Consider Homogenous Kuramoto model in this paper. In theorem 3.1, the author derive condition on $A$ such that all second-order critical points of $E(\theta)$ are in two opposite quadrants, by saying that $$|\sin(\theta_i-\theta_r)|\leq\frac{\sqrt{2}}{2},\forall i\in[n]$$.

To derive condition on $A$ in order to ensure this, they use order parameter:

since $\|r\||\sin(\theta_i-\theta_r)|=\Big|\sum_j(1-a_{ij})\sin(\theta_i-\theta_j)\Big|$, we can control $\|r\|$ and $\Big|\sum_j(1-a_{ij})\sin(\theta_i-\theta_j)\Big|$ for controlling $|\sin(\theta_i-\theta_r)|$ for all $i\in[n]$.

**Now I want to have a stronger result:**

For any second-order critical point $\theta$, its entries $\theta_i$ are in the same quadrant. Thus, in addition to above condition, we need to ensure

$$\cos(\theta_i-\theta_r)>0,\forall i\in[n]$$

**What I tried:**

Similar to the proof in theorem 3.1,

$\|r\|e^{-i(\theta_i-\theta_r)}=re^{i\theta_r}=\sum_je^{-i(\theta_i-\theta_j)}$

Take the real part, we have

$\|r\|\cos(\theta_i-\theta_r)=\sum_j\cos(\theta_i-\theta_j)$, for all $i\in[n]$.

Thus we need for all $i\in[n]$, $\cos(\theta_i-\theta_r)=\frac{\sum_j\cos(\theta_i-\theta_j)}{\|r\|}>0$.

Equivalently we need $\sum_j\cos(\theta_i-\theta_j)>0$ for all $i$.

Any insight is appreciated!