A closed subset $P$ of an $n$ -dimensional smooth manifold $X$ is called a submanifold with corners of dimension $k$ if any point $p\in P$ has an open neighborhood $U\ni p$ and a $C^{\infty}$ -diffeomorphism $\phi\colon U\stackrel{{\scriptstyle\sim}}{{\allowbreak\mathrel{\mathop{\hbox to 12.0pt{\rightarrowfill}}\limits}}}\mathbb{R}^{n}$ such that for some $r\geq 0$

	$\displaystyle\phi(p)=0,$
	$\displaystyle\phi(P\cap U)=\mathbb{R}^{r}_{\geq 0}\times\mathbb{R}^{k-r}\times 0_{\mathbb{R}^{n-k}}$

where $0_{\mathbb{R}^{n-k}}$ is the zero element of $\mathbb{R}^{n-k}$ . The set of submanifolds with corners is denoted by $\mathcal{P}(X)$ .

Denote $\widetilde{S}_{V}=R_{1}\bigcup R_{2}\bigcup R_{5}\bigcup R_{6}$ . We shall call a curve $\gamma$ a $\mu$ - vertical curve if $\gamma:[-2,2]\rightarrow\widetilde{S}_{V}$ is such that

(3.7)

$$\widetilde{\pi}\gamma(t)=(u(t),t)$$

where $\widetilde{\pi}:\widetilde{S}_{V}\rightarrow[-2,2]\times[-2,2]$ sends $\widetilde{S}_{V}$ bijectively into the square in the following way:

(3.8)

$$\widetilde{\pi}(x,y,z):=\left\{\begin{array}[]{lcr}(x,z)&\text{if}&(x,y,z)\in R_{1}\bigcup R_{2}\\ (y,z)&\text{if}&(x,y,z)\in R_{5}\bigcup R_{6}\end{array}\right.$$

and the continuous function $u:[-2,2]\rightarrow[-2,2]$ satisfies

(3.9)

$$|u(t_{1})-u(t_{2})|\leq\mu|t_{1}-t_{2}|.$$

Finally, a set $B\subset\widetilde{S}_{V}$ is a $\mu$ - vertical strip if there exist two $\mu$ -vertical curves $\gamma_{l}$ , $\gamma_{r}$ with $\widetilde{\pi}\gamma_{l}(t)=(u_{l}(t),t)$ and $\widetilde{\pi}\gamma_{r}(t)=(u_{r}(t),t)$ such that

(3.10)

$$\widetilde{\pi}B=\{(s,t)\in[-2,2]\times[-2,2]\,|\,\ u_{r}(t)\leq s\leq u_{r}(t)\},$$

we shall call the quantity $\text{diam}(B):=\max_{t\in[-2,2]}|u_{r}(t)-u_{l}(t)|$ the diameter of $B$ . Analogously we can define $\mu$ - horizontal curves and $\mu$ -horizontal strips.

The Hamming cube of dimension $d$ is defined as the set of all binary sequences of length d, that is its elements are of the form

$$\boldsymbol{x}=(0,1,1,0,1,\ldots,1)$$

and the distance between two strings is just the number of elements they don’t have in common divided by d:

$$\rho(\boldsymbol{x},\boldsymbol{y})=\frac{\sum_{i=1}^{d}{|x_{i}-y_{i}|}}{d}$$

This metric is known as the normalized Hamming distance. We will give the cube a uniform measure for this discussion.

Let $R$ and $S$ be idempotent rings. A (surjective) Morita context $(R,S,M,N,\psi,\phi)$ between $R$ and $S$ consists of unital bimodules ${}_{R}M_{S}$ and ${}_{S}N_{R}$ , a surjective $R$ -module homomorphism $\psi:M\otimes_{S}N\to R$ , and a surjective $S$ -module homomorphism $\phi:N\otimes_{R}M\to S$ satisfying

$$\phi(n\otimes m)n^{\prime}=n\psi(m\otimes n^{\prime})\qquad\text{ and }\qquad m^{\prime}\phi(n\otimes m)=\psi(m^{\prime}\otimes n)m$$

for every $m,m^{\prime}\in M$ and $n,n^{\prime}\in N$ . We say that $R$ and $S$ are Morita equivalent in the case that there exists a Morita context.

For each $\frac{r}{s}\in\mathbb{Q}_{\infty}$ we define its complexity by

$$\gamma(\tfrac{r}{s})=\lvert r\rvert+\lvert s\rvert+\lvert r+s\rvert.$$

We shall also say that the arc $\alpha_{p}^{\varepsilon}$ has complexity $\gamma(p)$ . The lowest complexity is $2$ and occurs if and only if $\frac{r}{s}\in\{-1,0,\infty\}$ .