= Solution
At fixed <particle number> $N$, write $n_1=n$ and $n_2=N-n$. The <contact interaction between two ring modes> has <energy eigenvalues>
$$
E(n)=\frac{g}{2L}\left[N(N-1)+2n(N-n)\right],\qquad n=0,1,\ldots,N.
$$
For $g>0$, the nonnegative product $n(N-n)$ is minimized at the endpoints. Consequently
$$
\boxed{E_{\min}=\frac{gN(N-1)}{2L},\qquad
|\psi_{\min}\rangle=c_1|N,0\rangle+c_2|0,N\rangle,\quad |c_1|^2+|c_2|^2=1.}
$$
For $N\geq1$ this is a two-dimensional degenerate <ground state> subspace. $N=0$ instead has the unique <Fock vacuum>. In this projected model, putting all particles into one traveling mode avoids the extra exchange contribution paid when both distinct modes are occupied.
For the alternative coefficients, put $\alpha=\beta=A<0$ and $\gamma=0$. Then $E(n)=A[N(N-1)-2n(N-n)]$ is again minimized at $n=0,N$, with $E_{\min}=AN(N-1)$. Evenness of $N$ is unnecessary for this result; the balanced <Fock state> maximizes this attractive-interaction <energy> instead of minimizing it.
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