${\mathscr{C}}$, along with some new moduli spaces. Schematically, the old and the new morphisms are given by the following diagrams (with grading indicated by horizontal levels):

$$\xymatrix @R=1cm @C=0.5cm {{\mathscr{C}}:&\ar[d]&&\ar[d]&&&{\overline}{{\mathscr{C}}}:&\ar[d]&&\bullet_f\ar[d]\ar[drr]\ar[dll]&&\ar[d]\\
&\bullet_x\ar[d]&&\bullet_y\ar[d]&&&&\bullet_{{\overline}{x}}\ar[d]&&\bullet_e\ar[drr]\ar[dll]&&\bullet_{{\overline}{y}}\ar[d]\\
&&&&&&&&&&&}$$

Specifically, the morphisms and moduli spaces are given as follows.

\[def:int\]The *intermediate flow category* ${\overline}{{\mathscr{C}}}$ is the flow category with objects $${\text{Ob}}({\overline}{{\mathscr{C}}})=\{{\overline}{a}\,|\,a\in{\text{Ob}}({\mathscr{C}})\}\cup\{e,f\},$$ where $|e|=i$ and $|f|=i+1$ and whose moduli spaces are given by$$\begin{aligned}
{\mathcal{M}}(e,{\overline}{b})&=&{\mathcal{M}}(x,b)\sqcup{\mathcal{M}}(y,b),\\
{\mathcal{M}}(f,{\overline}{b})&=&\left\{\begin{array}{lcl}\text{pt.}&&\text{if ${\overline}{b}={\overline}{x},e,{\overline}{y}$},\\
{\mathcal{M}}(e,{\overline}{b})\times[0,1]&&\text{otherwise}.\end{array}\right.\end{aligned}$$In all other cases ${\mathcal{M}}({\overline}{a},{\overline}{b})={\mathcal{M}}(a,b)$.

We now present a neat immersion and two choices of framing for ${\overline}{{\mathscr{C}}}$ relative to ${\textbf{d}}$. With some extra work, it is actually possible to neatly *embed* and frame ${\overline}{{\mathscr{C}}}$ relative to ${\textbf{d}}$. But this is not necessary here as we are free to eventually increase ${\textbf{d}}$ and then perturb the immersion to an embedding, without changing the associated stable homotopy type.

\[prop:immerse\]There exists a neat immersion and two framings $({\overline}{\iota},{\overline}{{\varphi}}(+))$, $({\overline}{\iota},{\overline}{{\varphi}}(-))$ of ${\overline}{{\mathscr{C}}}$, relative to ${\textbf{d}}$, which extend the neat embedding and framing $(\iota,{\varphi})$ of ${\mathscr{C}}$.

We will deal with the $({\overline}{\iota},{\overline}{{\varphi}}(+))$ version first and will suppress the “+” from the notation of ${\overline}{{\varphi}}(+)$ while we work on this case.

When ${\overline}{a}\neq e,f$, define the embeddings and framings ${\overline}{\iota}$ and ${\overline}{{\varphi}}$ of ${\mathcal{M}}({\overline}{a},{\overline}{b})$ by the $\iota$ 