then $$fr(I_{B,A})=1+\varepsilon_B.$$

3.  If $|a|=|x|$ and $a\not= x$, then the framings of components in ${\mathcal{M}}(a',b')$ are the same as in ${\mathcal{M}}(a,b)$. For $${\mathcal{M}}(x',b')={\mathcal{M}}(x,b) \sqcup {\mathcal{M}}(y,b)$$ the framing values in the new moduli space agree with the framing values in the old moduli spaces.

Assume $|a|=|x|+2$. In ${\mathscr{C}}_S$ the object $x'$ corresponds to $e$ from the intermediate flow category, and the intervals in ${\mathcal{M}}({\overline}{a},e)$ are of the form $\{-\} \times J$ where $-\in {\mathcal{M}}(f,e)$ from the intermediate flow category, and $J$ is an interval in the original ${\mathcal{M}}(a,x)$. By [@JLSmorse Proposition 3.6.6], $fr(\{-\}\times J)=fr(J)+1+\varepsilon_-$. This means $fr(\{-\}\times J)=fr(J)$.

For ${\mathcal{M}}(a',y')$ the intervals in ${\mathcal{M}}(a,y)$ do not change, but we also get intervals of the form $\{B\} \times J$ with $B\in {\mathcal{M}}(f,{\overline}{y})$ and $J$ in ${\mathcal{M}}(a,x)$. As before, $fr(\{B\}\times J) = fr(J) + 1 + \varepsilon_B$. Now if we use a $(-)$–slide, $\varepsilon_B=1$, and if we use a $(+)$–handle slide, $\varepsilon_B=0$. Hence we get (1).

Now assume $|a|=|x|+1$. In the intermediate flow category we get intervals $J_C$ and $J_B$ in ${\mathcal{M}}(f,{\overline}{b})$ coming from points $C\in {\mathcal{M}}(x,b)$ and $B\in {\mathcal{M}}(y,b)$. When passing to ${\mathscr{C}}_S$ the intervals $\{A\}\times J_C$ give a collar neighborhood to an interval in ${\mathcal{M}}(a,b)$ with endpoint $(C,A)\in {\mathcal{M}}(x,b)\times {\mathcal{M}}(a,x)$, which 