We need to show that for generators $x$ (index $k$) and $z$ (index $k - 2$): $\langle \partial^2 x, z \rangle = \sum_{\mu(y) = k-1} n(x, y) \cdot n(y, z) = 0 \pmod{2}.$

Consider the moduli space $\mathcal{M}(x, z)$ of Floer trajectories from $x$ to $z$. Since $\mu(x) - \mu(z) = 2$, this moduli space has virtual dimension 1 (after quotienting by the $\mathbb{R}$-translation).

Compactification: By Gromov compactness, $\overline{\mathcal{M}}(x, z)$ is a compact 1-manifold with boundary. Boundary points correspond to broken trajectories: pairs $(u_1, u_2)$ where $u_1 \in \mathcal{M}(x, y)$ and $u_2 \in \mathcal{M}(y, z)$ for some intermediate orbit $y$ with $\mu(y) = k - 1$.

Boundary structure: The boundary of the compact 1-manifold $\overline{\mathcal{M}}(x, z)$ is: $\partial \overline{\mathcal{M}}(x, z) = \bigsqcup_{\mu(y) = k-1} \mathcal{M}(x, y) \times \mathcal{M}(y, z).$

Counting: Since a compact 1-manifold with boundary has an even number of boundary points: $\sum_y n(x, y) \cdot n(y, z) \equiv 0 \pmod{2}.$

This holds for all $x, z$, proving $\partial^2 = 0$. $\square$