(families of) distributions are required by the pseudocalibration machinery in
order to define a candidate pseudoexpectation operator
$\widetilde{{\mathbb{E}}}$. For the Gaussian setting, we have the following
distributions.

######  Definition 3.1 (Gaussian PAP distributions).

The Gaussian PAP distributions are as follows.

  1. 1.

(Random distribution) $m$ i.i.d. vectors $d_{u}\sim{{\mathcal{N}}(0,I)}$.

  2. 2.

(Planted distribution) A vector $v$ is sampled uniformly from
$\left\\{\pm\frac{1}{\sqrt{n}}\right\\}^{n}$, as well as signs
$b_{u}\in_{\text{R}}\\{\pm 1\\}$, and $m$ vectors $d_{u}$ are drawn from
$\mathcal{N}(0,I)$ conditioned on $\langle d_{u},v\rangle=b_{u}$.

For the boolean setting, we have the following distributions.

######  Definition 3.2 (Boolean PAP distributions).

The boolean PAP distributions are as follows

  1. 1.

(Random distribution) $m$ i.i.d. vectors $d_{u}\in_{\text{R}}\\{-1,+1\\}^{n}$.

  2. 2.

(Planted distribution) A vector $v$ is sampled uniformly from
$\left\\{\pm\frac{1}{\sqrt{n}}\right\\}^{n}$, as well as signs
$b_{u}\in_{\text{R}}\\{\pm 1\\}$, and $m$ vectors $d_{u}$ are drawn from
$\left\\{\pm 1\right\\}^{n}$ conditioned on $\langle d_{u},v\rangle=b_{u}$.

####  2 Pseudocalibration technique

We will use the shorthand ${\mathbb{E}}_{\text{ra}}$ and
${\mathbb{E}}_{\text{pl}}$ for the expectation under the random and planted
distributions. Pseudocalibration gives a method for constructing a candidate
pseudoexpectation operator $\widetilde{{\mathbb{E}}}$. The idea behind
pseudocalibration is that
${\mathbb{E}}_{\text{ra}}\widetilde{{\mathbb{E}}}f(v)$ should match with
${\mathbb{E}}_{\text{pl}}f(v)$ for every low-degree test of the data
$t=t(d)=t(d_{1},\dots,d_{m})$,

${\mathbb{E}}_{\text{ra}}t(d)\widetilde{{\mathbb{E}}}f(v)={\mathbb{E}}_{\text{pl}}t(d)f(v).$

When pseudocalibrating, one can freely choose the “outer” basis in which to
express the polynomial $f(v)$, as well as the “inner” basis of low-degree
tests which should agree with the planted distribution. Though we attempted to
use alternate bases to simplify the analysis, ultimately we opted for the
standard choice of bases: a Fourier basis for the inner basis in each setting
(Hermite functions for the Gaussian setting, parity functions for the boolean
setting), and the coordinate basis $v^{I}$ for the outer basis.

When the inner basis is orthonormal under the random distribution (as a
Fourier basis is), the pseudocalibration condition gives a formula for the
coefficients of $\widetilde{{\mathbb{E}}}f(v)$ in the orthonormal basis
(though it only gives the coefficients of the low-degree functions $t(d)$).
Concretely, letting the inner basis be indexed by $\alpha\in{\mathcal{F}}$, as
a function of $d$ the pseudocalibration condition enforces

$\widetilde{{\mathbb{E}}}f(v)=\displaystyle\sum_{\begin{subarray}{c}\alpha\in{\mathcal{F}}:\\\
\left\lvert\alpha\right\rvert\leq
n^{\tau}\end{subarray}}\left({\mathbb{E}}_{\text{pl}}t_{\alpha}(d)f(v)\right)t_{\alpha}(d).$

Here we use “$\left\lvert\alpha\right\rvert\leq n^{\tau}$” to describe the set
of low-degree tests. The pseudocalibration condition does not prescribe any
coefficients for functions $t_{\alpha}(d)$ with
$\left\lvert\alpha\right\rvert>n^{\tau}$ and an economical choice is to set
these coefficients to zero.

When pseudocalibrating, our pseudoexpectation operator is guaranteed to be
linear, as the expression above is linear in $f$. It is guaranteed to satisfy
all constraints of the form “$f(v)=0$”. It will approximately satisfy
constraints of the form “$f(v,d)=0$”, though only up to truncation error.

######  Fact 3.3 (Proof in [sklowerbounds]).

If $p(v)$ is a polynomial which is uniformly zero on the planted distribution,
then $\widetilde{{\mathbb{E}}}[p]$ is the zero function. If $p(v,d)$ is a
polynomial which is uniformly zero on the planted distribution, then the only
nonzero Fourier coefficients of $\widetilde{{\mathbb{E}}}[p]$ are those with
size between $n^{\tau}\pm\deg_{d}(p)$.

Truncation introduces a tiny error in the constraints, which we are able to
handle in [sklowerbounds], omitted in this work for brevity.

For the pseudocalibration we truncate to only Fourier coefficients of size at
most $n^{\tau}$. The relationship between the parameters is $\delta\leq
c\tau\leq c^{\prime}\varepsilon$ where $c^{\prime}<c<1$ are absolute
constants. We will assume that they are sufficiently small for all our proofs
to go through.

Pseudocalibration also by default does not enforce the condition
$\widetilde{{\mathbb{E}}}[1]=1$. However, this is easily fixed by dividing the
operator by $\widetilde{{\mathbb{E}}}[1]$. As will be pointed out in Remark
4.9, w.h.p. in the unnormalized pseudocalibration,
$\widetilde{{\mathbb{E}}}[1]=1+\operatorname{o}_{n}(1)$ and so the error
introduced does not impact the statement of any lemmas.

####  3 Gaussian setting pseudocalibration

We start by computing the pseudocalibration for the Gaussian setting. Here the
natural choice of Fourier basis is the Hermite polynomials. Let
$\alpha\in({\mathbb{N}}^{n})^{m}$ denote a Hermite polynomial index. Define
$\alpha!:=\prod_{u,i}\alpha_{u,i}!$ and
$\left\lvert\alpha\right\rvert:=\sum_{u,i}\alpha_{u,i}$ and
$\left\lvert\alpha_{u}\right\rvert:=\sum_{i}\alpha_{u,i}$. We let
$h_{\alpha}(d_{1},\dots,d_{m})$ denote an unnormalized Hermite polynomial, so
that $h_{\alpha}/\sqrt{\alpha!}$ forms an orthonormal basis for polynomials in
the entries of the vectors $d_{1},\dots,d_{m}$, under the inner product
$\langle
p,q\rangle={\mathbb{E}}_{d_{1},\dots,d_{m}\sim{\mathcal{N}}(0,I)}[p\cdot q]$.

We can view $\alpha$ as an $m\times n$ matrix of natural numbers, and with
this view we also define $\alpha^{\intercal}\in({\mathbb{N}}^{m})^{n}$.

######  Lemma 3.4.

For any $I\subseteq[n]$, the pseudocalibration value is

$\widetilde{{\mathbb{E}}}v^{I}=\displaystyle\sum_{\begin{subarray}{c}\alpha:\left\lvert\alpha\right\rvert\leq
n^{\tau},\\\ \left\lvert\alpha_{u}\right\rvert\text{ even},\\\
\left\lvert(\alpha^{\intercal})_{i}\right\rvert\equiv I_{i}\;(\mod
2)\end{subarray}}\left(\prod_{u=1}^{m}h_{\left\lvert\alpha_{u}\right\rvert}(1)\right)\cdot\frac{1}{n^{\left\lvert
I\right\rvert/2+\left\lvert\alpha\right\rvert/2}}\cdot\frac{h_{\alpha}(d_{1},\dots,d_{m})}{\alpha!}.$

In words, the nonzero Fourier coefficients are those which have even row sums,
and whose column sums match the parity of $I$.

###### Proof.

The truncated pseudocalibrated value is defined to be

$\widetilde{{\mathbb{E}}}v^{I}=\displaystyle\sum_{\alpha:\left\lvert\alpha\right\rvert\leq
n^{\tau}}\frac{h_{\alpha}(d_{1},\dots,d_{m})}{\alpha!}\cdot{\mathbb{E}}_{\text{pl}}[h_{\alpha}(d_{1},\dots,d_{m})\cdot
v^{I}]$

So we set about to compute the planted moments. For this computation, the
following lemma is crucial. Here, we give a short proof of this lemma using
generating functions. For a different combinatorial proof, see
[sklowerbounds].

######  Lemma 3.5.

Let $\alpha\in{\mathbb{N}}^{n}$. When $v$ is fixed and $b$ is fixed (not
necessarily $\pm 1$) and $d\sim N(0,I)$ conditioned on $\langle
v,d\rangle=b\left\lVert v\right\rVert$,

${\mathbb{E}}_{d}[h_{\alpha}(d)]=\frac{v^{\alpha}}{\left\lVert
v\right\rVert^{\left\lvert\alpha\right\rvert}}\cdot
h_{\left\lvert\alpha\right\rvert}(b).$

###### Proof.

It suffices to prove the claim when $\left\lVert v\right\rVert=1$ since the
left-hand side is independent of $\left\lVert v\right\rVert$. Express
$d=bv+(I-vv^{\intercal})x$ where $x\sim N(0,I)$ is a standard normal variable.
Now we want

${\mathbb{E}}_{x\sim N(0,I)}h_{\alpha}\left(bv+(I-vv^{\intercal})x\right).$

The Hermite polynomial generating function is

$\displaystyle\sum_{\alpha\in{\mathbb{N}}^{n}}{\mathbb{E}}_{x\sim
N(0,I)}h_{\alpha}\left(bv+(I-vv^{\intercal})x\right)\frac{t^{\alpha}}{\alpha!}={\mathbb{E}}_{x}\exp\left(\langle
bv+(I-vv^{\intercal})x,t\rangle-\frac{\left\lVert
t\right\rVert_{2}^{2}}{2}\right)$
$=\int_{\mathbb{R}^{n}}\frac{1}{(2\pi)^{\frac{n}{2}}}\cdot\exp\left(\langle
bv+(I-vv^{\intercal})x,t\rangle-\frac{\left\lVert
t\right\rVert_{2}^{2}}{2}-\frac{\left\lVert
x\right\rVert_{2}^{2}}{2}\right)\;dx.$

Completing the square,

$\displaystyle=$
$\displaystyle\int_{\mathbb{R}^{n}}\frac{1}{(2\pi)^{\frac{n}{2}}}\cdot\exp\left(\langle
bv,t\rangle-\frac{\langle v,t\rangle^{2}}{2}-\frac{1}{2}\cdot\left\lVert
x-(t-\langle v,t\rangle v)\right\rVert_{2}^{2}\right)\;dx$ $\displaystyle=$
$\displaystyle\exp\left(\langle bv,t\rangle-\frac{\langle
v,t\rangle^{2}}{2}\right)$ $\displaystyle=$ $\displaystyle\exp\left(b\langle
v,t\rangle-\frac{1}{2}\cdot\langle v,t\rangle^{2}\right).$

How can we Taylor expand this in terms of $t$? The Taylor expansion of
$\exp(by-\frac{y^{2}}{2})$ is $\sum_{i=0}^{\infty}h_{i}(b)\frac{y^{i}}{i!}$.
That is, the $i$-th derivative in $y$ of $\exp(by-\frac{y^{2}}{2})$, evaluated
at 0, is $h_{i}(b)$. Using the chain rule with $y=\langle v,t\rangle$, the
$\alpha$-derivative in $t$ of our expression, evaluated at 0, is
$v^{\alpha}\cdot h_{\left\lvert\alpha\right\rvert}(b)$. This is the expression
we wanted when $\left\lVert v\right\rVert=1$, and along with the
aforementioned remark about homogeneity in $\left\lVert v\right\rVert$ this
completes the proof.

Now we can finish the calculation. To compute
${\mathbb{E}}_{\text{pl}}[h_{\alpha}(d_{1},\dots,d_{m})\cdot v^{I}]$,
marginalize $v$ and the $b_{u}$ and factor the conditionally independent
$b_{u}$ and $d_{u}$.

