smooth $(m+1)$–dimensional $\langle n+1\rangle$–manifold, given by setting $\partial_{n+1}(M\times[0,1])=M\times\{0,1\}$ and $\partial_i(M\times[0,1])=\partial_iM\times[0,1]$ otherwise.

\[lem:arc\]Suppose there is a neat immersion $\iota:M\looparrowright {\mathbb{E}}^{(d_k,\dots,d_{n+k})}$. For real numbers $u>0$, $v\geq 0$, let $$\gamma_{u,v}(t)=(s(t),2t-1)\in {\mathbb{R}}_+\times{\mathbb{R}}$$ be such that $(t-u)^2+s^2=v^2$. Then the function$$\iota':M\times[0,1]\looparrowright {\mathbb{E}}^{(d_k,\dots,d_{n+k})}\times {\mathbb{R}}_+\times{\mathbb{R}};\qquad \iota'(p,t):=(\iota(p),\gamma_{u,v}(t))$$ is a neat immersion.

Denote by $R:{\mathbb{E}}^{\textbf{d}}\to {\mathbb{E}}^{\textbf{d}}$ the reflection in the plane through the origin which is perpendicular to the first coordinate axis of ${\mathbb{R}}^{d_{n+k}}\subset {\mathbb{E}}^{\textbf{d}}$. Suppose $\iota$ extends to a neat immersion ${\varphi}:M\times[-{\varepsilon},{\varepsilon}]^A\looparrowright {\mathbb{E}}^{\textbf{d}}$, with $A=d_k+\dots+d_{n+k}$, then $-{\varphi}:=R\circ{\varphi}$ determines another possible framing for $\iota$. Define two possible boundary framings for $\iota'|_{M\times\{0,1\}}$by

1.  $$\begin{aligned}
    {\varphi}_1:M\times\{0,1\}\times[-{\varepsilon},{\varepsilon}]^{A}\times[-{\varepsilon},{\varepsilon}]&\looparrowright& {\mathbb{E}}^{\textbf{d}}\times{\mathbb{R}}_+\times{\mathbb{R}}\\
    \quad {\varphi}_1(p,0,\textbf{q},r)&=&({\varphi}(p,\textbf{q}),0, r-1)\\
    {\varphi}_1(p,1,\textbf{q},r)&=&({\varphi}(p,\textbf{q}),0,-r+1)\end{aligned}$$

2.  $$\begin{aligned}
    {\varphi}_2:M\times\{0,1\}\times[-{\varepsilon},{\varepsilon}]^{A}\times[-{\varepsilon},{\varepsilon}]&\looparrowright& {\mathbb{E}}^{\textbf{d}}\times{\mathbb{R}}_+\times{\mathbb{R}}\\
    \quad {\varphi}_2(p,0,\textbf{q},r)&=&({\varphi}(p,\textbf{q}),0, -r-1)\\
    {\varphi}_2(p,1,\textbf{q},r)&=&(-{\varphi}(p,\textbf{q}),0,-r+1)\end{aligned}$$

Then ${\varphi}_1$ and ${\varphi}_2$ respectively extend to framings for the whole of $\iota'$:$${\varphi}''_1,{\varphi}_2'':M\times[0,1]\times[-{\varepsilon},{\varepsilon}]^{A+1}\looparrowright {\mathbb{E}}^{\textbf{d}}\times{\mathbb{R}}_+\times{\mathbb{R}}.$$

Actually there is more than one way to extend ${\varphi}_2$ to the required framing ${\varphi}_2''$. The proof below uses a specific choice, which we shall carry with us to make calculations later. Precisely, we make the choice which conforms to one of the cases $\overline{{{}^+_+}{{}^+_-}}, \overline{{{}^+_-}{{}^+_+}}$ in the *coherent system of paths* described in [@MR3252965 Lemma 3.1], and in the preceding paragraphs there.

For a manifold with corners $N$, denote by $N^{(l)}$ the set of corner points of codimension 