= Solution
{c}
For $V\gg|t|$, the leading objective is to avoid adjacent electrons of the same <spin>. The unpolarized <Fermi sea>, although optimal within the preceding restricted family, has interaction <energy> per site $V(1/2-2/\pi^2)>0$. Its cost grows linearly with $V$, while states with no forbidden neighbouring pairs have zero interaction <energy>.
There are two independent <spinless fermion t-V chains>, one for each <spin>. On an even ring, avoiding all neighbouring pairs allows at most $M/2$ particles in each chain. Since the total is $M$, both bounds must be saturated, and each chain must alternate occupied and empty sites. Their two independent choices give \b[four limiting occupation patterns]:
$$
\boxed{\begin{gathered}
\uparrow,\downarrow,\uparrow,\downarrow,\ldots;\qquad
\downarrow,\uparrow,\downarrow,\uparrow,\ldots;\\
\uparrow\downarrow,0,\uparrow\downarrow,0,\ldots;\qquad
0,\uparrow\downarrow,0,\uparrow\downarrow,\ldots.
\end{gathered}}
$$
The first two are <Néel states> with one electron per site; the last two are <charge density waves> alternating doubly occupied and empty sites. Opposite-<spin> double occupation costs nothing in this particular <Hamiltonian>. Each species has alternating occupation in all four patterns.
These are the exact zero-interaction <ground states> at $t=0$, and the limiting ordered configurations as $V/|t|\to\infty$. At large finite $V/|t|$, <virtual hopping> admixes configurations with forbidden pairs, producing an <energy> correction of order $-M|t|^2/V$. A finite ring can have <superpositions> of the ordered configurations, with possible <quantum tunnelling> splittings; four exactly degenerate product <ground states> are not implied at finite hopping. The four ordered choices describe the strong-coupling <thermodynamic limit>, with <spontaneous symmetry breaking> understood.
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