. Decompose
into four components based on $\phi=(\alpha',\gamma')'$ and consider
the first component, and let $\Gamma_{isjt}=\bar{\mathcal{H}}_{\alpha\alpha,i,s}^{-1}+\bar{\mathcal{H}}_{\alpha\gamma,i,t}^{-1}+\bar{\mathcal{H}}_{\alpha\alpha,j,s}^{-1}+\bar{\mathcal{H}}_{\alpha\gamma,j,t}^{-1}$
\begin{align*}
\rVert\sum_{f}(\partial_{\alpha\alpha'\phi_{f}}\tilde{\mathcal{L}})\big[\bar{\mathcal{H}}^{-1}(\partial_{\beta\phi}\mathcal{L})\big]_{f}\rVert & =\rVert\sum_{f}(\partial_{\alpha\alpha'\phi_{f}}\tilde{\mathcal{L}})\big[\bar{\mathcal{H}}^{-1}(\partial_{\beta\phi}\mathcal{L})\big]_{f}\rVert\\
& =\rVert\frac{1}{N-1}\sum_{i}\sum_{s}\sum_{t\ne s}(\partial_{\alpha\alpha'\alpha_{i}}\tilde{\mathcal{L}})\big[\bar{\mathcal{H}}_{\alpha\alpha,i,s}^{-1}+\bar{\mathcal{H}}_{\alpha\gamma,i,t}^{-1}\big](\partial_{\beta\pi}\ell_{st})\\
& +\frac{1}{N-1}\sum_{i}\sum_{s}\sum_{t\ne s}(\partial_{\alpha\alpha'\gamma_{i}}\tilde{\mathcal{L}})\big[\bar{\mathcal{H}}_{\gamma\alpha,i,s}^{-1}+\bar{\mathcal{H}}_{\gamma\gamma,i,t}^{-1}\big](\partial_{\beta\pi}\ell_{st})\rVert\\
& =\max_{i}\lvert\frac{1}{(N-1)^{2}}\sum_{j\ne i}\sum_{s}\sum_{t\ne s}(\partial_{\pi^{3}}\tilde{\ell}_{ij})\Gamma_{isjt}(\partial_{\beta\pi}\ell_{st})\rvert\\
& \leq\max_{i}\lvert\frac{1}{(N-1)^{2}}\sum_{j\ne i}\sum_{s}\sum_{t\ne s}(\partial_{\pi^{3}}\tilde{\ell}_{ij})\Gamma_{isjt}(\partial_{\beta\pi}\ell_{st})\rvert\\
\end{align*}
Since the max over $(i,s,j,t)$ of $\Gamma_{isjt}=O_{p}(1)$ when
$s=i$, $t=i$, $j=s$, or $j=t$ and is $O_{p}(N^{-1})$ otherwise,
the above sum is $O_{p}(N^{-\frac{1}{2}+2\epsilon})$ using the bounds
in Lemma \ref{lem:S6}. The same bound can be shown for remaining
components of $\rVert\sum_{f}(\partial_{\phi\phi'\phi_{f}}\tilde{\mathcal{L}})\big[\bar{\mathcal{H}}^{-1}(\partial_{\beta\phi}\mathcal{L})\big]_{f}\rVert$.
Next, from Lemmas \ref{lem:tensor_bounds} and \ref{lem:H1_approx}
\begin{align*}
\rVert\sum_{f}(\partial_{\phi\phi'\phi_{f}}\bar{\mathcal{L}})\big[(\mathcal{H}^{-1}-\bar{\mathcal{H}}^{-1})(\partial_{\beta\phi}\mathcal{L})\big]_{f}\rVert & \leq\rVert\sum_{f}(\partial_{\phi\phi'\phi_{f}}\bar{\mathcal{L}})\big[\bar{\mathcal{H}}^{-1}\tilde{\mathcal{H}}\bar{\mathcal{H}}^{-1}(\partial_{\beta\phi}\mathcal{L})\big]_{f}\rVert\\
& +\rVert\sum_{f}(\partial_{\phi\phi'\phi_{f}}\bar{\mathcal{L}})\big[(\mathcal{H}^{-1}-\bar{\mathcal{H}}^{-1}-\bar{\mathcal{H}}^{-1}\tilde{\mathcal{H}}\bar{\mathcal{H}}^{-1})(\partial_{\beta\phi}\mathcal{L})\big]_{f}\rVert\\
& \leq O_{p}(N^{-\frac{1}{2}+4\epsilon})+\rVert\partial_{\phi\phi\phi}\bar{\mathcal{L}}\rVert\lVert\mathcal{H}^{-1}-\bar{\mathcal{H}}^{-1}-\bar{\mathcal{H}}^{-1}\tilde{\mathcal{H}}\bar{\mathcal{H}}^{-1}\rVert\lVert\partial_{\beta\phi}\mathcal{L}\rVert\\
& =O_{p}(N^{-\frac{1}{2}+6\epsilon})
\end{align*}
and $\rVert\sum_{f}(\partial_{\phi\phi'\phi_{f}}\bar{\mathcal{L}})\big[\mathcal{\bar{H}}^{-1}(\partial_{\beta\phi}\tilde{\mathcal{L}})\big]_{f}\rVert=O_{p}(N^{-1/2+2\epsilon})$
as in Lemma \ref{lem:tensor_bounds}. Since $\lVert W\rVert=O_{p}(N^{-1})$
and $\lVert W_{N}-\overline{W}_{N}\rVert=O_{p}(N^{-1/2+2\epsilon})$, this
gives the result.
\end{proof}
In the proofs below we will frequently replace $\mathcal{H}^{-1}$
and $W^{-1}$ by the following approximations.
\begin{lem}
\label{lem:H1_approx}Let Assumptions 1 and 2 hold. Then, for $k=0,1,2,3$
\begin{align*}
\lVert\mathcal{H}^{-1}-\sum_{j=0}^{k}(-1)^{j}(\bar{\mathcal{H}}^{-1}\tilde{\mathcal{H}})^{j}\bar{\mathcal{H}}^{-1}\rVert & =O_{p}(N^{-\frac{k+1}{2}+2(k+1)\epsilon})\\
W_N - \overline{W}_{N} &=O_{p}(N^{-\frac{1}{2}+2\epsilon}) \\
\lVert W_{N}^{-1}-\sum_{j=0}^{k}(-1)^{j}(\overline{W}_N^{-1}\tilde{W}_N)^{j}\overline{W}_N^{-1}\rVert & =O_{p}(N^{-\frac{k+1}{2}+2(k+1)\epsilon})
\end{align*}
\end{lem}
\begin{proof}
The approximations are based on the Neumann series expansion. Since
$\lVert\tilde{\mathcal{H}}\rVert=O_{p}(N^{-\frac{1}{2}+2\epsilon})=o_{p}(1)$,
the series converges with high probability for sufficiently large
$N$, and so we may write
\begin{align*}
\mathcal{H}^{-1} & =\sum_{j=0}^{\infty}(-1)^{j}(\bar{\mathcal{H}}^{-1}\tilde{\mathcal{H}})^{j}\bar{\mathcal{H}}^{-1}\\
& =\sum_{j=0}^{k}(-1)^{j}(\bar{\mathcal{H}}^{-1}\tilde{\mathcal{H}})^{j}\bar{\mathcal{H}}^{-1}+\sum_{j=k+1}^{\infty}(-1)^{j}(\bar{\mathcal{H}}^{-1}\tilde{\mathcal{H}})^{j}\bar{\mathcal{H}}^{-1}\\
& =\mathbf{H}_{k}+\zeta_{k+1}
\end{align*}
where $\mathbf{H}_{k}$ is the approximation for $\mathcal{H}^{-1}$
up to $k+1$ terms, and the approximation error satisfies
\begin{align*}
\lVert\zeta_{k+1}\rVert & =\lVert\sum_{j=k+1}^{\infty}(-1)^{j}(\bar{\mathcal{H}}^{-1}\tilde{\mathcal{H}})^{j}\bar{\mathcal{H}}^{-1}\rVert\\
& \leq\lVert(\bar{\mathcal{H}}^{-1}\tilde{\mathcal{H}})^{k+1}\rVert\lVert\sum_{j=0}^{\infty}(-1)^{j}(\bar{\mathcal{H}}^{-1}\tilde{\mathcal{H}})^{j}\bar{\mathcal{H}}^{-1}\rVert\\
& \leq\lVert\tilde{\mathcal{H}}\rVert^{k+1}\lVert\mathcal{\bar{H}}^{-1}\rVert^{k+1}\lVert\mathcal{H}^{-1}\rVert\\
& =O_{p}(N^{-\frac{k+1}{2}-2(k+1)\epsilon})
\end{align*}
For $W$, define $\overline{W}=\partial_{\beta\beta}\bar{\mathcal{L}}+\frac{1}{N}(\partial_{\beta\phi'}\bar{\mathcal{L}})\bar{\mathcal{H}}^{-1}(\partial_{\beta\phi}\bar{\mathcal{L}})$
and
\[
\tilde{W}_{N}=\frac{1}{N}\partial_{\beta\beta}\mathcal{\tilde{L}}+\frac{1}{N}\big((\partial_{\beta\phi'}\mathcal{L})\mathcal{H}^{-1}(\partial_{\beta\phi}\mathcal{L})-(\partial_{\beta\phi'}\bar{\mathcal{L}})\bar{\mathcal{H}}^{-1}(\partial_{\beta\phi}\bar{\mathcal{L}})\big)
\]
The first term in the approximation error $\frac{1}{N}\partial_{\beta\beta}\mathcal{\tilde{L}}=O_{p}(N^{-1})$
by Lemma \ref{lem:S6}. For the remaining term, we can decompose it
as
\begin{align*}
\frac{1}{N}(\partial_{\beta\phi'}\mathcal{L}) & \mathcal{H}^{-1}(\partial_{\beta\phi}\mathcal{L})-\frac{1}{N}(\partial_{\beta\phi'}\bar{\mathcal{L}})\bar{\mathcal{H}}^{-1}(\partial_{\beta\phi}\bar{\mathcal{L}})\\
& =\frac{1}{N}(\partial_{\beta\phi'}\tilde{\mathcal{L}})\bar{\mathcal{H}}^{-1}(\partial_{\beta\phi}\bar{\mathcal{L}})+\frac{1}{N}(\partial_{\beta\phi'}\bar{\mathcal{L}})\bar{\mathcal{H}}^{-1}(\partial_{\beta\phi}\tilde{\mathcal{L}})\\
& +\frac{1}{N}(\partial_{\beta\phi'}\tilde{\mathcal{L}})\bar{\mathcal{H}}^{-1}(\partial_{\beta\phi}\tilde{\mathcal{L}})+\frac{1}{N}(\partial_{\beta\phi'}\mathcal{L})\big(\mathcal{H}^{-1}-\bar{\mathcal{H}}^{-1}\big)(\partial_{\beta\phi}\mathcal{L})
\end{align*}
From Assumption B.1 of FW16, we have $\frac{1}{N}\lVert(\partial_{\beta\phi'}\tilde{\mathcal{L}})\bar{\mathcal{H}}^{-1}(\partial_{\beta\phi}\bar{\mathcal{L}})\rVert\leq O_{p}(N^{-1/2})$
and also that $\frac{1}{N}\lVert(\partial_{\beta\phi'}\tilde{\mathcal{L}})\bar{\mathcal{H}}^{-1}(\partial_{\beta\phi}\tilde{\mathcal{L}})\rVert\leq O_{p}(N^{-1})$.
Also
\begin{align*}
\frac{1}{N}\lVert(\partial_{\beta\phi'}\mathcal{L})\big(\mathcal{H}^{-1}-\bar{\mathcal{H}}^{-1}\big)(\partial_{\beta\phi}\mathcal{L})\rVert & \leq\frac{1}{N}\lVert\partial_{\beta\phi'}\mathcal{L}\rVert^{2}\lVert\mathcal{H}^{-1}-\bar{\mathcal{H}}^{-1}\rVert\\
& =O_{p}(N^{-\frac{1}{2}+2\epsilon})
\end{align*}
So we may write $\tilde{W}_{N}=O_{p}(N^{-\frac{1}{2}+2\epsilon})$,
and noting that $\overline{W}_{N}>0$ by Assumption 2, the approximation follows as for $\mathcal{H}^{-1}$.
\end{proof}
\begin{lem}
\label{lem:W_Op1}Let
\[
W_{N}=\frac{1}{N}W=\frac{1}{N}\partial_{\beta\beta}\mathcal{L}+\frac{1}{N}(\partial_{\beta\phi'}\mathcal{L})\mathcal{H}^{-1}(\partial_{\beta\phi}\mathcal{L})
\]
then, under Assumptions 1 and 2
\[
\lVert W_{N}\rVert=O_{p}(1)
\]
\end{lem}
\begin{proof}
Using the result in Lemma \ref{lem:H_tilde}, we have
\[
W_{N}=\frac{1}{N}W=\frac{1}{N}\partial_{\beta\beta}\mathcal{L}+\frac{1}{N}(\partial_{\beta\phi'}\mathcal{L})\bar{\mathcal{H}}^{-1}(\partial_{\beta\phi}\mathcal{L})
\]
The first term is $O_{p}(1)$, while the second term can be decomposed
into four parts
\begin{align*}
(\partial_{\beta\phi'}\mathcal{L})\mathcal{H}^{-1}(\partial_{\beta\phi}\mathcal{L}) & =(\partial_{\beta\alpha'}\mathcal{L})\mathcal{H}_{\alpha\alpha}^{-1}(\partial_{\beta\alpha}\mathcal{L})+(\partial_{\beta\alpha'}\mathcal{L})\mathcal{H}_{\alpha\gamma}^{-1}(\partial_{\beta\gamma}\mathcal{L})\\
& +(\partial_{\beta\gamma'}\mathcal{L})\mathcal{H}_{\gamma\alpha}^{-1}(\partial_{\beta\alpha}\mathcal{L})+(\partial_{\beta\gamma'}\mathcal{L})\mathcal{H}_{\gamma\gamma}^{-1}(\partial_{\beta\gamma}\mathcal{L})
\end{align*}
The first part is
\begin{align*}
\frac{1}{N}(\partial_{\beta\alpha'}\mathcal{L})\mathcal{H}_{\alpha\alpha}^{-1}(\partial_{\beta\alpha}\mathcal{L}) & =\frac{1}{N(N-1)^{2}}\sum_{i,s}(\mathcal{H}_{\alpha\alpha}^{-1})_{is}\sum_{j\ne i}\sum_{t\ne s}(\partial_{\beta\pi}\ell_{ij})(\partial_{\beta\pi}\ell_{st})\\
& =\frac{1}{N(N-1)^{2}}\sum_{i}(\mathcal{H}_{\alpha\alpha}^{-1})_{ii}\sum_{j\ne i}\sum_{t\ne i}(\partial_{\beta\pi}\ell_{ij})(\partial_{\beta\pi}\ell_{it})\\
& +\frac{1}{N^{2}(N-1)^{2}}\sum_{i}\sum_{s\ne i}N(\mathcal{H}_{\alpha\alpha}^{-1})_{is}\sum_{j\ne i}\sum_{t\ne s}(\partial_{\beta\pi}\ell_{ij})(\partial_{\beta\pi}\ell_{st})\\
& =O_{p}(1)
\end{align*}
where we use Lemma \ref{lem:H_approx}. Similar derivations for the
other parts give the result.
\end{proof}
\subsection{Products of matrices}
The next result demonstrates that the properties of $\mathcal{\bar{H}}^{-1}$ (a dominant diagonal and smaller off-diagonal terms) transfer to certain products of this matrix.
\begin{lem}
\label{lem:HAH_bound}Let Assumptions 1 and 2 hold, and define $A=\mathcal{\bar{H}}^{-1}M\mathcal{\bar{H}}^{-1}$,
where $M$ is a matrix of the form
\[
M=\begin{bmatrix}M_{\alpha\alpha} & M_{\alpha\gamma}\\
M_{\gamma\alpha} & M_{\gamma\gamma}
\end{bmatrix}
\]
with each block satisfying $\max_{i}\sum_{j}\vert M_{\alpha\alpha,ij}\vert=O_{p}(1)$.
Then $A_{ii}=O_{p}(1)$, and $\max_{i,j\ne i}\vert A_{ij}\vert=O_{p}(N^{-1})$.
\end{lem}
\begin{proof}
For $i\leq N$, we can write
\begin{align*}
\big[\mathcal{\bar{H}}^{-1}M\mathcal{\bar{H}}^{-1}\big]_{ij} & =(\bar{\mathcal{H}}_{\alpha\alpha}^{-1})_{i,\cdot}M_{\alpha\alpha'}(\bar{\mathcal{H}}_{\alpha\alpha}^{-1})_{\cdot,j}+(\bar{\mathcal{H}}_{\alpha\gamma}^{-1})_{i,\cdot}M_{\gamma\alpha'}(\bar{\mathcal{H}}_{\alpha\alpha}^{-1})_{\cdot,j}\\
& \quad+(\bar{\mathcal{H}}_{\alpha\alpha}^{-1})_{i,\cdot}M_{\alpha\gamma}(\bar{\mathcal{H}}_{\gamma\alpha}^{-1})_{\cdot,j}+(\bar{\mathcal{H}}_{\alpha\gamma}^{-1})_{i,\cdot}M_{\gamma\gamma'}(\bar{\mathcal{H}}_{\gamma\alpha}^{-1})_{\cdot,j}
\end{align*}
For the first component, we have that
\begin{align*}
\max_{i,j\ne i}\lvert(\bar{\mathcal{H}}_{\alpha\alpha}^{-1})_{i,\cdot}M_{\alpha\alpha'}(\bar{\mathcal{H}}_{\alpha\alpha}^{-1})_{\cdot,j}\rvert & =\max_{i,j\ne i}\lvert\sum_{s,t}(\bar{\mathcal{H}}_{\alpha\alpha}^{-1})_{i,s}(\bar{\mathcal{H}}_{\alpha\alpha}^{-1})_{t,j}M_{\alpha\alpha,st}\rvert\\
& \leq\max_{i,s\ne i}\vert(\bar{\mathcal{H}}_{\alpha\alpha}^{-1})_{i,s}\vert\max_{j,t\ne j}\vert(\bar{\mathcal{H}}_{\alpha\alpha}^{-1})_{t,j}\vert\max_{i,j}\sum_{s\ne i}\sum_{t\ne j}\vert M_{\alpha\alpha,st}\rvert\\
& +\max_{i,s\ne i}\vert(\bar{\mathcal{H}}_{\alpha\alpha}^{-1})_{i,s}\vert\max_{j}\vert(\bar{\mathcal{H}}_{\alpha\alpha}^{-1})_{j,j}\vert\max_{i,j}\sum_{s\ne i}\vert M_{\alpha\alpha,sj}\rvert\\
& +\max_{i}\vert(\bar{\mathcal{H}}_{\alpha\alpha}^{-1})_{i,i}\vert\max_{t,t\ne j}\vert(\bar{\mathcal{H}}_{\alpha\alpha}^{-1})_{t,j}\vert\max_{i,j}\sum_{t\ne j}\vert M_{\alpha\alpha,it}\rvert\\
& +\max_{i}\vert(\bar{\mathcal{H}}_{\alpha\alpha}^{-1})_{i,i}\vert^{2}\max_{i,j}\vert M_{\alpha\alpha,ij}\vert\\
& =1\{i=j\}\times O_{p}(1)+O_{p}(N^{-1})
\end{align*}
where the final line follows from the condition $\max_{i}\sum_{j}\vert M_{\alpha\alpha,ij}\vert=O_{p}(1)$,
$\max_{i}(\bar{\mathcal{H}}_{\alpha\alpha}^{-1})_{ii}=O_{p}(1)$,
and $\max_{i,j\ne i}(\bar{\mathcal{H}}_{\alpha\alpha}^{-1})_{ij}=O_{p}(N^{-1})$.
The next component is
\begin{align*}
\max_{i,j\ne i}\lvert(\bar{\mathcal{H}}_{\alpha\alpha}^{-1})_{i,\cdot}M_{\alpha\gamma'}(\bar{\mathcal{H}}_{\gamma\alpha}^{-1})_{\cdot,j}\rvert & =\max_{i,j\ne i}\lvert\sum_{s,t}(\bar{\mathcal{H}}_{\alpha\alpha}^{-1})_{i,s}(\bar{\mathcal{H}}_{\gamma\alpha}^{-1})_{t,j}M_{\alpha\gamma,st}\rvert\\
& \leq\max_{i,s\ne i}\vert(\bar{\mathcal{H}}_{\alpha\alpha}^{-1})_{i,s}\vert\max_{j,t}\vert(\bar{\mathcal{H}}_{\gamma\alpha}^{-1})_{t,j}\vert\max_{i,j}\sum_{s\ne i}\sum_{t}\vert M_{\alpha\gamma,st}\rvert\\
& +\max_{i}\vert(\bar{\mathcal{H}}_{\alpha\alpha}^{-1})_{i,i}\vert\max_{j,t}\vert(\bar{\mathcal{H}}_{\gamma\alpha}^{-1})_{t,j}\vert\max_{i,j}\sum_{t\ne j}\vert M_{\alpha\gamma,it}\rvert\\
& =O_{p}(N^{-1})
\end{align*}
where we use $\max_{i,j}(\bar{\mathcal{H}}_{\gamma\alpha}^{-1})_{ij}=O_{p}(N^{-1}).$
Similar logic applies to the remaining components of $A$.
\end{proof}
\begin{lem}
\label{lem:HEH_bound}Let Assumptions 1 and 2 hold, and define as
either $M=\mathcal{\bar{H}}^{-1}\bar{\mathcal{E}}\mathcal{\bar{H}}^{-1}$,
or $M=W\mathcal{\bar{H}}^{-1}\bar{\mathcal{H}}_{b}\mathcal{\bar{H}}^{-1}$.
Then $M_{ii}=O_{p}(1)$, and $\max_{i,j\ne i}\vert M_{ij}\vert=O_{p}(N^{-1})$.
\end{lem}
\begin{proof}
For the first result, it remains only to show that $\bar{\mathcal{E}}$
satisfies the conditions of Lemma \ref{lem:HAH_bound}. This follows
directly from the form of $\partial_{\beta\phi\phi'}\bar{\mathcal{L}}$
and Assumption 2, which implies $\vert\partial_{\beta\pi^{2}}\bar{\ell}_{ij}\vert\leq C$
for all $i$ and $j$. For $W\bar{\mathcal{H}}_{b}$, we have shown
the result holds for $\partial_{\beta\phi\phi'}\bar{\mathcal{L}}=\bar{\mathcal{E}}$
and so it remains to demonstrate the same for the term $\sum_{f}(\partial_{\phi\phi'\phi_{f}}\bar{\mathcal{L}}_{ij})\big[\bar{\mathcal{H}}^{-1}(\partial_{\beta\phi}\bar{\mathcal{L}})\big]_{f}$.
We have
\begin{align*}
\max_{i}&\sum_{j}\vert\sum_{f}(\partial_{\alpha\alpha'\phi_{f}}\bar{\mathcal{L}}_{ij})\big[\bar{\mathcal{H}}^{-1}(\partial_{\beta\phi}\bar{\mathcal{L}})\big]_{f}\vert \\
& \leq\max_{i}\frac{1}{N-1}\vert\sum_{j\ne i}(\partial_{\pi^{3}}\bar{\ell}_{ij})\big[\bar{\mathcal{H}}^{-1}(\partial_{\beta\phi}\bar{\mathcal{L}})\big]_{i}\vert\\
& +\max_{i}\frac{1}{N-1}\vert\sum_{j\ne i}(\partial_{\pi^{3}}\bar{\ell}_{ij})\big[\bar{\mathcal{H}}^{-1}(\partial_{\beta\phi}\bar{\mathcal{L}})\big]_{j}\vert\\
& =\max_{i}\frac{1}{(N-1)^{2}}\vert\sum_{s}\sum_{t\ne s}\sum_{j\ne i}(\partial_{\pi^{3}}\bar{\ell}_{ij})\big([\bar{\mathcal{H}}_{\alpha\alpha,is}^{-1}](\partial_{\beta\pi}\bar{\ell}_{st})+[\bar{\mathcal{H}}_{\alpha\gamma,is}^{-1}](\partial_{\beta\pi}\bar{\ell}_{ts})\big)\vert\\
& +\max_{i}\frac{1}{(N-1)^{2}}\vert\sum_{s}\sum_{t\ne s}\sum_{j\ne i}(\partial_{\pi^{3}}\bar{\ell}_{ij})\big([\bar{\mathcal{H}}_{\gamma\alpha,js}^{-1}](\partial_{\beta\pi}\bar{\ell}_{st})+[\bar{\mathcal{H}}_{\gamma\gamma,js}^{-1}](\partial_{\beta\pi}\bar{\ell}_{ts})\big)\vert\\
& =\max_{i}\frac{1}{(N-1)^{2}}\vert\sum_{s}\sum_{t\ne s}\sum_{j\ne i}(\partial_{\pi^{3}}\bar{\ell}_{ij})\Gamma_{isjt}(\partial_{\beta\pi}\bar{\ell}_{st})\vert
\end{align*}
Then, using the fact that $\Gamma_{isjt}=O_{p}(1)$ only when $i=s$,
$i=t$, $j=s$ or $j=t$ and $\vert\partial_{\beta\pi}\bar{\ell}_{st}\vert\leq C$
the term is clearly $O_{p}(1)$. The same steps can be used for the
$\partial_{\alpha\gamma'\phi_{f}}\bar{\mathcal{L}}$, $\partial_{\gamma\alpha'\phi_{f}}\bar{\mathcal{L}}$,
and $\partial_{\gamma\gamma'\phi_{f}}\bar{\mathcal{L}}$ components,
and hence $W\bar{\mathcal{H}}_{b}$ satisfies the required condition.
