Proof. Let $S$ be the number of pairs $(i, j)$ that satisfy the following conditions:
$$1 \le i < j \le 2023, \quad a_i a_j \ge 1.$$
Let $T$ be the number of elements among $a_1, a_2, \dots, a_{2023}$ that are not less than $1$. Then
$$100 = a_1 + a_2 + \dots + a_{2023} \ge T, \qquad (3)$$
$$10000 = (a_1 + a_2 + \dots + a_{2023})^2 = \sum_{i=1}^{2023} a_i^2 + 2 \sum_{1 \le i < j \le 2023} a_i a_j \ge T + 2S. \quad (4)$$
Adding the two equations and dividing by $2$, we get $N = S + T \le 5050$.
If equality holds, then equality in the above equation (2) holds, which means each $a_i^2$ is either $0$ or $1$. That is, among $a_1, a_2, \dots, a_{2023}$, exactly $100$ are $1$s and $1923$ are $0$s. This is a necessary and sufficient condition.