$\displaystyle{\mathbb{E}}_{\text{pl}}[h_{\alpha}(d_{1},\dots,d_{m})v^{I}]$
$\displaystyle={\mathbb{E}}_{v,b_{u}}v^{I}\prod_{u=1}^{m}{\mathbb{E}}_{d}\left[h_{\alpha_{u}}(d_{u})\mid
v,b_{u}\right]$
$\displaystyle={\mathbb{E}}_{v,b_{u}}v^{I}\cdot\prod_{u=1}^{m}\frac{v^{\alpha_{u}}}{\left\lVert
v\right\rVert^{\left\lvert\alpha_{u}\right\rvert}}\cdot
h_{\left\lvert\alpha_{u}\right\rvert}(b_{u})$
$\displaystyle(\text{\lx@cref{creftype~refnum}{lem:fixed-moments})}$
$\displaystyle=\left({\mathbb{E}}_{v}\frac{v^{I+\sum_{u=1}^{m}\alpha_{u}}}{\left\lVert
v\right\rVert^{\sum_{u=1}^{m}\left\lvert\alpha_{u}\right\rvert}}\right)\cdot\left(\prod_{u=1}^{m}{\mathbb{E}}_{b_{u}}h_{\left\lvert\alpha_{u}\right\rvert}(b_{u})\right)$

The Hermite polynomial expectations will be zero in expectation over $b_{u}$
if the degree is odd, and otherwise $b_{u}$ is raised to an even power and can
be replaced by 1. This requires that $\left\lvert\alpha_{u}\right\rvert$ is
even for all $u$. The norm $\left\lVert v\right\rVert$ is constantly $1$ and
can be dropped. The numerator will be $\frac{1}{n^{\left\lvert
I\right\rvert/2+\left\lvert\alpha\right\rvert/2}}$ if the parity of every
$\left\lvert(\alpha^{\intercal})_{i}\right\rvert$ matches $I_{i}$, and 0
otherwise. This completes the pseudocalibration calculation.

We can now write ${\mathcal{M}}$ in terms of graph matrices.

######  Definition 3.6.

Let $\mathcal{L}$ be the set of all proper shapes $\alpha$ with the following
properties

  * •

$U_{\alpha}$ and $V_{\alpha}$ only contain square vertices and
$|U_{\alpha}|,|V_{\alpha}|\leq n^{\delta}$

  * •

$W_{\alpha}$ has no degree $0$ vertices

  * •

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}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} }
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\leavevmode\hbox to9.48pt{\vbox to9.48pt{\pgfpicture\makeatletter\hbox{\hskip
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} }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-2.71138pt}{-2.63809pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
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}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} }
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}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}})+V_{\alpha}(\scalebox{0.85}{\footnotesize
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4.74158pt\lower-4.74158pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
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$\scalebox{0.85}{\footnotesize \leavevmode\hbox to9.48pt{\vbox
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}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} }
\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{{{}}}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}\in V(\alpha)$

  * •

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$\scalebox{0.85}{\footnotesize \leavevmode\hbox to12.49pt{\vbox
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}{{}{{{}}}{{}}{}{}{}{}{}{}{}{}{}{{}\pgfsys@moveto{6.0437pt}{0.0pt}\pgfsys@curveto{6.0437pt}{3.33788pt}{3.33788pt}{6.0437pt}{0.0pt}{6.0437pt}\pgfsys@curveto{-3.33788pt}{6.0437pt}{-6.0437pt}{3.33788pt}{-6.0437pt}{0.0pt}\pgfsys@curveto{-6.0437pt}{-3.33788pt}{-3.33788pt}{-6.0437pt}{0.0pt}{-6.0437pt}\pgfsys@curveto{3.33788pt}{-6.0437pt}{6.0437pt}{-3.33788pt}{6.0437pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{
} }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-3.62317pt}{-1.72221pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{$u$ }}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} }
\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{{{}}}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}\in V(\alpha)$

  * •

$|E(\alpha)|\leq n^{\tau}$

######  Remark 3.7.

Note that the shapes in $\mathcal{L}$ can have isolated vertices in
$U_{\alpha}\cap V_{\alpha}$.

######  Remark 3.8.

${\mathcal{L}}$ captures all the shapes that have nonzero coefficient when we
write ${\mathcal{M}}$ in terms of graph matrices. The constraint
$\deg(\scalebox{0.85}{\footnotesize \leavevmode\hbox to12.49pt{\vbox
to12.49pt{\pgfpicture\makeatletter\hbox{\hskip 6.2437pt\lower-6.2437pt\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{
{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}{{}{{{}}}{{}}{}{}{}{}{}{}{}{}{}{{}\pgfsys@moveto{6.0437pt}{0.0pt}\pgfsys@curveto{6.0437pt}{3.33788pt}{3.33788pt}{6.0437pt}{0.0pt}{6.0437pt}\pgfsys@curveto{-3.33788pt}{6.0437pt}{-6.0437pt}{3.33788pt}{-6.0437pt}{0.0pt}\pgfsys@curveto{-6.0437pt}{-3.33788pt}{-3.33788pt}{-6.0437pt}{0.0pt}{-6.0437pt}\pgfsys@curveto{3.33788pt}{-6.0437pt}{6.0437pt}{-3.33788pt}{6.0437pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{
} }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-3.62317pt}{-1.72221pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{$u$ }}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} }
\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{{{}}}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}})\geq 4$ arises
because pseudocalibration gives us that $\deg(\scalebox{0.85}{\footnotesize
\leavevmode\hbox to12.49pt{\vbox
to12.49pt{\pgfpicture\makeatletter\hbox{\hskip 6.2437pt\lower-6.2437pt\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{
{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}{{}{{{}}}{{}}{}{}{}{}{}{}{}{}{}{{}\pgfsys@moveto{6.0437pt}{0.0pt}\pgfsys@curveto{6.0437pt}{3.33788pt}{3.33788pt}{6.0437pt}{0.0pt}{6.0437pt}\pgfsys@curveto{-3.33788pt}{6.0437pt}{-6.0437pt}{3.33788pt}{-6.0437pt}{0.0pt}\pgfsys@curveto{-6.0437pt}{-3.33788pt}{-3.33788pt}{-6.0437pt}{0.0pt}{-6.0437pt}\pgfsys@curveto{3.33788pt}{-6.0437pt}{6.0437pt}{-3.33788pt}{6.0437pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{
} }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-3.62317pt}{-1.72221pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{$u$ }}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} }
\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{{{}}}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}})$ is even, $u$
cannot be isolated, and $h_{2}(1)=0$.

For a shape $\alpha$, we define

$\alpha!:=\prod_{e\in E(\alpha)}l(e)!$

Note that this equals the factorial of the corresponding index of the Hermite
polynomial for this shape.

######  Definition 3.9.

For any shape $\alpha$, if $\alpha\in\mathcal{L}$, define

$\lambda_{\alpha}:=\left(\prod_{\scalebox{0.85}{\footnotesize \leavevmode\hbox
to12.49pt{\vbox to12.49pt{\pgfpicture\makeatletter\hbox{\hskip
6.2437pt\lower-6.2437pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{
{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}{{}{{{}}}{{}}{}{}{}{}{}{}{}{}{}{{}\pgfsys@moveto{6.0437pt}{0.0pt}\pgfsys@curveto{6.0437pt}{3.33788pt}{3.33788pt}{6.0437pt}{0.0pt}{6.0437pt}\pgfsys@curveto{-3.33788pt}{6.0437pt}{-6.0437pt}{3.33788pt}{-6.0437pt}{0.0pt}\pgfsys@curveto{-6.0437pt}{-3.33788pt}{-3.33788pt}{-6.0437pt}{0.0pt}{-6.0437pt}\pgfsys@curveto{3.33788pt}{-6.0437pt}{6.0437pt}{-3.33788pt}{6.0437pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{
} }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-3.62317pt}{-1.72221pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{$u$ }}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} }
\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{{{}}}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}\in
V(\alpha)}h_{\deg(\scalebox{0.85}{\footnotesize \leavevmode\hbox
to12.49pt{\vbox to12.49pt{\pgfpicture\makeatletter\hbox{\hskip
6.2437pt\lower-6.2437pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{
{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}{{}{{{}}}{{}}{}{}{}{}{}{}{}{}{}{{}\pgfsys@moveto{6.0437pt}{0.0pt}\pgfsys@curveto{6.0437pt}{3.33788pt}{3.33788pt}{6.0437pt}{0.0pt}{6.0437pt}\pgfsys@curveto{-3.33788pt}{6.0437pt}{-6.0437pt}{3.33788pt}{-6.0437pt}{0.0pt}\pgfsys@curveto{-6.0437pt}{-3.33788pt}{-3.33788pt}{-6.0437pt}{0.0pt}{-6.0437pt}\pgfsys@curveto{3.33788pt}{-6.0437pt}{6.0437pt}{-3.33788pt}{6.0437pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{
} }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-3.62317pt}{-1.72221pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{$u$ }}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} }
\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{{{}}}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}})}(1)\right)\cdot\frac{1}{n^{(\left\lvert
U_{\alpha}\right\rvert+\left\lvert V_{\alpha}\right\rvert+\left\lvert
E(\alpha)\right\rvert)/2}}\cdot\frac{1}{\alpha!}$

Otherwise, define $\lambda_{\alpha}:=0$.

######  Corollary 3.10.

Modulo the footnote333Technically, the graph matrices $M_{\alpha}$ have rows
and columns indexed by all subsets of
${\mathcal{C}}_{m}\cup{\mathcal{S}}_{n}$. The submatrix with rows and columns
from $\binom{{\mathcal{S}}_{n}}{\leq D/2}$ equals the moment matrix for
$\widetilde{{\mathbb{E}}}$., ${\mathcal{M}}=\displaystyle\sum_{\text{shapes
}\alpha}\lambda_{\alpha}M_{\alpha}$.

####  4 Boolean setting pseudocalibration

We now present the pseudocalibration for the boolean setting. For the sequel,
we need notation for vectors on a slice of the boolean cube.

######  Definition 3.11 (Slice).

Let $v\in\\{\pm 1\\}^{n}$ and $\theta\in\mathbb{Z}$. The slice
$\operatorname{\mathcal{S}}_{v}(\theta)$ is defined as

$\operatorname{\mathcal{S}}_{v}(\theta)\coloneqq\\{d\in\\{\pm
1\\}^{n}~{}|~{}\langle v,d\rangle=\theta\\}.$

We use $\operatorname{\mathcal{S}}_{v}(\pm\theta)$ to denote
$\operatorname{\mathcal{S}}_{v}(\theta)\cup\operatorname{\mathcal{S}}_{v}(-\theta)$
and $\operatorname{\mathcal{S}}(\theta)$ to denote
$\operatorname{\mathcal{S}}_{v}(\theta)$ when $v$ is the all-ones vector.

######  Remark 3.12.

With our notation for the slice, the planted distribution in the boolean
setting can be equivalently described as

  1. 1.

Sample $v\in\\{\frac{\pm 1}{\sqrt{n}}\\}^{n}$ uniformly, and then

  2. 2.

Sample $d_{1},\dots,d_{m}$ independently and uniformly from
$\operatorname{\mathcal{S}}_{\sqrt{n}\cdot v}(\pm\sqrt{n})$.

The planted distribution doesn’t actually exist for every $n$, but this is
immaterial, as we can still define the pseudoexpectation via the same formula.

We will also need the expectation of monomials over the slice
$\operatorname{\mathcal{S}}(\sqrt{n})$ since they will appear in the
description of the pseudocalibrated Fourier coefficients.

######  Definition 3.13.

$e(k)\coloneqq{\mathbb{E}}_{x\in_{\text{R}}\mathcal{S}(\sqrt{n})}\left[x_{1}\cdots
x_{k}\right].$

We now compute the Fourier coefficients of
$\widetilde{{\mathbb{E}}}v^{\beta}$, where $\beta\in\mathbb{F}_{2}^{n}$. The
Fourier basis when $d_{1},\dots,d_{m}\in_{\text{R}}\\{\pm 1\\}^{n}$ is the set
of parity functions. Thus a character can be specified by
$\alpha\in({\mathbb{F}}_{2}^{n})^{m}$, where $\alpha$ is composed of $m$
vectors $\alpha_{1},\dots,\alpha_{m}\in\mathbb{F}_{2}^{n}$. More precisely,
the character $\chi_{\alpha}$ associated to $\alpha$ is defined as

$\chi_{\alpha}(d_{1},\dots,d_{m}):=\prod_{u=1}^{m}d_{u}^{\alpha_{u}}$

We denote by $\left\lvert\alpha\right\rvert$ the number of non-zero entries of
$\alpha$ and define $\left\lvert\alpha_{u}\right\rvert$ similarly. Thinking of
$\alpha$ as an $m\times n$ matrix with entries in ${\mathbb{F}}_{2}$, we also
define $\alpha^{\intercal}\in({\mathbb{F}}_{2}^{n})^{m}$.

