\bar{x} = (\bar{\alpha},\bar{\omega} , \bar{c}) \in X$ such that $ \bar{x} = \pi'_M(\bar{x} )$. Define $ H = X - \bar{x}$, and define $ \Delta_\omega \in {\mathbb{R}}$ such that $\Delta_\omega \geq \sup_{x \in H} |\pi_\omega (x) - \bar{\omega}| $.

Define $h,g_M^{i},g_M^{ii} $ to be functions of the form $ g_M : X \mapsto g_M(X) \in {\mathbb{R}}^M$ and define $g_\infty^i,g_\infty^{ii,a},g_\infty^{ii,b}$ to be functions of the form $ g_\infty: X \mapsto g_\infty(X) \in {\mathbb{R}}$ as follows:

$$\begin{aligned}
    [h(X)]_k := &\; \frac{2 C_0^2}{(s-1)M^{s-1} (M+k+1)^{s} }+  2 C_0 \sum_{j = M-k+1}^M  \frac{ |X|_j}{(j+k)^s} \label{eq:F_tail} \\
            [g_M^{i}(X)]_k  := &\;
    2 C_0  \Delta_\omega \sum_{j = M-k+1}^M  \frac{|X|_j}{(j+k)^{(s-1)}}    \nonumber \\
        &+ \frac{C_0^2  \Delta_\omega}{(s-2)(M+k+1)^sM^{(s-2)} }   + 
    \frac{C_0^2  \Delta_\omega }{(s-1)(M+k+1)^{(s-1)}M^{(s-1)} }   \label{eq:giM}  \\
        [g_M^{ii}(X)]_k  := & \;
    \frac{4 C_0^2 }{(s-1)(M+k+1)^s M^{s-1}}    + 
    2 C_0 \sum_{j = M-k+1}^M  \frac{|H|_j}{(j+k)^s}   \label{eq:giiM}  \\
        g_\infty^i(X) := &\;
    \max_{ M+1 \leq k \leq 2M } k^{s} \sum_{j=k-M}^{M}   |\bar{c}_j \bar{c}_{k-j}|    \label{eq:giInfty}\\
        g_\infty^{ii,a}(X) := &\; \max_{M+1 \leq k \leq 2 M}  k^s  \sum_{j=k-m}^M |H|_j |X|_{k-j}   \nonumber  \\
    &+ \frac{2 C_0^2 (2^s+1)}{(s-1)M^{s-1}} + C_0    \sum_{j=1}^{M} \left(|X|_j +|H|_j \right) \left( \left( \frac{M+j+1}{M+1} \right)^s +1 \right)    
    \label{def:giiB}    \\
        g_\infty^{ii,b}(X) := &\;  \frac{C_1^2 \gamma_{M+1}}{2} +    C_ 0C_1 \left( \frac{s-1}{(M+2)(s-2)} + \frac{s}{s-1} \right)  .       \label{def:giiC} 
    \end{aligned}$$

\[prop:CentralLemma\] Fix a cube $X$ 