= Solution
A magnetic field quantizes closed semiclassical orbits into <Landau levels>. Their passage through the <Fermi energy> modulates the <density of states> and scattering, producing Shubnikov-de Haas oscillations of resistivity. The fundamental frequency for a regular extremal orbit perpendicular to the field is
$$
\boxed{F=\frac{\hbar}{2\pi e}A_{\rm ext},\qquad
\Delta(1/B)=\frac{2\pi e}{\hbar A_{\rm ext}}.}
$$
Thus two distinct fundamentals ordinarily identify two inequivalent extremal areas, rather than two arbitrary points or two <spin> directions automatically giving two different geometric areas.
\b[The <particle-hole equality of tetragonal extremal orbit areas> is a special constraint of the exact half-filled nearest-neighbor band stated here.] For $B\parallel z$, its regular closed endpoint sections are the electron contour at $k_zc=\pi$ and the hole contour at $k_zc=0$ described in <6/b>. Translation $(k_xa,k_ya)\mapsto(k_xa+\pi,k_ya+\pi)$ changes the sign of both cosines and maps one enclosed pocket onto the other. Their areas are equal. Direct integration gives
$$
A_e a^2=\int_{-2\pi/3}^{2\pi/3}2\arccos[\tfrac12-\cos x]\,dx
=12.17169,\qquad A_h=A_e.
$$
Consequently this bare model predicts
$$
\boxed{\frac{F_e}{F_h}=1,}
$$
not two distinct regular fundamental frequencies. The intermediate $k_zc=\pi/2$ contour is a separatrix with a divergent <cyclotron period>, not another ordinary smooth extremal pocket to insert into the Onsager formula. The larger occupied-area slice at $k_z=0$ is the complement of a hole pocket on the Brillouin torus; counting that complement as a second enclosed planar orbit would give a spurious ratio.
For comparison, an additional small-$k_x,k_y$ parabolic approximation would replace the exact surface by $(k_xa)^2+(k_ya)^2\simeq4+\cos(k_zc)$ and suggest a rough belly-to-neck ratio $5/3$. That approximation is uncontrolled at half filling and erases the actual topology; it cannot be asserted as an exact prediction of the supplied cosine band. The observation of genuinely different frequencies in a real material therefore needs further information, such as longer-range hopping, a departure from half filling, extra bands, symmetry breaking or a resolved spin-dependent effect. The universal area-ratio inference is valid, but a nonunit numerical ratio is not determined consistently by the complete stated band assumptions alone.<Solid State Physics by David Tong>[§3.1.1, two spin states per band and monovalent filling; chapter 3, electron dynamics and quantum orbits; the equal-area result here follows directly from the supplied cosine band]{ref}
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