######  Lemma 3.14.

We have

$\widetilde{{\mathbb{E}}}v^{\beta}=\frac{1}{n^{\left\lvert\beta\right\rvert/2}}\sum_{\begin{subarray}{c}\alpha\colon\left\lvert\alpha\right\rvert\leq
n^{\tau},\\\ \left\lvert\alpha_{u}\right\rvert\text{ even},\\\
\left\lvert\alpha^{\intercal}_{i}\right\rvert\equiv\beta_{i}\;(\mod
2)\end{subarray}}\prod_{u=1}^{m}e(\left\lvert\alpha_{u}\right\rvert)\cdot\chi_{\alpha_{u}}(d_{u}).$

The set of nonzero coefficients has a similar structure as in the Gaussian
case: the rows of $\alpha$ must have an even number of entries, and the $i$-th
column must have parity matching $\beta_{i}$.

###### Proof.

Given $\alpha\in({\mathbb{F}}_{2}^{n})^{m}$ with
$\left\lvert\alpha\right\rvert\leq n^{\tau}$, the pseudocalibration equation
enforces by construction that

${\mathbb{E}}_{d_{1},\dots,d_{m}\in\\{\pm
1\\}^{n}}(\widetilde{{\mathbb{E}}}v^{\beta})(d_{1},\dots,d_{m})\cdot\chi_{\alpha}(d_{1},\dots,d_{m})={\mathbb{E}}_{\text{pl}}v^{\beta}\cdot\chi_{\alpha}(d_{1},\dots,d_{m}).$

Computing the RHS above yields

$\displaystyle{\mathbb{E}}_{v\in\\{\pm
1\\}^{n}}{\mathbb{E}}_{d_{1},\dots,d_{m}\in_{\text{R}}\operatorname{\mathcal{S}}_{v}(\pm\sqrt{n})}\left[v^{\beta}\prod_{u=1}^{m}\chi_{\alpha_{u}}(d_{u})\right]$
$\displaystyle={\mathbb{E}}_{v\in\\{\pm
1\\}^{n}}{\mathbb{E}}_{d_{1},\dots,d_{m}\in_{\text{R}}\operatorname{\mathcal{S}}(\pm\sqrt{n})}\left[v^{\beta}\prod_{u=1}^{m}\chi_{\alpha_{u}}(v)\chi_{\alpha_{u}}(d_{u})\right]$
$\displaystyle={\mathbb{E}}_{v\in\\{\pm
1\\}^{n}}\chi_{\alpha_{1}+\cdots+\alpha_{m}+\beta}(v){\mathbb{E}}_{d_{1},\dots,d_{m}\in\operatorname{\mathcal{S}}(\pm\sqrt{n})}\left[\prod_{i=1}^{m}\chi_{\alpha_{i}}(d_{i})\right]$
$\displaystyle=\mathbf{1}_{\left[\alpha_{1}+\cdots+\alpha_{m}=\beta\right]}\cdot\prod_{i=1}^{m}{\mathbb{E}}_{d_{i}\in\operatorname{\mathcal{S}}(\pm\sqrt{n})}\left[\chi_{\alpha_{i}}(d_{i})\right]$
$\displaystyle=\mathbf{1}_{\left[\alpha_{1}+\cdots+\alpha_{m}=\beta\right]}\cdot\prod_{i=1}^{m}\mathbf{1}_{\left[\left\lvert\alpha_{i}\right\rvert\equiv
0\pmod{2}\right]}\cdot\prod_{i=1}^{m}e(\left\lvert\alpha_{i}\right\rvert).$

Since we have a general expression for the Fourier coefficient of each
character, applying Fourier inversion concludes the proof.

We can now express the moment matrix in terms of graph matrices.

######  Definition 3.15.

Let ${\mathcal{L}}_{bool}$ be the set of shapes in ${\mathcal{L}}$ from
Definition 3.6 in which the edge labels are all 1.

######  Remark 3.16.

${\mathcal{L}}_{bool}$ captures all the shapes that have nonzero coefficient
when we write ${\mathcal{M}}$ in terms of graph matrices. Similar to Remark
3.8, since $e(2)=0$ (see 6.5), we have the same condition
$deg(\scalebox{0.85}{\footnotesize \leavevmode\hbox to12.49pt{\vbox
to12.49pt{\pgfpicture\makeatletter\hbox{\hskip 6.2437pt\lower-6.2437pt\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{
{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}{{}{{{}}}{{}}{}{}{}{}{}{}{}{}{}{{}\pgfsys@moveto{6.0437pt}{0.0pt}\pgfsys@curveto{6.0437pt}{3.33788pt}{3.33788pt}{6.0437pt}{0.0pt}{6.0437pt}\pgfsys@curveto{-3.33788pt}{6.0437pt}{-6.0437pt}{3.33788pt}{-6.0437pt}{0.0pt}\pgfsys@curveto{-6.0437pt}{-3.33788pt}{-3.33788pt}{-6.0437pt}{0.0pt}{-6.0437pt}\pgfsys@curveto{3.33788pt}{-6.0437pt}{6.0437pt}{-3.33788pt}{6.0437pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{
} }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-3.62317pt}{-1.72221pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{$u$ }}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} }
\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{{{}}}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}})\geq 4$ for shapes
in ${\mathcal{L}}_{bool}$.

######  Definition 3.17.

For all shapes $\alpha$, if $\alpha\in{\mathcal{L}}_{bool}$ define

$\lambda_{\alpha}:=\frac{1}{n^{(\left\lvert U_{\alpha}\right\rvert+\left\lvert
V_{\alpha}\right\rvert)/2}}\prod_{\scalebox{0.85}{\footnotesize
\leavevmode\hbox to12.49pt{\vbox
to12.49pt{\pgfpicture\makeatletter\hbox{\hskip 6.2437pt\lower-6.2437pt\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{
{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}{{}{{{}}}{{}}{}{}{}{}{}{}{}{}{}{{}\pgfsys@moveto{6.0437pt}{0.0pt}\pgfsys@curveto{6.0437pt}{3.33788pt}{3.33788pt}{6.0437pt}{0.0pt}{6.0437pt}\pgfsys@curveto{-3.33788pt}{6.0437pt}{-6.0437pt}{3.33788pt}{-6.0437pt}{0.0pt}\pgfsys@curveto{-6.0437pt}{-3.33788pt}{-3.33788pt}{-6.0437pt}{0.0pt}{-6.0437pt}\pgfsys@curveto{3.33788pt}{-6.0437pt}{6.0437pt}{-3.33788pt}{6.0437pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{
} }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-3.62317pt}{-1.72221pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{$u$ }}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} }
\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{{{}}}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}\in
V(\alpha)}e(\deg(\scalebox{0.85}{\footnotesize \leavevmode\hbox
to12.49pt{\vbox to12.49pt{\pgfpicture\makeatletter\hbox{\hskip
6.2437pt\lower-6.2437pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{
{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}{{}{{{}}}{{}}{}{}{}{}{}{}{}{}{}{{}\pgfsys@moveto{6.0437pt}{0.0pt}\pgfsys@curveto{6.0437pt}{3.33788pt}{3.33788pt}{6.0437pt}{0.0pt}{6.0437pt}\pgfsys@curveto{-3.33788pt}{6.0437pt}{-6.0437pt}{3.33788pt}{-6.0437pt}{0.0pt}\pgfsys@curveto{-6.0437pt}{-3.33788pt}{-3.33788pt}{-6.0437pt}{0.0pt}{-6.0437pt}\pgfsys@curveto{3.33788pt}{-6.0437pt}{6.0437pt}{-3.33788pt}{6.0437pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{
} }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-3.62317pt}{-1.72221pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{$u$ }}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} }
\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{{{}}}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}))$

Otherwise, let $\lambda_{\alpha}:=0$.

######  Corollary 3.18.

${\mathcal{M}}=\displaystyle\sum_{\text{shapes
}\alpha}\lambda_{\alpha}M_{\alpha}$

##### Unifying the analysis

It turns out that the analysis of the boolean setting mostly follows from the
analysis in the Gaussian setting. Initially, the boolean pseudocalibration is
essentially equal to the Gaussian pseudocalibration in which we have removed
all shapes containing at least one edge with a label $k\geq 2$. The
coefficients on the graph matrices will actually be slightly different, but
they both admit an upper bound that is sufficient for our purposes (see
Proposition 4.13 for the precise statement).

To unify the notation in our analysis, we conveniently set the edge functions
of the graphs in the boolean case to be

$h_{k}(x)=\left\\{\begin{array}[]{lr}1&\text{if }k=0\\\ x&\text{if }k=1\\\
0&\text{if }k\geq 2\end{array}\right.$

This choice of $h_{k}(x)$ preserves the fact that
$\\{h_{0}(x)=1,h_{1}(x)=x\\}$ is an orthogonal polynomial basis in the boolean
setting, while zeroing out graphs with larger labels.

During the course of the analysis, we may multiply two graph matrices and
produce graph matrices with improper parallel edges (so-called “intersections
terms”). For a fixed pair ${u,i}$ of vertices, parallel edges between $u$ and
$i$ with labels $l_{1},\dots,l_{s}$ correspond to the product of orthogonal
polynomials $\prod_{j=1}^{s}h_{l_{j}}(d_{u,i})\eqqcolon q(d_{u,i})$. We will
re-express this product as a linear combination of polynomials in the
orthogonal family, i.e.,
$q(d_{u,i})=\sum_{i=0}^{\textup{deg}(q)}\lambda_{i}\cdot h_{i}(d_{u,i})$ for
some coefficients $\lambda_{i}\in\mathbb{R}$. For the boolean case, the
polynomial $q(d_{u,i})$ will be either $h_{0}(d_{u,i})=1$ or
$h_{1}(d_{u,i})=d_{u,i}$. However, for the Gaussian setting there may be up to
$\textup{deg}(q)$ non-zero, potentially larger coefficients $\lambda_{i}$ for
the corresponding Hermite polynomials $h_{i}$. For the graphs that arise in
this way, we will always bound their contributions to ${\mathcal{M}}$ by
applying the triangle inequality and norm bounds. Since we show bounds using
the larger coefficients $\lambda_{i}$ from the Gaussian case, the same bounds
apply when using the 0/1 coefficients in the boolean case.

We will consider separate cases at any point where the analysis differs
between the two settings.

###  4 Proving PSD-ness

Looking at the shapes that make up ${\mathcal{M}}$, the trivial shape with $k$
square vertices contributes an identity matrix on the degree-$2k$ submatrix of
${\mathcal{M}}$. Our ultimate goal will be to bound all shapes against these
identity matrices.

######  Definition 4.1 (Block).

For $k,l\in\\{0,1,\dots,D/2\\}$, the $(k,l)$ block of ${\mathcal{M}}$ is the
submatrix with rows from $\binom{[n]}{k}$ and columns from $\binom{[n]}{l}$.
Note that when ${\mathcal{M}}$ is expressed as a sum of graph matrices, this
exactly restricts ${\mathcal{M}}$ to shapes $\alpha$ with $\left\lvert
U_{\alpha}\right\rvert=k$ and $\left\lvert V_{\alpha}\right\rvert=l$.

We define the parameter $\eta:=1/\sqrt{n}$. The trivial shapes live in the
diagonal blocks of ${\mathcal{M}}$, and on the $(k,k)$ block contribute a
factor of $\frac{1}{n^{k}}=\eta^{2k}$ on the diagonal. In principle, we could
make $\eta$ as small as we like444Though pseudocalibration truncation errors
may become nonnegligible for extremely tiny $\eta$. by considering the moments
of a rescaling of $v$ rather than $v$ itself. Counterintuitively, it will turn
out that the scaling helps us prove PSD-ness (see [sklowerbounds] for more
details). It turns out that pseudocalibrating $v$ as a unit vector
(equivalently, using $\eta=1/\sqrt{n}$) is sufficient for our analysis.

Towards the goal of bounding ${\mathcal{M}}$ by the identity terms, we will
bound the norm of matrices on each block of ${\mathcal{M}}$, and invoke the
following lemma to conclude PSD-ness.

######  Lemma 4.2.

Suppose a symmetric matrix ${\mathcal{A}}\in{\mathbb{R}}^{\binom{[n]}{\leq
D}\times\binom{[n]}{\leq D}}$ satisfies, for some parameter $\eta\in(0,1)$,

  1. 1.

For each $k\in\\{0,1,\dots,D\\}$, the $(k,k)$ block has minimum singular value
at least $\eta^{2k}(1-\frac{1}{D+1})$

  2. 2.

For each $k,l\in\\{0,1,\dots,D\\}$ such that $k\neq l$, the $(k,l)$ block has
norm at most $\frac{\eta^{k+l}}{D+1}$.

Then ${\mathcal{A}}\succeq 0$.