\end{proof}
\subsection{Further terms}
The remaing parts of this section provide bounds on a series of terms that appear in the expansion. These bounds are used in later sections to bound the expansion terrms in Section \ref{SA:expansion}.
\begin{lem}
\label{lem:element1}Let Assumptions 1 and 2 hold. Then, for $s+t \leq 5$\\ \\
(i) $\lVert(\partial_{\beta^{s}\phi'}\mathcal{L})\mathcal{H}^{-1}\mathcal{S}\rVert =O_{p}(1)$ \\\
(ii) $ \lVert(\partial_{\beta^{s}\phi'}\mathcal{L})\mathcal{H}^{-1}\mathcal{E}\mathcal{H}^{-1}\mathcal{S}\rVert =O_{p}(1)$\\
(iii) $ \lVert(\partial_{\beta^{s}\phi'}\mathcal{L})\mathcal{H}^{-1}(\partial_{\beta^{t}\phi}\mathcal{L})\rVert =O_{p}(N)$\\
(iv) $\lVert(\partial_{\beta^{s}\phi'}\mathcal{L})\mathcal{H}^{-1}\mathcal{E}\mathcal{H}^{-1}(\partial_{\beta^{t}\phi}\mathcal{L})\rVert =O_{p}(N)$
\end{lem}
\begin{proof}
Write
\begin{align*}
\lVert\mathcal{S}'\mathcal{H}^{-1}(\partial_{\beta^{s}\phi}\mathcal{L})\rVert & \leq\lVert\mathcal{S}'\mathcal{\bar{H}}^{-1}(\partial_{\beta^{s}\phi}\mathcal{L})\rVert+\lVert\mathcal{S}'(\mathcal{H}^{-1}-\mathcal{\bar{H}}^{-1})(\partial_{\beta^{s}\phi}\mathcal{L})\rVert\\
& \leq\lVert\mathcal{S}'\mathcal{\bar{H}}^{-1}(\partial_{\beta^{s}\phi}\mathcal{L})\rVert+o_{p}(1)
\end{align*}
Decomposing this term gives
\begin{align*}
\lVert\mathcal{S}'\bar{\mathcal{H}}^{-1}(\partial_{\beta^{s}\phi}\mathcal{L})\rVert & =\lVert(\partial_{\alpha'}\mathcal{L})\bar{\mathcal{H}}_{\alpha\alpha}^{-1}(\partial_{\beta^{s}\alpha}\mathcal{L})\rVert+\lVert(\partial_{\alpha}\mathcal{L})\bar{\mathcal{H}}_{\alpha\gamma}^{-1}(\partial_{\beta^{s}\gamma}\mathcal{L})\rVert\\
& \quad\lVert(\partial_{\gamma'}\mathcal{L})\bar{\mathcal{H}}_{\gamma\alpha}^{-1}(\partial_{\beta^{s}\alpha}\mathcal{L})\rVert+\lVert(\partial_{\gamma'}\mathcal{L})\bar{\mathcal{H}}_{\gamma\gamma}^{-1}(\partial_{\beta^{s}\gamma}\mathcal{L})\rVert
\end{align*}
the first term of which has second moment
\begin{align*}
&\bar{E}\Big[\lVert(\partial_{\alpha'}\mathcal{L})\bar{\mathcal{H}}_{\alpha\alpha}^{-1}(\partial_{\beta^{s}\alpha}\mathcal{L})\rVert^{2}\Big] \\
& =\frac{1}{(N-1)^{4}}\sum_{i,i'}\sum_{j,j'}\sum_{s\ne i}\sum_{s'\ne i'}\sum_{t\ne j}\sum_{t'\ne j'}(\bar{\mathcal{H}}_{\alpha\alpha}^{-1})_{st}(\bar{\mathcal{H}}_{\alpha\alpha}^{-1})_{s't'}\bar{E}\big[(\partial_{\pi}\ell_{is})(\partial_{\beta^{s}\pi}\ell_{jt})(\partial_{\pi}\ell_{i's'})(\partial_{\beta^{s}\pi}\ell_{j't'})\big]\\
& =\frac{1}{(N-1)^{4}}\sum_{i,i'}\sum_{j,j'}\sum_{s\ne\{i,j\}}\sum_{s'\ne\{i',j'\}}(\bar{\mathcal{H}}_{\alpha\alpha}^{-1})_{ss}(\bar{\mathcal{H}}_{\alpha\alpha}^{-1})_{s's'}\bar{E}\big[(\partial_{\pi}\ell_{is})(\partial_{\beta^{s}\pi}\ell_{js})(\partial_{\pi}\ell_{i's'})(\partial_{\beta^{s}\pi}\ell_{j's'})\big]\\
& +\frac{2}{(N-1)^{4}}\sum_{i,i'}\sum_{j,j'}\sum_{s\ne\{i,t\}}\sum_{s'\ne\{i',j'\}}\sum_{t\ne j}(\bar{\mathcal{H}}_{\alpha\alpha}^{-1})_{st}(\bar{\mathcal{H}}_{\alpha\alpha}^{-1})_{s's'}\bar{E}\big[(\partial_{\pi}\ell_{is})(\partial_{\beta^{s}\pi}\ell_{jt})(\partial_{\pi}\ell_{i's'})(\partial_{\beta^{s}\pi}\ell_{j's'})\big]\\
& +\frac{2}{(N-1)^{4}}\sum_{i,i'}\sum_{j,j'}\sum_{s\ne\{i,t\}}\sum_{s'\ne\{i',t'\}}\sum_{t\ne j}\sum_{t'\ne j'}(\bar{\mathcal{H}}_{\alpha\alpha}^{-1})_{st}(\bar{\mathcal{H}}_{\alpha\alpha}^{-1})_{s't'}\bar{E}\big[(\partial_{\pi}\ell_{is})(\partial_{\beta^{s}\pi}\ell_{jt})(\partial_{\pi}\ell_{i's'})(\partial_{\beta^{s}\pi}\ell_{s't'})\big]
\end{align*}
Consider the first term above. Since $\bar{E}[\partial_{\pi}\ell_{is}]=0$,
we must have either: $(i',s')\in\{(i,s),(s,i)\}$, or $(i,s)\in\{(j,s),(s,j)\}$
and $(i',s')\in\{(j',s'),(s',j')\}$. In either case, there are at
most four unique subscripts in the summation, so that the sum is $O_{p}(1)$.
A similar argument applied to the remaining terms, combined with the
fact that $(\bar{\mathcal{H}}_{\alpha\alpha}^{-1})_{ss}=O_{p}(1)$
while $(\bar{\mathcal{H}}_{\alpha\alpha}^{-1})_{st}=O_{p}(N^{-1})$
for $s\ne t$, gives $O_{p}(1)$ for the whole term.
The second result is identical to the first, using Lemma \ref{lem:HEH_bound}
and $\lVert\tilde{\mathcal{E}}\rVert=o_{p}(1)$. For the third result,
we have $\mathcal{F}^{s,t}=(\partial_{\beta^{s}\phi'}\mathcal{L})\mathcal{H}^{-1}(\partial_{\beta^{t}\phi}\mathcal{L})$.
For $s=t=1$ we have
\begin{align*}
\lVert(\partial_{\beta^{s}\phi'}\mathcal{L})\mathcal{H}^{-1}(\partial_{\beta^{t}\phi}\mathcal{L})\rVert & \leq\lVert(\partial_{\beta^{s}\phi'}\mathcal{L})\bar{\mathcal{H}}^{-1}(\partial_{\beta^{t}\phi}\mathcal{L})\rVert+\lVert(\partial_{\beta^{s}\phi'}\mathcal{L})\big(\mathcal{H}^{-1}-\bar{\mathcal{H}}^{-1}\big)(\partial_{\beta^{t}\phi}\mathcal{L})\rVert\\
& \leq\lVert(\partial_{\beta^{s}\phi'}\mathcal{L})\bar{\mathcal{H}}^{-1}(\partial_{\beta^{t}\phi}\mathcal{L})\rVert+\lVert\partial_{\beta^{s}\phi'}\mathcal{L}\rVert\lVert\mathcal{H}^{-1}-\bar{\mathcal{H}}^{-1}\rVert\lVert\partial_{\beta^{t}\phi}\mathcal{L}\rVert\\
& \leq\lVert(\partial_{\beta^{s}\phi'}\mathcal{L})\bar{\mathcal{H}}^{-1}(\partial_{\beta^{t}\phi}\mathcal{L})\rVert+O_{p}(N^{-1/2+3/q})
\end{align*}
from lemma \ref{lem:S6} (iii) and lemma \ref{lem:H_tilde}. We can decompose the first term as
\begin{align*}
\lVert(\partial_{\beta\phi'}\mathcal{L})\bar{\mathcal{H}}^{-1}(\partial_{\beta\phi}\mathcal{L})\rVert & =\lVert(\partial_{\beta\alpha'}\mathcal{L})\bar{\mathcal{H}}_{\alpha\alpha}^{-1}(\partial_{\beta\alpha}\mathcal{L})\rVert+\lVert(\partial_{\beta\alpha}\mathcal{L})\bar{\mathcal{H}}_{\alpha\gamma}^{-1}(\partial_{\beta\gamma}\mathcal{L})\rVert\\
& \quad\lVert(\partial_{\beta\gamma'}\mathcal{L})\bar{\mathcal{H}}_{\gamma\alpha}^{-1}(\partial_{\beta\alpha}\mathcal{L})\rVert+\lVert(\partial_{\beta\gamma'}\mathcal{L})\bar{\mathcal{H}}_{\gamma\gamma}^{-1}(\partial_{\beta\gamma}\mathcal{L})\rVert
\end{align*}
the first term of which is
\begin{align*}
&\bar{E}\Big[\lVert\frac{1}{N}(\partial_{\beta\alpha'}\mathcal{L})\bar{\mathcal{H}}_{\alpha\alpha}^{-1}(\partial_{\beta\alpha}\mathcal{L})\rVert^{2}\Big] \\
& =\frac{1}{N^{2}(N-1)^{4}}\sum_{i,i'}\sum_{j,j'}\sum_{s\ne i}\sum_{s'\ne i'}\sum_{t\ne j}\sum_{t'\ne j'}(\bar{\mathcal{H}}_{\alpha\alpha}^{-1})_{st}(\bar{\mathcal{H}}_{\alpha\alpha}^{-1})_{s't'}\bar{E}\big[(\partial_{\beta\pi}\ell_{is})(\partial_{\beta\pi}\ell_{jt})(\partial_{\beta\pi}\ell_{i's'})(\partial_{\beta\pi}\ell_{j't'})\big]\\
& =\frac{1}{N^{2}(N-1)^{4}}\sum_{i,i'}\sum_{j,j'}\sum_{s\ne\{i,j\}}\sum_{s'\ne\{i',j'\}}(\bar{\mathcal{H}}_{\alpha\alpha}^{-1})_{ss}(\bar{\mathcal{H}}_{\alpha\alpha}^{-1})_{s's'}\bar{E}\big[(\partial_{\beta\pi}\ell_{is})(\partial_{\beta\pi}\ell_{js})(\partial_{\beta\pi}\ell_{i's'})(\partial_{\beta\pi}\ell_{j's'})\big]\\
& +\frac{2}{N^{2}(N-1)^{4}}\sum_{i,i'}\sum_{j,j'}\sum_{s\ne\{i,t\}}\sum_{s'\ne\{i',j'\}}\sum_{t\ne j}(\bar{\mathcal{H}}_{\alpha\alpha}^{-1})_{st}(\bar{\mathcal{H}}_{\alpha\alpha}^{-1})_{s's'}\bar{E}\big[(\partial_{\beta\pi}\ell_{is})(\partial_{\beta\pi}\ell_{jt})(\partial_{\beta\pi}\ell_{i's'})(\partial_{\beta\pi}\ell_{s's'})\big]\\
& +\frac{2}{N^{2}(N-1)^{4}}\sum_{i,i'}\sum_{j,j'}\sum_{s\ne\{i,t\}}\sum_{s'\ne\{i',t'\}}\sum_{t\ne j}\sum_{t'\ne j'}(\bar{\mathcal{H}}_{\alpha\alpha}^{-1})_{st}(\bar{\mathcal{H}}_{\alpha\alpha}^{-1})_{s't'}\bar{E}\big[(\partial_{\beta\pi}\ell_{is})(\partial_{\beta\pi}\ell_{jt})(\partial_{\beta\pi}\ell_{i's'})(\partial_{\beta\pi}\ell_{s't'})\big]\\
& =O_{p}(1)
\end{align*}
where the final line follows from the fact that $(\bar{\mathcal{H}}_{\alpha\alpha}^{-1})_{ss}=O_{p}(1)$
while $(\bar{\mathcal{H}}_{\alpha\alpha}^{-1})_{st}=O_{p}(N^{-1})$
for $s\ne t$. The remaining terms are $O_{p}(N)$ by similar reasoning.
The final result can be shown in the same way as the third result
using Lemma \ref{lem:HEH_bound}.
\end{proof}
\begin{lem}
\label{lem:tensor_bounds}Let Assumptions 1 and 2 hold. For $r+s\leq3$
and for $t_{1}=\{0,1\},t_{2}=\{1,2\}$\\ \\
(i) $\lVert\sum_{f}(\partial_{\beta^{r}\phi\phi'\phi_{f}}\mathcal{L})\big[\mathcal{H}^{-1}(\partial_{\beta^{s}\phi}\mathcal{L})\big]_{f}\rVert =O_{p}(N^{4\epsilon})$\\
(ii) $\lVert\sum_{f,g}(\partial_{\beta^{t_{1}}\phi\phi'\phi_{f}\phi_{g}}\mathcal{L})\big[\mathcal{H}^{-1}(\partial_{\beta\phi}\mathcal{L})\big]_{f}\big[\mathcal{H}^{-1}(\partial_{\beta^{t_{2}}\phi}\mathcal{L})\big]_{g}\rVert =O_{p}(N^{6\epsilon})$\\
(iii) $\lVert\sum_{e,f,g}(\partial_{\phi\phi'\phi_{f}\phi_{g}}\mathcal{L})\big[\mathcal{H}^{-1}(\partial_{\beta\phi}\mathcal{L})\big]_{e}\big[\mathcal{H}^{-1}(\partial_{\beta\phi}\mathcal{L})\big]_{f}\big[\mathcal{H}^{-1}(\partial_{\beta\phi}\mathcal{L})\big]_{g}\rVert =O_{p}(N^{8\epsilon}$)\\
(iv) $\lVert\sum_{f}(\partial_{\phi\phi'\phi_{f}}\mathcal{L})\big[\mathcal{H}^{-1}\mathcal{S}\big]_{f}\rVert =O_{p}(N^{-\frac{1}{2}+2\epsilon})$\\
(v) $\lVert\sum_{e,f}(\partial_{\phi\phi'\phi_{e}\phi_{f}}\mathcal{L})\big[\mathcal{H}^{-1}(\partial_{\beta\phi}\mathcal{L})\big]_{f}\big[\mathcal{H}^{-1}\mathcal{S}\big]_{e}\rVert =O_{p}(N^{-\frac{1}{2}+4\epsilon})$\\
(vi) $\lVert\sum_{f}(\partial_{\beta^{r}\phi\phi'\phi_{f}}\mathcal{L})\big[\mathcal{H}^{-1}\tilde{\mathcal{H}}\mathcal{H}^{-1}(\partial_{\beta^{s}\phi}\mathcal{L})\big]_{f}\rVert =O_{p}(N^{-\frac{1}{2}+4\epsilon})$\\
(vii) $\lVert\sum_{e,f,g}(\partial_{\phi\phi'\phi_{f}\phi_{g}}\mathcal{L})\big[\mathcal{H}^{-1}(\partial_{\beta\phi}\mathcal{L})\big]_{e}\big[\mathcal{H}^{-1}(\partial_{\beta\phi}\mathcal{L})\big]_{f}\big[\mathcal{H}^{-1}\mathcal{S}\big]_{g}\rVert =O_{p}(N^{-\frac{1}{2}+6\epsilon})$
\end{lem}
\begin{proof}
The proofs are substantially similar for each of the terms and so we
provide details for the second statement only, with $t=1$. We can
again decompose this term into
\begin{align*}
\lVert & \sum_{f,g}(\partial_{\phi\phi'\phi_{f}\phi_{g}}\mathcal{L})\big[\mathcal{H}^{-1}(\partial_{\beta\phi}\mathcal{L})\big]_{f}\big[\mathcal{H}^{-1}(\partial_{\beta\phi}\mathcal{L})\big]_{g}\rVert\\
& \leq\lVert\sum_{f,g}(\partial_{\phi\phi'\phi_{f}\phi_{g}}\mathcal{L})\big[\mathcal{\bar{H}}^{-1}(\partial_{\beta\phi}\mathcal{L})\big]_{f}\big[\mathcal{\bar{H}}^{-1}(\partial_{\beta\phi}\mathcal{L})\big]_{g}\rVert\\
& +\lVert\sum_{f,g}(\partial_{\phi\phi'\phi_{f}\phi_{g}}\mathcal{L})\big[\mathcal{\bar{H}}^{-1}\tilde{\mathcal{H}}\mathcal{\bar{H}}^{-1}(\partial_{\beta\phi}\mathcal{L})\big]_{f}\big[\mathcal{\bar{H}}^{-1}(\partial_{\beta\phi}\mathcal{L})\big]_{g}\rVert\\
& +\lVert\sum_{f,g}(\partial_{\phi\phi'\phi_{f}\phi_{g}}\mathcal{L})\big[\mathcal{\bar{H}}^{-1}(\partial_{\beta\phi}\mathcal{L})\big]_{f}\big[\mathcal{\bar{H}}^{-1}\tilde{\mathcal{H}}\mathcal{\bar{H}}^{-1}(\partial_{\beta\phi}\mathcal{L})\big]_{g}\rVert\\
& +O_{p}(N^{6\epsilon})
\end{align*}
with the bound on the remainder following from
\begin{align*}
\lVert & \sum_{f,g}(\partial_{\phi\phi'\phi_{f}\phi_{g}}\mathcal{L})\big[\big(\mathcal{H}^{-1}-\mathcal{\bar{H}}^{-1}+\mathcal{\bar{H}}^{-1}\tilde{\mathcal{H}}\mathcal{\bar{H}}^{-1}\big)(\partial_{\beta\phi}\mathcal{L})\big]_{f}\big[\mathcal{H}^{-1}(\partial_{\beta\phi}\mathcal{L})\big]_{g}\rVert\\
& \leq\lVert\partial_{\phi^{4}}\mathcal{L}\rVert\cdot\lVert\mathcal{H}^{-1}-\mathcal{\bar{H}}^{-1}+\mathcal{\bar{H}}^{-1}\tilde{\mathcal{H}}\mathcal{\bar{H}}^{-1}\rVert\cdot\lVert\partial_{\beta\phi}\mathcal{L}\rVert^{2}\cdot\lVert\mathcal{H}^{-1}\rVert
\end{align*}
and Lemma \ref{lem:H1_approx}.
Focussing on the first term, we can decompose the matrix $\partial_{\phi\phi'\phi_{f}\phi_{g}}\mathcal{L}$
into four parts as $\partial_{\alpha\alpha'\phi_{f}\phi_{g}}\mathcal{L}$,
$\partial_{\alpha\gamma'\phi_{f}\phi_{g}}\mathcal{L}$ etc. The first
of these is
\begin{align*}
\lVert & \sum_{f,g}(\partial_{\alpha\alpha'\phi_{f}\phi_{g}}\mathcal{L})\big[\mathcal{\bar{H}}^{-1}(\partial_{\beta\phi}\mathcal{L})\big]_{f}\big[\mathcal{\bar{H}}^{-1}(\partial_{\beta\phi}\mathcal{L})\big]_{g}\rVert\\
& \leq\lVert\sum_{f=1}^{N}(\partial_{\alpha\alpha'\alpha_{f}\alpha_{f}}\mathcal{L})\big[\mathcal{\bar{H}}_{\alpha,\cdot}^{-1}(\partial_{\beta\phi}\mathcal{L})\big]_{f}\big[\mathcal{\bar{H}}_{\alpha,\cdot}^{-1}(\partial_{\beta\phi}\mathcal{L})\big]_{f}\rVert\\
& +2\lVert\sum_{f,g=1}^{N}(\partial_{\alpha\alpha'\alpha_{f}\gamma_{g}}\mathcal{L})\big[\mathcal{\bar{H}}_{\alpha,\cdot}^{-1}(\partial_{\beta\phi}\mathcal{L})\big]_{f}\big[\mathcal{\bar{H}}_{\gamma,\cdot}^{-1}(\partial_{\beta\phi}\mathcal{L})\big]_{g}\rVert\\
& +\lVert\sum_{f=1}^{N}(\partial_{\alpha\alpha'\gamma_{f}\gamma_{f}}\mathcal{L})\big[\mathcal{\bar{H}}_{\gamma,\cdot}^{-1}(\partial_{\beta\phi}\mathcal{L})\big]_{f}\big[\mathcal{\bar{H}}_{\gamma,\cdot}^{-1}(\partial_{\beta\phi}\mathcal{L})\big]_{f}\rVert\\
& =\max_{i}\lvert\frac{1}{(N-1)^{3}}\sum_{j\ne i}\partial_{\pi^{4}}\ell_{ij}\big(\sum_{k}\sum_{l\ne k}((\bar{\mathcal{H}}_{\alpha\alpha}^{-1})_{ik}+(\bar{\mathcal{H}}_{\alpha\gamma}^{-1})_{il})\partial_{\beta\pi}\ell_{kl}\big)^{2}\rvert\\
& +\max_{i}\lvert\frac{1}{(N-1)^{3}}\sum_{j\ne i}\partial_{\pi^{4}}\ell_{ij}\big(\sum_{k}\sum_{l\ne k}((\bar{\mathcal{H}}_{\alpha\alpha}^{-1})_{ik}+(\bar{\mathcal{H}}_{\alpha\gamma}^{-1})_{il})\partial_{\beta\pi}\ell_{kl}\big)\\
& \quad\times\big(\sum_{k}\sum_{l\ne k}((\bar{\mathcal{H}}_{\gamma\alpha}^{-1})_{jk}+(\bar{\mathcal{H}}_{\gamma\gamma}^{-1})_{jl})\partial_{\beta\pi}\ell_{kl}\big)\rvert\\
& +\max_{i}\lvert\frac{1}{(N-1)^{3}}\sum_{j\ne i}\partial_{\pi^{4}}\ell_{ij}\big(\sum_{k}\sum_{l\ne k}((\bar{\mathcal{H}}_{\gamma\alpha}^{-1})_{jk}+(\bar{\mathcal{H}}_{\gamma\gamma}^{-1})_{jl})\partial_{\beta\pi}\ell_{kl}\big)^{2}\rvert
\end{align*}
Considering the first of this new set of terms
\begin{align*}
\max_{i} & \lvert\frac{1}{(N-1)^{3}}\sum_{j\ne i}\partial_{\pi^{4}}\ell_{ij}\big(\sum_{k}\sum_{l\ne k}((\bar{\mathcal{H}}_{\alpha\alpha}^{-1})_{ik}+(\bar{\mathcal{H}}_{\alpha\gamma}^{-1})_{il})\partial_{\beta\pi}\ell_{kl}\big)^{2}\rvert\\
& =\max_{i}\lvert\frac{1}{(N-1)^{3}}\sum_{j\ne i}\partial_{\pi^{4}}\ell_{ij}\big(\sum_{k}\sum_{l\ne k}\sum_{s}\sum_{t\ne s}((\bar{\mathcal{H}}_{\alpha\alpha}^{-1})_{ik}(\bar{\mathcal{H}}_{\alpha\alpha}^{-1})_{is}+(\bar{\mathcal{H}}_{\alpha\alpha}^{-1})_{ik}(\bar{\mathcal{H}}_{\alpha\gamma}^{-1})_{it}\\
& +(\bar{\mathcal{H}}_{\alpha\gamma}^{-1})_{il}(\bar{\mathcal{H}}_{\alpha\alpha}^{-1})_{is}+(\bar{\mathcal{H}}_{\alpha\gamma}^{-1})_{il}(\bar{\mathcal{H}}_{\alpha\gamma}^{-1})_{it})(\partial_{\beta\pi}\ell_{kl})(\partial_{\beta\pi}\ell_{st})\big)\rvert\\
& =\max_{i}\lvert\frac{1}{(N-1)^{3}}\sum_{j\ne i}\partial_{\pi^{4}}\ell_{ij}\big(\sum_{l\ne i}\sum_{t\ne i}(\bar{\mathcal{H}}_{\alpha\alpha}^{-1})_{ii}(\bar{\mathcal{H}}_{\alpha\alpha}^{-1})_{ii}(\partial_{\beta\pi}\ell_{il})(\partial_{\beta\pi}\ell_{it})\big)\rvert\\
& +2\max_{i}\lvert\frac{1}{N(N-1)^{3}}\sum_{j\ne i}\partial_{\pi^{4}}\ell_{ij}\big(\sum_{l\ne k}\sum_{s\ne i}\sum_{t\ne s}((\bar{\mathcal{H}}_{\alpha\alpha}^{-1})_{ii}(N\bar{\mathcal{H}}_{\alpha\alpha}^{-1})_{is})(\partial_{\beta\pi}\ell_{il})(\partial_{\beta\pi}\ell_{st})\big)\rvert\\
& +\max_{i}\lvert\frac{1}{N^{2}(N-1)^{3}}\sum_{j\ne i}\partial_{\pi^{4}}\ell_{ij}\big(\sum_{k\ne i}\sum_{l\ne k}\sum_{s\ne i}\sum_{t\ne s}((N\bar{\mathcal{H}}_{\alpha\alpha}^{-1})_{ik}(N\bar{\mathcal{H}}_{\alpha\alpha}^{-1})_{is})(\partial_{\beta\pi}\ell_{kl})(\partial_{\beta\pi}\ell_{st})\big)\rvert\\
& +2\max_{i}\lvert\frac{1}{N(N-1)^{3}}\sum_{j\ne i}\partial_{\pi^{4}}\ell_{ij}\big(\sum_{l\ne i}\sum_{s}\sum_{t\ne s}(\bar{\mathcal{H}}_{\alpha\alpha}^{-1})_{ii}(N\bar{\mathcal{H}}_{\alpha\gamma}^{-1})_{it}(\partial_{\beta\pi}\ell_{il})(\partial_{\beta\pi}\ell_{st})\big)\rvert\\
& +2\max_{i}\lvert\frac{1}{N^{2}(N-1)^{3}}\sum_{j\ne i}\partial_{\pi^{4}}\ell_{ij}\big(\sum_{k\ne i}\sum_{l\ne k}\sum_{s}\sum_{t\ne s}(N\bar{\mathcal{H}}_{\alpha\alpha}^{-1})_{ik}(N\bar{\mathcal{H}}_{\alpha\gamma}^{-1})_{it}(\partial_{\beta\pi}\ell_{kl})(\partial_{\beta\pi}\ell_{st})\big)\rvert\\
& +\max_{i}\lvert\frac{1}{N^{2}(N-1)^{3}}\sum_{j\ne i}\partial_{\pi^{4}}\ell_{ij}\big(\sum_{k}\sum_{l\ne k}\sum_{s}\sum_{t\ne s}(N\bar{\mathcal{H}}_{\alpha\gamma}^{-1})_{il}(N\bar{\mathcal{H}}_{\alpha\gamma}^{-1})_{it}(\partial_{\beta\pi}\ell_{kl})(\partial_{\beta\pi}\ell_{st})\big)\rvert\\
& =O_{p}(N^{6\epsilon})
\end{align*}
where the final line applies Lemma \ref{lem:S6} to each term, e.g
\begin{align*}
\max_{i} & \lvert\frac{1}{(N-1)^{3}}\sum_{j\ne i}\partial_{\pi^{4}}\ell_{ij}\big(\sum_{l\ne i}\sum_{t\ne i}(\bar{\mathcal{H}}_{\alpha\alpha}^{-1})_{ii}(\bar{\mathcal{H}}_{\alpha\alpha}^{-1})_{ii}(\partial_{\beta\pi}\ell_{il})(\partial_{\beta\pi}\ell_{it})\big)\rvert\\
& \leq\max_{i}\lvert(\bar{\mathcal{H}}_{\alpha\alpha}^{-1})_{ii}\rvert^{2}\times\max_{i}\lvert\frac{1}{N-1}\sum_{j\ne i}\partial_{\pi^{4}}\ell_{ij}\rvert\\
& \times\max_{i}\lvert\frac{1}{N-1}\sum_{j\ne i}\partial_{\beta\pi}\ell_{ij}\rvert\times\max_{i}\lvert\frac{1}{N-1}\sum_{j\ne i}\partial_{\beta\pi}\ell_{ij}\rvert\\
& =O_{p}(1)\times O_{p}(N^{2\epsilon})\times O_{p}(N^{2\epsilon})\times O_{p}(N^{2\epsilon})
\end{align*}
The remaining terms in $\partial_{\phi\phi'\phi_{f}\phi_{g}}\mathcal{L}$
can be dealt with identically. \\
The terms related to $\sum_{f,g}(\partial_{\phi\phi'\phi_{f}\phi_{g}}\mathcal{L})\big[\mathcal{\bar{H}}^{-1}\tilde{\mathcal{H}}\mathcal{\bar{H}}^{-1}(\partial_{\beta\phi}\mathcal{L})\big]_{f}\big[\mathcal{\bar{H}}^{-1}(\partial_{\beta\phi}\mathcal{L})\big]_{g}$
also have nearly identical proofs, applying Lemma \ref{lem:HEH_bound}
so that $\mathcal{\bar{H}}^{-1}\tilde{\mathcal{H}}\mathcal{\bar{H}}^{-1}$
behaves similarly to $\mathcal{\bar{H}}^{-1}$ in the proof. The proofs
for terms withe $[\mathcal{H}^{-1}\mathcal{S}]$ or $\tilde{\mathcal{H}}$
are essentially the same, but use the fact the one of the terms in
the sum is mean zero, e.g.
\begin{align*}
\bar{E}\Big[\big(\max_{i}\lvert\frac{1}{\sqrt{N-1}}\sum_{j\ne i}\partial_{\pi}\ell_{ij}\rvert\big)^{2}\Big] & \leq\sum_{i}\bar{E}\Big[\big(\lvert\frac{1}{\sqrt{N-1}}\sum_{j\ne i}\partial_{\pi}\ell_{ij}\rvert\big)^{2}\Big]\\
& \leq CNN^{-1}=O_{p}(1)
\end{align*}
and hence $\max_{i}\lvert\frac{1}{N-1}\sum_{j\ne i}\partial_{\pi}\ell_{ij}\rvert=O_{p}(N^{-1/2})$.
