= Solution
The first two <vectors> displace every particle identically in the horizontal or vertical direction. Every relative <displacement> is zero, so every <spring> has zero extension to first order. The third <vector> is $(y_i,-x_i)$ evaluated at the equilibrium points, which is an infinitesimal clockwise <rotation>. For any edge the relative <displacement> is perpendicular to that edge, again giving zero extension.
These are the <rigid-motion zero modes of a spring framework>: \b[two <translations> and one infinitesimal <rotation>]. Each has $Kq=0$, so $\boxed{\omega=0}$. In the linearized dynamics its amplitude has the form $A+Bt$, expressing free rigid motion rather than a restoring oscillation. Exact finite rigid <rotations> also preserve every <spring> length; the mode <vector> is their first-order tangent.
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