###### Proof.

We need to show that for all vectors $x$, ${x^{\intercal}}{{\mathcal{A}}}x\geq
0$. Given a vector $x$, let $x_{0},\dots,x_{D}$ be its components in blocks
$0,\dots,D$. Observe that

$\displaystyle{x^{\intercal}}{{\mathcal{A}}}x\geq\sum_{k\in[0,D]}{\eta^{2k}\left(1-\frac{1}{D+1}\right)\left\lVert
x_{k}\right\rVert^{2}}-\sum_{k\neq
l\in[0,D]}{\frac{\eta^{k+l}}{D+1}\left\lVert x_{k}\right\rVert\left\lVert
x_{l}\right\rVert}$ $\displaystyle=(\left\lVert
x_{0}\right\rVert,{\eta}\left\lVert
x_{1}\right\rVert,\dots,{\eta}^{D}\left\lVert
x_{D}\right\rVert)\begin{pmatrix}1-\frac{1}{D+1}&-\frac{1}{D+1}&\cdots&-\frac{1}{D+1}\\\
-\frac{1}{D+1}&1-\frac{1}{D+1}&\cdots&-\frac{1}{D+1}\\\
\vdots&\vdots&\ddots&\vdots\\\
-\frac{1}{D+1}&-\frac{1}{D+1}&\cdots&1-\frac{1}{D+1}\end{pmatrix}\begin{pmatrix}\left\lVert
x_{0}\right\rVert\\\ {\eta}\left\lVert x_{1}\right\rVert\\\ \vdots\\\
{\eta}^{D}\left\lVert x_{D}\right\rVert\end{pmatrix}\geq 0.$

We start by defining spiders, which are special shapes $\alpha$ that we will
handle separately in the decomposition of ${\mathcal{M}}$. Informally, these
contain special substructures which allow their norm bounds not to be
negligible with respect to the identity matrix. We then show that shapes which
are not spiders have bounded norms.

######  Definition 4.3 (Left Spider).

A left spider is a proper shape
$\alpha=(V(\alpha),E(\alpha),U_{\alpha},V_{\alpha})$ with the property that
there exist two distinct square vertices $\scalebox{0.85}{\footnotesize
\leavevmode\hbox to9.48pt{\vbox to9.48pt{\pgfpicture\makeatletter\hbox{\hskip
4.74158pt\lower-4.74158pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
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}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{
{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{
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} }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-2.71138pt}{-2.63809pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{$i$ }}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} }
\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{{{}}}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}},\scalebox{0.85}{\footnotesize
\leavevmode\hbox to11.48pt{\vbox
to11.48pt{\pgfpicture\makeatletter\hbox{\hskip 5.73787pt\lower-5.73787pt\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{
{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}{{}{}{}{}{}{{}}{}{{}{}}{{}{}}{}{{}{}}{}{{}{}}{}{{}\pgfsys@moveto{5.53787pt}{5.53787pt}\pgfsys@lineto{-5.53787pt}{5.53787pt}\pgfsys@lineto{-5.53787pt}{-5.53787pt}\pgfsys@lineto{5.53787pt}{-5.53787pt}\pgfsys@closepath\pgfsys@stroke\pgfsys@invoke{
} }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-3.20953pt}{-1.86032pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{$j$ }}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} }
\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{{{}}}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}\in U_{\alpha}$ of
degree $1$ and a circle vertex $\scalebox{0.85}{\footnotesize \leavevmode\hbox
to12.49pt{\vbox to12.49pt{\pgfpicture\makeatletter\hbox{\hskip
6.2437pt\lower-6.2437pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{
{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}{{}{{{}}}{{}}{}{}{}{}{}{}{}{}{}{{}\pgfsys@moveto{6.0437pt}{0.0pt}\pgfsys@curveto{6.0437pt}{3.33788pt}{3.33788pt}{6.0437pt}{0.0pt}{6.0437pt}\pgfsys@curveto{-3.33788pt}{6.0437pt}{-6.0437pt}{3.33788pt}{-6.0437pt}{0.0pt}\pgfsys@curveto{-6.0437pt}{-3.33788pt}{-3.33788pt}{-6.0437pt}{0.0pt}{-6.0437pt}\pgfsys@curveto{3.33788pt}{-6.0437pt}{6.0437pt}{-3.33788pt}{6.0437pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{
} }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-3.62317pt}{-1.72221pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{$u$ }}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} }
\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{{{}}}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}\in V(\alpha)$ such
that $E(\alpha)$ contains the edges $(\scalebox{0.85}{\footnotesize
\leavevmode\hbox to9.48pt{\vbox to9.48pt{\pgfpicture\makeatletter\hbox{\hskip
4.74158pt\lower-4.74158pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{
{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}{{}{}{}{}{}{{}}{}{{}{}}{{}{}}{}{{}{}}{}{{}{}}{}{{}\pgfsys@moveto{4.54158pt}{4.54158pt}\pgfsys@lineto{-4.54158pt}{4.54158pt}\pgfsys@lineto{-4.54158pt}{-4.54158pt}\pgfsys@lineto{4.54158pt}{-4.54158pt}\pgfsys@closepath\pgfsys@stroke\pgfsys@invoke{
} }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-2.71138pt}{-2.63809pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{$i$ }}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} }
\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{{{}}}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}},\scalebox{0.85}{\footnotesize
\leavevmode\hbox to12.49pt{\vbox
to12.49pt{\pgfpicture\makeatletter\hbox{\hskip 6.2437pt\lower-6.2437pt\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{
{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}{{}{{{}}}{{}}{}{}{}{}{}{}{}{}{}{{}\pgfsys@moveto{6.0437pt}{0.0pt}\pgfsys@curveto{6.0437pt}{3.33788pt}{3.33788pt}{6.0437pt}{0.0pt}{6.0437pt}\pgfsys@curveto{-3.33788pt}{6.0437pt}{-6.0437pt}{3.33788pt}{-6.0437pt}{0.0pt}\pgfsys@curveto{-6.0437pt}{-3.33788pt}{-3.33788pt}{-6.0437pt}{0.0pt}{-6.0437pt}\pgfsys@curveto{3.33788pt}{-6.0437pt}{6.0437pt}{-3.33788pt}{6.0437pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{
} }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-3.62317pt}{-1.72221pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{$u$ }}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} }
\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{{{}}}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}})$ and
$(\scalebox{0.85}{\footnotesize \leavevmode\hbox to11.48pt{\vbox
to11.48pt{\pgfpicture\makeatletter\hbox{\hskip 5.73787pt\lower-5.73787pt\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{
{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}{{}{}{}{}{}{{}}{}{{}{}}{{}{}}{}{{}{}}{}{{}{}}{}{{}\pgfsys@moveto{5.53787pt}{5.53787pt}\pgfsys@lineto{-5.53787pt}{5.53787pt}\pgfsys@lineto{-5.53787pt}{-5.53787pt}\pgfsys@lineto{5.53787pt}{-5.53787pt}\pgfsys@closepath\pgfsys@stroke\pgfsys@invoke{
} }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-3.20953pt}{-1.86032pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{$j$ }}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} }
\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{{{}}}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}},\scalebox{0.85}{\footnotesize
\leavevmode\hbox to12.49pt{\vbox
to12.49pt{\pgfpicture\makeatletter\hbox{\hskip 6.2437pt\lower-6.2437pt\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{
{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}{{}{{{}}}{{}}{}{}{}{}{}{}{}{}{}{{}\pgfsys@moveto{6.0437pt}{0.0pt}\pgfsys@curveto{6.0437pt}{3.33788pt}{3.33788pt}{6.0437pt}{0.0pt}{6.0437pt}\pgfsys@curveto{-3.33788pt}{6.0437pt}{-6.0437pt}{3.33788pt}{-6.0437pt}{0.0pt}\pgfsys@curveto{-6.0437pt}{-3.33788pt}{-3.33788pt}{-6.0437pt}{0.0pt}{-6.0437pt}\pgfsys@curveto{3.33788pt}{-6.0437pt}{6.0437pt}{-3.33788pt}{6.0437pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{
} }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-3.62317pt}{-1.72221pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{$u$ }}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} }
\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{{{}}}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}})$ (these are
necessarily the only edges incident to $i$ and $j$ ).

The vertices $i$ and $j$ are called the end vertices of $\alpha$. Because of
degree parity, the end vertices must lie in
$U_{\alpha}\setminus(U_{\alpha}\cap V_{\alpha})$.

######  Definition 4.4 (Right spider).

A shape $\alpha=(V(\alpha),E(\alpha),U_{\alpha},V_{\alpha})$ is a right spider
if $\alpha^{\intercal}=(V(\alpha),E(\alpha),V_{\alpha},U_{\alpha})$ is a left
spider. The end vertices of $\alpha^{\intercal}$ are also called the end
vertices of $\alpha$.

######  Definition 4.5 (Spider).

A shape $\alpha$ is a spider if it is either a left spider or a right spider.

######  Remark 4.6.

A spider can have many pairs of end vertices. For each possible spider shape,
we single out a pair of end vertices, so that in what follows we can discuss
“the” end vertices of the spider.

####  1 Non-spiders are negligible

For non-spiders, we will now show that their norm is small. We point out that
this norm bound on non-spiders critically relies on the assumption $m\leq
n^{3/2-\varepsilon}$.

######  Lemma 4.7.

If $\alpha\in{\mathcal{L}}$ is not a trivial shape and not a spider, then

$\frac{1}{n^{|E(\alpha)|/2}}n^{\frac{w(V(\alpha))-w(S_{\min})}{2}}\leq\frac{1}{n^{\Omega(\varepsilon|E(\alpha)|)}}$

where $S_{min}$ is the minimum vertex separator of $\alpha$.

###### Proof.

The idea behind the proof is as follows. Each square vertex which is not in
the minimum vertex separator contributes $\sqrt{n}$ to the norm bound while
each circle vertex which is not in the minimum vertex separator contributes
$\sqrt{m}$. To compensate for this, we will try and take the factor of
$\frac{1}{\sqrt{n}}$ from each edge and distribute it among its two endpoints
so that each square vertex which is not in the minimum vertex separator is
assigned a factor of $\frac{1}{\sqrt{n}}$ or smaller and each circle vertex
which is not in the minimum vertex separator is assigned a factor of
$\frac{1}{\sqrt{m}}$ or smaller.

######  Remark 4.8.

Instead of using the minimum vertex separator, we will actually use a set $S$
of square vertices such that $w(S)\leq w(S_{\min})$. For details, see the
actual distribution scheme below.

To motivate the distribution scheme which we use, we first give two attempts
which don’t quite work. For simplicity, for these first two attempts we assume
that $U_{\alpha}\cap V_{\alpha}=\emptyset$ as vertices in $U_{\alpha}\cap
V_{\alpha}$ can essentially be ignored.

  1. Attempt 1: Take each edge and assign a factor of $\frac{1}{\sqrt[4]{n}}$ to its square endpoint and a factor of $\frac{1}{\sqrt[8]{m}}$ to its circle endpoint.

With this distribution scheme, since each circle vertex has degree at least
$4$, each circle vertex is assigned a factor of $\frac{1}{\sqrt{m}}$ or
smaller. Since each square vertex in $W_{\alpha}$ has degree at least $2$,
each square vertex in $W_{\alpha}$ is assigned a factor of
$\frac{1}{\sqrt{n}}$ or smaller. However, square vertices in $U_{\alpha}\cup
V_{\alpha}$ may only have degree $1$ in which case they are assigned a factor
of $\frac{1}{\sqrt[4]{n}}$ which is not small enough.

To fix this issue, we can have all of the edges which are incident to a square
vertex in $U_{\alpha}\cup V_{\alpha}$ give their entire factor of
$\frac{1}{\sqrt{n}}$ to the square vertex.

######  Remark 4.9.

For analyzing $\widetilde{{\mathbb{E}}}[1]$, this first attempt works as
$U_{\alpha}=V_{\alpha}=\emptyset$. Thus, as long as $m\leq n^{2-\varepsilon}$,
with high probability
$\widetilde{{\mathbb{E}}}[1]=1\pm\operatorname{o}_{n}(1)$ .

  2. Attempt 2: For each edge which is between a square vertex in $U_{\alpha}\cup V_{\alpha}$ and a circle vertex, we assign a factor of $\frac{1}{\sqrt{n}}$ to the square vertex and nothing to the circle vertex. For all other edges, we assign a factor of $\frac{1}{\sqrt[4]{n}}$ to its square endpoint and a factor of $\frac{1}{\sqrt[6]{m}}$ to its circle endpoint (which we can do because $m\leq n^{\frac{3}{2}-\varepsilon}$).