\end{proof}
\begin{lem}
\label{lem:tensors2}Let Assumptions 1 and 2 hold. Then, $s+t \leq 5$\\ \\
(i) $\lVert\sum_{f}(\partial_{\phi\phi'\phi_{f}}\mathcal{L})\big[\mathcal{H}^{-1}\mathcal{H}_{b}\mathcal{H}^{-1}(\partial_{\beta\beta\phi}\mathcal{L})\big]_{f}\rVert =O_{p}(N^{-1+8\epsilon})$\\
(ii) $(\partial_{\beta^{s}\phi'}\mathcal{L})\mathcal{H}^{-1}\mathcal{H}_{b}\mathcal{H}^{-1}\mathcal{E}^{r}\mathcal{H}^{-1}(\partial_{\beta^{t}\phi}\mathcal{L}) =O_{p}(1)$\\
(iii) $\mathcal{S}'\mathcal{H}^{-1}\sum_{f}(\partial_{\beta\phi\phi'\phi_{f}}\mathcal{L})\big[\mathcal{H}^{-1}(\partial_{\beta\phi}\mathcal{L})\big]_{f}\mathcal{H}^{-1}(\partial_{\beta\phi}\mathcal{L}) =O_{p}(1)$\\
(iv) $\mathcal{S}'\mathcal{H}^{-1}\sum_{f}(\partial_{\beta\phi\phi'\phi_{f}}\mathcal{L})\big[\mathcal{H}^{-1}(\partial_{\beta\phi}\mathcal{L})\big]_{f}\mathcal{H}^{-1}\mathcal{S} =O_{p}(1)$\\
(v) $\mathcal{S}'\mathcal{H}^{-1}\sum_{e,f}(\partial_{\phi\phi'\phi_{e}\phi_{f}}\mathcal{L})\big[\mathcal{H}^{-1}(\partial_{\beta\phi}\mathcal{L})\big]_{e}\big[\mathcal{H}^{-1}(\partial_{\beta\phi}\mathcal{L})\big]_{f}\mathcal{H}^{-1}(\partial_{\beta\phi}\mathcal{L}) =O_{p}(N^{4\epsilon})$\\
(vi) $\lVert\sum_{g}\mathcal{S}'\mathcal{H}^{-1}\mathcal{H}_{b}\mathcal{H}^{-1}\mathcal{H}_{s_{g}}\mathcal{H}^{-1}\mathcal{S}\mathcal{S}_{g}\rVert =O_{p}(N^{-\frac{3}{2}+4\epsilon})$\\
(vii) $ \lVert\mathcal{S}'\mathcal{H}^{-1}W^{-1}\sum_{f}(\partial_{\phi\phi'\phi_{f}}\mathcal{L})\big[\mathcal{H}^{-1}\mathcal{H}_{bb}\mathcal{H}^{-1}(\partial_{\beta\phi}\mathcal{L})\big]_{f}\mathcal{H}^{-1}\mathcal{S}\rVert =O_{p}(N^{-3+14\epsilon})$\\
(vii) $\lVert\sum_{g,h}\mathcal{S}'\mathcal{H}^{-1}\mathcal{H}_{b}\mathcal{H}^{-1}\mathcal{H}_{s_{g}s_{h}}\mathcal{H}^{-1}\mathcal{S}\mathcal{S}_{g}\mathcal{S}_{h}\rVert =O_{p}(N^{-2+10\epsilon})$
\end{lem}
\begin{proof}
For expression 1 we have
\begin{align*}
\lVert\sum_{f}(\partial_{\phi\phi'\phi_{f}}\mathcal{L})\big[\mathcal{H}^{-1}\mathcal{H}_{b}\mathcal{H}^{-1}(\partial_{\beta\beta\phi}\mathcal{L})\big]_{f}\rVert & \leq\lVert\sum_{f}(\partial_{\phi\phi'\phi_{f}}\mathcal{L})\big[\mathcal{H}^{-1}\mathcal{\bar{H}}_{b}\mathcal{H}^{-1}(\partial_{\beta\beta\phi}\mathcal{L})\big]_{f}\rVert\\
& +\lVert\sum_{f}(\partial_{\phi\phi'\phi_{f}}\mathcal{L})\big[\mathcal{H}^{-1}(\mathcal{H}_{b}-\bar{\mathcal{H}}_{b})\mathcal{H}^{-1}(\partial_{\beta\beta\phi}\mathcal{L})\big]_{f}\rVert\\
& =\lVert\sum_{f}(\partial_{\phi\phi'\phi_{f}}\mathcal{L})\big[\mathcal{H}^{-1}\mathcal{\bar{H}}_{b}\mathcal{H}^{-1}(\partial_{\beta\beta\phi}\mathcal{L})\big]_{f}\rVert+O_{p}(N^{-1+8\epsilon})\\
& =O_{p}(N^{-1+8\epsilon})
\end{align*}
using Lemmas \ref{lem:H_tilde} and \ref{lem:HEH_bound} combined
with Lemma \ref{lem:tensor_bounds}. Next
\begin{align*}
\lVert(\partial_{\beta^{s}\phi'}\mathcal{L})\mathcal{H}^{-1}\mathcal{H}_{b}\mathcal{H}^{-1}\mathcal{E}^{r}\mathcal{H}^{-1}(\partial_{\beta^{t}\phi}\mathcal{L})\rVert & \leq\lVert(\partial_{\beta^{s}\phi'}\mathcal{L})\mathcal{H}^{-1}\mathcal{\bar{H}}_{b}\mathcal{H}^{-1}\mathcal{E}^{r}\mathcal{H}^{-1}(\partial_{\beta^{t}\phi}\mathcal{L})\rVert\\
& +\lVert(\partial_{\beta^{s}\phi'}\mathcal{L})\mathcal{H}^{-1}(\mathcal{H}_{b}-\mathcal{\bar{H}}_{b})\mathcal{H}^{-1}\mathcal{E}^{r}\mathcal{H}^{-1}(\partial_{\beta^{t}\phi}\mathcal{L})\rVert\\
& =O_{p}(1)+O_{p}(N^{-\frac{1}{2}+6\epsilon})=O_{p}(1)
\end{align*}
For the third and fourth terms, note that $\sum_{f}(\partial_{\beta\phi\phi'\phi_{f}}\mathcal{L})\big[\mathcal{H}^{-1}(\partial_{\beta\phi}\mathcal{L})\big]_{f}$
satisfies the same conditions as $W\mathcal{H}_{b}$ in the proof
in Lemmas \ref{lem:H_tilde} and \ref{lem:HEH_bound}, and hence we
can apply the results of those lemmas along with the result in Lemma
\ref{lem:tensor_bounds}.
The fifth result can be shown similarly. First note that
\begin{align*}
\lVert & \sum_{e,f}(\partial_{\phi\phi'\phi_{e}\phi_{f}}\mathcal{L})\big[\mathcal{H}^{-1}(\partial_{\beta\phi}\mathcal{L})\big]_{e}\big[\mathcal{H}^{-1}(\partial_{\beta\phi}\mathcal{L})\big]_{f}-\sum_{e,f}(\partial_{\phi\phi'\phi_{e}\phi_{f}}\bar{\mathcal{L}})\big[\mathcal{\bar{H}}^{-1}(\partial_{\beta\phi}\bar{\mathcal{L}})\big]_{e}\big[\bar{\mathcal{H}}^{-1}(\partial_{\beta\phi}\bar{\mathcal{L}})\big]_{f}\rVert\\
& =O_{p}(N^{-\frac{1}{2}+4\epsilon})
\end{align*}
following the proofs in Lemma \ref{lem:tensor_bounds}. Then, to apply
Lemma \ref{lem:HEH_bound}, let $i\leq N$ and so
\begin{align*}
\Big[\sum_{e,f}(\partial_{\phi\phi'\phi_{e}\phi_{f}}\bar{\mathcal{L}})\big[\mathcal{\bar{H}}^{-1}(\partial_{\beta\phi}\bar{\mathcal{L}})\big]_{e}\big[\bar{\mathcal{H}}^{-1}(\partial_{\beta\phi}\bar{\mathcal{L}})\big]_{f}\Big]_{ii} & =(\partial_{\alpha_{i}^{4}}\bar{\mathcal{L}})\big[\mathcal{\bar{H}}^{-1}(\partial_{\beta\phi}\bar{\mathcal{L}})\big]_{i}\big[\bar{\mathcal{H}}^{-1}(\partial_{\beta\phi}\bar{\mathcal{L}})\big]_{i}\\
& +2\sum_{j\ne i}(\partial_{\alpha_{i}^{3}\gamma_{j}}\bar{\mathcal{L}})\big[\mathcal{\bar{H}}^{-1}(\partial_{\beta\phi}\bar{\mathcal{L}})\big]_{i}\big[\bar{\mathcal{H}}^{-1}(\partial_{\beta\phi}\bar{\mathcal{L}})\big]_{j}\\
& +\sum_{j\ne i}(\partial_{\alpha_{i}^{2}\gamma_{j}^{2}}\bar{\mathcal{L}})\big[\mathcal{\bar{H}}^{-1}(\partial_{\beta\phi}\bar{\mathcal{L}})\big]_{j}\big[\bar{\mathcal{H}}^{-1}(\partial_{\beta\phi}\bar{\mathcal{L}})\big]_{j}
\end{align*}
and since $\big[\mathcal{\bar{H}}^{-1}(\partial_{\beta\phi}\bar{\mathcal{L}})\big]_{i}\leq C$,
$\partial_{\alpha_{i}^{4}}\bar{\mathcal{L}}\leq C$ and $\sum_{j\ne i}(\partial_{\pi^{4}}\bar{\mathcal{\ell}}_{ij})\leq C$
we have \\
$\max_{i}\vert\Big[\sum_{e,f}(\partial_{\phi\phi'\phi_{e}\phi_{f}}\bar{\mathcal{L}})\big[\mathcal{\bar{H}}^{-1}(\partial_{\beta\phi}\bar{\mathcal{L}})\big]_{e}\big[\bar{\mathcal{H}}^{-1}(\partial_{\beta\phi}\bar{\mathcal{L}})\big]_{f}\Big]_{ii}\vert=O_{p}(1)$
as required (other components of the matrix can be shown in the same
way). This then gives
\[
\mathcal{S}'\mathcal{H}^{-1}\sum_{e,f}(\partial_{\phi\phi'\phi_{e}\phi_{f}}\bar{\mathcal{L}})\big[\mathcal{\bar{H}}^{-1}(\partial_{\beta\phi}\bar{\mathcal{L}})\big]_{e}\big[\bar{\mathcal{H}}^{-1}(\partial_{\beta\phi}\bar{\mathcal{L}})\big]_{f}\mathcal{H}^{-1}(\partial_{\beta\phi}\mathcal{L})=O_{p}(1)
\]
and so
\begin{align*}
&\vert\mathcal{S}'\mathcal{H}^{-1}\sum_{e,f}(\partial_{\phi\phi'\phi_{e}\phi_{f}}\mathcal{L})\big[\mathcal{H}^{-1}(\partial_{\beta\phi}\mathcal{L})\big]_{e}\big[\mathcal{H}^{-1}(\partial_{\beta\phi}\mathcal{L})\big]_{f}\mathcal{H}^{-1}(\partial_{\beta\phi}\mathcal{L})\vert \\
& \leq\lVert\mathcal{S}'\mathcal{H}^{-1}\rVert\lVert\mathcal{H}^{-1}(\partial_{\beta\phi}\mathcal{L})\rVert O_{p}(N^{-1/2+4\epsilon})+O_{p}(1) =O_{p}(N^{4\epsilon})
\end{align*}
The 6th and 7th terms can be bounded as
\begin{align*}
\lVert\sum_{g}\mathcal{S}'\mathcal{H}^{-1}\mathcal{H}_{b}\mathcal{H}^{-1}\mathcal{H}_{s_{g}}\mathcal{H}^{-1}\mathcal{S}\mathcal{S}_{g}\rVert & \leq\lVert\sum_{g}\mathcal{S}'\bar{\mathcal{H}}^{-1}\bar{\mathcal{H}}_{b}\mathcal{\bar{H}}^{-1}\mathcal{H}_{s_{g}}\mathcal{H}^{-1}\mathcal{S}\mathcal{S}_{g}\rVert\\
& +\lVert\sum_{g}\mathcal{S}'(\mathcal{H}^{-1}\mathcal{H}_{b}\mathcal{H}^{-1}-\bar{\mathcal{H}}^{-1}\bar{\mathcal{H}}_{b}\mathcal{\bar{H}}^{-1})\mathcal{H}_{s_{g}}\mathcal{H}^{-1}\mathcal{S}\mathcal{S}_{g}\rVert\\
& \leq N^{1-1/q}\lVert\mathcal{H}^{-1}\rVert_{q}\lVert\bar{\mathcal{H}}^{-1}\bar{\mathcal{H}}_{b}\mathcal{\bar{H}}^{-1}\rVert_{q}\lVert\sum_{g}\mathcal{H}_{s_{g}}\mathcal{S}_{g}\rVert_{q}\lVert\mathcal{S}\rVert_{q}^{2}\\
& +N^{1-1/q}\lVert\mathcal{H}^{-1}\mathcal{H}_{b}\mathcal{H}^{-1}-\bar{\mathcal{H}}^{-1}\bar{\mathcal{H}}_{b}\mathcal{\bar{H}}^{-1}\rVert\lVert\mathcal{H}^{-1}\rVert_{q}\lVert\sum_{g}\mathcal{H}_{s_{g}}\mathcal{S}_{g}\rVert_{q}\lVert\mathcal{S}\rVert_{q}^{2}\\
& =O_{p}(N^{-\frac{3}{2}+4\epsilon})
\end{align*}
\begin{align*}
\lVert & \mathcal{S}'\mathcal{H}^{-1}W^{-1}\sum_{f}(\partial_{\phi\phi'\phi_{f}}\mathcal{L})\big[\mathcal{H}^{-1}\mathcal{H}_{bb}\mathcal{H}^{-1}(\partial_{\beta\phi}\mathcal{L})\big]_{f}\mathcal{H}^{-1}\mathcal{S}\rVert\\
& \leq N^{1-1/q}\lVert W^{-1}\rVert\lVert\partial_{\phi\phi\phi}\mathcal{L}\rVert_{q}\lVert\mathcal{H}^{-1}\rVert_{q}^{4}\lVert\mathcal{H}_{bb}\rVert_{q}\lVert\mathcal{S}\rVert_{q}^{2}\lVert\partial_{\beta\phi}\mathcal{L}\rVert_{q}\\
& =O_{p}(N^{-3+14\epsilon})
\end{align*}
Finally, for term 8 we have
\begin{align*}
\lVert\sum_{g,h}\mathcal{S}'\mathcal{H}^{-1}\mathcal{H}_{b}\mathcal{H}^{-1}\mathcal{H}_{s_{g}s_{h}}\mathcal{H}^{-1}\mathcal{S}\mathcal{S}_{g}\mathcal{S}_{h}\rVert & \leq\lVert\sum_{g,h}\mathcal{S}'\mathcal{\bar{H}}^{-1}\mathcal{\bar{H}}_{b}\mathcal{\bar{H}}^{-1}\mathcal{H}_{s_{g}s_{h}}\mathcal{H}^{-1}\mathcal{S}\mathcal{S}_{g}\mathcal{S}_{h}\rVert\\
& +\lVert\sum_{g,h}\mathcal{S}'(\mathcal{H}^{-1}\mathcal{H}_{b}\mathcal{H}^{-1}-\mathcal{\bar{H}}^{-1}\mathcal{\bar{H}}_{b}\mathcal{\bar{H}}^{-1})\mathcal{H}_{s_{g}s_{h}}\mathcal{H}^{-1}\mathcal{S}\mathcal{S}_{g}\mathcal{S}_{h}\rVert\\
& \leq N^{1-\frac{1}{q}}O_{p}(N^{-1})\lVert\mathcal{S}\rVert_{q}^{4}\lVert\mathcal{H}_{ss}\rVert_{q}\\
& =O_{p}(N^{-2+\frac{3}{q}+4\epsilon})=O_{p}(N^{-2+10\epsilon})
\end{align*}
\end{proof}
\section{\label{sec:bounds_ind}Bounds on individual components}
This section provides bounds on the terms $\mathcal{P}$, $\mathcal{R}$, $\mathcal{F}$, $\mathcal{G}$, $W$, and $\mathcal{H}$, and their derivatives. These form the basis for bounding the terms in the asymptotic expansion. The bounds follow from the definition of the terms (see Section \ref{sec:expressions_ind}) as well as the bounds presented in the previous section.
\subsection{Bounds on $\mathcal{F}, \mathcal{P}, \mathcal{R}$ terms}
\begin{lem}
\label{lem:F}Let Assumptions 1 and 2 hold. Then we have, for $s,t\in\{1,2,3\}$
and $s+t\leq5$,
\begin{align*}
\lVert\mathcal{F}^{s,t}\rVert & =O_{p}(N)\\
\lVert\mathcal{F}^{s,(r),t}\rVert & =O_{p}(N)\\
\lVert\mathcal{F}_{b}^{s,t}\rVert & =O_{p}(1)\\
\lVert\mathcal{F}_{b}^{s,(r),t}\rVert & =O_{p}(1)\\
\lVert\mathcal{S}'\mathcal{F}_{s}^{s,t}\rVert & =O_{p}(1)\\
\lVert\mathcal{F}_{bb}^{s,t}\rVert & =O_{p}(N^{-1+8\epsilon})\\
\lVert\mathcal{S}'\mathcal{F}_{ss'}\mathcal{S}\rVert & =O_{p}(1)
\end{align*}
\end{lem}
\begin{proof}
The first two results follow directly from Lemma \ref{lem:HEH_bound}.
For the first derivatives, we have
\begin{align*}
\lVert\mathcal{F}_{b}^{s,t}\rVert & \leq\lVert W^{-1}\rVert\Big(\lVert\mathcal{F}^{s+1,t}\rVert+\lVert\mathcal{F}^{1,(s),t}\rVert+\lVert\mathcal{F}^{s,t+1}\rVert+\lVert\mathcal{F}^{s,(t),1}\rVert\Big)\\
& \quad+\lVert(\partial_{\beta^{s}\phi'}\mathcal{L})\mathcal{H}^{-1}\mathcal{H}_{b}\mathcal{H}^{-1}(\partial_{\beta^{t}\phi}\mathcal{L})\rVert\\
& =O_{p}(1)
\end{align*}
from Lemma \ref{lem:tensors2} (the result for $\mathcal{F}_{b}^{s,(r),t}$
follows similarly). Also
\begin{align*}
\lVert\mathcal{S}'\mathcal{F}_{s}^{s,t}\rVert & \leq\lVert W^{-1}\rVert\lVert\mathcal{S}'\big(\mathcal{G}^{1,s+1}+\mathcal{G}\mathcal{E}^{s}\mathcal{H}^{-1}\big)(\partial_{\beta^{t}\phi}\mathcal{L})\rVert+\lVert\mathcal{S}'\mathcal{H}^{-1}\mathcal{E}^{s}\mathcal{H}^{-1}(\partial_{\beta^{t}\phi}\mathcal{L})\rVert\\
& \quad+\lVert W^{-1}\rVert\lVert\mathcal{S}'\big(\mathcal{G}^{t+1,1}+\mathcal{H}^{-1}\mathcal{E}^{t}\mathcal{G}\big)(\partial_{\beta^{s}\phi}\mathcal{L})\rVert+\lVert\mathcal{S}'\mathcal{H}^{-1}\mathcal{E}^{t}\mathcal{H}^{-1}(\partial_{\beta^{s}\phi'}\mathcal{L})\rVert\\
& \quad+\lVert(\partial_{\beta^{s}\phi'}\mathcal{L})\mathcal{H}^{-1}\sum_{g}\mathcal{H}_{s_{g}}\mathcal{S}_{g}\mathcal{H}^{-1}(\partial_{\beta^{t}\phi}\mathcal{L})\rVert\\
& =O_{p}(1)
\end{align*}
which follows from results in Lemma \ref{lem:element1} and the bound
\begin{align*}
\lVert &(\partial_{\beta^{s}\phi'}\mathcal{L})\mathcal{H}^{-1}\sum_{g}\mathcal{H}_{s_{g}}\mathcal{S}_{g}\mathcal{H}^{-1}(\partial_{\beta^{t}\phi}\mathcal{L})\rVert \\
& \leq\lVert W^{-1}(\partial_{\beta\phi'}\mathcal{L})\mathcal{H}^{-1}\mathcal{S}\cdot(\partial_{\beta^{s}\phi'}\mathcal{L})\mathcal{H}^{-1}(\partial_{\beta\phi\phi'}\mathcal{L})\mathcal{H}^{-1}(\partial_{\beta^{t}\phi}\mathcal{L})\rVert\\
& +\lVert(\partial_{\beta^{s}\phi'}\mathcal{L})\mathcal{H}^{-1}\sum_{f}(\partial_{\phi\phi'\phi_{f}}\mathcal{L})[\mathcal{H}^{-1}\mathcal{S}]_{f}\mathcal{H}^{-1}(\partial_{\beta^{t}\phi}\mathcal{L})\rVert\\
& +\lVert W^{-1}(\partial_{\beta\phi'}\mathcal{L})\mathcal{H}^{-1}\mathcal{S}\cdot(\partial_{\beta^{s}\phi'}\mathcal{L})\mathcal{H}^{-1}\sum_{f}(\partial_{\phi\phi'\phi_{f}}\mathcal{L})\big[(\partial_{\beta\phi'}\mathcal{L})\mathcal{H}^{-1}\big]_{f}\mathcal{H}^{-1}(\partial_{\beta^{t}\phi}\mathcal{L})\rVert\\
& =O_{p}(1)
\end{align*}
where we use Lemma \ref{lem:HEH_bound} on the term $\sum_{f}(\partial_{\phi\phi'\phi_{f}}\bar{\mathcal{L}})\big[(\partial_{\beta\phi'}\bar{\mathcal{L}})\bar{\mathcal{H}}^{-1}\big]_{f}$ and apply Lemma \ref{lem:element1}.
The result for $\mathcal{F}_{bb}^{s,t}$ follows similarly by applying
results in Lemmas \ref{lem:tensor_bounds} and \ref{lem:tensors2}
along with Lemma \ref{lem:HEH_bound} (note that Lemma \ref{lem:HEH_bound}
may be applied repeatedly since the product of two matrices with $O_{p}(1)$
diagonal and $O_{p}(N^{-1})$ off-diagonals has this same property).
The final result follows from applications of the same bounds to the
expression for $\mathcal{F}_{ss}$.
