a non-transitive $h$-vertex [$\mathcal{T}_{cliq}$-forcing]{} tournament then $$\label{equation: degree control} 
            \sum _{u \in V(H)} {\big (d^+_H(u) - d^-_H(u) \big )^2} 
                    \leq 
            h(h-1).$$

We start by determining the coefficient of $x^2$ in ${q_H}$. We claim it is equal to $$\begin{aligned}
\label{eqn:x2-coefficient}
    \frac {1}{6} \times \Big ( \sum _{u \in V(H)} 
    {\big (d^+_H(u) - d^-_H(u) \big )^2} - {h(h-1)} \Big ).\end{aligned}$$

Given $e \in E(H)$ let $I_e$ denote the function $I_e: S_h \to \{-1,1\}$ with $I_e(\sigma ) = 1$ if $e \in F_{\sigma ,H}$ and $I_e(\sigma ) = -1$ if $e \in 
    B_{\sigma ,H}$. Then we have $$\label{equation: diff_expression_p_H}
            {q_H}(x) 
                    = 
            {\mathbb E} _{\sigma \sim S_h} \Big ( (1+x)^{|F_{\sigma ,H}|} (1-x)^{|B_{\sigma, H }|} 
            \Big ) 
                    = 
            {\mathbb E} _{\sigma \sim S_h} \Big ( \prod _{e \in E(H)}(1+I_e(\sigma ) x) 
            \Big ).$$ This implies that the coefficient of $x^2$ in ${q_H}(x)$ is $$\label{equation: x^2-expression}
            \sum _{\{e,e'\} \in \binom {E(H)}{2}} \Big ( {\mathbb E}_{\sigma \sim S_h} 
             I_e(\sigma )I_{e'}(\sigma ) \Big ).$$ The contribution of each pair $\{e,e'\} \in \binom {E(H)}{2}$ to this sum is determined by how these edges meet. If $e \cap e' = \emptyset $ then the contribution to the sum is $0$. If $e$ and $e'$ are both out-edges of a single 