With this distribution scheme, each square vertex is assigned a factor of
$\frac{1}{\sqrt{n}}$. Since $\alpha$ is not a spider, no circle vertex is
adjacent to two vertices in $U_{\alpha}$ or $V_{\alpha}$. Thus, any circle
vertex which is not adjacent to both a square vertex in $U_{\alpha}$ and a
square vertex in $V_{\alpha}$ must be adjacent to at least $3$ square vertices
in $W_{\alpha}$ and is thus assigned a factor of $\frac{1}{\sqrt{m}}$ or
smaller. However, we can have circle vertices which are adjacent to both a
square vertex in $U_{\alpha}$ and a square vertex in $V_{\alpha}$. These
circle vertices may be assigned a factor of $\frac{1}{\sqrt[3]{m}}$, which is
not small enough.

To fix this, observe that whenever we have a circle vertex which is adjacent
to both a square vertex in $U_{\alpha}$ and a square vertex in $V_{\alpha}$,
this gives a path of length $2$ from $U_{\alpha}$ to $V_{\alpha}$. Any vertex
separator must contain one of the vertices in this path, so we can put one of
these two square vertices in $S$ and not assign it a factor of
$\frac{1}{\sqrt{n}}$.

  3. Actual distribution scheme: Based on these observations, we use the following distribution scheme. Here we are no longer assuming that $U_{\alpha}\cap V_{\alpha}$ is empty.

    1. 1.

Choose a set of square vertices $S\subseteq U_{\alpha}\cup V_{\alpha}$ as
follows. Start with $S=U_{\alpha}\cap V_{\alpha}$. Whenever we have a circle
vertex which is adjacent to both a square vertex in $U_{\alpha}\setminus
V_{\alpha}$ and a square vertex in $V_{\alpha}\setminus U_{\alpha}$, put one
of these two square vertices in $S$ (this choice is arbitrary). Observe that
$w(S)\leq w(S_{\min})$

    2. 2.

For each edge which is incident to a square vertex in $S$, assign a factor of
$\frac{1}{\sqrt[3]{m}}$ to its circle endpoint and nothing to this square.

    3. 3.

For each edge which is incident to a square vertex in $(U_{\alpha}\cup
V_{\alpha})\setminus S$, assign a factor of $\frac{1}{\sqrt{n}}$ to the square
vertex and nothing to the circle vertex.

    4. 4.

For all other edges, assign a factor of $\frac{1}{\sqrt[4]{n}}$ to its square
endpoint and a factor of $\frac{1}{\sqrt[6]{m}}$ to its circle endpoint.

Now each square vertex which is not in $S$ is assigned a factor of
$\frac{1}{\sqrt{n}}$ and since $\alpha$ is not a spider, all circle vertices
are assigned a factor of $\frac{1}{\sqrt{m}}$ or smaller.

We now make this argument formal.

Let ${\mathcal{C}}_{\alpha}$ and ${\mathcal{S}}_{\alpha}$ be the set of circle
vertices and the set of square vertices in $\alpha$ respectively. We have
$n^{\frac{w(V(\alpha))-w(S_{\min})}{2}}\leq
n^{0.5|{\mathcal{S}}_{\alpha}\setminus
S_{min}|+(0.75-\frac{\varepsilon}{2})|{\mathcal{C}}_{\alpha}\setminus
S_{min}|}$. So, it suffices to prove that

$|E(\alpha)|-|{\mathcal{S}}_{\alpha}\setminus
S_{min}|-(1.5-\varepsilon)|{\mathcal{C}}_{\alpha}\setminus
S_{min}|\geq\Omega(\varepsilon|E(\alpha)|)$

Let $Q=U_{\alpha}\cap V_{\alpha},P=(U_{\alpha}\cup V_{\alpha})\setminus Q$ and
let $P^{\prime}$ be the set of vertices of $P$ that have degree $1$ and are
not in $S_{min}$. Let $E_{1}$ be the set of edges incident to $P^{\prime}$ and
let $E_{2}=E(\alpha)\setminus E_{1}$.

For each vertex $i$ (resp. $u$ ), let the number of edges of $E_{2}$ incident
to it be $\deg^{\prime}(\scalebox{0.85}{\footnotesize \leavevmode\hbox
to9.48pt{\vbox to9.48pt{\pgfpicture\makeatletter\hbox{\hskip
4.74158pt\lower-4.74158pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{
{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}{{}{}{}{}{}{{}}{}{{}{}}{{}{}}{}{{}{}}{}{{}{}}{}{{}\pgfsys@moveto{4.54158pt}{4.54158pt}\pgfsys@lineto{-4.54158pt}{4.54158pt}\pgfsys@lineto{-4.54158pt}{-4.54158pt}\pgfsys@lineto{4.54158pt}{-4.54158pt}\pgfsys@closepath\pgfsys@stroke\pgfsys@invoke{
} }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-2.71138pt}{-2.63809pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{$i$ }}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} }
\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{{{}}}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}})$ (resp.
$\deg^{\prime}(\scalebox{0.85}{\footnotesize \leavevmode\hbox to12.49pt{\vbox
to12.49pt{\pgfpicture\makeatletter\hbox{\hskip 6.2437pt\lower-6.2437pt\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{
{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}{{}{{{}}}{{}}{}{}{}{}{}{}{}{}{}{{}\pgfsys@moveto{6.0437pt}{0.0pt}\pgfsys@curveto{6.0437pt}{3.33788pt}{3.33788pt}{6.0437pt}{0.0pt}{6.0437pt}\pgfsys@curveto{-3.33788pt}{6.0437pt}{-6.0437pt}{3.33788pt}{-6.0437pt}{0.0pt}\pgfsys@curveto{-6.0437pt}{-3.33788pt}{-3.33788pt}{-6.0437pt}{0.0pt}{-6.0437pt}\pgfsys@curveto{3.33788pt}{-6.0437pt}{6.0437pt}{-3.33788pt}{6.0437pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{
} }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-3.62317pt}{-1.72221pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{$u$ }}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} }
\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{{{}}}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}})$). Since $\alpha$
is bipartite, we have that $|E_{2}|=\sum_{\scalebox{0.85}{\footnotesize
\leavevmode\hbox to9.48pt{\vbox to9.48pt{\pgfpicture\makeatletter\hbox{\hskip
4.74158pt\lower-4.74158pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{
{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}{{}{}{}{}{}{{}}{}{{}{}}{{}{}}{}{{}{}}{}{{}{}}{}{{}\pgfsys@moveto{4.54158pt}{4.54158pt}\pgfsys@lineto{-4.54158pt}{4.54158pt}\pgfsys@lineto{-4.54158pt}{-4.54158pt}\pgfsys@lineto{4.54158pt}{-4.54158pt}\pgfsys@closepath\pgfsys@stroke\pgfsys@invoke{
} }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-2.71138pt}{-2.63809pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{$i$ }}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} }
\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{{{}}}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}\in{\mathcal{S}}_{\alpha}}\deg^{\prime}(\scalebox{0.85}{\footnotesize
\leavevmode\hbox to9.48pt{\vbox to9.48pt{\pgfpicture\makeatletter\hbox{\hskip
4.74158pt\lower-4.74158pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{
{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}{{}{}{}{}{}{{}}{}{{}{}}{{}{}}{}{{}{}}{}{{}{}}{}{{}\pgfsys@moveto{4.54158pt}{4.54158pt}\pgfsys@lineto{-4.54158pt}{4.54158pt}\pgfsys@lineto{-4.54158pt}{-4.54158pt}\pgfsys@lineto{4.54158pt}{-4.54158pt}\pgfsys@closepath\pgfsys@stroke\pgfsys@invoke{
} }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-2.71138pt}{-2.63809pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{$i$ }}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} }
\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{{{}}}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}})=\sum_{\scalebox{0.85}{\footnotesize
\leavevmode\hbox to12.49pt{\vbox
to12.49pt{\pgfpicture\makeatletter\hbox{\hskip 6.2437pt\lower-6.2437pt\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{
{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}{{}{{{}}}{{}}{}{}{}{}{}{}{}{}{}{{}\pgfsys@moveto{6.0437pt}{0.0pt}\pgfsys@curveto{6.0437pt}{3.33788pt}{3.33788pt}{6.0437pt}{0.0pt}{6.0437pt}\pgfsys@curveto{-3.33788pt}{6.0437pt}{-6.0437pt}{3.33788pt}{-6.0437pt}{0.0pt}\pgfsys@curveto{-6.0437pt}{-3.33788pt}{-3.33788pt}{-6.0437pt}{0.0pt}{-6.0437pt}\pgfsys@curveto{3.33788pt}{-6.0437pt}{6.0437pt}{-3.33788pt}{6.0437pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{
} }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-3.62317pt}{-1.72221pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{$u$ }}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} }
\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{{{}}}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}\in{\mathcal{C}}_{\alpha}}\deg^{\prime}(\scalebox{0.85}{\footnotesize
\leavevmode\hbox to12.49pt{\vbox
to12.49pt{\pgfpicture\makeatletter\hbox{\hskip 6.2437pt\lower-6.2437pt\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{
{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}{{}{{{}}}{{}}{}{}{}{}{}{}{}{}{}{{}\pgfsys@moveto{6.0437pt}{0.0pt}\pgfsys@curveto{6.0437pt}{3.33788pt}{3.33788pt}{6.0437pt}{0.0pt}{6.0437pt}\pgfsys@curveto{-3.33788pt}{6.0437pt}{-6.0437pt}{3.33788pt}{-6.0437pt}{0.0pt}\pgfsys@curveto{-6.0437pt}{-3.33788pt}{-3.33788pt}{-6.0437pt}{0.0pt}{-6.0437pt}\pgfsys@curveto{3.33788pt}{-6.0437pt}{6.0437pt}{-3.33788pt}{6.0437pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{
} }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-3.62317pt}{-1.72221pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{$u$ }}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} }
\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{{{}}}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}})$. We get that

$|E(\alpha)|=|E_{1}|+|E_{2}|=|P^{\prime}|+\frac{1}{2}(\sum_{\scalebox{0.85}{\footnotesize
\leavevmode\hbox to9.48pt{\vbox to9.48pt{\pgfpicture\makeatletter\hbox{\hskip
4.74158pt\lower-4.74158pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{
{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}{{}{}{}{}{}{{}}{}{{}{}}{{}{}}{}{{}{}}{}{{}{}}{}{{}\pgfsys@moveto{4.54158pt}{4.54158pt}\pgfsys@lineto{-4.54158pt}{4.54158pt}\pgfsys@lineto{-4.54158pt}{-4.54158pt}\pgfsys@lineto{4.54158pt}{-4.54158pt}\pgfsys@closepath\pgfsys@stroke\pgfsys@invoke{
} }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-2.71138pt}{-2.63809pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{$i$ }}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} }
\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{{{}}}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}\in{\mathcal{S}}_{\alpha}}\deg^{\prime}(\scalebox{0.85}{\footnotesize
\leavevmode\hbox to9.48pt{\vbox to9.48pt{\pgfpicture\makeatletter\hbox{\hskip
4.74158pt\lower-4.74158pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{
{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}{{}{}{}{}{}{{}}{}{{}{}}{{}{}}{}{{}{}}{}{{}{}}{}{{}\pgfsys@moveto{4.54158pt}{4.54158pt}\pgfsys@lineto{-4.54158pt}{4.54158pt}\pgfsys@lineto{-4.54158pt}{-4.54158pt}\pgfsys@lineto{4.54158pt}{-4.54158pt}\pgfsys@closepath\pgfsys@stroke\pgfsys@invoke{
} }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-2.71138pt}{-2.63809pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{$i$ }}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} }
\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{{{}}}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}})+\sum_{\scalebox{0.85}{\footnotesize
\leavevmode\hbox to12.49pt{\vbox
to12.49pt{\pgfpicture\makeatletter\hbox{\hskip 6.2437pt\lower-6.2437pt\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{
{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}{{}{{{}}}{{}}{}{}{}{}{}{}{}{}{}{{}\pgfsys@moveto{6.0437pt}{0.0pt}\pgfsys@curveto{6.0437pt}{3.33788pt}{3.33788pt}{6.0437pt}{0.0pt}{6.0437pt}\pgfsys@curveto{-3.33788pt}{6.0437pt}{-6.0437pt}{3.33788pt}{-6.0437pt}{0.0pt}\pgfsys@curveto{-6.0437pt}{-3.33788pt}{-3.33788pt}{-6.0437pt}{0.0pt}{-6.0437pt}\pgfsys@curveto{3.33788pt}{-6.0437pt}{6.0437pt}{-3.33788pt}{6.0437pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{
} }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-3.62317pt}{-1.72221pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{$u$ }}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} }
\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{{{}}}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}\in{\mathcal{C}}_{\alpha}}\deg^{\prime}(\scalebox{0.85}{\footnotesize
\leavevmode\hbox to12.49pt{\vbox
to12.49pt{\pgfpicture\makeatletter\hbox{\hskip 6.2437pt\lower-6.2437pt\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{
{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}{{}{{{}}}{{}}{}{}{}{}{}{}{}{}{}{{}\pgfsys@moveto{6.0437pt}{0.0pt}\pgfsys@curveto{6.0437pt}{3.33788pt}{3.33788pt}{6.0437pt}{0.0pt}{6.0437pt}\pgfsys@curveto{-3.33788pt}{6.0437pt}{-6.0437pt}{3.33788pt}{-6.0437pt}{0.0pt}\pgfsys@curveto{-6.0437pt}{-3.33788pt}{-3.33788pt}{-6.0437pt}{0.0pt}{-6.0437pt}\pgfsys@curveto{3.33788pt}{-6.0437pt}{6.0437pt}{-3.33788pt}{6.0437pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{
} }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-3.62317pt}{-1.72221pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{$u$ }}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} }
\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{{{}}}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}))$