\end{proof}
\begin{lem}
\label{lem:R}Let Assumptions 1 and 2 hold. Then we have, for $a\in\{1,2,3\}$,
\begin{align*}
\lVert\mathcal{R}^{a}\rVert_{q} & =O_{p}(N^{2\epsilon})\\
\lVert\mathcal{R}_{b}^{a}\rVert_{q} & =O_{p}(N^{-1+6\epsilon})\\
\lVert\mathcal{R}_{s}^{a}\rVert_{q} & =O_{p}(N^{4\epsilon})\\
\lVert\mathcal{R}_{bs}^{a}\rVert_{q} & =O_{p}(N^{-1+8\epsilon})\\
\lVert\mathcal{R}_{ss}^{a}\rVert_{q} & =O_{p}(N^{6\epsilon})\\
\lVert\mathcal{R}_{bss}^{a}\rVert_{q} & =O_{p}(N^{-1+10\epsilon})
\end{align*}
\end{lem}
\begin{proof}
Application of the individual bounds shown in this section along with the Cauchy-Schwarz and triangle inequalities give the results. For example,
\begin{align*}
\lVert\mathcal{R}_{s}^{a}\rVert_{q} & \leq\Big(\lVert W^{-1}\rVert_{q}\lVert\mathcal{R}^{1}\rVert_{q}\lVert\mathcal{R}^{a+1}\rVert_{q}\\
& +\lVert\mathcal{E}^{a}\rVert_{q}\lVert\mathcal{H}^{-1}\rVert_{q}^{2}+\lVert W^{-1}\rVert_{q}\lVert\mathcal{R}^{1}\rVert_{q}^{2}\lVert\mathcal{E}^{a}\rVert_{q}\lVert\mathcal{H}^{-1}\rVert_{q}\\
& +\lVert\mathcal{R}^{a}\rVert_{q}\lVert\mathcal{H}^{-1}\rVert_{q}\lVert\mathcal{H}_{s}\rVert_{q}\Big)\\
& =O_{p}(N^{4\epsilon})
\end{align*}
\end{proof}
\begin{lem}
\label{lem:P}Let Assumptions 1 and 2 hold. Then we have, for $a,r\in\{1,2,3\}$,
\begin{align*}
\lVert\mathcal{P}^{r}\rVert_{q} & =O_{p}(N^{2\epsilon}) & \lVert\mathcal{P}^{(a,r)}\rVert_{q} & =O_{p}(N^{4\epsilon})\\
\lVert\mathcal{P}_{b}^{r}\rVert_{q} & =O_{p}(N^{-1+6\epsilon}) & \lVert\mathcal{P}_{b}^{(a,r)}\rVert_{q} & =O_{p}(N^{-1+8\epsilon})\\
\lVert\mathcal{P}_{s}^{r}\rVert_{q} & =O_{p}(N^{4\epsilon}) & \lVert\mathcal{P}_{s}^{(a,r)}\rVert_{q} & =O_{p}(N^{6\epsilon})\\
\lVert\mathcal{P}_{bs_{}}^{r}\rVert_{q} & =O_{p}(N^{-1+8\epsilon}) & \lVert\mathcal{P}_{bs}^{(a,r)}\rVert & =O_{p}(N^{-1+10\epsilon})\\
\lVert\mathcal{P}_{ss}^{r}\rVert_{q} & =O_{p}(N^{6\epsilon}) & \lVert\mathcal{P}_{ss}^{(a,r)}\rVert_{q} & =O_{p}(N^{8\epsilon})\\
\lVert\mathcal{P}_{bss}^{r}\rVert_{q} & =O_{p}(N^{-1+10\epsilon}) & \lVert\mathcal{P}_{bss}^{(a,r)}\rVert_{q} & =O_{p}(N^{-1+12\epsilon})
\end{align*}
\begin{align*}
\lVert\mathcal{P}_{s_{}}^{r}\rVert_{q} & \leq\lVert W^{-1}\rVert_{q}\lVert\mathcal{R}^{1}\rVert_{q}\lVert\mathcal{P}_{}^{r+1}\rVert_{q}\\
& +\lVert\partial_{\beta^{r}\phi^{4}}\mathcal{L}\rVert_{q}\lVert\mathcal{H}^{-1}\rVert_{q}^{2}\\
& +\lVert W^{-1}\rVert_{q}\lVert\partial_{\beta^{r}\phi^{4}}\mathcal{L}\rVert_{q}\lVert\mathcal{H}^{-1}\rVert_{q}\lVert\mathcal{R}^{1}\rVert_{q}^{2}\\
& +\lVert\partial_{\beta^{r}\phi^{3}}\mathcal{L}\rVert_{q}\lVert\mathcal{H}^{-1}\rVert_{q}^{2}\lVert\mathcal{H}_{s}\rVert_{q}\\
& =O_{p}(N^{4\epsilon})
\end{align*}
\end{lem}
\begin{proof}
As above, application of the individual bounds shown in this section along with the Cauchy-Schwarz and triangle inequalities give the results. For example,
\begin{align*}
\lVert\mathcal{P}_{s_{}}^{r}\rVert_{q} & \leq\lVert W^{-1}\rVert_{q}\lVert\mathcal{R}^{1}\rVert_{q}\lVert\mathcal{P}_{}^{r+1}\rVert_{q}\\
& +\lVert\partial_{\beta^{r}\phi^{4}}\mathcal{L}\rVert_{q}\lVert\mathcal{H}^{-1}\rVert_{q}^{2}\\
& +\lVert W^{-1}\rVert_{q}\lVert\partial_{\beta^{r}\phi^{4}}\mathcal{L}\rVert_{q}\lVert\mathcal{H}^{-1}\rVert_{q}\lVert\mathcal{R}^{1}\rVert_{q}^{2}\\
& +\lVert\partial_{\beta^{r}\phi^{3}}\mathcal{L}\rVert_{q}\lVert\mathcal{H}^{-1}\rVert_{q}^{2}\lVert\mathcal{H}_{s}\rVert_{q}\\
& =O_{p}(N^{4\epsilon})
\end{align*}
\end{proof}
\subsection{Bounds on $W$ terms}
\begin{lem}
\label{lem:W}Let Assumptions 1 and 2 hold. Then
\begin{align*}
\lVert W\rVert & =O_{p}(N) & \lVert W_{b}\rVert & =O_{p}(N^{4\epsilon})\\
\lVert\mathcal{S}'W_{s}\rVert & =O_{p}(1) & \lVert W_{s}\rVert_{q} &=O_{p}(N^{1/q})\\
\lVert W_{bb}\rVert & =O_{p}(N^{-1+8\epsilon}) & \lVert W_{bs}\rVert_{q} &=O_{p}(N^{-1+6\epsilon})\\
\lVert\mathcal{S}'W_{bs}\rVert & =O_{p}(N^{-1+4\epsilon}) & \lVert W_{ss}\rVert_{q} & =O_{p}(N^{6\epsilon})\\
\lVert W_{bss}\rVert_{q} & =O_{p}(N^{-1+12\epsilon})
\end{align*}
\end{lem}
\begin{proof}
The first result follows from the expression for $W$ and the result
in Lemma \ref{lem:F}.
\begin{align*}
\lVert W_{b}\rVert & \leq\lVert W^{-1}\rVert\lVert\partial_{\beta\beta\beta}\mathcal{L}\rVert+\lVert W^{-1}\rVert\lVert\mathcal{F}^{2,1}\rVert+\lVert\mathcal{F}_{b}\rVert\\
& =O_{p}(N^{4\epsilon})
\end{align*}
For the next result we have
\begin{align*}
\lVert\mathcal{S}'W_{s}\rVert & \leq\lVert W^{-1}\partial_{\beta\beta\beta}\mathcal{L}\rVert\lVert\mathcal{S}'\mathcal{H}^{-1}(\partial_{\beta\phi}\mathcal{L})\rVert+\lVert\mathcal{S}'\Big[\mathcal{H}^{-1}+W^{-1}\mathcal{G}\big](\partial_{\beta\beta\phi'}\mathcal{L})\rVert+\lVert\mathcal{S}'\mathcal{F}_{s}\rVert\\
& =O_{p}(1)
\end{align*}
Similarly, the remaining results simply follow from the definitions
of the terms and the bounds on individual components that have already
been derived.
\end{proof}
\subsection{Bounds on $\mathcal{H}$ terms}
\begin{lem}
\label{lem:H}Let Assumptions 1 and 2 hold. Then,\\ \\
(i) $\lVert\mathcal{H}_{b}\rVert =O_{p}(N^{-1+4\epsilon})$ \\
(ii) $\lVert\mathcal{H}_{s'}\mathcal{S}\rVert =O_{p}(N^{-\frac{1}{2}+2\epsilon})$ and $\lVert\mathcal{H}_{s}\rVert_{q}=O_{p}(N^{2\epsilon})$ \\
(iii) $\lVert\mathcal{H}_{bb}\rVert =O_{p}(N^{-2+8\epsilon})$ and $\lVert\mathcal{S}'\mathcal{H}^{-1}\mathcal{H}_{bb}\mathcal{H}^{-1}(\partial_{\beta\phi}\mathcal{L})\rVert=O_{p}(N^{-2+4\epsilon})$\\
(iv) $\lVert\sum_{g}\mathcal{S}'\mathcal{H}^{-1}\mathcal{H}_{bs_{g}}\mathcal{H}^{-1}\mathcal{S}\mathcal{S}_{g}\rVert =O_{p}(N^{-\frac{3}{2}+2\epsilon})$ and $\lVert\mathcal{H}_{bs}\rVert_{q}=O_{p}(N^{-1+4\epsilon})$\\
(v) $\lVert\mathcal{H}_{ss}\rVert_{q} =O_{p}(N^{4\epsilon})$\\
(vi) $\lVert\mathcal{S}'\mathcal{H}^{-1}\mathcal{H}_{bbb}\mathcal{H}^{-1}\mathcal{S}\rVert =O_{p}(N^{-3+14\epsilon})$ and $\lVert\mathcal{H}_{bbb}\rVert_{q}=O_{p}(N^{-3+12\epsilon})$\\
(vii) $\lVert\sum_{g}\mathcal{S}'\mathcal{H}^{-1}\mathcal{H}_{bbs_{g}}\mathcal{H}^{-1}\mathcal{S}\mathcal{S}_{g}\rVert =O_{p}(N^{-5/2+10\epsilon})$\\
(viii) $\lVert\mathcal{H}_{bss}\rVert_{q}=O_{p}(N^{-1+8\epsilon})$ and $\lVert\mathcal{H}_{bbs}\rVert_{q}=O_{p}(N^{-2+8\epsilon})$\\
(ix) $\lVert\mathcal{H}_{bsss}\rVert_{q} =O_{p}(N^{-1+12\epsilon})$ and $\lVert\mathcal{H}_{sss}\rVert_{q}=O_{p}(N^{6\epsilon})$
\end{lem}
\begin{proof}
For (i), application of Lemma \ref{lem:tensor_bounds} and the fact that $\frac{1}{N}W>0$
by assumption gives.
\begin{align*}
\lVert\mathcal{H}_{b}\rVert & \leq\lVert W^{-1}\rVert\big(\lVert\partial_{\beta\phi\phi'}\mathcal{L}\rVert+\lVert\sum_{f}(\partial_{\phi\phi'\phi_{f}}\mathcal{L})\big[\mathcal{H}^{-1}(\partial_{\beta\phi}\mathcal{L})\big]_{f}\rVert\big)\\
& =O_{p}(N^{-1})\big(O_{p}(N^{2\epsilon})+O_{p}(N^{4\epsilon})\big)=O_{p}(N^{-1+4\epsilon})
\end{align*}
For (ii), we have
\begin{align*}
\sum_{g}\mathcal{H}_{s_{g}}\mathcal{S}_{g} & =W^{-1}(\partial_{\beta\phi'}\mathcal{L})\mathcal{H}^{-1}\mathcal{S}(\partial_{\beta\phi\phi'}\mathcal{L})\\
& +\sum_{f}(\partial_{\phi\phi'\phi_{f}}\mathcal{L})[\mathcal{H}^{-1}\mathcal{S}]_{f}\\
& +W^{-1}\sum_{f}(\partial_{\phi\phi'\phi_{f}}\mathcal{L})\big[(\partial_{\beta\phi'}\mathcal{L})\mathcal{H}^{-1}\big]_{f}(\partial_{\beta\phi'}\mathcal{L})\mathcal{H}^{-1}\mathcal{S}
\end{align*}
and hence, using $\lVert W^{-1}\rVert =O_p(N^{-1})$, an applying Lemma \ref{lem:S6} (iii) and Lemma \ref{lem:element1} we get
\begin{align*}
\lVert\sum_{g}\mathcal{H}_{s_{g}}\mathcal{S}_{g}\rVert & \leq\lVert W^{-1}\rVert\lVert(\partial_{\beta\phi'}\mathcal{L})\mathcal{H}^{-1}\mathcal{S}\rVert\lVert\partial_{\beta\phi\phi'}\mathcal{L}\rVert+\lVert\sum_{f}(\partial_{\phi\phi'\phi_{f}}\mathcal{L})[\mathcal{H}^{-1}S]_{f}\rVert\\
& +\lVert W^{-1}\rVert\lVert\sum_{f}(\partial_{\phi\phi'\phi_{f}}\mathcal{L})\big[(\partial_{\beta\phi'}\mathcal{L})\mathcal{H}^{-1}\big]_{f}\rVert\lVert(\partial_{\beta\phi'}\mathcal{L})\mathcal{H}^{-1}\mathcal{S}\rVert\\
& =O_{p}(N^{-1+2\epsilon})+O_{p}(N^{-1/2+2\epsilon})+O_{p}(N^{-1+4\epsilon})\\
& =O_{p}(N^{-1/2+2\epsilon})
\end{align*}
The second statement follows from Lemma \ref{lem:S6} (iii) and the definition of $\mathcal{H}_s$.
For (iii), we have from the expression for $\mathcal{H}_{bb}$ that
\begin{align*}
\lVert \mathcal{H}_{bb}\rVert & \leq \lVert W^{-1}\rVert \lVert W_{b} \lVert \mathcal{H}_{b}+ \lVert W^{-1} \lVert \mathcal{E}_{b}^{1}\rVert\\
&+ \lVert W^{-2}\rVert \lVert \sum_{f}(\partial_{\beta\phi\phi'\phi_{f}}\mathcal{L})\big[\mathcal{H}^{-1}(\partial_{\beta\phi}\mathcal{L})\big]_{f}\rVert\\
& +\lVert W^{-2} \rVert \lVert \sum_{e,f}(\partial_{\phi\phi'\phi_{e}\phi_{f}}\mathcal{L})\big[\mathcal{H}^{-1}(\partial_{\beta\phi}\mathcal{L})\big]_{e}\big[\mathcal{H}^{-1}(\partial_{\beta\phi}\mathcal{L})\big]_{f}\rVert\\
& +\lVert W^{-1}\rVert \lVert\sum_{f}(\partial_{\phi\phi'\phi_{f}}\mathcal{L})\big[\mathcal{H}^{-1}\mathcal{H}_{b}\mathcal{H}^{-1}(\partial_{\beta\phi}\mathcal{L})\big]_{f}\rVert\\
& +\lVert W^{-2} \rVert\lVert\sum_{f}(\partial_{\phi\phi'\phi_{f}}\mathcal{L})\big[\mathcal{H}^{-1}(\partial_{\beta\beta\phi}\mathcal{L})\big]_{f}\rVert\\
& +\lVert W^{-2}\rVert \lVert\sum_{f}(\partial_{\phi\phi'\phi_{f}}\mathcal{L})\big[\mathcal{H}^{-1}\mathcal{E}^{1}\mathcal{H}^{-1}(\partial_{\beta\phi}\mathcal{L})\big]_{f}\rVert\\
&=O_p(N^{-2+8\epsilon})
\end{align*}
where the final line comes frrom applying Lemmas \ref{lem:tensors2} and \ref{lem:tensors2} as well as \ref{lem:W}. Similarly, each remaining term can be bound using the lemmas in Section \ref{sec:basic_lemmas} and this section, the expressions given for each term and application of the Cauchy-Schwarz and triangle inequalities.
\end{proof}
\subsection{Bounds on $\mathcal{G}$ terms}
\begin{lem}
\label{lem:G}Let Assumptions 1 and 2 hold. Then,$s+t \leq 5$\\ \\
(i) $\lVert\mathcal{S}'\mathcal{G}^{s,t}\mathcal{S}\rVert =O_{p}(1)$\\
(ii) $\lVert\mathcal{S}'\mathcal{G}_{b}^{s,t}\mathcal{S}\rVert =O_{p}(N^{-1})$\\
(iii) $\lVert\sum_{g}\mathcal{S}'\mathcal{G}_{s_{g}}^{s,t}\mathcal{S}\mathcal{S}_{g}\rVert =O_{p}(N^{-\frac{1}{2}+4\epsilon})$\\
(iv) $\lVert\mathcal{S}'\mathcal{G}_{bb}\mathcal{S}\rVert =O_{p}(N^{-2+4\epsilon})$\\
(v) $\lVert\sum_{g}\mathcal{S}'\mathcal{G}_{bs_{g}}\mathcal{S}\mathcal{S}_{g}\rVert =O_{p}(N^{-\frac{3}{2}+4\epsilon})$\\
(vi) $\lVert\mathcal{S}'\mathcal{G}_{bbb}\mathcal{S}\rVert =O_{p}(N^{-\frac{5}{2}+14\epsilon})$
\end{lem}
\begin{proof}
(i) Writing $\lVert\mathcal{S}'\mathcal{G}^{s,t}\mathcal{S}\rVert\leq\lVert\mathcal{S}'\mathcal{H}^{-1}(\partial_{\beta^s\phi}\mathcal{L})\rVert\lVert\mathcal{S}'\mathcal{H}^{-1}(\partial_{\beta^t\phi}\mathcal{L})\rVert$,
the result follows from Lemma \ref{lem:element1}. For (ii) we have
\begin{align*}
\lVert\mathcal{S}'\mathcal{G}_{b}\mathcal{S}\rVert & \leq\lVert\mathcal{S}'\mathcal{H}^{-1}\mathcal{H}_{b}\mathcal{G}\mathcal{S}\rVert+\lVert\mathcal{S}'\mathcal{G}\mathcal{H}_{b}\mathcal{H}^{-1}\mathcal{S}\rVert+\lVert W^{-1}\mathcal{S}'\mathcal{G}^{2,1}\mathcal{S}\rVert\\
& +\lVert W^{-1}\mathcal{S}'\mathcal{H}^{-1}\mathcal{E}\mathcal{G}\mathcal{S}\rVert+\lVert W^{-1}\mathcal{S}'\mathcal{G}^{1,2}\mathcal{S}\rVert+\lVert W^{-1}\mathcal{S}'\mathcal{G}\mathcal{E}\mathcal{H}^{-1}\mathcal{S}\rVert\\
& =O_{p}(N^{-1})
\end{align*}
where the final line applies results from Lemma \ref{lem:element1} (we show for $s=t=1$ here but results apply for $s,t$ as in the lemma),
e.g.
\begin{align*}
\rVert\mathcal{S}'\mathcal{H}^{-1}\mathcal{H}_{b}\mathcal{G}\mathcal{S}\rVert & \leq\rVert\mathcal{S}'\mathcal{H}^{-1}\mathcal{H}_{b}\mathcal{H}^{-1}(\partial_{\beta\phi'}\mathcal{L})\rVert\lVert(\partial_{\beta\phi'}\mathcal{L})\mathcal{H}^{-1}\mathcal{S}\rVert\\
& \leq\rVert\mathcal{S}'\mathcal{H}^{-1}\mathcal{\bar{H}}_{b}\mathcal{H}^{-1}(\partial_{\beta\phi'}\mathcal{L})\rVert O_{p}(1)+o_{p}(1)\\
& =O_{p}(N^{-1})
\end{align*}
For (iii)
\begin{align*}
\lVert\sum_{g}\mathcal{S}'\mathcal{G}_{s_{g}}\mathcal{S}\mathcal{S}_{g}\rVert & \leq\lVert\sum_{g}\mathcal{S}'\mathcal{H}^{-1}\mathcal{H}_{s_{g}}\mathcal{G}\mathcal{S}\mathcal{S}_{g}\rVert+\lVert\sum_{g}\mathcal{S}'\mathcal{G}\mathcal{H}_{s_{g}}\mathcal{H}^{-1}\mathcal{S}\mathcal{S}_{g}\rVert\\
& +\lVert W^{-1}\mathcal{S}'(\mathcal{G}^{2,1}+\mathcal{G}^{1,2})\mathcal{S}\rVert\lVert\mathcal{S}'\mathcal{H}^{-1}(\partial_{\beta\phi}\mathcal{L})\rVert\\
& +\lVert\mathcal{S}'\mathcal{H}^{-1}\mathcal{E}^{1}\mathcal{H}^{-1}\mathcal{S}\lVert\rVert(\partial_{\beta^{t}\phi'}\mathcal{L})\mathcal{H}^{-1}\mathcal{S}\rVert\\
& +\lVert\mathcal{S}'\mathcal{H}^{-1}(\partial_{\beta^{s}\phi}\mathcal{L})\rVert\lVert\mathcal{S}'\mathcal{H}^{-1}\mathcal{E}^{1}\mathcal{H}^{-1}\mathcal{S}\rVert\\
& +\lVert W^{-1}\mathcal{S}'\mathcal{H}^{-1}\mathcal{E}^{1}\mathcal{G}\mathcal{S}\rVert\lVert\mathcal{S}'\mathcal{H}^{-1}(\partial_{\beta\phi}\mathcal{L})\rVert\\
& +\lVert W^{-1}\mathcal{S}'\mathcal{G}\mathcal{E}^{1}\mathcal{H}^{-1}\mathcal{S}\rVert\lVert\mathcal{S}'\mathcal{H}^{-1}(\partial_{\beta\phi}\mathcal{L})\rVert\\
& =O_{p}(N^{2\epsilon})+O_{p}(N^{2\epsilon})+O_{p}(N^{-1})+O_{p}(1)\\
& =O_{p}(N^{2\epsilon})
\end{align*}
by application of Lemma \ref{lem:element1} and the bounds on individual
components, e.g.
\begin{align*}
\lVert\sum_{g}\mathcal{S}'\mathcal{H}^{-1}\mathcal{H}_{s_{g}}\mathcal{G}\mathcal{S}\mathcal{S}_{g}\rVert & \leq N^{1-\frac{2}{q}}\lVert\mathcal{H}_{s}\rVert_{q}\lVert\mathcal{H}^{-1}\rVert_{q}\lVert\mathcal{S}\rVert_{q}^{3}\\
& =O_{p}(N^{-\frac{1}{2}+\frac{1}{q}+2\epsilon})=O_{p}(N^{-\frac{1}{2}+4\epsilon})
\end{align*}
The remaining components can be bounded similarly using Lemmas \ref{lem:element1} to \ref{lem:tensors2}, but are not shown here due to the length of the expressions for second and third order derivatives of $\mathcal{G}$.
\end{proof}
\section{\label{sec:full_expansion} Bounds for expansion terms}
Finally, we make use of the bounds in the previous section to give bounds on the asymptotic expansion terms, which justifies the expansion in \ref{sec:beta_expansion}. All results follow through applcation of the expressions derived in Section \ref{sec:expressions_ind},
along with the bounds from Section \ref{sec:bounds_ind}, and application
of the Cauchy-Schwarz/triangle inequalities.
\subsection{Third derivative terms}
\begin{align*}
\lVert\partial_{bbb}\mathcal{L}^{*}\rVert & \leq\lVert W^{-2}\rVert\lVert W_{b}\rVert\\
& =O_{p}(N^{-2+4\epsilon})\\
\lVert(\partial_{bbs}\mathcal{L}^{*})\mathcal{S}\rVert & \leq\lVert W^{-2}\rVert\lVert W_{s}\mathcal{S}\rVert\\
& =O_{p}(N^{-2})\\
\lVert\mathcal{S}'(\partial_{bss'}\mathcal{L}^{*})\mathcal{S}\rVert & \leq\lVert\mathcal{S}'\mathcal{H}^{-1}\mathcal{H}_{b}\mathcal{H}^{-1}\mathcal{S}\rVert-\lVert W^{-2}\rVert\lVert W_{b}\rVert\lVert\mathcal{S}'\mathcal{G}\mathcal{S}\rVert+\lVert W^{-1}\rVert\lVert\mathcal{S}'\mathcal{G}_{b}\mathcal{S}\rVert\\
& =O_{p}(N^{-1+4\epsilon})+O_{p}(N^{-2+4\epsilon})+O_{p}(N^{-\frac{3}{2}+4\epsilon})\\
& =O_{p}(N^{-1+4\epsilon})\\
\lVert\sum_{g}\mathcal{S}'(\partial_{ss's_{g}}\mathcal{L}^{*})\mathcal{S}\mathcal{S}_{g}\rVert & =\lVert\sum_{g}\mathcal{S}'\mathcal{H}^{-1}\mathcal{H}_{s_{g}}\mathcal{H}^{-1}\mathcal{S}\mathcal{S}_{g}\rVert-\lVert W^{-2}\rVert\lVert\sum_{g}W_{s_{g}}\mathcal{S}_{g}\rVert\lVert\mathcal{S}'\mathcal{G}\mathcal{S}\rVert \\
&+\lVert W^{-1}\rVert\lVert\sum_{g}\mathcal{S}'\mathcal{G}_{s_{g}}\mathcal{S}\mathcal{S}_{g}\rVert\\
& =O_{p}(N^{-1/2+2\epsilon})+O_{p}(N^{-3/2+2\epsilon})+O_{p}(N^{-1+2\epsilon})\\
& =O_{p}(N^{-1/2+2\epsilon})
\end{align*}
\subsection{Fourth derivative terms}
Using the bounds provided in Section \ref{sec:bounds_ind} we find
\begin{align*}
\lVert\partial_{b^{4}}\mathcal{L}^{*}\rVert & \leq2\lVert W^{-3}\rVert\lVert W_{b}^{2}\rVert+\lVert W^{-2}\lVert\lVert W_{bb}\rVert\\
& =O_{p}(N^{-3+8\epsilon})=o_{p}(N^{-2})\\
\lVert(\partial_{bbbs'}\mathcal{L}^{*})\mathcal{S}\rVert & \leq2\lVert W^{-3}\rVert\lVert W_{b}\rVert\lVert W_{s'}\mathcal{S}\rVert+\lVert W^{-2}\rVert\lVert W_{bs'}\mathcal{S}\rVert\\
& =O_{p}(N^{-\frac{5}{2}+4\epsilon})+O_{p}(N^{-3+4\epsilon})\\
& =O_{p}(N^{-\frac{5}{2}+4\epsilon})=o_{p}(N^{-2})\\
\lVert\mathcal{S}'(\partial_{bbss'}\mathcal{L}_{(1)}^{*})\mathcal{S}\rVert & \leq\lVert2\mathcal{S}'\mathcal{H}^{-1}\mathcal{H}_{b}\mathcal{H}^{-1}\mathcal{H}_{b}\mathcal{H}^{-1}\mathcal{S}\rVert+\lVert\mathcal{S}'\mathcal{H}^{-1}\mathcal{H}_{bb}\mathcal{H}^{-1}\mathcal{S}\rVert\\
& =O_{p}(N^{-2})+O_{p}(N^{-2+8\epsilon})\\
\lVert\mathcal{S}'(\partial_{bbss'}\mathcal{L}_{(2)}^{*})\mathcal{S}\rVert & \leq\lVert2\mathcal{S}'W^{-3}W_{b}^{2}\mathcal{G}\mathcal{S}\rVert+\lVert\mathcal{S}'W^{-2}W_{bb}\mathcal{G}\mathcal{S}\rVert+\lVert\mathcal{S}'W^{-2}W_{b}\mathcal{G}_{b}\mathcal{S}\rVert\\
& +\lVert\mathcal{S}'W^{-1}\mathcal{G}_{bb}\mathcal{S}\rVert\\
& =O_{p}(N^{-3+8\epsilon})+O_{p}(N^{-3+8\epsilon})+O_{p}(N^{-3+4\epsilon})=o_{p}(N^{-2})\\
\lVert\sum_{g}\mathcal{S}'\partial_{bss's_{g}}\mathcal{L}_{(1)}^{*}\mathcal{S}\mathcal{S}_{g}\rVert & \leq\lVert\sum_{g}\mathcal{S}'\mathcal{H}^{-1}\mathcal{H}_{b}\mathcal{H}^{-1}\mathcal{H}_{s_{g}}\mathcal{H}^{-1}\mathcal{S}\mathcal{S}_{g}\rVert+\lVert\sum_{g}\mathcal{S}'\mathcal{H}^{-1}\mathcal{H}_{s_{g}}\mathcal{H}^{-1}\mathcal{H}_{b}\mathcal{H}^{-1}\mathcal{S}\mathcal{S}_{g}\rVert\\
& +\lVert\sum_{g}\mathcal{S}'\mathcal{H}^{-1}\mathcal{H}_{bs_{g}}\mathcal{H}^{-1}\mathcal{S}\mathcal{S}_{g}\rVert\\
& =O_{p}(N^{-\frac{3}{2}+4\epsilon})+O_{p}(N^{-\frac{3}{2}+2\epsilon})+O_{p}(N^{-\frac{3}{2}+2\epsilon})\\
\lVert\sum_{g}\mathcal{S}'\partial_{bss's_{g}}\mathcal{L}_{(2)}^{*}\mathcal{S}\mathcal{S}_{g}\rVert & \leq\lVert2\sum_{g}W^{-3}W_{b}W_{s_{g}}\mathcal{S}'\mathcal{G}\mathcal{S}\mathcal{S}_{g}\rVert+\lVert\sum_{g}W^{-2}W_{bs_{g}}\mathcal{S}'\mathcal{G}\mathcal{S}\mathcal{S}_{g}\rVert\\
& +\lVert\sum_{g}W^{-2}W_{s_{g}}\mathcal{S}'\mathcal{G}_{b}\mathcal{S}\mathcal{S}_{g}\rVert\\
& +\lVert\sum_{g}W^{-2}W_{b}\mathcal{S}'\mathcal{G}_{s_{g}}\mathcal{S}\mathcal{S}_{g}\rVert+\lVert\sum_{g}W^{-1}\mathcal{S}'\mathcal{G}_{bs_{g}}\mathcal{S}\mathcal{S}_{g}\rVert\\
& =O_{p}(N^{-\frac{5}{2}+6\epsilon})+O_{p}(N^{-3+4\epsilon})+O_{p}(N^{-\frac{5}{2}+2\epsilon})\\
& +O_{p}(N^{-\frac{5}{2}+6\epsilon})+O_{p}(N^{-\frac{5}{2}+6\epsilon})\\
& =o_{p}(N^{-2})
\end{align*}
\subsection{Fifth derivative terms}
Application of the bounds provided in Section \ref{sec:bounds_ind}
gives
\begin{align*}
\lVert\mathcal{S}'\partial_{bbbss'}\mathcal{L}_{(1)}^{*}\mathcal{S}\rVert & \leq\lVert6\mathcal{S}'\mathcal{H}^{-1}\mathcal{H}_{b}\mathcal{H}^{-1}\mathcal{H}_{b}\mathcal{H}^{-1}\mathcal{H}_{b}\mathcal{H}^{-1}\mathcal{S}\rVert\\
& +3\lVert\mathcal{S}'\mathcal{H}^{-1}\mathcal{H}_{bb}\mathcal{H}^{-1}\mathcal{H}_{b}\mathcal{H}^{-1}\mathcal{S}\rVert+3\lVert\mathcal{S}'\mathcal{H}^{-1}\mathcal{H}_{b}\mathcal{H}^{-1}\mathcal{H}_{bb}\mathcal{H}^{-1}\mathcal{S}\rVert\\
& +\lVert\mathcal{S}'\mathcal{H}^{-1}\mathcal{H}_{bbb}\mathcal{H}^{-1}\mathcal{S}\rVert\\
& =O_{p}(N^{-3})+O_{p}(N^{-3+12\epsilon})+O_{p}(N^{-3+14\epsilon})=o_{p}(N^{-2})
\end{align*}
\begin{align*}
\lVert\sum_{g}\mathcal{S}'\partial_{bbss's_{g}}\mathcal{L}_{(1)}^{*}\mathcal{S}\mathcal{S}_{g}\rVert & =6\lVert\sum_{g}\mathcal{S}'\mathcal{H}^{-1}\mathcal{H}_{s_{g}}\mathcal{H}^{-1}\mathcal{H}_{b}\mathcal{H}^{-1}\mathcal{H}_{b}\mathcal{H}^{-1}\mathcal{S}\mathcal{S}_{g}\rVert\\
& +4\lVert\sum_{g}\mathcal{S}'\mathcal{H}^{-1}\mathcal{H}_{bs_{g}}\mathcal{H}^{-1}\mathcal{H}_{b}\mathcal{H}^{-1}\mathcal{S}\mathcal{S}_{g}\rVert\\
& +2\lVert\sum_{g}\mathcal{S}'\mathcal{H}^{-1}\mathcal{H}_{s_{g}}\mathcal{H}^{-1}\mathcal{H}_{bb}\mathcal{H}^{-1}\mathcal{S}\mathcal{S}_{g}\rVert+\\
& +\lVert\sum_{g}\mathcal{S}'\mathcal{H}^{-1}\mathcal{H}_{bbs_{g}}\mathcal{H}^{-1}\mathcal{S}\mathcal{S}_{g}\rVert\\
& =O_{p}(N^{-\frac{5}{2}+10\epsilon})+O_{p}(N^{-\frac{5}{2}+6\epsilon})+O_{p}(N^{-\frac{5}{2}+10\epsilon})\\
& =o_{p}(N^{-2})
\end{align*}
\begin{align*}
\lVert\sum_{g,h}\mathcal{S}'\partial_{bss's_{g}s_{h}}\mathcal{L}_{(1)}^{*}\mathcal{S}\mathcal{S}_{g}\mathcal{S}_{h}\rVert & \leq6\lVert\sum_{g,h}\mathcal{S}'\mathcal{H}^{-1}\mathcal{H}_{s_{h}}\mathcal{H}^{-1}\mathcal{H}_{b}\mathcal{H}^{-1}\mathcal{H}_{s_{g}}\mathcal{H}^{-1}\mathcal{S}\mathcal{S}_{g}\mathcal{S}_{h}\rVert\\
& +4\lVert\sum_{g,h}\mathcal{S}'\mathcal{H}^{-1}\mathcal{H}_{bs_{h}}\mathcal{H}^{-1}\mathcal{H}_{s_{g}}\mathcal{H}^{-1}\mathcal{S}\mathcal{S}_{g}\mathcal{S}_{h}\rVert\\
& +2\lVert\sum_{g,h}\mathcal{S}'\mathcal{H}^{-1}\mathcal{H}_{b}\mathcal{H}^{-1}\mathcal{H}_{s_{g}s_{h}}\mathcal{H}^{-1}\mathcal{S}\mathcal{S}_{g}\mathcal{S}_{h}\rVert\\
& +\lVert\sum_{g,h}\mathcal{S}'\mathcal{H}^{-1}\mathcal{H}_{bs_{g}s_{h}}\mathcal{H}^{-1}\mathcal{S}\mathcal{S}_{g}\mathcal{S}_{h}\rVert\\
& =O_{p}(N^{-2+8\epsilon})+O_{p}(N^{-2+4\epsilon})+O_{p}(N^{-2+10\epsilon})+O_{p}(N^{-2+10\epsilon})
\end{align*}
\subsection{Sixth derivative terms}
\begin{align*}
\lVert\sum_{f,g,h}\mathcal{S}'\partial_{bss's_{f}s_{g}s_{h}}\mathcal{L}_{(1)}^{*}\mathcal{S}\mathcal{S}_{f}\mathcal{S}_{g}\mathcal{S}_{h}\rVert & \leq C\Big(\lVert\sum_{f,g,h}\mathcal{S}'\mathcal{H}^{-1}\mathcal{H}_{s_{f}}\mathcal{H}^{-1}\mathcal{H}_{b}\mathcal{H}^{-1}\mathcal{H}_{s_{g}}\mathcal{H}^{-1}\mathcal{H}_{s_{h}}\mathcal{H}^{-1}\mathcal{S}\mathcal{S}_{f}\mathcal{S}_{g}\mathcal{S}_{h}\rVert\\
& +\lVert\sum_{f,g,h}\mathcal{S}'\mathcal{H}^{-1}\mathcal{H}_{bs_{f}}\mathcal{H}^{-1}\mathcal{H}_{s_{g}}\mathcal{H}^{-1}\mathcal{H}_{s_{h}}\mathcal{H}^{-1}\mathcal{S}\mathcal{S}_{f}\mathcal{S}_{g}\mathcal{S}_{h}\rVert\\
& +\lVert\sum_{f,g,h}\mathcal{S}'\mathcal{H}^{-1}\mathcal{H}_{b}\mathcal{H}^{-1}\mathcal{H}_{s_{f}s_{g}}\mathcal{H}^{-1}\mathcal{H}_{s_{h}}\mathcal{H}^{-1}\mathcal{S}\mathcal{S}_{f}\mathcal{S}_{g}\mathcal{S}_{h}\rVert\\
& +\lVert\sum_{f,g,h}\mathcal{S}'\mathcal{H}^{-1}\mathcal{H}_{bs_{g}s_{f}}\mathcal{H}^{-1}\mathcal{H}_{s_{h}}\mathcal{H}^{-1}\mathcal{S}\mathcal{S}_{f}\mathcal{S}_{g}\mathcal{S}_{h}\rVert\\
& +\lVert\sum_{f,g,h}\mathcal{S}'\mathcal{H}^{-1}\mathcal{H}_{bs_{g}}\mathcal{H}^{-1}\mathcal{H}_{s_{h}s_{f}}\mathcal{H}^{-1\mathcal{S}\mathcal{S}_{f}\mathcal{S}_{g}\mathcal{S}_{h}}\rVert\\
& +\lVert\sum_{f,g,h}\mathcal{S}'\mathcal{H}^{-1}\mathcal{H}_{b}\mathcal{H}^{-1}\mathcal{H}_{s_{f}s_{g}s_{h}}\mathcal{H}^{-1}\mathcal{S}\mathcal{S}_{f}\mathcal{S}_{g}\mathcal{S}_{h}\rVert\\
& +\lVert\sum_{f,g,h}\mathcal{S}'\mathcal{H}^{-1}\mathcal{H}_{s_{f}s_{g}s_{h}}\mathcal{H}^{-1}\mathcal{S}\mathcal{S}_{f}\mathcal{S}_{g}\mathcal{S}_{h}\rVert\Big)\\
& =o_{p}(N^{-2})
\end{align*}
\section{Jackknife expansion } \label{SA:expansion_betak}
\subsection{\label{subsec:leaveout_expansion_beta}Asymptotic expansion in leave-out
samples}
In order to derive results for the jackknife estimator for $\beta$, we first establish
the asymptotic expansion for $\widehat{\beta}_{(k)}$,
i.e. the leave-out sample parameter estimate. The expansions can
be derived identically to the full sample estimators, replacing $\ell_{ij}$
in the objective function with $\frac{N-1}{N-2}\ell_{ij}1_{ij}^{k}$
where $1_{ij}^{k}$ is an indicator variable that is equal to one
whenever the observation $(i,j)$ is included in the $k$-th leave-out
sample, and is zero when that observation has been dropped. In comparing
the asymptotic expansions of the full-sample and leave-out sample
estimators, we will replace $\mathcal{H}^{-1}$ and $W_{N}^{-1}$
terms with their conditional expectations. For this purpose, we first
state a new version of Lemma \ref{lem:H_tilde} for the leave-out
sample.
