and ${\varphi}$ as in ${\mathscr{C}}$.

For ${\varepsilon}$ small, define the embeddings ${\overline}{\iota}_{f,{\overline}{x}}$ ${\overline}{\iota}_{f,e}$, ${\overline}{\iota}_{f,{\overline}{y}}$ as the points $$(4{\varepsilon},0,\dots,0), \quad(0,0,\dots,0), \quad(-4{\varepsilon},0,\dots,0)\in{\mathbb{R}}^{d_i}$$respectively. Extend these embeddings to framed embeddings ${\overline}{{\varphi}}_{f,{\overline}{x}}$ ${\overline}{{\varphi}}_{f,e}$, ${\overline}{{\varphi}}_{f,{\overline}{y}}$ in the obvious way, by taking the product of ${\varepsilon}$ neighbourhoods in each co-ordinate of ${\mathbb{R}}^{d_i}$. There are exactly two ways to frame an embedded point in any Euclidean space, which we will denote “$+$” and “$-$”. Declare ${\overline}{{\varphi}}_{f,{\overline}{x}}$ and ${\overline}{{\varphi}}_{f,{\overline}{y}}$ to both be $+$ framings and ${\overline}{{\varphi}}_{f,e}$ to be a $-$.

Fixing an object ${\overline}{b}$ with $|{\overline}{b}|<i$, we will now immerse and frame ${\mathcal{M}}(e,{\overline}{b})={\mathcal{M}}(x,b)\sqcup{\mathcal{M}}(y,b)$. To do this we will simply superimpose the embeddings and framings given by $\iota$ and ${\varphi}$. Note that this might introduce immersion points. Specifically, define the neat immersion$${\overline}{\iota}_{e,{\overline}{b}}:{\mathcal{M}}(e,{\overline}{b})\looparrowright{\mathbb{E}}^{\textbf{d}}[|e|:|{\overline}{b}|];\qquad p\mapsto\left\{\begin{array}{lcl}\iota_{x,b}(p)&&\text{if $p\in {\mathcal{M}}(x,b)$,}\\ \iota_{y,b}(p)&&\text{if $p\in {\mathcal{M}}(y,b)$.}\end{array}\right.$$And similarly, for ${\varphi}_{x,b},{\varphi}_{y,b}$, we obtain the framing ${\overline}{{\varphi}}_{e,{\overline}{b}}$.

We will now immerse and frame ${\mathcal{M}}(f,{\overline}{b})$, where ${\overline}{b}\neq {\overline}{x}, {\overline}{y},e$. First apply Lemma \[lem:arc\], case (1) to $\iota_{x,b}$, setting $j=i-k-1$ and $(u,v)=(2{\varepsilon},2{\varepsilon})$, to obtain neat embeddings $\iota_{x,b}'$, ${\varphi}_{x,b}'$ and ${\varphi}_{x,b}''$. Now apply Lemma \[lem:arc\], case (1) a second time, this time to $\iota_{y,b}$, setting $j=i-k-1$ and $(u,v)=(-2{\varepsilon},2{\varepsilon})$, to obtain neat embeddings $\iota_{y,b}'$, ${\varphi}_{y,b}'$ and ${\varphi}_{y,b}''$. In order to use these functions to define 