We also have $|S_{\alpha}\setminus
S_{min}|\leq|P^{\prime}|+|{\mathcal{S}}_{\alpha}\cap
W_{\alpha}|+|{\mathcal{S}}_{\alpha}\cap(P\setminus
P^{\prime})|\leq|P^{\prime}|+\frac{1}{2}\sum_{\scalebox{0.85}{\footnotesize
\leavevmode\hbox to9.48pt{\vbox to9.48pt{\pgfpicture\makeatletter\hbox{\hskip
4.74158pt\lower-4.74158pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{
{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}{{}{}{}{}{}{{}}{}{{}{}}{{}{}}{}{{}{}}{}{{}{}}{}{{}\pgfsys@moveto{4.54158pt}{4.54158pt}\pgfsys@lineto{-4.54158pt}{4.54158pt}\pgfsys@lineto{-4.54158pt}{-4.54158pt}\pgfsys@lineto{4.54158pt}{-4.54158pt}\pgfsys@closepath\pgfsys@stroke\pgfsys@invoke{
} }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-2.71138pt}{-2.63809pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{$i$ }}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} }
\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{{{}}}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}\in{\mathcal{S}}_{\alpha}}\deg^{\prime}(\scalebox{0.85}{\footnotesize
\leavevmode\hbox to9.48pt{\vbox to9.48pt{\pgfpicture\makeatletter\hbox{\hskip
4.74158pt\lower-4.74158pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{
{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}{{}{}{}{}{}{{}}{}{{}{}}{{}{}}{}{{}{}}{}{{}{}}{}{{}\pgfsys@moveto{4.54158pt}{4.54158pt}\pgfsys@lineto{-4.54158pt}{4.54158pt}\pgfsys@lineto{-4.54158pt}{-4.54158pt}\pgfsys@lineto{4.54158pt}{-4.54158pt}\pgfsys@closepath\pgfsys@stroke\pgfsys@invoke{
} }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-2.71138pt}{-2.63809pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{$i$ }}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} }
\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{{{}}}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}})$ because each
square vertex outside $P^{\prime}\cup Q$ has degree at least $2$ and is not
incident to any edge in $E_{1}$. So, it suffices to prove

$\frac{1}{2}\sum_{\scalebox{0.85}{\footnotesize \leavevmode\hbox
to12.49pt{\vbox to12.49pt{\pgfpicture\makeatletter\hbox{\hskip
6.2437pt\lower-6.2437pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{
{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}{{}{{{}}}{{}}{}{}{}{}{}{}{}{}{}{{}\pgfsys@moveto{6.0437pt}{0.0pt}\pgfsys@curveto{6.0437pt}{3.33788pt}{3.33788pt}{6.0437pt}{0.0pt}{6.0437pt}\pgfsys@curveto{-3.33788pt}{6.0437pt}{-6.0437pt}{3.33788pt}{-6.0437pt}{0.0pt}\pgfsys@curveto{-6.0437pt}{-3.33788pt}{-3.33788pt}{-6.0437pt}{0.0pt}{-6.0437pt}\pgfsys@curveto{3.33788pt}{-6.0437pt}{6.0437pt}{-3.33788pt}{6.0437pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{
} }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-3.62317pt}{-1.72221pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{$u$ }}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} }
\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{{{}}}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}\in{\mathcal{C}}_{\alpha}}\deg^{\prime}(\scalebox{0.85}{\footnotesize
\leavevmode\hbox to12.49pt{\vbox
to12.49pt{\pgfpicture\makeatletter\hbox{\hskip 6.2437pt\lower-6.2437pt\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{
{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}{{}{{{}}}{{}}{}{}{}{}{}{}{}{}{}{{}\pgfsys@moveto{6.0437pt}{0.0pt}\pgfsys@curveto{6.0437pt}{3.33788pt}{3.33788pt}{6.0437pt}{0.0pt}{6.0437pt}\pgfsys@curveto{-3.33788pt}{6.0437pt}{-6.0437pt}{3.33788pt}{-6.0437pt}{0.0pt}\pgfsys@curveto{-6.0437pt}{-3.33788pt}{-3.33788pt}{-6.0437pt}{0.0pt}{-6.0437pt}\pgfsys@curveto{3.33788pt}{-6.0437pt}{6.0437pt}{-3.33788pt}{6.0437pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{
} }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-3.62317pt}{-1.72221pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{$u$ }}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} }
\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{{{}}}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}})-(1.5-\varepsilon)|{\mathcal{C}}_{\alpha}\setminus
S_{min}|\geq\Omega(\varepsilon|E(\alpha)|)$

Now, observe that each $\scalebox{0.85}{\footnotesize \leavevmode\hbox
to12.49pt{\vbox to12.49pt{\pgfpicture\makeatletter\hbox{\hskip
6.2437pt\lower-6.2437pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{
{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}{{}{{{}}}{{}}{}{}{}{}{}{}{}{}{}{{}\pgfsys@moveto{6.0437pt}{0.0pt}\pgfsys@curveto{6.0437pt}{3.33788pt}{3.33788pt}{6.0437pt}{0.0pt}{6.0437pt}\pgfsys@curveto{-3.33788pt}{6.0437pt}{-6.0437pt}{3.33788pt}{-6.0437pt}{0.0pt}\pgfsys@curveto{-6.0437pt}{-3.33788pt}{-3.33788pt}{-6.0437pt}{0.0pt}{-6.0437pt}\pgfsys@curveto{3.33788pt}{-6.0437pt}{6.0437pt}{-3.33788pt}{6.0437pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{
} }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-3.62317pt}{-1.72221pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{$u$ }}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} }
\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{{{}}}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}\in{\mathcal{C}}_{\alpha}$
is incident to at most two edges in $E_{1}$. This is because if it were
adjacent to at least $3$ edges in $E_{1}$, then either $u$ is adjacent to at
least two vertices of degree $1$ in $U_{\alpha}$ or $u$ is adjacent to at
least two vertices of degree $1$ in $V_{\alpha}$. However, this cannot happen
since $\alpha$ is not a spider. This implies that
$\deg^{\prime}(\scalebox{0.85}{\footnotesize \leavevmode\hbox to12.49pt{\vbox
to12.49pt{\pgfpicture\makeatletter\hbox{\hskip 6.2437pt\lower-6.2437pt\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{
{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}{{}{{{}}}{{}}{}{}{}{}{}{}{}{}{}{{}\pgfsys@moveto{6.0437pt}{0.0pt}\pgfsys@curveto{6.0437pt}{3.33788pt}{3.33788pt}{6.0437pt}{0.0pt}{6.0437pt}\pgfsys@curveto{-3.33788pt}{6.0437pt}{-6.0437pt}{3.33788pt}{-6.0437pt}{0.0pt}\pgfsys@curveto{-6.0437pt}{-3.33788pt}{-3.33788pt}{-6.0437pt}{0.0pt}{-6.0437pt}\pgfsys@curveto{3.33788pt}{-6.0437pt}{6.0437pt}{-3.33788pt}{6.0437pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{
} }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-3.62317pt}{-1.72221pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{$u$ }}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} }
\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{{{}}}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}})\geq\deg(\scalebox{0.85}{\footnotesize
\leavevmode\hbox to12.49pt{\vbox
to12.49pt{\pgfpicture\makeatletter\hbox{\hskip 6.2437pt\lower-6.2437pt\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{
{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}{{}{{{}}}{{}}{}{}{}{}{}{}{}{}{}{{}\pgfsys@moveto{6.0437pt}{0.0pt}\pgfsys@curveto{6.0437pt}{3.33788pt}{3.33788pt}{6.0437pt}{0.0pt}{6.0437pt}\pgfsys@curveto{-3.33788pt}{6.0437pt}{-6.0437pt}{3.33788pt}{-6.0437pt}{0.0pt}\pgfsys@curveto{-6.0437pt}{-3.33788pt}{-3.33788pt}{-6.0437pt}{0.0pt}{-6.0437pt}\pgfsys@curveto{3.33788pt}{-6.0437pt}{6.0437pt}{-3.33788pt}{6.0437pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{
} }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-3.62317pt}{-1.72221pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{$u$ }}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} }
\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{{{}}}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}})-2$.