\begin{lem}
\label{lem:HW_approx_leaveout}Let Assumptions 1 and 2 hold, and let
$\mathcal{H}_{(k)}=-\frac{1}{N-2}\sum_{i}\sum_{j\ne i}\partial_{\phi\phi'}\ell_{ij}1_{ij}^{k}$.
Then, for $s=0,1,2,3$, $t=2,3,4,5$ and $s+t\leq6$
\begin{align*}
\lVert\mathcal{H}_{(k)}-\bar{\mathcal{H}}\rVert & =O_{p}(N^{-\frac{1}{2}+2\epsilon})\\
\lVert\partial_{\beta^{s}\phi^{t}}\mathcal{L}_{(k)}-\partial_{\beta^{s}\phi^{t}}\bar{\mathcal{L}}\rVert & =O_{p}(N^{-\frac{1}{2}+2\epsilon})
\end{align*}
and
\begin{align*}
\lVert\mathcal{H}_{(k)}^{-1}-\bar{\mathcal{H}}^{-1}(\tilde{\mathcal{H}}_{(k)}\bar{\mathcal{H}}^{-1})^{k}\rVert & =O_{p}(N^{-\frac{k+1}{2}+2(k+1)\epsilon})\\
\lVert W_{N,(k)}^{-1}-\bar{W}_{N}^{-1}(\tilde{W}_{N,(k)}\bar{W}_{N}^{-1})^{k}\rVert & =O_{p}(N^{-\frac{k+1}{2}+2(k+1)\epsilon})
\end{align*}
\end{lem}
\begin{proof}
Following the proof for the full-sample matrix, first decompose the
second derivative matrix as
\[
\lVert\mathcal{H}_{(k)}-\bar{\mathcal{H}}\rVert\leq\lVert\partial_{\alpha\alpha}\mathcal{L}_{(k)}-\partial_{\alpha\alpha}\bar{\mathcal{L}}\rVert+2\lVert\partial_{\alpha\gamma}\mathcal{L}_{(k)}-\partial_{\alpha\gamma}\bar{\mathcal{L}}\rVert+\lVert\partial_{\gamma\gamma}\mathcal{L}_{(k)}-\partial_{\gamma\gamma}\bar{\mathcal{L}}\rVert
\]
Define $\partial_{\pi^{2}}\tilde{\ell}_{(k),ij}=\frac{N-1}{N-2}\partial_{\pi^{2}}\ell_{ij}1_{ij}^{k}-\partial_{\pi^{2}}\bar{\ell}_{ij}$.
To bound the first term, first note that
\begin{align*}
\bar{E}\Big[\max_{i}\Big(\frac{1}{N-1}\sum_{j\ne i}\partial_{\pi^{2}}\tilde{\ell}_{(k),ij}\Big)^{q}\Big] & \leq\frac{1}{(N-1)^{q}}\sum_{i}\bar{E}\Big[\Big(\sum_{j\ne i}\partial_{\pi^{2}}\tilde{\ell}_{(k),ij}\Big)^{q}\Big]\\
& \leq C\frac{1}{(N-1)^{q}}\sum_{i}\bar{E}\Big[\Big(\sum_{j\ne i}\partial_{\pi^{2}}\tilde{\ell}_{ij}1_{ij}^{k}\Big)^{q}\Big] \\
&+C\frac{1}{(N-1)^{q}}\sum_{i}\bar{E}\Big[\Big(\sum_{j\ne i}\partial_{\pi^{2}}\bar{\ell}_{ij}(1-1_{ij}^{k})\Big)^{q}\Big]\\
& \leq C\frac{1}{(N-1)^{q}}\sum_{i}\bar{E}\Big[\Big(\partial_{\pi^{2}}\bar{\ell}_{ii_{k}^{*}}\Big)^{q}\Big]+O_{p}(N^{1-q/2})\\
& =O_{p}(N^{1-q/2})
\end{align*}
where $i_{k}^{*}$ is the receiver such that $1_{ii_{k}^{*}}^{k}=0$.
Then we can apply the same steps as in the proof of Lemma \ref{lem:H_tilde}
to give
\begin{align*}
\bar{E}\lVert\partial_{\alpha\alpha}\mathcal{L}_{(k)}-\partial_{\alpha\alpha}\bar{\mathcal{L}}\rVert^{q} & =O_{p}(N^{1-q/2})
\end{align*}
and hence $\lVert\partial_{\alpha\alpha}\mathcal{L}_{(k)}-\partial_{\alpha\alpha}\bar{\mathcal{L}}\rVert=O_{p}(N^{-\frac{1}{2}+\frac{1}{q}})$
and similarly for $\lVert\partial_{\gamma\gamma}\mathcal{L}_{(k)}-\partial_{\gamma\gamma}\bar{\mathcal{L}}\rVert$.
The bound $\lVert\partial_{\alpha\gamma}\mathcal{L}_{(k)}-\partial_{\alpha\gamma}\bar{\mathcal{L}}\rVert=O_{p}(N^{-\frac{1}{2}+\frac{1}{q}})$
follows similarly, which gives $\lVert\mathcal{H}_{(k)}-\bar{\mathcal{H}}\rVert=O_{p}(N^{-\frac{1}{2}+\frac{1}{q}})=O_{p}(N^{-\frac{1}{2}+2\epsilon})$.
The second result follows in the same way. We similarly show that
\begin{align*}
\lVert\partial_{\beta\phi'}\tilde{\mathcal{L}}_{(k)}\rVert & \leq\lVert\frac{1}{N-1}\sum_{i}\sum_{j\ne i}\partial_{\beta\phi'}\tilde{\ell}_{ij}1_{ij}^{k}\rVert+\lVert\frac{1}{N-1}\sum_{i}\partial_{\beta\phi'}\bar{\ell}_{(k),ii_{k}^{*}}\rVert\\
& =O_{p}(\lVert\partial_{\beta\phi}\tilde{\mathcal{L}}\rVert)
\end{align*}
since $\big[\frac{1}{N-1}\sum_{i}\partial_{\beta\phi'}\bar{\ell}_{(k),ii_{k}^{*}}\big]_{s}=\frac{1}{N-1}\partial_{\beta\pi}\bar{\ell}_{(k),ss_{k}^{*}}$
for $s=1,\dots,N$ and $\frac{1}{N-1}\partial_{\beta\pi}\bar{\ell}_{(k),s_{k}^{\dagger}s}$
(where $s_{k}^{\dagger}$ is the sender for which $(s_{k}^{\dagger},s)$
is dropped in leave-out sample $k$) for $s=N+1,\dots,2N$. Then,
using these results and applying the same steps as in Lemma \ref{lem:H1_approx},
we can show that the same approximations hold in the leave-out samples.
\end{proof}
Using this result, the first-order expansion for $\widehat{\beta}_{(k)}$
can be shown to be given by
\begin{align*}
NW_{N,(k)}(\widehat{\beta}_{(k)}-\beta) & =(\partial_{\beta}\mathcal{L}_{(k)})+(\partial_{\beta\phi'}\mathcal{L}_{(k)})\mathcal{H}_{(k)}^{-1}\mathcal{S}_{(k)}+\frac{1}{2}\mathcal{S}_{(k)}'\mathcal{H}_{(k)}^{-1}(\partial_{\beta\phi\phi'}\mathcal{L}_{(k)})\mathcal{H}_{(k)}^{-1}\mathcal{S}_{(k)}\\
& \quad+\frac{1}{2}\mathcal{S}_{(k)}'\mathcal{H}_{(k)}^{-1}\sum_{f}(\partial_{\phi\phi'\phi_{f}}\mathcal{L}_{(k)})\big[\mathcal{H}_{(k)}^{-1}(\partial_{\beta\phi}\mathcal{L}_{(k)})\big]_{f}\mathcal{H}_{(k)}^{-1}\mathcal{S}_{(k)}+o_{p}(1)\\
& =(\partial_{\beta}\mathcal{L}_{(k)})+(\partial_{\beta\phi'}\mathcal{\bar{L}})\mathcal{\bar{H}}^{-1}\mathcal{S}_{(k)}+(\partial_{\beta\phi'}\mathcal{\bar{L}})\mathcal{\bar{H}}^{-1}\tilde{\mathcal{H}}_{(k)}\mathcal{\bar{H}}^{-1}\mathcal{S}_{(k)}\\
& \quad+\frac{1}{2}\mathcal{S}_{(k)}'\bar{\mathcal{H}}^{-1}(\partial_{\beta\phi\phi'}\bar{\mathcal{L}})\bar{\mathcal{H}}^{-1}\mathcal{S}_{(k)}\\
& \quad+\frac{1}{2}\mathcal{S}_{(k)}'\bar{\mathcal{H}}^{-1}\sum_{f}(\partial_{\phi\phi'\phi_{f}}\bar{\mathcal{L}})\big[\bar{\mathcal{H}}^{-1}(\partial_{\beta\phi}\bar{\mathcal{L}})\big]_{f}\bar{\mathcal{H}}^{-1}\mathcal{S}_{(k)}+o_{p}(1)
\end{align*}
Since $\lVert W_{N,(k)}-\bar{W}_{N}\rVert=o_{p}(1)$ by Lemma \ref{lem:HW_approx_leaveout},
the same expansion up to first order applies to $N\bar{W}_{N}(\widehat{\beta}_{(k)}-\beta)$.
\subsection{Jackknife results for higher-order terms}
In the main appendix, the first-order terms of the jackknife estimator
$\widehat{\beta}_{J}$ are derived. Here we show that the remaining
terms up to $O_{p}(N^{-1})$ in the expansion for $N(\widehat{\beta}_{J}-\beta)$
are in fact $o_{p}(1)$ and so do not affect the asymptotic distribution
of the estimator. As can be seen in Section \ref{sec:full_expansion},
there is an extremely large number of terms in the asymptotic expansion
that must be considered. However, inspection of the terms shows that
they share a common structure, that is, they can be expressed as V-statistics
of a certain order that depend on sums of the derivatives of $\ell_{ij}$
up to sixth order. Here we prove the result for an example term, and
discuss how this same proof can be used to show that the remaining
terms will also be $o_{p}(1)$.
Consider the expansion term $\mathcal{S}'(\partial_{bbss'}\mathcal{L}_{(1)}^{*})\mathcal{S}\mathcal{S}_{\beta}$,
which contains the term
\[
\mathcal{S}'\mathcal{H}^{-1}\mathcal{H}_{bb}\mathcal{H}^{-1}\mathcal{S}\mathcal{S}_{\beta}
\]
Expanding out this terms gives
\begin{align*}
\mathcal{S}'\mathcal{H}^{-1}\mathcal{H}_{bb}\mathcal{H}^{-1}\mathcal{S}\mathcal{S}_{\beta} & =-W^{-2}W_{b}\mathcal{S}'\mathcal{H}^{-1}\sum_{f}(\partial_{\phi\phi'\phi_{f}}\mathcal{L})\big[\mathcal{H}^{-1}(\partial_{\beta\phi}\mathcal{L})\big]_{f}\mathcal{H}^{-1}\mathcal{S}\mathcal{S}_{\beta}\\
& -W^{-2}W_{b}\mathcal{S}'\mathcal{H}^{-1}\mathcal{E}\mathcal{H}^{-1}\mathcal{S}\mathcal{S}_{\beta}\\
& +W^{-2}\mathcal{S}'\mathcal{H}^{-1}\mathcal{E}^{2}\mathcal{H}^{-1}\mathcal{S}\mathcal{S}_{\beta}\\
& +W^{-2}\mathcal{S}'\mathcal{H}^{-1}\sum_{f}(\partial_{\beta\phi\phi'\phi_{f}}\mathcal{L})\big[\mathcal{H}^{-1}(\partial_{\beta\phi}\mathcal{L})\big]_{f}\mathcal{H}^{-1}\mathcal{S}\mathcal{S}_{\beta}\\
& -W^{-2}\mathcal{S}'\mathcal{H}^{-1}\sum_{f}(\partial_{\beta\phi\phi'\phi_{f}}\mathcal{L})\big[\mathcal{H}^{-1}(\partial_{\beta\phi}\mathcal{L})\big]_{f}\mathcal{H}^{-1}\mathcal{S}\mathcal{S}_{\beta}\\
& -W^{-2}\mathcal{S}'\mathcal{H}^{-1}\sum_{e,f}(\partial_{\phi\phi'\phi_{e}\phi_{f}}\mathcal{L})\big[\mathcal{H}^{-1}(\partial_{\beta\phi}\mathcal{L})\big]_{e}\big[\mathcal{H}^{-1}(\partial_{\beta\phi}\mathcal{L})\big]_{f}\mathcal{H}^{-1}\mathcal{S}\mathcal{S}_{\beta}\\
& -W^{-2}\mathcal{S}'\mathcal{H}^{-1}\sum_{f}(\partial_{\phi\phi'\phi_{f}}\mathcal{L})\big[\mathcal{H}^{-1}\mathcal{E}\mathcal{H}^{-1}(\partial_{\beta\phi}\mathcal{L})\big]_{f}\mathcal{H}^{-1}\mathcal{S}\mathcal{S}_{\beta}\\
& -W^{-2}\mathcal{S}'\mathcal{H}^{-1}\sum_{e,f}(\partial_{\phi\phi'\phi_{f}}\mathcal{L})\big[\mathcal{H}^{-1}(\partial_{\phi\phi'\phi_{e}}\mathcal{L})\mathcal{H}^{-1}(\partial_{\beta\phi}\mathcal{L})\big]_{f}\big[\mathcal{H}^{-1}(\partial_{\beta\phi}\mathcal{L})\big]_{e}\mathcal{H}^{-1}\mathcal{S}\mathcal{S}_{\beta}\\
& -W^{-2}\mathcal{S}'\mathcal{H}^{-1}\sum_{f}(\partial_{\phi\phi'\phi_{f}}\mathcal{L})\big[\mathcal{H}^{-1}(\partial_{\beta\beta\phi}\mathcal{L})\big]_{f}\mathcal{H}^{-1}\mathcal{S}\mathcal{S}_{\beta}\\
& -W^{-2}\mathcal{S}'\mathcal{H}^{-1}\sum_{f}(\partial_{\phi\phi'\phi_{f}}\mathcal{L})\big[\mathcal{H}^{-1}\mathcal{E}\mathcal{H}^{-1}(\partial_{\beta\phi}\mathcal{L})\big]_{f}\mathcal{H}^{-1}\mathcal{S}\mathcal{S}_{\beta}
\end{align*}
Take the first term in this expression. We can replace $W$ with $\bar{W}+(W-\bar{W})$,
and similarly for $W_{b}$, $\mathcal{H}^{-1}$ and $\partial_{\phi\phi'\phi_{f}}\mathcal{L}$
to give
\[
\bar{W}^{-2}\bar{W}_{b}\mathcal{S}'\bar{\mathcal{H}}^{-1}\sum_{f}(\partial_{\phi\phi'\phi_{f}}\bar{\mathcal{L}})\big[\bar{\mathcal{H}}^{-1}(\partial_{\beta\phi}\mathcal{L})\big]_{f}\mathcal{\bar{H}}^{-1}\mathcal{S}\mathcal{S}_{\beta}+o_{p}(N^{-2})
\]
This result holds identically for the leave-out samples, replacing
$\mathcal{S}$ with $\mathcal{S}_{(k)}$ and similarly for $\partial_{\beta\phi}\mathcal{L}$
and $\mathcal{S}_{\beta}$. Ignoring the $\bar{W}^{-2}\bar{W}_{b}$
term for the moment, which will not be affected by the jackknifing,
we can decompose the above sum further into the components of $\phi=(\alpha,\gamma)$,
for example the first of these terms would be
\begin{align*}
\mathcal{S}_{\alpha}' & \bar{\mathcal{H}}_{\alpha\alpha}^{-1}\sum_{f=1}^{N}(\partial_{\alpha\alpha'\alpha_{f}}\bar{\mathcal{L}})\big[\bar{\mathcal{H}}_{\alpha\alpha}^{-1}(\partial_{\beta\alpha}\mathcal{L})+\bar{\mathcal{H}}_{\alpha\gamma}^{-1}(\partial_{\beta\gamma}\mathcal{L})\big]_{f}\mathcal{\bar{H}}_{\alpha\alpha}^{-1}\mathcal{S}_{\alpha}\mathcal{S}_{\beta}\\
= & \mathcal{S}_{\beta}\sum_{i}\sum_{j}\sum_{s}\sum_{t}\mathcal{S}_{\alpha_{i}}[\bar{\mathcal{H}}_{\alpha\alpha}^{-1}]_{ij}[\partial_{\alpha\alpha'\alpha_{j}}\bar{\mathcal{L}}]_{jj}\big([\bar{\mathcal{H}}_{\alpha\alpha}^{-1}]_{jt}(\partial_{\beta\alpha_{t}}\mathcal{L})+[\bar{\mathcal{H}}_{\alpha\gamma}^{-1}]_{jt}(\partial_{\beta\gamma_{t}}\mathcal{L})\big)[\mathcal{\bar{H}}_{\alpha\alpha}^{-1}]_{js}\mathcal{S}_{\alpha_{s}}\\
= & \mathcal{S}_{\beta}\frac{1}{(N-1)^{3}}\sum_{j}\sum_{i,i'\ne i}\sum_{s,s'\ne s}\sum_{t,t'\ne t}[\bar{\mathcal{H}}_{\alpha\alpha}^{-1}]_{ij}[\mathcal{\bar{H}}_{\alpha\alpha}^{-1}]_{js}[\partial_{\alpha\alpha'\alpha_{j}}\bar{\mathcal{L}}]_{jj} \\
&\qquad\times (\partial_{\pi}\ell_{i,i'})(\partial_{\pi}\ell_{s,s'})\big([\bar{\mathcal{H}}_{\alpha\alpha}^{-1}]_{jt}(\partial_{\beta\pi}\ell_{t,t'})+[\bar{\mathcal{H}}_{\alpha\gamma}^{-1}]_{jt}(\partial_{\beta\pi}\ell_{t',t})\big)\\
= & \frac{1}{(N-1)^{4}}\sum_{j}\sum_{i,i'\ne i}\sum_{s,s'\ne s}\sum_{t,t'\ne t}\sum_{r,r'\ne r}[\bar{\mathcal{H}}_{\alpha\alpha}^{-1}]_{ij}[\mathcal{\bar{H}}_{\alpha\alpha}^{-1}]_{js}[\partial_{\alpha\alpha'\alpha_{j}}\bar{\mathcal{L}}]_{jj} \\
&\qquad\times (\partial_{\pi}\ell_{i,i'})(\partial_{\pi}\ell_{s,s'})\big([\bar{\mathcal{H}}_{\alpha\alpha}^{-1}]_{jt}(\partial_{\beta\pi}\ell_{t,t'})+[\bar{\mathcal{H}}_{\alpha\gamma}^{-1}]_{jt}(\partial_{\beta\pi}\ell_{t',t})\big)(\partial_{\beta}\ell_{r,r'})
\end{align*}
Define
\begin{align*}
Q_{ist}^{\alpha} & =\sum_{j}[\bar{\mathcal{H}}_{\alpha\alpha}^{-1}]_{ij}[\mathcal{\bar{H}}_{\alpha\alpha}^{-1}]_{js}[\partial_{\alpha\alpha'\alpha_{j}}\bar{\mathcal{L}}]_{jj}[\bar{\mathcal{H}}_{\alpha\alpha}^{-1}]_{jt}\\
Q_{ist}^{\gamma} & =\sum_{j}[\bar{\mathcal{H}}_{\alpha\alpha}^{-1}]_{ij}[\mathcal{\bar{H}}_{\alpha\alpha}^{-1}]_{js}[\partial_{\alpha\alpha'\alpha_{j}}\bar{\mathcal{L}}]_{jj}[\bar{\mathcal{H}}_{\alpha\gamma}^{-1}]_{jt}
\end{align*}
and note that, by Lemma \ref{lem:H_approx}, we have $\max_{i\ne s\ne t}Q_{ist}=O_{p}(N^{-2})$,
while the max such that two of $(i,s,t)$ are equal is $O_{p}(N^{-1})$
and $Q_{iii}=O_{p}(1)$. We can write the term now as
\begin{align*}
& \frac{1}{(N-1)^{4}}\sum_{i,i'\ne i}\sum_{s,s'\ne s}\sum_{t,t'\ne t}\sum_{r,r'\ne r}Q_{ist}^{\alpha}(\partial_{\pi}\ell_{i,i'})(\partial_{\pi}\ell_{s,s'})(\partial_{\beta\pi}\ell_{t,t'})(\partial_{\beta}\ell_{r,r'})\\
& +\frac{1}{(N-1)^{4}}\sum_{i,i'\ne i}\sum_{s,s'\ne s}\sum_{t,t'\ne t}\sum_{r,r'\ne r}Q_{ist}^{\gamma}(\partial_{\pi}\ell_{i,i'})(\partial_{\pi}\ell_{s,s'})(\partial_{\beta\pi}\ell_{t',t})(\partial_{\beta}\ell_{r,r'})
\end{align*}
which is the sum of two V-statistic type terms, of fourth order. We
now consider the effect of the jackknife operation on these terms.
