a neat immersion ${\overline}{\iota}_{f,{\overline}{b}}$, we will need to increase the range of $\iota_{y,b}', \iota_{y,b}'$ by a factor or ${\mathbb{R}}^{d_i-1}$. We do so by simply setting the new $d_i-1$ co-ordinates in the range of each neat embedding to 0. In an abuse of notation, use the same symbols for the new functions - i.e. from now on write$$\begin{aligned}
\iota_{x,b}':&{\mathcal{M}}(x,b)\times[0,1]\looparrowright {\mathbb{E}}^{\textbf{d}}[|x|:|b|]\times{\mathbb{R}}_+\times{\mathbb{R}}\times{\mathbb{R}}^{d_i-1}= {\mathbb{E}}^{\textbf{d}}[|f|:|{\overline}{b}|],\\
\iota_{y,b}':&{\mathcal{M}}(y,b)\times[0,1]\looparrowright {\mathbb{E}}^{\textbf{d}}[|y|:|b|]\times{\mathbb{R}}_+\times{\mathbb{R}}\times{\mathbb{R}}^{d_i-1}= {\mathbb{E}}^{\textbf{d}}[|f|:|{\overline}{b}|].\end{aligned}$$ Using this, define a neat immersion of ${\mathcal{M}}(f,{\overline}{b})$$${\overline}{\iota}_{f,{\overline}{b}}:{\mathcal{M}}(e,{\overline}{b})\times[0,1]\looparrowright{\mathbb{E}}^{\textbf{d}}[|f|,|{\overline}{b}|];\quad (p,q)\mapsto\left\{\begin{array}{lcl}\iota_{x,b}'(p,q)&&\text{if $p\in {\mathcal{M}}(x,b)$,}\\ \iota_{y,b}'(p,q)&&\text{if $p\in {\mathcal{M}}(y,b)$.}\end{array}\right.$$To build a framing ${\overline}{{\varphi}}_{f,{\overline}{b}}$, we will need to modify *both* the domain and range of the neat embeddings ${\varphi}_{x,b}''$, ${\varphi}_{y,b}''$. Specifically, we must extend the domains of these neat embeddings by a factor of $[-{\varepsilon},{\varepsilon}]^{d_i-1}$, and we must extend the ranges by a factor of ${\mathbb{R}}^{d_i-1}$. We do this by simply extending ${\varphi}_{x,b}''$ and ${\varphi}_{y,b}''$ by the obvious inclusion $[-{\varepsilon},{\varepsilon}]^{d_i-1}\subset{\mathbb{R}}^{d_i-1}$. Again we abuse notation and use the same symbols for the extended functions:$$\begin{aligned}
{\varphi}_{x,b}'':&{\mathcal{M}}(x,b)\times[0,1]\times[-{\varepsilon},{\varepsilon}]^A\looparrowright {\mathbb{E}}^{\textbf{d}}[|x|:|b|]\times{\mathbb{R}}_+\times{\mathbb{R}}\times{\mathbb{R}}^{d_i-1},\\
{\varphi}_{y,b}'':&{\mathcal{M}}(y,b)\times[0,1]\times[-{\varepsilon},{\varepsilon}]^A\looparrowright {\mathbb{E}}^{\textbf{d}}[|y|:|b|]\times{\mathbb{R}}_+\times{\mathbb{R}}\times{\mathbb{R}}^{d_i-1},\end{aligned}$$where $A=d_{|{\overline}{b}|}+\dots+d_{|f|-1}$. So, at last, we have a neat immersion: $$\begin{array}{rcl}{\overline}{{\varphi}}_{f,{\overline}{b}}:{\mathcal{M}}(e,{\overline}{b})\times[0,1]\times[-{\varepsilon},{\varepsilon}]^A&\looparrowright&{\mathbb{E}}^{\textbf{d}}[|f|,|{\overline}{b}|]\\(p,q,\textbf{t})&\mapsto&\left\{\begin{array}{lcl}{\varphi}_{x,b}''(p,q.\textbf{t})&&\text{if $p\in {\mathcal{M}}(x,b)$,}\\ {\varphi}_{y,b}''(p,q,\textbf{t})&&\text{if $p\in {\mathcal{M}}(y,b)$.}\end{array}\right.\end{array}$$Note that the use of the construction from Lemma \[lem:arc\] is consistent with our choice of $+$ and $-$ framings ${\overline}{{\varphi}}_{f,{\overline}{x}}$ ${\overline}{{\varphi}}_{f,e}$, ${\overline}{{\varphi}}_{f,{\overline}{y}}$ from earlier. The “+” case is completed.

Now we 