Note moreover that if $\scalebox{0.85}{\footnotesize \leavevmode\hbox
to12.49pt{\vbox to12.49pt{\pgfpicture\makeatletter\hbox{\hskip
6.2437pt\lower-6.2437pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{
{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}{{}{{{}}}{{}}{}{}{}{}{}{}{}{}{}{{}\pgfsys@moveto{6.0437pt}{0.0pt}\pgfsys@curveto{6.0437pt}{3.33788pt}{3.33788pt}{6.0437pt}{0.0pt}{6.0437pt}\pgfsys@curveto{-3.33788pt}{6.0437pt}{-6.0437pt}{3.33788pt}{-6.0437pt}{0.0pt}\pgfsys@curveto{-6.0437pt}{-3.33788pt}{-3.33788pt}{-6.0437pt}{0.0pt}{-6.0437pt}\pgfsys@curveto{3.33788pt}{-6.0437pt}{6.0437pt}{-3.33788pt}{6.0437pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{
} }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-3.62317pt}{-1.72221pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{$u$ }}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} }
\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{{{}}}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}\in{\mathcal{C}}_{\alpha}\setminus
S_{min}$, we have that $\deg^{\prime}(\scalebox{0.85}{\footnotesize
\leavevmode\hbox to12.49pt{\vbox
to12.49pt{\pgfpicture\makeatletter\hbox{\hskip 6.2437pt\lower-6.2437pt\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{
{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}{{}{{{}}}{{}}{}{}{}{}{}{}{}{}{}{{}\pgfsys@moveto{6.0437pt}{0.0pt}\pgfsys@curveto{6.0437pt}{3.33788pt}{3.33788pt}{6.0437pt}{0.0pt}{6.0437pt}\pgfsys@curveto{-3.33788pt}{6.0437pt}{-6.0437pt}{3.33788pt}{-6.0437pt}{0.0pt}\pgfsys@curveto{-6.0437pt}{-3.33788pt}{-3.33788pt}{-6.0437pt}{0.0pt}{-6.0437pt}\pgfsys@curveto{3.33788pt}{-6.0437pt}{6.0437pt}{-3.33788pt}{6.0437pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{
} }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-3.62317pt}{-1.72221pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{$u$ }}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} }
\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{{{}}}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}})\geq\deg(\scalebox{0.85}{\footnotesize
\leavevmode\hbox to12.49pt{\vbox
to12.49pt{\pgfpicture\makeatletter\hbox{\hskip 6.2437pt\lower-6.2437pt\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{
{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}{{}{{{}}}{{}}{}{}{}{}{}{}{}{}{}{{}\pgfsys@moveto{6.0437pt}{0.0pt}\pgfsys@curveto{6.0437pt}{3.33788pt}{3.33788pt}{6.0437pt}{0.0pt}{6.0437pt}\pgfsys@curveto{-3.33788pt}{6.0437pt}{-6.0437pt}{3.33788pt}{-6.0437pt}{0.0pt}\pgfsys@curveto{-6.0437pt}{-3.33788pt}{-3.33788pt}{-6.0437pt}{0.0pt}{-6.0437pt}\pgfsys@curveto{3.33788pt}{-6.0437pt}{6.0437pt}{-3.33788pt}{6.0437pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{
} }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-3.62317pt}{-1.72221pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{$u$ }}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} }
\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{{{}}}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}})-1$. This is
because, building on the preceding argument,
$\deg^{\prime}(\scalebox{0.85}{\footnotesize \leavevmode\hbox to12.49pt{\vbox
to12.49pt{\pgfpicture\makeatletter\hbox{\hskip 6.2437pt\lower-6.2437pt\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{
{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}{{}{{{}}}{{}}{}{}{}{}{}{}{}{}{}{{}\pgfsys@moveto{6.0437pt}{0.0pt}\pgfsys@curveto{6.0437pt}{3.33788pt}{3.33788pt}{6.0437pt}{0.0pt}{6.0437pt}\pgfsys@curveto{-3.33788pt}{6.0437pt}{-6.0437pt}{3.33788pt}{-6.0437pt}{0.0pt}\pgfsys@curveto{-6.0437pt}{-3.33788pt}{-3.33788pt}{-6.0437pt}{0.0pt}{-6.0437pt}\pgfsys@curveto{3.33788pt}{-6.0437pt}{6.0437pt}{-3.33788pt}{6.0437pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{
} }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-3.62317pt}{-1.72221pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{$u$ }}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} }
\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{{{}}}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}})=\deg(\scalebox{0.85}{\footnotesize
\leavevmode\hbox to12.49pt{\vbox
to12.49pt{\pgfpicture\makeatletter\hbox{\hskip 6.2437pt\lower-6.2437pt\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{
{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}{{}{{{}}}{{}}{}{}{}{}{}{}{}{}{}{{}\pgfsys@moveto{6.0437pt}{0.0pt}\pgfsys@curveto{6.0437pt}{3.33788pt}{3.33788pt}{6.0437pt}{0.0pt}{6.0437pt}\pgfsys@curveto{-3.33788pt}{6.0437pt}{-6.0437pt}{3.33788pt}{-6.0437pt}{0.0pt}\pgfsys@curveto{-6.0437pt}{-3.33788pt}{-3.33788pt}{-6.0437pt}{0.0pt}{-6.0437pt}\pgfsys@curveto{3.33788pt}{-6.0437pt}{6.0437pt}{-3.33788pt}{6.0437pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{
} }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-3.62317pt}{-1.72221pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{$u$ }}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} }
\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{{{}}}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}})-2$ can only happen
if there exist $\scalebox{0.85}{\footnotesize \leavevmode\hbox to9.48pt{\vbox
to9.48pt{\pgfpicture\makeatletter\hbox{\hskip 4.74158pt\lower-4.74158pt\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{
{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}{{}{}{}{}{}{{}}{}{{}{}}{{}{}}{}{{}{}}{}{{}{}}{}{{}\pgfsys@moveto{4.54158pt}{4.54158pt}\pgfsys@lineto{-4.54158pt}{4.54158pt}\pgfsys@lineto{-4.54158pt}{-4.54158pt}\pgfsys@lineto{4.54158pt}{-4.54158pt}\pgfsys@closepath\pgfsys@stroke\pgfsys@invoke{
} }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-2.71138pt}{-2.63809pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{$i$ }}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} }
\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{{{}}}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}\in
U_{\alpha},\scalebox{0.85}{\footnotesize \leavevmode\hbox to11.48pt{\vbox
to11.48pt{\pgfpicture\makeatletter\hbox{\hskip 5.73787pt\lower-5.73787pt\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{
{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}{{}{}{}{}{}{{}}{}{{}{}}{{}{}}{}{{}{}}{}{{}{}}{}{{}\pgfsys@moveto{5.53787pt}{5.53787pt}\pgfsys@lineto{-5.53787pt}{5.53787pt}\pgfsys@lineto{-5.53787pt}{-5.53787pt}\pgfsys@lineto{5.53787pt}{-5.53787pt}\pgfsys@closepath\pgfsys@stroke\pgfsys@invoke{
} }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-3.20953pt}{-1.86032pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{$j$ }}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} }
\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{{{}}}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}\in V_{\alpha}$ such
that $(\scalebox{0.85}{\footnotesize \leavevmode\hbox to9.48pt{\vbox
to9.48pt{\pgfpicture\makeatletter\hbox{\hskip 4.74158pt\lower-4.74158pt\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{
{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}{{}{}{}{}{}{{}}{}{{}{}}{{}{}}{}{{}{}}{}{{}{}}{}{{}\pgfsys@moveto{4.54158pt}{4.54158pt}\pgfsys@lineto{-4.54158pt}{4.54158pt}\pgfsys@lineto{-4.54158pt}{-4.54158pt}\pgfsys@lineto{4.54158pt}{-4.54158pt}\pgfsys@closepath\pgfsys@stroke\pgfsys@invoke{
} }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-2.71138pt}{-2.63809pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{$i$ }}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} }
\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{{{}}}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}},\scalebox{0.85}{\footnotesize
\leavevmode\hbox to12.49pt{\vbox
to12.49pt{\pgfpicture\makeatletter\hbox{\hskip 6.2437pt\lower-6.2437pt\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{
{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}{{}{{{}}}{{}}{}{}{}{}{}{}{}{}{}{{}\pgfsys@moveto{6.0437pt}{0.0pt}\pgfsys@curveto{6.0437pt}{3.33788pt}{3.33788pt}{6.0437pt}{0.0pt}{6.0437pt}\pgfsys@curveto{-3.33788pt}{6.0437pt}{-6.0437pt}{3.33788pt}{-6.0437pt}{0.0pt}\pgfsys@curveto{-6.0437pt}{-3.33788pt}{-3.33788pt}{-6.0437pt}{0.0pt}{-6.0437pt}\pgfsys@curveto{3.33788pt}{-6.0437pt}{6.0437pt}{-3.33788pt}{6.0437pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{
} }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-3.62317pt}{-1.72221pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{$u$ }}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} }
\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{{{}}}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}),(\scalebox{0.85}{\footnotesize
\leavevmode\hbox to11.48pt{\vbox
to11.48pt{\pgfpicture\makeatletter\hbox{\hskip 5.73787pt\lower-5.73787pt\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{
{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}{{}{}{}{}{}{{}}{}{{}{}}{{}{}}{}{{}{}}{}{{}{}}{}{{}\pgfsys@moveto{5.53787pt}{5.53787pt}\pgfsys@lineto{-5.53787pt}{5.53787pt}\pgfsys@lineto{-5.53787pt}{-5.53787pt}\pgfsys@lineto{5.53787pt}{-5.53787pt}\pgfsys@closepath\pgfsys@stroke\pgfsys@invoke{
} }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-3.20953pt}{-1.86032pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{$j$ }}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} }
\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{{{}}}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}},\scalebox{0.85}{\footnotesize
\leavevmode\hbox to12.49pt{\vbox
to12.49pt{\pgfpicture\makeatletter\hbox{\hskip 6.2437pt\lower-6.2437pt\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{
{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}{{}{{{}}}{{}}{}{}{}{}{}{}{}{}{}{{}\pgfsys@moveto{6.0437pt}{0.0pt}\pgfsys@curveto{6.0437pt}{3.33788pt}{3.33788pt}{6.0437pt}{0.0pt}{6.0437pt}\pgfsys@curveto{-3.33788pt}{6.0437pt}{-6.0437pt}{3.33788pt}{-6.0437pt}{0.0pt}\pgfsys@curveto{-6.0437pt}{-3.33788pt}{-3.33788pt}{-6.0437pt}{0.0pt}{-6.0437pt}\pgfsys@curveto{3.33788pt}{-6.0437pt}{6.0437pt}{-3.33788pt}{6.0437pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{
} }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-3.62317pt}{-1.72221pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{$u$ }}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} }
\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{{{}}}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}})\in E_{1}$. But
then, note that we have $\scalebox{0.85}{\footnotesize \leavevmode\hbox
to9.48pt{\vbox to9.48pt{\pgfpicture\makeatletter\hbox{\hskip
4.74158pt\lower-4.74158pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{
{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}{{}{}{}{}{}{{}}{}{{}{}}{{}{}}{}{{}{}}{}{{}{}}{}{{}\pgfsys@moveto{4.54158pt}{4.54158pt}\pgfsys@lineto{-4.54158pt}{4.54158pt}\pgfsys@lineto{-4.54158pt}{-4.54158pt}\pgfsys@lineto{4.54158pt}{-4.54158pt}\pgfsys@closepath\pgfsys@stroke\pgfsys@invoke{
} }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-2.71138pt}{-2.63809pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{$i$ }}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} }
\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{{{}}}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}},\scalebox{0.85}{\footnotesize
\leavevmode\hbox to11.48pt{\vbox
to11.48pt{\pgfpicture\makeatletter\hbox{\hskip 5.73787pt\lower-5.73787pt\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{
{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}{{}{}{}{}{}{{}}{}{{}{}}{{}{}}{}{{}{}}{}{{}{}}{}{{}\pgfsys@moveto{5.53787pt}{5.53787pt}\pgfsys@lineto{-5.53787pt}{5.53787pt}\pgfsys@lineto{-5.53787pt}{-5.53787pt}\pgfsys@lineto{5.53787pt}{-5.53787pt}\pgfsys@closepath\pgfsys@stroke\pgfsys@invoke{
} }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-3.20953pt}{-1.86032pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{$j$ }}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} }
\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{{{}}}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}\not\in S_{min}$ by
definition of $P^{\prime}$ and also, $\scalebox{0.85}{\footnotesize
\leavevmode\hbox to12.49pt{\vbox
to12.49pt{\pgfpicture\makeatletter\hbox{\hskip 6.2437pt\lower-6.2437pt\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{
{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}{{}{{{}}}{{}}{}{}{}{}{}{}{}{}{}{{}\pgfsys@moveto{6.0437pt}{0.0pt}\pgfsys@curveto{6.0437pt}{3.33788pt}{3.33788pt}{6.0437pt}{0.0pt}{6.0437pt}\pgfsys@curveto{-3.33788pt}{6.0437pt}{-6.0437pt}{3.33788pt}{-6.0437pt}{0.0pt}\pgfsys@curveto{-6.0437pt}{-3.33788pt}{-3.33788pt}{-6.0437pt}{0.0pt}{-6.0437pt}\pgfsys@curveto{3.33788pt}{-6.0437pt}{6.0437pt}{-3.33788pt}{6.0437pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{
} }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-3.62317pt}{-1.72221pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{$u$ }}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} }
\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{{{}}}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}\not\in S_{min}$ by
assumption. This means that there is a path from $U_{\alpha}$ to $V_{\alpha}$
which does not pass through $S_{min}$, which is a contradiction.

Finally, we set $\varepsilon$ small enough such that the following
inequalities are true, both of which follow from the fact that
$\deg(\scalebox{0.85}{\footnotesize \leavevmode\hbox to12.49pt{\vbox
to12.49pt{\pgfpicture\makeatletter\hbox{\hskip 6.2437pt\lower-6.2437pt\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{
{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}{{}{{{}}}{{}}{}{}{}{}{}{}{}{}{}{{}\pgfsys@moveto{6.0437pt}{0.0pt}\pgfsys@curveto{6.0437pt}{3.33788pt}{3.33788pt}{6.0437pt}{0.0pt}{6.0437pt}\pgfsys@curveto{-3.33788pt}{6.0437pt}{-6.0437pt}{3.33788pt}{-6.0437pt}{0.0pt}\pgfsys@curveto{-6.0437pt}{-3.33788pt}{-3.33788pt}{-6.0437pt}{0.0pt}{-6.0437pt}\pgfsys@curveto{3.33788pt}{-6.0437pt}{6.0437pt}{-3.33788pt}{6.0437pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{
} }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-3.62317pt}{-1.72221pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{$u$ }}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} }
\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{{{}}}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}})\geq 4$ for all
$\scalebox{0.85}{\footnotesize \leavevmode\hbox to12.49pt{\vbox
to12.49pt{\pgfpicture\makeatletter\hbox{\hskip 6.2437pt\lower-6.2437pt\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{
{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}{{}{{{}}}{{}}{}{}{}{}{}{}{}{}{}{{}\pgfsys@moveto{6.0437pt}{0.0pt}\pgfsys@curveto{6.0437pt}{3.33788pt}{3.33788pt}{6.0437pt}{0.0pt}{6.0437pt}\pgfsys@curveto{-3.33788pt}{6.0437pt}{-6.0437pt}{3.33788pt}{-6.0437pt}{0.0pt}\pgfsys@curveto{-6.0437pt}{-3.33788pt}{-3.33788pt}{-6.0437pt}{0.0pt}{-6.0437pt}\pgfsys@curveto{3.33788pt}{-6.0437pt}{6.0437pt}{-3.33788pt}{6.0437pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{
} }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-3.62317pt}{-1.72221pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{$u$ }}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} }
\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{{{}}}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}\in{\mathcal{C}}_{\alpha}$.

