= Solution
An <optical pupil> ripple of spatial period $\Lambda$ diffracts starlight into two sidebands displaced by $\theta\simeq\lambda/\Lambda$ along the ripple's <spatial frequency> vector. Substituting the axial Nyquist period gives the <deformable-mirror control radius>
$$
\boxed{\theta_{\max}=\frac{N\lambda}{2D}.}
$$
The square actuator lattice has component cutoffs $|u_x|,|u_y|\le N/2$, hence an ideal square corrected region with this half-width in each <photodetector> direction. Its diagonal reaches $\sqrt2$ times the axial radius; an isotropic conservative restriction is the inscribed disk. Using $\lambda/D$ as the scale of a <spatial resolution element>, the square is about $N$ elements on a side and $N^2$ in area; the disk has about $\pi N^2/4$ elements. These are angular area counts, not the quadrant coefficient count in part (ii).
A circular illuminated <optical pupil> has fewer active actuator degrees of freedom than the complete square array, and influence functions alter the usable boundary. Moreover one real <wave phase> <deformable mirror> produces conjugately related corrections at opposite speckles: arbitrary complex-field correction of both independent <wave amplitude> and <wave phase> errors generally needs additional control, or a restricted half-plane. The square therefore describes ideal spatial-frequency access, rather than a guarantee that every intensity element inside it can be independently set to zero. The Fourier-domain square control region is derived in https://arxiv.org/abs/astro-ph/0510597[Bordé and Traub's speckle-nulling analysis].
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