Mean-zero projection of the regular phase vector

Assume countable choice. Let $G=(V,E)$ be a finite connected undirected graph, with $n=\mathrm{\mid }V\mathrm{\mid }>0$, symmetric binary adjacency matrix $A$, and self-loops allowed and counted once in each row sum. Suppose every row sum is $d\ge 0$, let $\alpha \ge 0$, and assume $\mathrm{\parallel }\mathrm{\Delta }\mathrm{\parallel }\le \alpha d$, where $\mathrm{\Delta }=A-(d\mathrm{/}n)J$ and $J$ is the all-ones matrix; the norm is the Euclidean operator norm. Let $\theta$ be a stable state, meaning a local minimum on the phase torus of ${\mathcal{E}}_G(\theta )=\frac{1}{2}{\sum }_{x,y}{A}_{xy}(1-\cos ({\theta }_x-{\theta }_y))$. Choose representatives in $(-\pi ,\pi ]$ after common rotation so that ${\rho }_1\ge 0$ is real, where ${\rho }_k=n^{-1}{\sum }_x{e}^{ik{\theta }_x}$. Put $X={\rho }_1^2$, $r=\mathrm{\mid }{\rho }_2\mathrm{\mid }$, and $M=n^{-1}{\sum }_{x}H({\theta }_x)$, where $H(t)={\sin }^2\mathrm{\mid }t\mathrm{\mid }$ for $\mathrm{\mid }t\mathrm{\mid }\le \pi \mathrm{/}2$ and $H(t)=1$ for $\pi \mathrm{/}2\le \mathrm{\mid }t\mathrm{\mid }\le \pi$.

For $f_x=e^{i{\theta }_x}$ and $g=f-{\rho }_1\mathbf{1}$, one has $\mathrm{\Delta }f=\mathrm{\Delta }g$, $\mathrm{\parallel }g{\mathrm{\parallel }}_2^2=n(1-X)$, and $\mathrm{\parallel }\mathrm{\Delta }f{\mathrm{\parallel }}_2^2\le {\alpha }^2{d}^{2}n(1-X)$.