  1. 1.

For any $\scalebox{0.85}{\footnotesize \leavevmode\hbox to12.49pt{\vbox
to12.49pt{\pgfpicture\makeatletter\hbox{\hskip 6.2437pt\lower-6.2437pt\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{
{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}{{}{{{}}}{{}}{}{}{}{}{}{}{}{}{}{{}\pgfsys@moveto{6.0437pt}{0.0pt}\pgfsys@curveto{6.0437pt}{3.33788pt}{3.33788pt}{6.0437pt}{0.0pt}{6.0437pt}\pgfsys@curveto{-3.33788pt}{6.0437pt}{-6.0437pt}{3.33788pt}{-6.0437pt}{0.0pt}\pgfsys@curveto{-6.0437pt}{-3.33788pt}{-3.33788pt}{-6.0437pt}{0.0pt}{-6.0437pt}\pgfsys@curveto{3.33788pt}{-6.0437pt}{6.0437pt}{-3.33788pt}{6.0437pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{
} }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-3.62317pt}{-1.72221pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{$u$ }}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} }
\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{{{}}}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}\in{\mathcal{C}}_{\alpha}\cap
S_{min}$, we have $\frac{\deg(\scalebox{0.85}{\footnotesize \leavevmode\hbox
to12.49pt{\vbox to12.49pt{\pgfpicture\makeatletter\hbox{\hskip
6.2437pt\lower-6.2437pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{
{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}{{}{{{}}}{{}}{}{}{}{}{}{}{}{}{}{{}\pgfsys@moveto{6.0437pt}{0.0pt}\pgfsys@curveto{6.0437pt}{3.33788pt}{3.33788pt}{6.0437pt}{0.0pt}{6.0437pt}\pgfsys@curveto{-3.33788pt}{6.0437pt}{-6.0437pt}{3.33788pt}{-6.0437pt}{0.0pt}\pgfsys@curveto{-6.0437pt}{-3.33788pt}{-3.33788pt}{-6.0437pt}{0.0pt}{-6.0437pt}\pgfsys@curveto{3.33788pt}{-6.0437pt}{6.0437pt}{-3.33788pt}{6.0437pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{
} }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-3.62317pt}{-1.72221pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{$u$ }}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} }
\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{{{}}}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}})-2}{2}\geq\frac{\varepsilon}{10}\deg(\scalebox{0.85}{\footnotesize
\leavevmode\hbox to12.49pt{\vbox
to12.49pt{\pgfpicture\makeatletter\hbox{\hskip 6.2437pt\lower-6.2437pt\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{
{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}{{}{{{}}}{{}}{}{}{}{}{}{}{}{}{}{{}\pgfsys@moveto{6.0437pt}{0.0pt}\pgfsys@curveto{6.0437pt}{3.33788pt}{3.33788pt}{6.0437pt}{0.0pt}{6.0437pt}\pgfsys@curveto{-3.33788pt}{6.0437pt}{-6.0437pt}{3.33788pt}{-6.0437pt}{0.0pt}\pgfsys@curveto{-6.0437pt}{-3.33788pt}{-3.33788pt}{-6.0437pt}{0.0pt}{-6.0437pt}\pgfsys@curveto{3.33788pt}{-6.0437pt}{6.0437pt}{-3.33788pt}{6.0437pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{
} }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-3.62317pt}{-1.72221pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{$u$ }}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} }
\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{{{}}}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}})$.

  2. 2.

For any $\scalebox{0.85}{\footnotesize \leavevmode\hbox to12.49pt{\vbox
to12.49pt{\pgfpicture\makeatletter\hbox{\hskip 6.2437pt\lower-6.2437pt\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{
{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}{{}{{{}}}{{}}{}{}{}{}{}{}{}{}{}{{}\pgfsys@moveto{6.0437pt}{0.0pt}\pgfsys@curveto{6.0437pt}{3.33788pt}{3.33788pt}{6.0437pt}{0.0pt}{6.0437pt}\pgfsys@curveto{-3.33788pt}{6.0437pt}{-6.0437pt}{3.33788pt}{-6.0437pt}{0.0pt}\pgfsys@curveto{-6.0437pt}{-3.33788pt}{-3.33788pt}{-6.0437pt}{0.0pt}{-6.0437pt}\pgfsys@curveto{3.33788pt}{-6.0437pt}{6.0437pt}{-3.33788pt}{6.0437pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{
} }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-3.62317pt}{-1.72221pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{$u$ }}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} }
\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{{{}}}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}\in{\mathcal{C}}_{\alpha}\setminus
S_{min}$, we have $\frac{\deg(\scalebox{0.85}{\footnotesize \leavevmode\hbox
to12.49pt{\vbox to12.49pt{\pgfpicture\makeatletter\hbox{\hskip
6.2437pt\lower-6.2437pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{
{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}{{}{{{}}}{{}}{}{}{}{}{}{}{}{}{}{{}\pgfsys@moveto{6.0437pt}{0.0pt}\pgfsys@curveto{6.0437pt}{3.33788pt}{3.33788pt}{6.0437pt}{0.0pt}{6.0437pt}\pgfsys@curveto{-3.33788pt}{6.0437pt}{-6.0437pt}{3.33788pt}{-6.0437pt}{0.0pt}\pgfsys@curveto{-6.0437pt}{-3.33788pt}{-3.33788pt}{-6.0437pt}{0.0pt}{-6.0437pt}\pgfsys@curveto{3.33788pt}{-6.0437pt}{6.0437pt}{-3.33788pt}{6.0437pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{
} }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-3.62317pt}{-1.72221pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{$u$ }}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} }
\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{{{}}}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}})-1}{2}-1.5+\varepsilon\geq\frac{\varepsilon}{10}\deg(\scalebox{0.85}{\footnotesize
\leavevmode\hbox to12.49pt{\vbox
to12.49pt{\pgfpicture\makeatletter\hbox{\hskip 6.2437pt\lower-6.2437pt\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{
{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}{{}{{{}}}{{}}{}{}{}{}{}{}{}{}{}{{}\pgfsys@moveto{6.0437pt}{0.0pt}\pgfsys@curveto{6.0437pt}{3.33788pt}{3.33788pt}{6.0437pt}{0.0pt}{6.0437pt}\pgfsys@curveto{-3.33788pt}{6.0437pt}{-6.0437pt}{3.33788pt}{-6.0437pt}{0.0pt}\pgfsys@curveto{-6.0437pt}{-3.33788pt}{-3.33788pt}{-6.0437pt}{0.0pt}{-6.0437pt}\pgfsys@curveto{3.33788pt}{-6.0437pt}{6.0437pt}{-3.33788pt}{6.0437pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{
} }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-3.62317pt}{-1.72221pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{$u$ }}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} }
\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{{{}}}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}})$.

Using this, we get

$\displaystyle\frac{1}{2}\sum_{\scalebox{0.85}{\footnotesize \leavevmode\hbox
to12.49pt{\vbox to12.49pt{\pgfpicture\makeatletter\hbox{\hskip
6.2437pt\lower-6.2437pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{
{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}{{}{{{}}}{{}}{}{}{}{}{}{}{}{}{}{{}\pgfsys@moveto{6.0437pt}{0.0pt}\pgfsys@curveto{6.0437pt}{3.33788pt}{3.33788pt}{6.0437pt}{0.0pt}{6.0437pt}\pgfsys@curveto{-3.33788pt}{6.0437pt}{-6.0437pt}{3.33788pt}{-6.0437pt}{0.0pt}\pgfsys@curveto{-6.0437pt}{-3.33788pt}{-3.33788pt}{-6.0437pt}{0.0pt}{-6.0437pt}\pgfsys@curveto{3.33788pt}{-6.0437pt}{6.0437pt}{-3.33788pt}{6.0437pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{
} }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-3.62317pt}{-1.72221pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{$u$ }}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} }
\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{{{}}}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}\in{\mathcal{C}}_{\alpha}}\deg^{\prime}(\scalebox{0.85}{\footnotesize
\leavevmode\hbox to12.49pt{\vbox
to12.49pt{\pgfpicture\makeatletter\hbox{\hskip 6.2437pt\lower-6.2437pt\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{
{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}{{}{{{}}}{{}}{}{}{}{}{}{}{}{}{}{{}\pgfsys@moveto{6.0437pt}{0.0pt}\pgfsys@curveto{6.0437pt}{3.33788pt}{3.33788pt}{6.0437pt}{0.0pt}{6.0437pt}\pgfsys@curveto{-3.33788pt}{6.0437pt}{-6.0437pt}{3.33788pt}{-6.0437pt}{0.0pt}\pgfsys@curveto{-6.0437pt}{-3.33788pt}{-3.33788pt}{-6.0437pt}{0.0pt}{-6.0437pt}\pgfsys@curveto{3.33788pt}{-6.0437pt}{6.0437pt}{-3.33788pt}{6.0437pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{
} }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-3.62317pt}{-1.72221pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{$u$ }}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} }
\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{{{}}}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}})$
$\displaystyle-(1.5-\varepsilon)|{\mathcal{C}}_{\alpha}\setminus S_{min}|$
$\displaystyle\geq\sum_{\scalebox{0.85}{\footnotesize \leavevmode\hbox
to12.49pt{\vbox to12.49pt{\pgfpicture\makeatletter\hbox{\hskip
6.2437pt\lower-6.2437pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{
{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}{{}{{{}}}{{}}{}{}{}{}{}{}{}{}{}{{}\pgfsys@moveto{6.0437pt}{0.0pt}\pgfsys@curveto{6.0437pt}{3.33788pt}{3.33788pt}{6.0437pt}{0.0pt}{6.0437pt}\pgfsys@curveto{-3.33788pt}{6.0437pt}{-6.0437pt}{3.33788pt}{-6.0437pt}{0.0pt}\pgfsys@curveto{-6.0437pt}{-3.33788pt}{-3.33788pt}{-6.0437pt}{0.0pt}{-6.0437pt}\pgfsys@curveto{3.33788pt}{-6.0437pt}{6.0437pt}{-3.33788pt}{6.0437pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{
} }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-3.62317pt}{-1.72221pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{$u$ }}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} }
\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{{{}}}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}\in{\mathcal{C}}_{\alpha}\cap
S_{min}}\frac{\deg(\scalebox{0.85}{\footnotesize \leavevmode\hbox
to12.49pt{\vbox to12.49pt{\pgfpicture\makeatletter\hbox{\hskip
6.2437pt\lower-6.2437pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{
{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}{{}{{{}}}{{}}{}{}{}{}{}{}{}{}{}{{}\pgfsys@moveto{6.0437pt}{0.0pt}\pgfsys@curveto{6.0437pt}{3.33788pt}{3.33788pt}{6.0437pt}{0.0pt}{6.0437pt}\pgfsys@curveto{-3.33788pt}{6.0437pt}{-6.0437pt}{3.33788pt}{-6.0437pt}{0.0pt}\pgfsys@curveto{-6.0437pt}{-3.33788pt}{-3.33788pt}{-6.0437pt}{0.0pt}{-6.0437pt}\pgfsys@curveto{3.33788pt}{-6.0437pt}{6.0437pt}{-3.33788pt}{6.0437pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{
} }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-3.62317pt}{-1.72221pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{$u$ }}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} }
\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{{{}}}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}})-2}{2}+\sum_{\scalebox{0.85}{\footnotesize
\leavevmode\hbox to12.49pt{\vbox
to12.49pt{\pgfpicture\makeatletter\hbox{\hskip 6.2437pt\lower-6.2437pt\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{
{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}{{}{{{}}}{{}}{}{}{}{}{}{}{}{}{}{{}\pgfsys@moveto{6.0437pt}{0.0pt}\pgfsys@curveto{6.0437pt}{3.33788pt}{3.33788pt}{6.0437pt}{0.0pt}{6.0437pt}\pgfsys@curveto{-3.33788pt}{6.0437pt}{-6.0437pt}{3.33788pt}{-6.0437pt}{0.0pt}\pgfsys@curveto{-6.0437pt}{-3.33788pt}{-3.33788pt}{-6.0437pt}{0.0pt}{-6.0437pt}\pgfsys@curveto{3.33788pt}{-6.0437pt}{6.0437pt}{-3.33788pt}{6.0437pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{
} }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-3.62317pt}{-1.72221pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{$u$ }}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} }
\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{{{}}}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}\in{\mathcal{C}}_{\alpha}\setminus
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