= Solution
Write <spatial frequency> in cycles across the <optical pupil> as $u=D/\Lambda$. The previous result gives a maximum axial <spatial frequency> $N/2$. If one counts one representative per positive-quadrant integer lattice point inside an isotropic disk $u_x^2+u_y^2\le(N/2)^2$, the continuum area estimate is
$$
\boxed{M_{\rm quadrant}\simeq\frac14\pi\left(\frac N2\right)^2=\frac{\pi N^2}{16}.}
$$
This is the geometric counting convention producing the printed formula. Lattice boundaries, the zero-frequency piston and axis points give lower-order corrections, so the formula is not an exact integer count.
The mode definition in the PDF is $D=j\Lambda/2$, hence $u=j/2$ and $j_{\max}=N$. It does not itself specify a two-dimensional lattice, its orientation, its independent <wave phase> components or a circular rather than square <spatial frequency> cutoff. Full-cycle integer frequencies correspond only to the even values of $j$ in that definition. A general real <wavefront error> also needs both <sine> and <cosine> quadratures, and distinct oblique orientations cannot all be identified with a single positive quadrant: the <Fourier transform> of a real <wavefront> has conjugate coefficients at $\mathbf u$ and $-\mathbf u$, rather than identifying every independent sign change of a component.
For comparison, a circular <spatial frequency> disk has approximately $\pi N^2/4$ signed full-cycle lattice points, or $\pi N^2/8$ conjugate pairs; two real <wave phase> components per pair restore approximately $\pi N^2/4$ real degrees of freedom. A square <spatial frequency> support gives a different count. The circular physical <optical pupil> similarly has only about $\pi N^2/4$ illuminated actuators. \b[The printed $M$ is a quadrant area estimate under an extra counting convention; it is not a uniquely determined count of all independently correctable <wavefront> modes from the stated one-dimensional mode condition.]
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