Consider the first of the terms (the result will clearly be identical
for both terms) and note that $Q_{ist}^{\alpha}$ is fixed across
leave-out samples, so that the $k$-th leave-out version of this term
is
\[
\frac{1}{(N-2)^{4}}\sum_{i,i'\ne i}\sum_{s,s'\ne s}\sum_{t,t'\ne t}\sum_{r,r'\ne r}Q_{ist}^{\alpha}(\partial_{\pi}\ell_{i,i'})(\partial_{\pi}\ell_{s,s'})(\partial_{\beta\pi}\ell_{t,t'})(\partial_{\beta}\ell_{r,r'})1_{i,i'}^{k}1_{s,s'}^{k}1_{t,t'}^{k}1_{r,r'}^{k}
\]
The compute the average over the $N-1$ leave-out samples first define
\[
n(i,i',s,s',t,t',r,r')=N-1-\sum_{k}1_{i,i'}^{k}1_{s,s'}^{k}1_{t,t'}^{k}1_{r,r'}^{k}
\]
which counts the number of leave-out samples in which all four observations
appear -- this number depends on whether the four observations appear
in the same set $\mathcal{I}_{k}$, or in two, three of four different
sets. We can write the average as
\begin{align*}
\frac{1}{N-1} & \sum_{k}\frac{1}{(N-2)^{4}}\sum_{i,i'\ne i}\sum_{s,s'\ne s}\sum_{t,t'\ne t}\sum_{r,r'\ne r}Q_{ist}^{\alpha}(\partial_{\pi}\ell_{i,i'})(\partial_{\pi}\ell_{s,s'})(\partial_{\beta\pi}\ell_{t,t'})(\partial_{\beta}\ell_{r,r'})1_{i,i'}^{k}1_{s,s'}^{k}1_{t,t'}^{k}1_{r,r'}^{k}\\
& =\frac{1}{(N-1)(N-2)^{4}}\sum_{i,i'\ne i}\sum_{s,s'\ne s}\sum_{t,t'\ne t}\sum_{r,r'\ne r}Q_{ist}^{\alpha}(\partial_{\pi}\ell_{i,i'})(\partial_{\pi}\ell_{s,s'})(\partial_{\beta\pi}\ell_{t,t'})(\partial_{\beta}\ell_{r,r'}) \\
&\qquad\times \big(N-1-n(i,i',s,s',t,t',r,r')\big)
\end{align*}
The jackknifed term is then equal to
\begin{align*}
\vert\mathcal{J}\vert= & \lvert\frac{N-1}{(N-1)^{4}}\sum_{i,i'\ne i}\sum_{s,s'\ne s}\sum_{t,t'\ne t}\sum_{r,r'\ne r}Q_{ist}^{\alpha}(\partial_{\pi}\ell_{i,i'})(\partial_{\pi}\ell_{s,s'})(\partial_{\beta\pi}\ell_{t,t'})(\partial_{\beta}\ell_{r,r'})\\
- & \frac{N-2}{(N-1)(N-2)^{4}}\sum_{i,i'\ne i}\sum_{s,s'\ne s}\sum_{t,t'\ne t}\sum_{r,r'\ne r}Q_{ist}^{\alpha}(\partial_{\pi}\ell_{i,i'})(\partial_{\pi}\ell_{s,s'})(\partial_{\beta\pi}\ell_{t,t'})(\partial_{\beta}\ell_{r,r'}) \\
&\qquad\times \big(N-1-n(i,i',s,s',t,t',r,r')\big)\rvert\\
\leq & \frac{C}{N^{3}}\sum_{j=1}^{4}\lvert\sum_{i,i'\ne i}\sum_{s,s'\ne s}\sum_{t,t'\ne t}\sum_{r,r'\ne r}Q_{ist}^{\alpha}(\partial_{\pi}\ell_{i,i'})(\partial_{\pi}\ell_{s,s'})(\partial_{\beta\pi}\ell_{t,t'})(\partial_{\beta}\ell_{r,r'})1\{n(i,i',s,s',t,t',r,r')=j\}\rvert
\end{align*}
since $(\frac{1}{(N-1)^{3}}-\frac{N-1-n(i,i',s,s',t,t',r,r')}{(N-1)(N-2)^{3}})=O(N^{-3})$.
To bound the summation, we decompose the sum based on the order of
$Q_{ist}^{\alpha}$, using the shorthand notation $1_{n=j}=1\{n(i,i',s,s',t,t',r,r')=j\}$.
For some $j$ we have
\begin{align*}
\vert & \sum_{i,i'\ne i}\sum_{s,s'\ne s}\sum_{t,t'\ne t}\sum_{r,r'\ne r}Q_{ist}^{\alpha}(\partial_{\pi}\ell_{i,i'})(\partial_{\pi}\ell_{s,s'})(\partial_{\beta\pi}\ell_{t,t'})(\partial_{\beta}\ell_{r,r'})1_{n=j}\vert\\
& \leq\vert\sum_{i,i'\ne i}\sum_{s\ne i,s'\ne i}\sum_{t\ne(i,s),t'\ne t}\sum_{r,r'\ne r}Q_{ist}^{\alpha}(\partial_{\pi}\ell_{i,i'})(\partial_{\pi}\ell_{s,s'})(\partial_{\beta\pi}\ell_{t,t'})(\partial_{\beta}\ell_{r,r'})1_{n=j}\vert\\
& +\vert\sum_{i,i'\ne i}\sum_{s'\ne i}\sum_{t\ne i,t'\ne t}\sum_{r,r'\ne r}Q_{iit}^{\alpha}(\partial_{\pi}\ell_{i,i'})(\partial_{\pi}\ell_{i,s'})(\partial_{\beta\pi}\ell_{t,t'})(\partial_{\beta}\ell_{r,r'})1_{n=j}\vert\\
& +\vert\sum_{i,i'\ne i}\sum_{s\ne i,s'\ne s}\sum_{t'\ne i}\sum_{r,r'\ne r}Q_{isi}^{\alpha}(\partial_{\pi}\ell_{i,i'})(\partial_{\pi}\ell_{s,s'})(\partial_{\beta\pi}\ell_{i,t'})(\partial_{\beta}\ell_{r,r'})1_{n=j}\vert\\
& +\vert\sum_{i,i'\ne i}\sum_{s\ne i,s'\ne s}\sum_{t'\ne s}\sum_{r,r'\ne r}Q_{iss}^{\alpha}(\partial_{\pi}\ell_{i,i'})(\partial_{\pi}\ell_{s,s'})(\partial_{\beta\pi}\ell_{s,t'})(\partial_{\beta}\ell_{r,r'})1_{n=j}\vert\\
& +\vert\sum_{i,i'\ne i}\sum_{s'\ne i}\sum_{t'\ne i}\sum_{r,r'\ne r}Q_{iii}^{\alpha}(\partial_{\pi}\ell_{i,i'})(\partial_{\pi}\ell_{i,s'})(\partial_{\beta\pi}\ell_{i,t'})(\partial_{\beta}\ell_{r,r'})1_{n=j}\vert
\end{align*}
The first term is
\begin{align*}
\bar{E}\Big[ & \Big(\sum_{i,i'\ne i}\sum_{s\ne i,s'\ne i}\sum_{t\ne(i,s),t'\ne t}\sum_{r,r'\ne r}Q_{ist}^{\alpha}(\partial_{\pi}\ell_{i,i'})(\partial_{\pi}\ell_{s,s'})(\partial_{\beta\pi}\ell_{t,t'})(\partial_{\beta}\ell_{r,r'})1_{n=j}\Big)^{2}\Big]\\
= & \sum_{i,i'\ne i}\sum_{s\ne i,s'\ne i}\sum_{t\ne(i,s),t'\ne t}\sum_{r,r'\ne r}\sum_{j,j'\ne j}\sum_{a\ne j,a'\ne j}\sum_{b\ne(j,a),b'\ne b}\sum_{c,c'\ne c}Q_{ist}^{\alpha}Q_{jab}^{\alpha}\\
& \times\bar{E}\big[(\partial_{\pi}\ell_{i,i'})(\partial_{\pi}\ell_{s,s'})(\partial_{\beta\pi}\ell_{t,t'})(\partial_{\beta}\ell_{r,r'})(\partial_{\pi}\ell_{j,j'})(\partial_{\pi}\ell_{a,a'})(\partial_{\beta\pi}\ell_{b,b'})(\partial_{\beta}\ell_{c,c'})\big]1_{n=j}1_{n'=j}
\end{align*}
Since we have $\max_{i\ne s\ne t}Q_{ist}=O_{p}(N^{-2})$, and $\bar{E}[\partial_{\pi}\ell_{ii'}]=\bar{E}[\partial_{\beta}\ell_{ii'}]=0$,
this term is $O_{p}(N^{6})$. Similarly, for the final term we have
\begin{align*}
\bar{E}\Big[\Big( & \sum_{i,i'\ne i}\sum_{s'\ne i}\sum_{t'\ne i}\sum_{r,r'\ne r}Q_{iii}^{\alpha}(\partial_{\pi}\ell_{i,i'})(\partial_{\pi}\ell_{i,s'})(\partial_{\beta\pi}\ell_{i,t'})(\partial_{\beta}\ell_{r,r'})1_{n=j}\Big)^{2}\Big]\\
= & \sum_{i,i'\ne i}\sum_{s'\ne i}\sum_{t'\ne i}\sum_{r,r'\ne r}\sum_{j,j'\ne j}\sum_{a'\ne j}\sum_{b'\ne j}\sum_{c,c'\ne c}Q_{iii}^{\alpha}Q_{jjj}^{\alpha}\\
& \times\bar{E}\big[(\partial_{\pi}\ell_{i,i'})(\partial_{\pi}\ell_{i,s'})(\partial_{\beta\pi}\ell_{i,t'})(\partial_{\beta}\ell_{r,r'})(\partial_{\pi}\ell_{j,j'})(\partial_{\pi}\ell_{j,a'})(\partial_{\beta\pi}\ell_{j,b'})(\partial_{\beta}\ell_{c,c'})\big]\\
= & O_{p}(N^{8})
\end{align*}
and hence the final term is $O_{p}(N^{4})$. The remaining terms can
be shown to also be $O_{p}(N^{4})$. This gives $\mathcal{J}=O_{p}(N)$.
Nearly identical steps apply to the remaining components of the term
$\mathcal{S}'\bar{\mathcal{H}}^{-1}\sum_{f}(\partial_{\phi\phi'\phi_{f}}\bar{\mathcal{L}})\big[\bar{\mathcal{H}}^{-1}(\partial_{\beta\phi}\mathcal{L})\big]_{f}\mathcal{\bar{H}}^{-1}\mathcal{S}\mathcal{S}_{\beta}$
and adding back the terms $\bar{W}^{-2}\bar{W}_{b}$, we find that
the jackknife operator applied to the full term satisfies
\[
\mathcal{J}\Big[\bar{W}^{-2}\bar{W}_{b}\mathcal{S}'\bar{\mathcal{H}}^{-1}\sum_{f}(\partial_{\phi\phi'\phi_{f}}\bar{\mathcal{L}})\big[\bar{\mathcal{H}}^{-1}(\partial_{\beta\phi}\mathcal{L})\big]_{f}\mathcal{\bar{H}}^{-1}\mathcal{S}\mathcal{S}_{\beta}\Big]=o_{p}(1).
\]
Each of the remaining terms in $\mathcal{S}'\mathcal{H}^{-1}\mathcal{H}_{bb}\mathcal{H}^{-1}\mathcal{S}\mathcal{S}_{\beta}$
can similarly be shown to have the same V-statistic like structure,
up to fifth-order. Inspection of the expressions in Section \ref{sec:full_expansion}
shows that an essentially identical proof can be used for any of these
terms. That is, we first replace matrices with expansions around their
conditional expectations, and the expand the terms to give sums of
products of terms like the above. The jackknifed versions of these
sums can then straightforwardly be shown to have the same asymptotic
order as the original terms, and hence each of the jackknifed terms
is $o_{p}(1)$ as required.
\subsection{Results for weighted jackknife} \label{subsec:weighted_proof}
In order to complete the proof of Theorem 1 in Section \ref{subsec:thm1proof},
we must show:
(1) $(N-1)\lVert W_{J}-W_{N}\rVert=o_{p}(1)$
(2) $(N-1)(\widehat{W}_{J}-W_{J})(\widehat{\beta}-\beta)-(N-2)\frac{1}{N-1}\sum_{k}(\widehat{W}_{(k)}-W_{(k)})(\widehat{\beta}_{(k)}-\beta)=o_{p}(N^{-1})$
(3) $\lVert \widehat{W}_J - \overline{W}_N\rVert=o_p(1)$
\subsubsection*{(1) $(N-1)(W_{J}-W_{N})=o_{p}(1)$}
Expanding out this term we get
\begin{align*}
&(N-1)(W_{J}-W_{N}) \\
&=(N-1)\frac{1}{N}\Big\{\frac{1}{N-1}\sum_{k}\partial_{\beta\beta}\mathcal{L}_{(k)}\\
&+\frac{1}{N-1}\sum_{k}(\partial_{\beta\phi'}\mathcal{L}_{(k)})\mathcal{H}_{(k)}^{-1}(\partial_{\beta\phi}\mathcal{L}_{(k)})-\partial_{\beta\beta}\mathcal{L}-(\partial_{\beta\phi'}\mathcal{L})\mathcal{H}^{-1}(\partial_{\beta\phi}\mathcal{L})\Big\}\\
& =\frac{N-1}{N}\Big\{\frac{1}{N-1}\sum_{k}(\partial_{\beta\phi'}\mathcal{L}_{(k)})\mathcal{H}_{(k)}^{-1}(\partial_{\beta\phi}\mathcal{L}_{(k)})-(\partial_{\beta\phi'}\mathcal{L})\mathcal{H}^{-1}(\partial_{\beta\phi}\mathcal{L})\Big\}\\
& =\frac{N-1}{N}\Big\{\frac{1}{N-1}\sum_{k}(\partial_{\beta\phi'}\mathcal{L}_{(k)})\Big(\mathcal{\bar{H}}^{-1}-\mathcal{\bar{H}}^{-1}\tilde{\mathcal{H}}_{(k)}\mathcal{\bar{H}}^{-1}+\mathcal{\bar{H}}^{-1}\tilde{\mathcal{H}}_{(k)}\mathcal{\bar{H}}^{-1}\tilde{\mathcal{H}}_{(k)}\mathcal{\bar{H}}^{-1}\Big)(\partial_{\beta\phi}\mathcal{L}_{(k)})\\
&-(\partial_{\beta\phi'}\mathcal{L})\Big(\mathcal{\bar{H}}^{-1}-\mathcal{\bar{H}}^{-1}\tilde{\mathcal{H}}\mathcal{\bar{H}}^{-1}+
\mathcal{\bar{H}}^{-1}\tilde{\mathcal{H}}\mathcal{\bar{H}}^{-1}\tilde{\mathcal{H}}\mathcal{\bar{H}}^{-1}\Big)(\partial_{\beta\phi}\mathcal{L})\Big\} +o_p(1)
\end{align*}
where the remainder term is $o_p(1)$ since $\lVert\mathcal{H}^{-1}-\mathcal{\bar{H}}^{-1}+\mathcal{\bar{H}}^{-1}\tilde{\mathcal{H}}\mathcal{\bar{H}}^{-1}-\mathcal{\bar{H}}^{-1}\tilde{\mathcal{H}}\mathcal{\bar{H}}^{-1}\tilde{\mathcal{H}}\mathcal{\bar{H}}^{-1}\rVert=O_{p}(N^{-\frac{3}{2}+6\epsilon})$
by Lemma \ref{lem:H1_approx}, $\lVert\partial_{\beta\phi'}\mathcal{L}\rVert=O_{p}(N^{1/2})$,
and similarly for the leaveout versions.
Further expanding out the first of these approximation terms gives
\begin{align*}
\frac{1}{N-1} & \sum_{k}(\partial_{\beta\phi'}\mathcal{L}_{(k)})\mathcal{\bar{H}}^{-1}(\partial_{\beta\phi}\mathcal{L}_{(k)})-(\partial_{\beta\phi'}\mathcal{L})\mathcal{\bar{H}}^{-1}(\partial_{\beta\phi}\mathcal{L})\\
= & \frac{1}{N-1}\sum_{k}\frac{1}{(N-2)^{2}}\sum_{i}\sum_{j\ne i}\sum_{s}\sum_{t\ne s}\Gamma_{isjt}(\partial_{\beta\pi}\ell_{ij})(\partial_{\beta\pi}\ell_{st})1_{ij}^{k}1_{st}^{k}\\
& -\frac{1}{(N-1)^{2}}\sum_{i}\sum_{j\ne i}\sum_{s}\sum_{t\ne s}\Gamma_{isjt}(\partial_{\beta\pi}\ell_{ij})(\partial_{\beta\pi}\ell_{st})\\
= & \frac{1}{(N-1)^{2}}\sum_{i}\sum_{j\ne i}\sum_{s}\sum_{t\ne s}\Gamma_{isjt}(\partial_{\beta\pi}\ell_{ij})(\partial_{\beta\pi}\ell_{st})\Big(\frac{(N-1)}{(N-2)^{2}}\sum_{k}1_{ij}^{k}1_{st}^{k}-1\Big)\\
= & \frac{1}{(N-1)^{2}(N-2)}\sum_{i}\sum_{j\ne i}\sum_{s}\sum_{t\ne s}\Gamma_{isjt}(\partial_{\beta\pi}\ell_{ij})(\partial_{\beta\pi}\ell_{st})I_{(ij)(st)}^{1}\\
& -\frac{1}{(N-1)^{2}(N-2)^{2}}\sum_{i}\sum_{j\ne i}\sum_{s}\sum_{t\ne s}\Gamma_{isjt}(\partial_{\beta\pi}\ell_{ij})(\partial_{\beta\pi}\ell_{st})(1-I_{(ij)(st)}^{1}) \\
&=o_p(1)
\end{align*}
since $\Gamma_{isjt}$ is $O_p(1)$ only when either $i=s$ or $j=t$, and is $O_p(N^{-1})$ otherwise.
Expanding out the next term in the approximation gives
\begin{align*}
\frac{1}{N-1} & \sum_{k}(\partial_{\beta\phi'}\mathcal{L}_{(k)})\mathcal{\bar{H}}^{-1}\tilde{\mathcal{H}}_{(k)}\mathcal{\bar{H}}^{-1}(\partial_{\beta\phi}\mathcal{L}_{(k)})-(\partial_{\beta\phi'}\mathcal{L})\mathcal{\bar{H}}^{-1}\tilde{\mathcal{H}}\mathcal{\bar{H}}^{-1}(\partial_{\beta\phi}\mathcal{L})\\
= & \frac{1}{N-1}\sum_{k}\frac{1}{(N-2)^{3}}\sum_{i,j,s}\sum_{i'\ne i}\sum_{j'\ne j}\sum_{s'\ne s}\Gamma_{ii'jj'}\Gamma_{jj'ss'}(\partial_{\beta\pi}\ell_{ii'})(\partial_{\beta\pi}\ell_{ss'})(\partial_{\pi^{2}}\ell_{jj'})1_{ii'}^{k}1_{jj'}^{k}1_{ss'}^{k}\\
& -\frac{1}{(N-1)^{3}}\sum_{i,j,s}\sum_{i'\ne i}\sum_{j'\ne j}\sum_{s'\ne s}\Gamma_{ii'jj'}\Gamma_{jj'ss'}(\partial_{\beta\pi}\ell_{ii'})(\partial_{\beta\pi}\ell_{ss'})(\partial_{\pi^{2}}\ell_{jj'})\\
= & \frac{1}{(N-1)^{3}}\sum_{i,j,s}\sum_{i'\ne i}\sum_{j'\ne j}\sum_{s'\ne s}\Gamma_{ii'jj'}\Gamma_{jj'ss'}(\partial_{\beta\pi}\ell_{ii'})(\partial_{\beta\pi}\ell_{ss'})(\partial_{\pi^{2}}\ell_{jj'})\Big(\frac{(N-1)^{2}}{(N-2)^{3}}\sum_{k}1_{ii'}^{k}1_{jj'}^{k}1_{ss'}^{k}-1\Big)\\
= & O(N^{-1})\frac{1}{(N-1)^{3}}\sum_{i,j,s}\sum_{i'\ne i}\sum_{j'\ne j}\sum_{s'\ne s}\Gamma_{ii'jj'}\Gamma_{jj'ss'}(\partial_{\beta\pi}\ell_{ii'})(\partial_{\beta\pi}\ell_{ss'})(\partial_{\pi^{2}}\ell_{jj'})I_{(ii')(ss')(jj')}^{1}\\
& +O(N^{-1})\frac{1}{(N-1)^{3}}\sum_{i,j,s}\sum_{i'\ne i}\sum_{j'\ne j}\sum_{s'\ne s}\Gamma_{ii'jj'}\Gamma_{jj'ss'}(\partial_{\beta\pi}\ell_{ii'})(\partial_{\beta\pi}\ell_{ss'})(\partial_{\pi^{2}}\ell_{jj'})I_{(ii')(ss')(jj')}^{2}\\
& +O(N^{-2})\frac{1}{(N-1)^{3}}\sum_{i,j,s}\sum_{i'\ne i}\sum_{j'\ne j}\sum_{s'\ne s}\Gamma_{ii'jj'}\Gamma_{jj'ss'}(\partial_{\beta\pi}\ell_{ii'})(\partial_{\beta\pi}\ell_{ss'})(\partial_{\pi^{2}}\ell_{jj'})I_{(ii')(ss')(jj')}^{3}\\
= & o_{p}(1)
\end{align*}
where the third equality follows from the fact that $\big(\frac{(N-1)^{2}}{(N-2)^{3}}\sum_{k}1_{ii'}^{k}1_{jj'}^{k}1_{ss'}^{k}-1\big)$ is $O(N^{-1})$ whenever $\sum_{k}1_{ii'}^{k}1_{jj'}^{k}1_{ss'}^{k}$ is equal to $(N-2)$ or $(N-3)$, but is $O(N^{-2})$ when the three observations span three different $\mathcal{I}_k$, while the
final equality is due to the fact that: (1)
$\Gamma_{ii'jj'}\Gamma_{jj'ss'}$ is $O_{p}(1)$ only if one of $i=j,i'=j'$ holds and one of $j=s,j'=s'$ holds, and is $O_{p}(N^{-1})$ if only
one of the two holds, and is $O_{p}(N^{-2})$ if neither hold, and (2) when $I^1_{(ii')(ss')(jj')}=1$, the indices $s'$ and $j'$ are fixed given $i'$, while for $I^2_{(ii')(ss')(jj')}=1$, choosing two of these fixes the third. Similar steps show that
\[
\frac{1}{N-1}\sum_{k}(\partial_{\beta\phi'}\mathcal{L}_{(k)})\mathcal{\bar{H}}^{-1}\tilde{\mathcal{H}}_{(k)}\mathcal{\bar{H}}^{-1}\tilde{\mathcal{H}}_{(k)}\mathcal{\bar{H}}^{-1}(\partial_{\beta\phi}\mathcal{L}_{(k)})-(\partial_{\beta\phi'}\mathcal{L})\mathcal{\bar{H}}^{-1}\tilde{\mathcal{H}}\mathcal{\bar{H}}^{-1}\tilde{\mathcal{H}}\mathcal{\bar{H}}^{-1}(\partial_{\beta\phi}\mathcal{L})=o_{p}(1)
\]
This gives the required result.
