Step 1: Action functional. Fixed points of $\phi = \phi_1^H$ correspond to 1-periodic orbits of $\dot{x} = X_H(t, x)$, which are critical points of the action functional $\mathcal{A}_H(\gamma) = -\int_0^1 \gamma^*\lambda + \int_0^1 H(t, \gamma(t)) \, dt$ on the loop space.

Step 2: Floer complex. The Floer chain complex $CF_*(H; \mathbb{Z}_2)$ is the $\mathbb{Z}_2$-vector space generated by fixed points of $\phi$ (1-periodic orbits of $X_H$). The differential $\partial$ counts index-1 connecting orbits: solutions $u: \mathbb{R} \times S^1 \to T^{2n}$ of the Floer equation $\partial_s u + J(\partial_t u - X_H(t, u)) = 0$ with $\lim_{s \to \pm\infty} u(s, \cdot) = \gamma_\pm$.

Step 3: Invariance. The Floer homology $HF_*(H) = H_*(CF_*, \partial)$ is independent of $H$ and $J$ (by continuation maps). For $H = 0$, the Floer complex reduces to the Morse complex of a small Morse function on $T^{2n}$, giving $HF_*(0) \cong H_*(T^{2n}; \mathbb{Z}_2)$.

Step 4: Conclusion. Since $HF_*(H) \cong H_*(T^{2n}; \mathbb{Z}_2)$ and $\dim CF_* \geq \dim HF_*$: the number of fixed points $\geq \sum_k \dim H_k(T^{2n}; \mathbb{Z}_2) = 2^{2n}$. $\square$