\subsubsection*{(2) $(N-1)(\widehat{W}_{J}-W_{J})(\widehat{\beta}-\beta)-(N-2)\frac{1}{N-1}\sum_{k}(\widehat{W}_{(k)}-W_{(k)})(\widehat{\beta}_{(k)}-\beta)=o_{p}(N^{-1})$}
We first provide a first-order expansion for $\widehat{W}$. Recall
that we can write $\widehat{W}=\big(\partial_{bb}\mathcal{L}^{*}(0,0)\big)^{-1}$,
and so letting $\mathcal{W}^{*}(b,s)=\big(\partial_{bb}\mathcal{L}^{*}(b,s)\big)^{-1}$
and $\mathcal{W}^{*}=\big(\partial_{bb}\mathcal{L}^{*}(\mathcal{S}_{\beta},\mathcal{S})\big)^{-1}$
we can expand to give
\begin{align*}
\widehat{W} & =\mathcal{W}^{*}+(\partial_{b}\mathcal{W}^{*})\mathcal{S_{\beta}}+(\partial_{s'}\mathcal{W}^{*})\mathcal{S}\\
& +\frac{1}{2}\mathcal{S}'(\partial_{ss'}\mathcal{W}^{*})\mathcal{S}+o_{p}(N^{-1})\\
\widehat{W}_{N}-W_{N} & =\frac{1}{N}(\partial_{b}\mathcal{W}^{*})\mathcal{S_{\beta}}+\frac{1}{N}(\partial_{s'}\mathcal{W}^{*})\mathcal{S}+\frac{1}{2}\frac{1}{N}\mathcal{S}'(\partial_{ss'}\mathcal{W}^{*})\mathcal{S}+o_{p}(N^{-1})
\end{align*}
Up to first order we find
\begin{align*}
N(\widehat{W}_{N}-W_{N}) & =(\partial_{b}\mathcal{W}^{*})\mathcal{S_{\beta}}+(\partial_{s'}\mathcal{W}^{*})\mathcal{S}+\frac{1}{2}\mathcal{S}'(\partial_{ss'}\mathcal{W}^{*})\mathcal{S}+o_{p}(1)\\
& =\bar{W}_{b}\mathcal{S_{\beta}}+\bar{W}_{s'}\mathcal{S}+\bar{W}^{2}\mathcal{S}'\mathcal{\bar{H}}^{-1}\mathcal{\bar{H}}_{b}\mathcal{\bar{H}}^{-1}\mathcal{\bar{H}}_{b}\mathcal{\bar{H}}^{-1}\mathcal{S}\\
& \quad-\frac{1}{2}\bar{W}^{2}\mathcal{S}'\mathcal{\bar{H}}^{-1}\mathcal{\bar{H}}_{bb}\mathcal{\bar{H}}^{-1}\mathcal{S}+o_{p}(1)
\end{align*}
and similarly for the leaveout versions $\widehat{W}_{(k)}$. We can
then write
\begin{align*}
N(\widehat{W}_{J}-W_{J}) &= N\frac{1}{N-1}\sum_k (\widehat{W}_{(k)}-W_{(k)}) \\ &=\frac{1}{N-1}\sum_{k}\Big(\bar{W}_{b}\mathcal{S}_{\beta,(k)}+\bar{W}_{s'}\mathcal{S}_{(k)}+\bar{W}^{2}\mathcal{S}_{(k)}'\mathcal{\bar{H}}^{-1}\mathcal{\bar{H}}_{b}\mathcal{\bar{H}}^{-1}\mathcal{\bar{H}}_{b}\mathcal{\bar{H}}^{-1}\mathcal{S}_{(k)}\\
& \quad-\frac{1}{2}\bar{W}^{2}\mathcal{S}_{(k)}'\mathcal{\bar{H}}^{-1}\mathcal{\bar{H}}_{bb}\mathcal{\bar{H}}^{-1}\mathcal{S}_{(k)}\Big)+o_{p}(1)\\
&= \frac{1}{N-1}\sum_k \big( V_{1,(k)} + V_{2,(k)} + V_{3,(k)} + V_{4,(k)} \big)+o_p(1) \\
N(\widehat{\beta}-\beta) & =\bar{W}_{N}^{-1}\big(U^{(0)}+U^{(1)}\big)+o_{p}(1)
\end{align*}
Using these expansions (and the equivalent leaveout versions) we can write
\begin{align*}
&(N-1)(\widehat{W}_{J}-W_{J})(\widehat{\beta}-\beta)-(N-2)\frac{1}{N-1}\sum_{k}(\widehat{W}_{(k)}-W_{(k)})(\widehat{\beta}_{(k)}-\beta) \\
&= \frac{N-1}{N^2} \frac{1}{N-1}\sum_k \big( V_{1,(k)} + V_{2,(k)} + V_{3,(k)} + V_{4,(k)} \big)\bar{W}_{N}^{-1}\big(U^{(0)}+U^{(1)}\big) \\
&\quad - \frac{N-2}{N^2} \frac{1}{N-1}\sum_k \big( V_{1,(k)} + V_{2,(k)} + V_{3,(k)} + V_{4,(k)} \big)\bar{W}_{N}^{-1}\big(U^{(0)}_{(k)}+U^{(1)}_{(k)}\big) + o_p(N^{-1})
\end{align*}
Following similar steps to that of the higher-order jackknife expansion,
we can show that each of the terms in the product $N(\widehat{W}_{J}-W_{J})\times(\widehat{\beta}-\beta)$
is $o_{p}(1)$ when jackknifed. For example, take the first term in
the expansion of both $N(\widehat{W}_{J}-W_{J})$ and $(\widehat{\beta}-\beta)$
and apply the jackknife to give
\begin{align*}
(N-1) & \frac{1}{N}\frac{1}{N-1}\sum_{k}V_{1,(k)}\frac{1}{N}\bar{W}_{N}^{-1}U^{(0)}-(N-2)\frac{1}{N-1}\sum_{k}\frac{1}{N}V_{1,(k)}\frac{1}{N}\bar{W}_{N}^{-1}U_{(k)}^{(0)}\\
= & \frac{1}{N^{2}}(N-1)\bar{W}_{b}\mathcal{S}_{\beta}\bar{W}_{N}^{-1}\big(\partial_{\beta}\mathcal{L}+(\partial_{\beta\phi'}\bar{\mathcal{L}})\bar{\mathcal{H}}^{-1}\mathcal{S}\big)\\
& -\frac{1}{N^{2}}(N-2)\frac{1}{N-1}\sum_{k}\bar{W}_{b}\mathcal{S}_{\beta,(k)}\bar{W}_{N}^{-1}\big(\partial_{\beta}\mathcal{L}_{(k)}+(\partial_{\beta\phi'}\bar{\mathcal{L}})\bar{\mathcal{H}}^{-1}\mathcal{S}_{(k)}\big)\\
&= \frac{1}{N^2} \mathbf{J}\big[ \bar{W}_{b}\bar{W}_{N}^{-1}\mathcal{S}_{\beta}\big(\partial_{\beta}\mathcal{L}+(\partial_{\beta\phi'}\bar{\mathcal{L}})\bar{\mathcal{H}}^{-1}\mathcal{S}\big) \big]\\
&= o_p(N^{-2})
\end{align*}
where the final line follows from application of Lemma \ref{lem:jack_r0} in the main appendix. Similarly, each of the remaining terms can be shown to be jackknife versions of products of up to four random sums, all of which are $o_p(N^{-1})$, from which the result follows.
\paragraph*{(3) $\lVert \widehat{W}_J - \overline{W}_N\rVert=o_p(1)$}
For the final result, we can apply: (a) the first result $\lVert W_{J}-W_{N}\rVert=o_p (1)$, (b) $\lVert \widehat{W}_{J}-W_{J}\rVert=o_p (1)$ (which following from parameter consistency and the fact that $W$ is a smooth function of parameters), and (c) $\lVert W_{N}-\overline{W}_{N}\rVert=o_p (1)$ from Lemma \ref{lem:H1_approx}, along with the triangle inequality to give the result.
\section{Asymptotic expansion for average effect} \label{app:avg_effect}
\subsection{Expansion for $\widehat{\phi}$}
In order to derive an expansion for the average effect parameter,
we first give an expansion for the fixed effect parameters $\phi=(\alpha',\gamma')'$.
Since we have $\partial_{s}\mathcal{L}^{*}(0,0)=-\widehat{\phi}$
and $\partial_{s}\mathcal{L}^{*}(\mathcal{S}_{\beta},\mathcal{S})=-\phi_{0}$,
we take a Taylor expansion to give
\begin{align*}\widehat{\phi}-\phi_{0} & =(\partial_{bs}\mathcal{L}^{*})\mathcal{S}_{\beta}+(\partial_{ss'}\mathcal{L}^{*})\mathcal{S}\\
& -\frac{1}{2}(\partial_{bbs}\mathcal{L}^{*})\mathcal{S}_{\beta}^{2}\\
& -(\partial_{bss'}\mathcal{L}^{*})\mathcal{S}\mathcal{S}_{\beta}\\
& -\frac{1}{2}\sum_{g}(\partial_{ss's_{g}}\mathcal{L}^{*})\mathcal{S}\mathcal{S}_{g}\\
& +\frac{1}{2}\sum_{g}(\partial_{bss's_{g}}\mathcal{L}_{(1)}^{*})\mathcal{S}_{g}\mathcal{S}\mathcal{S}_{\beta}\\
& +\frac{1}{6}\sum_{f,g}(\partial_{ss's_{f}s_{g}}\mathcal{L}_{(1)}^{*})\mathcal{S}_{f}\mathcal{S}_{g}\mathcal{S}\\
& -\frac{1}{24}\sum_{e,f,g}(\partial_{ss's_{e}s_{f}s_{g}}\mathcal{L}_{(1)}^{*})\mathcal{S}\mathcal{S}_{e}\mathcal{S}_{f}\mathcal{S}_{g}\\
& +\tilde{R}_{\phi}
\end{align*}
where the remainder is
\begin{align*}\tilde{R}_{\phi} & =\frac{1}{6}\big(\partial_{bbbs}\mathcal{L}^{*}(\bar{b},\bar{s})\big)\mathcal{S}_{\beta}^{3}\\
& +\frac{1}{2}\big(\partial_{bbss'}\mathcal{L}^{*}(\bar{b},\bar{s})\big)\mathcal{S}\mathcal{S}_{\beta}^{2}\\
& +\frac{1}{2}\sum_{g}\big(\partial_{bss's_{g}}\mathcal{L}_{(2)}^{*}(\bar{b},\bar{s})\big)\mathcal{S}\mathcal{S}_{g}\mathcal{S}_{\beta}\\
& +\frac{1}{6}\sum_{f,g}\big(\partial_{ss's_{f}s_{g}}\mathcal{L}_{(2)}^{*}(\bar{b},\bar{s})\big)\mathcal{S}\mathcal{S}_{f}\mathcal{S}_{g}\\
& -\frac{1}{4}\sum_{g}\big(\partial_{bbss's_{g}}\mathcal{L}_{(1)}^{*}(\bar{b},\bar{s})\big)\mathcal{S}\mathcal{S}_{g}\mathcal{S}_{\beta}^{2}\\
& -\frac{1}{6}\sum_{g,h}\big(\partial_{bss's_{g}s_{h}}\mathcal{L}_{(1)}^{*}(\bar{b},\bar{s})\big)\mathcal{S}\mathcal{S}_{g}\mathcal{S}_{h}\mathcal{S}_{\beta}\\
& +\frac{1}{120}\sum_{e,f,g,h}(\partial_{ss's_{e}s_{f}s_{g}s_{h}}\mathcal{L}_{(1)}^{*})\mathcal{S}\mathcal{S}_{e}\mathcal{S}_{f}\mathcal{S}_{g}\mathcal{S}_{h}
\end{align*}
\begin{lem}
Let Assumptions 1 and 2 hold. Then the remainder term in the expansion
for $\widehat{\phi}$ satisfies
\begin{align*}
\lVert\tilde{R}_{\phi}\rVert_{q} & = O_{p}(N^{-5/2+18\epsilon})
\end{align*}
\end{lem}
\begin{proof}
The result follows from application of the bounds in \textbackslash ref\{sec:bounds\_ind\}
to the expressions derived in \textbackslash ref\{sec:expressions\_ind\}.
For the final term, we have that
\begin{align*}\lVert\partial_{ss's_{e}s_{f}s_{g}s_{h}}\mathcal{L}_{(1)}^{*}\rVert_{q} & \leq36\lVert\mathcal{H}^{-1}\mathcal{H}_{s_{e}}\mathcal{H}^{-1}\mathcal{H}_{s_{f}}\mathcal{H}^{-1}\mathcal{H}_{s_{g}}\mathcal{H}^{-1}\mathcal{H}_{s_{h}}\mathcal{H}^{-1}\rVert_{q}\\
& +9\lVert\mathcal{H}^{-1}\mathcal{H}_{s_{e}s_{f}}\mathcal{H}^{-1}\mathcal{H}_{s_{g}}\mathcal{H}^{-1}\mathcal{H}_{s_{h}}\mathcal{H}^{-1}\rVert_{q}\\
& +9\lVert\mathcal{H}^{-1}\mathcal{H}_{s_{f}}\mathcal{H}^{-1}\mathcal{H}_{s_{e}s_{g}}\mathcal{H}^{-1}\mathcal{H}_{s_{h}}\mathcal{H}^{-1}\rVert_{q}\\
& +9\lVert\mathcal{H}^{-1}\mathcal{H}_{s_{f}}\mathcal{H}^{-1}\mathcal{H}_{s_{g}}\mathcal{H}^{-1}\mathcal{H}_{s_{e}s_{h}}\mathcal{H}^{-1}\rVert_{q}\\
& +6\lVert\mathcal{H}^{-1}\mathcal{H}_{s_{e}}\mathcal{H}^{-1}\mathcal{H}_{s_{f}s_{g}}\mathcal{H}^{-1}\mathcal{H}_{s_{h}}\mathcal{H}^{-1}\rVert_{q}\\
& +2\lVert\mathcal{H}^{-1}\mathcal{H}_{s_{e}s_{f}s_{g}}\mathcal{H}^{-1}\mathcal{H}_{s_{h}}\mathcal{H}^{-1}\rVert_{q}\\
& +2\lVert\mathcal{H}^{-1}\mathcal{H}_{s_{f}s_{g}}\mathcal{H}^{-1}\mathcal{H}_{s_{e}s_{h}}\mathcal{H}^{-1}\rVert_{q}\\
& +6\lVert\mathcal{H}^{-1}\mathcal{H}_{s_{e}}\mathcal{H}^{-1}\mathcal{H}_{s_{g}}\mathcal{H}^{-1}\mathcal{H}_{s_{f}s_{h}}\mathcal{H}^{-1}\rVert_{q}\\
& +2\lVert\mathcal{H}^{-1}\mathcal{H}_{s_{e}s_{g}}\mathcal{H}^{-1}\mathcal{H}_{s_{f}s_{h}}\mathcal{H}^{-1}\rVert_{q}\\
& +2\lVert\mathcal{H}^{-1}\mathcal{H}_{s_{g}}\mathcal{H}^{-1}\mathcal{H}_{s_{e}s_{f}s_{h}}\mathcal{H}^{-1}\rVert_{q}\\
& +6\lVert\mathcal{H}^{-1}\mathcal{H}_{s_{e}}\mathcal{H}^{-1}\mathcal{H}_{s_{f}}\mathcal{H}^{-1}\mathcal{H}_{s_{g}s_{h}}\mathcal{H}^{-1}\rVert_{q}\\
& +2\lVert\mathcal{H}^{-1}\mathcal{H}_{s_{e}s_{f}}\mathcal{H}^{-1}\mathcal{H}_{s_{g}s_{h}}\mathcal{H}^{-1}\rVert_{q}\\
& +2\lVert\mathcal{H}^{-1}\mathcal{H}_{s_{f}}\mathcal{H}^{-1}\mathcal{H}_{s_{e}s_{g}s_{h}}\mathcal{H}^{-1}\rVert_{q}\\
& +2\lVert\mathcal{H}^{-1}\mathcal{H}_{s_{e}}\mathcal{H}^{-1}\mathcal{H}_{s_{f}s_{g}s_{h}}\mathcal{H}^{-1}\rVert_{q}\\
& +\lVert\mathcal{H}^{-1}\mathcal{H}_{s_{e}s_{f}s_{g}s_{h}}\mathcal{H}^{-1}\rVert_{q}
\end{align*}
so that $\lVert\partial_{ssssss}\mathcal{L}_{(1)}^{*}\rVert_{q}=O_{p}(N^{8\epsilon})$.
We then have
\begin{align*}
\lVert\sum_{e,f,g,h}\big(\partial_{ss's_{e}s_{f}s_{g}s_{h}}\mathcal{L}_{(1)}^{*}(\bar{b},\bar{s})\big)\mathcal{S}\mathcal{S}_{e}\mathcal{S}_{f}\mathcal{S}_{g}\mathcal{S}_{h}\rVert_{q} & \leq\lVert\partial_{sssss}\mathcal{L}^{*}\rVert_{q}\lVert\mathcal{S}\rVert_{q}^{5}\\
& =O_{p}(N^{-5/2+18\epsilon})
\end{align*}
\end{proof}
\subsection{\label{subsec:avg_effect_bounds}Bounds for derivation of expansion}
In order to provide bounds on the terms in the asymptotic expansion
for the average effect, we first state some additional bounds on terms
that are used in the expansion for $\widehat{\phi}$. The first set
of bounds shows that $N\partial_{\phi}\bar{\Delta}$ has $O_{p}(1)$
elements, while $N\partial_{\phi\phi'}\bar{\Delta}$ is a matrix
that satisfies the conditions in Lemma \ref{lem:HAH_bound}. In addition,
we show that $N\partial_{\phi}\tilde{\Delta}$ satisfies the conditions
in Lemma \ref{lem:S6}, while $N\partial_{\phi\phi'}\tilde{\Delta}$
satisfies a bound like that in Lemma \ref{lem:H_tilde}.
The next lemmas are used to bound the jackknifed versions of the new
forms of product terms appearing in the expansion for the average
effects.
\begin{lem} \label{lem:Delta_12}
Let $\lambda$ be a set of $r$ observations
$(i,j)$ involving $p$ unique agents, and $\Lambda_{N}$ be the collection
of all such $\lambda$ formed by permuting the agents in $\lambda$.
Then, under Assumption 3,
(i) $\partial_{\phi}\bar{\Delta}$ has $O_{p}(N^{-1})$ elements
(ii) $\partial_{\alpha\alpha'}\bar{\Delta}$, $\partial_{\alpha\gamma'}\bar{\Delta}$,
$\partial_{\gamma\alpha'}\bar{\Delta}$, and $\partial_{\gamma\gamma'}\bar{\Delta}$
each have $O_{p}(N^{-1})$ diagonal elements and $O_{p}(N^{-2})$
off-diagonal elements
\end{lem}
\begin{proof}
Let $\lambda_{\alpha}$ denote the set of $p_{\alpha}$ sender agents
in the observations within $\lambda$, and $\lambda_{\gamma}$ the
set of $p_{\gamma}$ receiving agents. There are $\vert\Lambda_{N}\vert=\frac{N!}{(N-p)!}$
ways of selecting the $p$ agents in $\lambda$. Among these permutations,
agent $i$ is a sender $p_{\alpha}\frac{(N-1)!}{(N-p)!}$ times, while
node $j$ is the receiver $p_{\gamma}\frac{(N-1)!}{(N-p)!}$ times.
Using this, the first derivatives of $\bar{\Delta}$ with respect
to the fixed effects are
\begin{align*}
\partial_{\alpha_{i}}\bar{\Delta} & =\frac{1}{\vert\Lambda_{N}\vert}\sum_{\lambda:i\in\lambda_{\alpha}}\partial_{\alpha_{i}}\bar{m}_{\lambda}=O_{p}(N^{-1})\\
\partial_{\gamma_{i}}\bar{\Delta} & =\frac{1}{\vert\Lambda_{N}\vert}\sum_{\lambda:i\in\lambda_{\gamma}}\partial_{\gamma_{i}}\bar{m}_{\lambda}=O_{p}(N^{-1})
\end{align*}
where the $O_{p}(N^{-1})$ statements come from the fact that $p_{\alpha}\frac{(N-1)!}{(N-p)!}/\frac{N!}{(N-p)!}=p_{\alpha}/N$.
An identical result applies to the diagonal elements of $\partial_{\phi\phi'}\bar{\Delta}$,
i.e. $\partial_{\alpha_{i}\alpha_{i}}\bar{\Delta}=O_{p}(N^{-1})$
and $\partial_{\gamma_{j}\gamma_{j}}\bar{\Delta}=O_{p}(N^{-1})$
since they are sums over the same sets of $\lambda$. Also, if the
presence of $i$ as a sender agent implies that $i$ is also a receiver
in $\lambda$ (e.g. the cyclic triangle $\{(i,j),(j,k),(k,i)\}$)
then it will be the case that $\partial_{\alpha_{i}\gamma_{i}}\bar{\Delta}=O_{p}(N^{-1})$
also (if this is not true it will be lower order).
Next, consider the off-diagonal components of $\partial_{\phi\phi'}\bar{\Delta}$.
If $p_{\alpha}=1$ then $\partial_{\alpha_{i}\alpha_{j}}\bar{\Delta}=0$,
otherwise, there are ${p_{\alpha} \choose 2}\frac{(N-2)!}{(N-p)!}$
permutations that contain both $i$ and $j$ as senders. Similarly,
for $p_{\gamma}\geq2$, there are ${p_{\gamma} \choose 2}\frac{(N-2)!}{(N-p)!}$
permutations that contain both $i$ and $j$ as receivers. Finally,
there are \emph{at most} $p_{\alpha}p_{\gamma}\frac{(N-2)!}{(N-p)!}$
permutations in which $i$ is a sender and $j$ a receiver (this is
an upper bound since with $i$ in a particular sender position, not
all receiver positions may be valid for $j$). This, along with Assumption
3, gives the results
\begin{align*}
\partial_{\alpha_{i}\alpha_{j}}\bar{\Delta} & =O_{p}(N^{-2})\\
\partial_{\alpha_{i}\gamma_{j}}\bar{\Delta} & =O_{p}(N^{-2})\\
\partial_{\gamma_{i}\gamma_{j}}\bar{\Delta} & =O_{p}(N^{-2})
\end{align*}
which demonstrates the lemma.
\end{proof}
\begin{lem}
\label{lem:delta_bounds}Let Assumptions 1, 2 and 3 hold, for $s=\{0,1,2,3\}$
\begin{align*}
N\partial_{\beta^{s}\alpha_{i}}\tilde{\Delta} & =O_{p}(N^{-1/2})\\
\max_{i}\vert N\partial_{\beta^{s}\alpha_{i}}\tilde{\Delta}\vert & =O_{p}(N^{-1/2+2\epsilon})
\end{align*}
and hence,
\begin{align*}
\lVert\partial_{\beta^{s}\phi}\Delta\rVert_{q} & =O_{p}(N^{-1+2\epsilon})\\
\lVert\partial_{\beta^{s}\phi}\tilde{\Delta}\rVert_{q} & =O_{p}(N^{-\frac{3}{2}+2\epsilon})\\
\lVert\partial_{\beta^{s}\phi\phi'}\tilde{\Delta}\rVert_{q} & =O_{p}(N^{-\frac{3}{2}+4\epsilon})\\
\lVert\partial_{\beta^{s}\phi\phi'}\Delta\rVert_{q} & =O_{p}(N^{-1+2\epsilon})\\
\lVert\partial_{\beta\phi\phi\phi}\tilde{\Delta}\rVert_{q} & =O_{p}(N^{-\frac{3}{2}+4\epsilon})\\
\lVert\partial_{\beta\phi\phi\phi}\Delta\rVert_{q} & =O_{p}(N^{-1+2\epsilon})\\
\lVert\partial_{\phi\phi\phi\phi}\tilde{\Delta}\rVert_{q} & =O_{p}(N^{-\frac{3}{2}+4\epsilon})\\
\lVert\partial_{\phi\phi\phi\phi}\Delta\rVert_{q} & =O_{p}(N^{-1+2\epsilon}) \\
\lVert\partial_{\phi\phi\phi\phi\phi}\Delta\rVert_{q} & =O_{p}(N^{-1+2\epsilon})
\end{align*}
\end{lem}
\begin{proof}
For the first claim, note that
\[
\bar{E}\big[(N\partial_{\beta^{s}\alpha_{i}}\tilde{\Delta})^{q}\big]=\frac{N^{q}}{\vert\Lambda_{N}\vert^{q}}\sum_{\lambda_{1}:i\in\lambda_{\alpha}}\cdots\sum_{\lambda_{q}:i\in\lambda_{\alpha}}\bar{E}[(\partial_{\beta^{s}\alpha_{i}}\tilde{m}_{\lambda_{1}})\cdots(\partial_{\beta^{s}\alpha_{i}}\tilde{m}_{\lambda_{q}})]
\]
Since terms are independent unless they share a common dyad, each
of the $\lambda_{j}$ must share a dyad with at least one other $\lambda_{k}$.
The number of $\lambda$ that contain a particular dyad is of order
$N^{-2}$ smaller than $\vert\Lambda_{N}\vert$ and hence the above
term must be at most $O_{p}(N^{-q/2})$, and hence $N\partial_{\beta^{s}\alpha_{i}}\tilde{\Delta}=O_{p}(N^{-1/2})$.
This also implies
\begin{align*}
\bar{E}\big[\max_{i}\vert N\partial_{\beta^{s}\alpha_{i}}\tilde{\Delta}\vert^{q}\big] & \leq\sum_{i}\bar{E}\big[\vert N\partial_{\beta^{s}\alpha_{i}}\tilde{\Delta}\vert^{q}\big]\\
& =O_{p}(N^{1-q/2})\\
\max_{i}\vert N\partial_{\beta^{s}\alpha_{i}}\tilde{\Delta}\vert & =O_{p}(N^{-\frac{1}{2}+\frac{1}{q}})=O_{p}(N^{-\frac{1}{2}+2\epsilon})
\end{align*}
The next results follow from the first two by
\begin{align*}
\bar{E}\big[\lVert\partial_{\beta^{s}\phi}\Delta\rVert_{q}^{q}\big] & \leq\bar{E}\big[\sum_{i}\vert\frac{1}{\vert\Lambda_{N}\vert}\sum_{\lambda\in\lambda_{\alpha}}m_{\beta^{s}\alpha_{i}}(\lambda)\vert^{q}\big]+\bar{E}\big[\sum_{i}\vert\frac{1}{\vert\Lambda_{N}\vert}\sum_{\lambda\in\lambda_{\gamma}}m_{\beta^{s}\gamma_{i}}(\lambda)\vert^{q}\big]\\
& \leq\sum_{i}\frac{\vert\Lambda_{\alpha}\vert^{q-1}}{\vert\Lambda_{N}\vert^{q}}\sum_{\lambda\in\lambda_{\alpha}}\bar{E}\big[\vert m_{\beta^{s}\alpha_{i}}(\lambda)\vert^{q}\big]+\sum_{i}\frac{\vert\Lambda_{\alpha}\vert^{q-1}}{\vert\Lambda_{N}\vert}\sum_{\lambda\in\lambda_{\gamma}}\bar{E}\big[\vert m_{\beta^{s}\gamma_{i}}(\lambda)\vert^{q}\big]\\
& =O_{p}(N^{1-q})\\
\lVert\partial_{\beta^{s}\phi}\Delta\rVert_{q} & =O_{p}(N^{-1+\frac{1}{q}})
\end{align*}
and
\begin{align*}
\bar{E}\big[\lVert\partial_{\beta^{s}\phi}\tilde{\Delta}\rVert_{q}^{q}\big] & \leq\sum_{i}\bar{E}\big[\vert\partial_{\beta^{s}\alpha_{i}}\tilde{\Delta}\vert^{q}\big]+\sum_{i}\bar{E}\big[\vert\partial_{\beta^{s}\gamma_{i}}\tilde{\Delta}\vert^{q}\big]\\
& =O_{p}(N^{1-\frac{3q}{2}})\\
\lVert\partial_{\beta^{s}\phi}\tilde{\Delta}\rVert_{q} & =O_{p}(N^{-\frac{3}{2}+\frac{1}{q}})
\end{align*}
For the second result, we focus on the block $\partial_{\beta^{s}\alpha\alpha'}\tilde{\Delta}$,
with other blocks shown similarly. First, we have
\[
\bar{E}\big[(N\partial_{\beta^{s}\alpha_{i}\alpha_{j}}\tilde{\Delta})^{q}\big]=\frac{N^{q}}{\vert\Lambda_{N}\vert^{q}}\sum_{\lambda_{1}:\{i,j\}\in\lambda_{\alpha}}\cdots\sum_{\lambda_{q}:\{i,j\}\in\lambda_{\alpha}}\bar{E}[(\partial_{\beta^{s}\alpha_{i}\alpha_{j}}\tilde{m}_{\lambda_{1}})\cdots(\partial_{\beta^{s}\alpha_{i}\alpha_{j}}\tilde{m}_{\lambda_{q}})]
\]
Again, terms are independent unless they share a common dyad. Given
this, and the fact that there are $O(N^{-2}\vert\Lambda_{N}\vert)$
$\lambda$ that contain both $\alpha_{i}$ and $\alpha_{j}$ (and
of order $N$ fewer that share a common dyad), the above summation
is $O_{p}(N^{-\frac{3}{2}q})$. Using this result, we then have
\begin{align*}
\bar{E}\big[(\max_{i,j\ne i}N\partial_{\beta^{s}\alpha_{i}\alpha_{j}}\tilde{\Delta})^{q}\big] & \leq\sum_{i}\sum_{j\ne i}\bar{E}\big[(N\partial_{\beta^{s}\alpha_{i}\alpha_{j}}\tilde{\Delta})^{q}\big]\\
& =O_{p}(N^{2-\frac{3}{2}q})
\end{align*}
and so $\max_{i,j\ne i}(N\partial_{\beta^{s}\alpha_{i}\alpha_{j}}\tilde{\Delta})=O_{p}(N^{-\frac{3}{2}+2/q})$.
Similarly to the second statement of this lemma, we can show $\max_{i}(N\partial_{\beta^{s}\alpha_{i}\alpha_{i}}\tilde{\Delta})=O_{p}(N^{-\frac{1}{2}+1/q})$.
These bounds then imply (using Lemma S.4 in FW16)
\begin{align*}
\lVert\partial_{\beta^{s}\alpha\alpha'}\tilde{\Delta}\rVert_{q} & \leq\lVert\partial_{\beta^{s}\alpha\alpha'}\tilde{\Delta}\rVert_{\infty}\\
& =\max_{i}\vert\sum_{j}\partial_{\beta^{s}\alpha_{i}\alpha_{j}}\tilde{\Delta}\vert\\
& \leq\max_{i}\vert\partial_{\beta^{s}\alpha_{i}\alpha_{i}}\tilde{\Delta}\vert+N\max_{i\ne j}\vert\partial_{\beta^{s}\alpha_{i}\alpha_{j}}\tilde{\Delta}\vert\\
& =O_{p}(N^{-\frac{3}{2}+\frac{2}{q}})
\end{align*}
and similarly for the remaining blocks of $\partial_{\beta^{s}\phi\phi'}\tilde{\Delta}$.
Similarly, is is straightforward to show that $\bar{E}\big[(\partial_{\alpha_{i}\alpha_{i}}\Delta)^{q}\big]=O_{p}(N^{-q})$
and $\bar{E}\big[(\sum_{j\ne i}\vert\partial_{\alpha_{i}\alpha_{j}}\Delta\vert)^{q}\big]=O_{p}(N^{-q})$
for $i\ne j$, and hence
\begin{align*}
\max_{i}\vert\partial_{\alpha_{i}\alpha_{i}}\Delta\vert & =O_{p}(N^{-1+\frac{1}{q}})\\
\max_{i}\sum_{j\ne i}\vert\partial_{\alpha_{i}\alpha_{j}}\Delta\vert & =O_{p}(N^{-1+\frac{1}{q}})
\end{align*}
from which the final result follows, since $\lVert\partial_{\alpha\alpha}\Delta\rVert_{q}\leq\lVert\partial_{\alpha\alpha}\Delta\rVert_{\infty}$,
and the same can be shown for the remaining components of $\partial_{\phi\phi'}\Delta$.
Finally, the result for $\partial_{\beta\phi\phi\phi}\mathcal{L}$
and $\partial_{\phi\phi\phi\phi}\mathcal{L}$, we have
\begin{align*}
\lVert\partial_{\beta\phi\phi\phi}\Delta\rVert_{q} & =\lVert\sum_{g}(\partial_{\phi\phi\phi_{g}}\Delta)\rVert_{q}\\
& \leq\lVert\sum_{g}(\partial_{\phi\phi\phi_{g}}\Delta)\rVert_{\infty}\\
& =\max_{e}\vert\sum_{f}\sum_{g}(\partial_{\phi_{e}\phi_{f}\phi_{g}}\Delta)\vert\\
& \leq\max_{e}\vert(\partial_{\phi_{e}\phi_{e}\phi_{e}}\Delta)\vert+\max_{e}\vert\sum_{f\ne e}(\partial_{\phi_{e}\phi_{f}\phi_{f}}\Delta)\vert\\
& +\max_{e}\vert\sum_{f\ne e}(\partial_{\phi_{e}\phi_{e}\phi_{f}}\Delta)\vert+\max_{e}\vert\sum_{f\ne g\ne e}(\partial_{\phi_{e}\phi_{f}\phi_{g}}\Delta)\vert\\
& =O_{p}(N^{-1+1/q})
\end{align*}
since we have that derivatives with respect to $j$ distinct indices
in $\phi$ result in a sum that is $O_{p}(N^{-j})$. The bound for
$\partial_{\phi\phi\phi\phi}\mathcal{L}$ can be shown similarly. The results for $\partial_{\beta\phi\phi\phi}\Delta$ and $\partial_{\phi\phi\phi\phi}\Delta$ follow identically to $\partial_{\beta\phi\phi}\Delta$.
\end{proof}
\subsection{Asymptotic expansion}
An expansion of the average effects estimator is
\begin{align*}
\widehat{\Delta}-\Delta & =(\partial_{\beta}\Delta)(\widehat{\beta}-\beta_{0})\\
& +(\partial_{\phi'}\Delta)(\widehat{\phi}-\phi_{0})\\
& \quad+\frac{1}{2}(\partial_{\beta\beta}\Delta)(\widehat{\beta}-\beta_{0})^{2}\\
& +(\partial_{\beta\phi'}\Delta)(\widehat{\phi}-\phi_{0})(\widehat{\beta}-\beta_{0})\\
& \quad+\frac{1}{2}(\widehat{\phi}-\phi_{0})'(\partial_{\phi\phi'}\Delta)(\widehat{\phi}-\phi_{0})\\
& \quad+\frac{1}{2}(\widehat{\phi}-\phi_{0})'(\partial_{\beta\phi\phi'}\Delta)(\widehat{\phi}-\phi_{0})(\widehat{\beta}-\beta_{0})\\
& \quad+\frac{1}{6}\sum_{g}(\widehat{\phi}-\phi_{0})'(\partial_{\phi\phi'\phi_{g}}\Delta)(\widehat{\phi}-\phi_{0})(\widehat{\phi}_{g}-\phi_{0,g})\\
& \quad+\frac{1}{24}\sum_{g,h}(\widehat{\phi}-\phi_{0})'\big(\partial_{\phi\phi'\phi_{g}\phi_{h}}\Delta\big)(\widehat{\phi}-\phi_{0})(\widehat{\phi}_{g}-\phi_{0,g})(\widehat{\phi}_{h}-\phi_{0,h})\\
& \quad+\tilde{R}_{\Delta}
\end{align*}
where the remainder term satisfies
\begin{align*}\tilde{R}_{\Delta} & =\frac{1}{6}\big(\partial_{\beta\beta\beta}\Delta(\bar{\beta},\bar{\phi})\big)(\widehat{\beta}-\beta_{0})^{3}\\
& +\frac{1}{2}(\partial_{\beta\beta\phi'}\Delta)(\widehat{\phi}-\phi_{0})(\widehat{\beta}-\beta_{0})^{2}\\
& \quad+\frac{1}{6}\big(\partial_{\beta\beta\beta\phi'}\Delta(\bar{\beta},\bar{\phi})\big)(\widehat{\phi}-\phi_{0})(\widehat{\beta}-\beta_{0})^{3}\\
& \quad+\frac{1}{4}(\widehat{\phi}-\phi_{0})'\big(\partial_{\beta\beta\phi\phi'}\Delta(\bar{\beta},\bar{\phi})\big)(\widehat{\phi}-\phi_{0})(\widehat{\beta}-\beta_{0})^{2}\\
& \quad+\frac{1}{6}\sum_{g}(\widehat{\phi}-\phi_{0})'\big(\partial_{\beta\phi\phi'\phi_{g}}\Delta(\bar{\beta},\bar{\phi})\big)(\widehat{\phi}-\phi_{0})(\widehat{\phi}_{g}-\phi_{0,g})(\widehat{\beta}-\beta_{0})\\
& \quad+\frac{1}{120}\sum_{f,g,h}(\widehat{\phi}-\phi_{0})'\big(\partial_{\phi\phi'\phi_{f}\phi_{g}\phi_{h}}\Delta(\bar{\beta},\bar{\phi})\big)(\widehat{\phi}-\phi_{0})(\widehat{\phi}_{f}-\phi_{0,f})(\widehat{\phi}_{g}-\phi_{0,g})(\widehat{\phi}_{h}-\phi_{0,h})\\
& =o_{p}(N^{-2})
\end{align*}
Substitution of the expansion for $\widehat{\beta}$ up to order $N^{-2}$
and the expansion for $\widehat{\phi}$ shown above then gives an expansion with remainder $o_p(N^{-2})$, since for example
\begin{align*}
\lVert(\partial_{\phi'}\Delta) \tilde{R}_\phi \rVert_{q} & \leq N^{1-2/q}\lVert\partial_{\phi'}\Delta\rVert_{q}\lVert\tilde{R}_\phi\rVert_{q}\\
& =O_p(N^{-1/q})\rVert_{q}\lVert\tilde{R}_\phi\rVert_{q}=O_p(N^{-5/2+16\epsilon})=o_{p}(N^{-2})
\end{align*}
It follows that the expansion up to first-order is
\begin{align*}
N(\widehat{\Delta}-\Delta) & =(\partial_{\beta}\bar{\Delta})N(\widehat{\beta}-\beta_{0})+N(\partial_{\phi'}\Delta)(\partial_{bs}\mathcal{L}^{*})\mathcal{S}_{\beta}\\
& \quad+N(\partial_{\phi'}\Delta)(\partial_{ss'}\mathcal{L}^{*})\mathcal{S}-\frac{1}{2}N(\partial_{\phi'}\Delta)\sum_{g}(\partial_{ss's_{g}}\mathcal{L}^{*})\mathcal{S}\mathcal{S}_{g}\\
& \quad+\frac{1}{2}N\mathcal{S}'(\partial_{ss'}\mathcal{L}^{*})(\partial_{\phi\phi'}\Delta)(\partial_{ss'}\mathcal{L}^{*})\mathcal{S}+o_{p}(1)\\
\\
& =(\partial_{\beta}\bar{\Delta})\bar{W}_{N}^{-1}\big(U^{(0)}+U^{(1)}\big)\\
& \quad+\bar{W}_{N}^{-1}(\partial_{\beta\phi'}\bar{\mathcal{L}})\bar{\mathcal{H}}^{-1}(\partial_{\phi}\bar{\Delta})\mathcal{S}_{\beta}\\
& \quad+N(\partial_{\phi}\bar{\Delta})\mathcal{\bar{H}}^{-1}\mathcal{S}+N(\partial_{\phi}\tilde{\Delta})\mathcal{\bar{H}}^{-1}\mathcal{S}+N(\partial_{\phi}\bar{\Delta})\mathcal{\bar{H}}^{-1}\tilde{\mathcal{H}}\mathcal{\bar{H}}^{-1}\mathcal{S}\\
& \quad+W_{N}^{-1}(\partial_{\phi}\bar{\Delta})\mathcal{\bar{H}}^{-1}(\partial_{\beta\phi}\bar{\mathcal{L}})\Big((\partial_{\beta\phi'}\bar{\mathcal{L}})\mathcal{\bar{H}}^{-1}\mathcal{S}+(\partial_{\beta\phi'}\tilde{\mathcal{L}})\mathcal{\bar{H}}^{-1}\mathcal{S}+(\partial_{\beta\phi'}\bar{\mathcal{L}})\mathcal{\bar{H}}^{-1}\tilde{\mathcal{H}}\mathcal{\bar{H}}^{-1}\mathcal{S}\Big)\\
& \quad+\frac{1}{2}N\mathcal{S}'\bar{\mathcal{H}}^{-1}\sum_{f}(\partial_{\phi\phi'\phi_{f}}\bar{\mathcal{L}})[\mathcal{\bar{H}}^{-1}(\partial_{\phi'}\bar{\Delta})]_{f}\mathcal{\bar{H}}^{-1}\mathcal{S}\\
& \quad+\frac{1}{2}\bar{W}^{-1}(\partial_{\phi'}\bar{\Delta})\mathcal{\bar{H}}^{-1}(\partial_{\beta\phi}\mathcal{\bar{L}})\mathcal{S}'\bar{\mathcal{H}}^{-1}\sum_{f}(\partial_{\phi\phi'\phi_{f}}\bar{\mathcal{L}})[\mathcal{\bar{H}}^{-1}(\partial_{\phi'}\bar{\Delta})]_{f}\mathcal{\bar{H}}^{-1}\mathcal{S}\\
& \quad+\frac{1}{2}N\mathcal{S}'\bar{\mathcal{H}}^{-1}(\partial_{\phi\phi'}\bar{\Delta})\bar{\mathcal{H}}^{-1}\mathcal{S}+o_{p}(1)\\
\\
& =\Big((\partial_{\beta}\bar{\Delta})+(\partial_{\beta\phi'}\bar{\mathcal{L}})\bar{\mathcal{H}}^{-1}(\partial_{\phi}\bar{\Delta})\Big)\bar{W}_{N}^{-1}\big(U^{(0)}+U^{(1)}\big)\\
& \quad+N(\partial_{\phi}\bar{\Delta})\mathcal{\bar{H}}^{-1}\mathcal{S}+N(\partial_{\phi}\tilde{\Delta})\mathcal{\bar{H}}^{-1}\mathcal{S}-N(\partial_{\phi}\bar{\Delta})\mathcal{\bar{H}}^{-1}\tilde{\mathcal{H}}\mathcal{\bar{H}}^{-1}\mathcal{S}\\
& \quad+\frac{1}{2}N\mathcal{S}'\bar{\mathcal{H}}^{-1}\Big((\partial_{\phi\phi'}\bar{\Delta})+\sum_{f}(\partial_{\phi\phi'\phi_{f}}\bar{\mathcal{L}})[\mathcal{\bar{H}}^{-1}(\partial_{\phi'}\bar{\Delta})]_{f}\Big)\bar{\mathcal{H}}^{-1}\mathcal{S}+o_{p}(1)
\end{align*}
The expansion for the leave-out estimator follows similarly, replacing $m_\lambda$ with $\frac{N-1}{N-r-1} m_\lambda 1^k_\lambda$. Note that the same bounds on $(N\partial_{\beta^{s}\alpha_{i}}\tilde{\Delta})$ derivied in Lemma \ref{lem:delta_bounds} also apply to $(N\partial_{\beta^{s}\alpha_{i}}\tilde{\Delta}_{(k)})$, so that we may replace the $\partial_{\beta}\Delta_{(k)}$ and $\partial_{\alpha}\Delta_{(k)}$ terms with $\partial_{\beta}\bar{\Delta}$ and $\partial_{\alpha}\bar{\Delta}$ in the first-order expansion. This gives the expansion for $N(\widehat{\Delta}_{(k)} - \Delta_{(k)})$ as shown in Lemma 7.
\subsection{Jackknifing higher-order terms for average effect}
Analogously to the jackknife of the common parameter $\widehat{\beta},$
we demonstrate the effect of the jackknife for just one higher-order
term in the expansion for $\widehat{\Delta}$. Inspection of this
expansion shows that the structure of each of the terms is similar,
i.e. a V-statistic like term of some order, multiplied by $(\partial_{\phi'}\Delta)$
or $(\partial_{\phi\phi'}\Delta)$. The bound for the jackknife
version of each of these many terms is therefore highly similar to
the specific term we demonstrate here. Consider the expansion term
$(\partial_{\phi'}\Delta)(\partial_{bss'}\mathcal{L}_{(1)}^{*})\mathcal{S}\mathcal{S}_{\beta}$,
which contains the term
\[
(\partial_{\phi'}\Delta)\mathcal{H}^{-1}\mathcal{H}_{b}\mathcal{H}^{-1}\mathcal{S}\mathcal{S}_{\beta}
\]
Expanding out this terms gives
\begin{align*}
(\partial_{\phi'}\Delta)\mathcal{H}^{-1}\mathcal{H}_{b}\mathcal{H}^{-1}\mathcal{S}\mathcal{S}_{\beta} & =W^{-1}(\partial_{\phi'}\Delta)\mathcal{H}^{-1}(\partial_{\beta\phi\phi'}\mathcal{L})\mathcal{H}^{-1}\mathcal{S}\mathcal{S}_{\beta}\\
& +W^{-1}(\partial_{\phi'}\Delta)\mathcal{H}^{-1}\sum_{f}(\partial_{\phi\phi'\phi_{f}}\mathcal{L})\big[\mathcal{H}^{-1}(\partial_{\beta\phi}\mathcal{L})\big]_{f}\mathcal{H}^{-1}\mathcal{S}\mathcal{S}_{\beta}
\end{align*}
Take the first term in this expression. Replacing $W$ with $\bar{W}+(W-\bar{W})$,
and similarly for $\mathcal{H}^{-1}$ and $\partial_{\beta\phi\phi'}\mathcal{L}$
we get
\[
\frac{1}{N}\bar{W}_{N}^{-1}(\partial_{\phi'}\Delta)\mathcal{\bar{H}}^{-1}(\partial_{\beta\phi\phi'}\bar{\mathcal{L}})\bar{\mathcal{H}}^{-1}\mathcal{S}\mathcal{S}_{\beta}+o_{p}(N^{-2})
\]
This result holds identically for the leave-out samples, replacing
$\mathcal{S}$ with $\mathcal{S}_{(k)}$ and similarly for $\partial_{\phi'}\Delta$
and $\mathcal{S}_{\beta}$. Ignoring the $\frac{1}{N}\bar{W}_{N}^{-1}$
term for the moment, which will not be affected by the jackknifing,
and letting $\bar{M}=\mathcal{\bar{H}}^{-1}(\partial_{\beta\phi\phi'}\bar{\mathcal{L}})\bar{\mathcal{H}}^{-1}$,
we can decompose the above sum further into the components of $\phi=(\alpha,\gamma)$,
for example the first of these terms would be
\begin{align*}
(\partial_{\alpha'}\Delta)\bar{M}_{\alpha\alpha}\mathcal{S}_{\alpha}\mathcal{S}_{\beta} & =\frac{1}{\lvert\Lambda_{N}\rvert}\frac{1}{(N-1)^{2}}\sum_{i}\sum_{\lambda:i\in\lambda_{\alpha}}\sum_{s}\sum_{t\ne s}\sum_{j}\sum_{l\ne j}M_{is}(\partial_{\alpha_{i}}m_{\lambda})(\partial_{\pi}\ell_{st})(\partial_{\beta}\ell_{jl})
\end{align*}
and note that, by Lemma \ref{lem:HAH_bound}, we have $\max_{i\ne s}\bar{M}_{is}=O_{p}(N^{-1})$,
and $\max_{i}\bar{M}_{ii}=O_{p}(1)$. This is again a V-statistic
like term, and we will show that the jackknifed version of this term
is $o_{p}(1)$. The $k$-th leave-out version of this term is
\begin{align*}
\frac{N-1}{N-r-1}&(\partial_{\alpha'}\Delta_{N,(k)})\bar{M}_{\alpha\alpha}\mathcal{S}_{\alpha,(k)}\mathcal{S}_{\beta,(k)} \\
& =\frac{1}{\lvert\Lambda_{N}\rvert}\frac{N-1}{(N-2)^{2}(N-r-1)}\sum_{i}\sum_{\lambda:i\in\lambda_{\alpha}}\sum_{s}\sum_{t\ne s}\sum_{j}\sum_{l\ne j}M_{is}(\partial_{\alpha_{i}}m_{\lambda})(\partial_{\pi}\ell_{st})(\partial_{\beta}\ell_{jl})1_{\lambda}^{k}1_{st}^{k}1_{jl}^{k}
\end{align*}
The jackknifed term is then
\begin{align*}
\mathcal{J}= & (N-1)(\partial_{\alpha'}\Delta)\bar{M}_{\alpha\alpha}\mathcal{S}_{\alpha}\mathcal{S}_{\beta}-\frac{N-2}{N-1}\sum_{k}(\partial_{\alpha'}\Delta_{N,(k)})\bar{M}_{\alpha\alpha}\mathcal{S}_{\alpha,(k)}\mathcal{S}_{\beta,(k)}\\
= & \frac{1}{\lvert\Lambda_{N}\rvert}\frac{1}{N-1}\sum_{i}\sum_{\lambda:i\in\lambda_{\alpha}}\sum_{s}\sum_{t\ne s}\sum_{j}\sum_{l\ne j}M_{is}(\partial_{\alpha_{i}}m_{\lambda})(\partial_{\pi}\ell_{st})(\partial_{\beta}\ell_{jl})\\
& -\frac{1}{\lvert\Lambda_{N}\rvert}\frac{1}{(N-2)(N-r-1)}\sum_{i}\sum_{\lambda:i\in\lambda_{\alpha}}\sum_{s}\sum_{t\ne s}\sum_{j}\sum_{l\ne j}M_{is}(\partial_{\alpha_{i}}m_{\lambda})(\partial_{\pi}\ell_{st})(\partial_{\beta}\ell_{jl})\big(\sum_{k}1_{\lambda}^{k}1_{st}^{k}1_{jl}^{k}\big)\\
= & \frac{1}{\lvert\Lambda_{N}\rvert}\sum_{i}\sum_{\lambda:i\in\lambda_{\alpha}}\sum_{s}\sum_{t\ne s}\sum_{j}\sum_{l\ne j}M_{is}(\partial_{\alpha_{i}}m_{\lambda})(\partial_{\pi}\ell_{st})(\partial_{\beta}\ell_{jl})\Big(\frac{1}{N-1}-\frac{\sum_{k}1_{\lambda}^{k}1_{st}^{k}1_{jl}^{k}}{(N-2)(N-r-1)}\Big)
\end{align*}
Note that, since $\lambda$ contains $r$ observations, the set $\{\lambda,(s,t),(j,l)\}$
spans at most $r+2$ different sets $\mathcal{I}_{k}$ and so $\sum_{k}1_{\lambda}^{k}1_{st}^{k}1_{jl}^{k}\in\{N-r-3,\dots,N-2\}$,
depending on how many of the $\mathcal{I}_{k}$ the observations are
contained in. Then, letting $1_{n}$ be an indicator variable that
is equal to one when the set $\{\lambda,(s,t),(j,l)\}$ covers $n$
different $\mathcal{I}_{k},$ we can write
\begin{align*}
\vert\mathcal{J}\vert & \leq\sum_{n=1}^{r+2}\vert\frac{1}{\lvert\Lambda_{N}\rvert}\sum_{i}\sum_{\lambda:i\in\lambda_{\alpha}}\sum_{s}\sum_{t\ne s}\sum_{j}\sum_{l\ne j}M_{is}(\partial_{\alpha_{i}}m_{\lambda})(\partial_{\pi}\ell_{st})(\partial_{\beta}\ell_{jl})\Big(\frac{1}{N-1}-\frac{N-n-1}{(N-2)(N-r-1)}\Big)1_{n}\vert\\
& \leq C\frac{1}{N^{2}}\sum_{n=1}^{r+2}\vert\frac{1}{\lvert\Lambda_{N}\rvert}\sum_{i}\sum_{\lambda:i\in\lambda_{\alpha}}\sum_{s}\sum_{t\ne s}\sum_{j}\sum_{l\ne j}M_{is}(\partial_{\alpha_{i}}m_{\lambda})(\partial_{\pi}\ell_{st})(\partial_{\beta}\ell_{jl})1_{n}\vert\\
& \leq C\frac{1}{N^{2}}\sum_{n=1}^{r+2}\vert\frac{1}{\lvert\Lambda_{N}\rvert}\sum_{i}\sum_{\lambda:i\in\lambda_{\alpha}}\sum_{s}\sum_{t\ne s}\sum_{j}\sum_{l\ne j}M_{is}(\partial_{\alpha_{i}}m_{\lambda})(\partial_{\pi}\ell_{st})(\partial_{\beta}\ell_{jl})1_{n}\vert
\end{align*}
The second moment of the RHS is
\begin{align*}
&\bar{E}\Big[ \Big(\frac{1}{\lvert\Lambda_{N}\rvert}\sum_{i}\sum_{\lambda:i\in\lambda_{\alpha}}\sum_{s}\sum_{t\ne s}\sum_{j}\sum_{l\ne j}M_{is}(\partial_{\alpha_{i}}m_{\lambda})(\partial_{\pi}\ell_{st})(\partial_{\beta}\ell_{jl})1_{n}\Big)^{2}\Big]\\
&= \frac{1}{\lvert\Lambda_{N}\rvert^{2}}\sum_{i,i',s,s',j,j'}\sum_{\substack{\lambda:i\in\lambda_{\alpha}\\
\lambda':i'\in\lambda_{\alpha}}} \sum_{t\ne s}\sum_{t'\ne s'}\sum_{l\ne j}\sum_{l'\ne j'}M_{is}M_{i's'}\bar{E}\Big[(\partial_{\alpha_{i}}m_{\lambda})(\partial_{\alpha_{i'}}m_{\lambda'})(\partial_{\pi}\ell_{st})(\partial_{\pi}\ell_{s't'})(\partial_{\beta}\ell_{jl})(\partial_{\beta}\ell_{j'l'})1_{n}1_{n'}\Big]\\
&= \frac{1}{\lvert\Lambda_{N}\rvert^{2}}\sum_{i,i',j,j'}\sum_{\substack{\lambda:i\in\lambda_{\alpha}\\
\lambda':i'\in\lambda_{\alpha}}} \sum_{t\ne i}\sum_{t'\ne i'}\sum_{l\ne j}\sum_{l'\ne j'}M_{ii}M_{i'i'}\bar{E}\Big[(\partial_{\alpha_{i}}m_{\lambda})(\partial_{\alpha_{i'}}m_{\lambda'})(\partial_{\pi}\ell_{it})(\partial_{\pi}\ell_{i't'})(\partial_{\beta}\ell_{jl})(\partial_{\beta}\ell_{j'l'})1_{n}1_{n'}\Big]\\
& +2\frac{1}{\lvert\Lambda_{N}\rvert^{2}}\sum_{i,i',j,j'}\sum_{\substack{\lambda:i\in\lambda_{\alpha}\\
\lambda':i'\in\lambda_{\alpha}}} \sum_{s\ne i}\sum_{\substack{t\ne s,t'\ne i'\\
l\ne j,l'\ne j'}}M_{is}M_{i'i'}\bar{E}\Big[(\partial_{\alpha_{i}}m_{\lambda})(\partial_{\alpha_{i'}}m_{\lambda'})(\partial_{\pi}\ell_{st})(\partial_{\pi}\ell_{i't'})(\partial_{\beta}\ell_{jl})(\partial_{\beta}\ell_{j'l'})1_{n}1_{n'}\Big]\\
& +\frac{1}{\lvert\Lambda_{N}\rvert^{2}}\sum_{i,i',j,j'}\sum_{\substack{\lambda:i\in\lambda_{\alpha}\\
\lambda':i'\in\lambda_{\alpha}}} \sum_{s\ne i,s'\ne i'}\sum_{\substack{t\ne s,t'\ne s'\\
l\ne j,l'\ne j'}} M_{is}M_{i's'}\bar{E}\Big[(\partial_{\alpha_{i}}m_{\lambda})(\partial_{\alpha_{i'}}m_{\lambda'})(\partial_{\pi}\ell_{st})(\partial_{\pi}\ell_{s't'})(\partial_{\beta}\ell_{jl})(\partial_{\beta}\ell_{j'l'})1_{n}1_{n'}\Big]
\end{align*}
Since $\bar{E}[\partial_{\pi}\ell_{ij}]=\bar{E}[\partial_{\beta}\ell_{ij}]=0$,
while $\max_{i\ne s}\bar{M}_{is}=O_{p}(N^{-1})$, and $\max_{i}\bar{M}_{ii}=O_{p}(1)$,
we can conclude that the above sum is $O_{p}(N^{2})$, and hence $\vert\mathcal{J}\vert=O_{p}(N^{-1})$.
The same steps can be shown for the remaining elements of this term,
from which we then conclude that the jackknifed version of $(\partial_{\phi'}\Delta)\mathcal{H}^{-1}\mathcal{H}_{b}\mathcal{H}^{-1}\mathcal{S}\mathcal{S}_{\beta}$
is $O_{p}(N^{-2})$, and hence does not show up in the asymptotic
distribution of $\widehat{\Delta}_{J}$. Inspection of the expansion
for $\widehat{\Delta}$ shows that the remaining terms of the
expansion share this similar V-statistic like structure, and so similar
proofs could be applied to each of the terms.
This then gives the result
\[
(N-1)(\widehat{\Delta}-\Delta)-(N-2)\frac{1}{N-1}\sum_{k}(\widehat{\Delta}_{(k)}-\Delta_{(k)})=\frac{1}{N-1}\sum_{i}\sum_{j\ne i}h_{ij}+o_{p}(1).
\]
\end{document}