Quantization of harmonic periods on product manifolds $T^2 \times T^2$ under vanishing off-diagonal form decomposition

Summarize the problem I am investigating the period integrals of harmonic forms on a smooth 4-manifold with product topology $M \cong T^2 \times T^2$, equipped with real angular coordinates $(\theta_1, \theta_2) \in [0, 2\pi)^2$ and phase coordinates $(\phi_1, \phi_2) \in [0, 2\pi)^2$. Specifically, I am trying to determine the analytical conditions under which the integral of a smooth $(k,k)$-form over a closed $2k$-cycle collapses strictly to rational values. Provide details and any research To avoid boundary degeneracies near $r \to 0$, consider a pseudo-Riemannian metric tensor $g_{ij}$ defined over $\mathcal{M} \cong T^2 \times T^2 \times [\epsilon, \infty)$ for a fixed parameter $\epsilon > 0$: $ds^2 = (R_1 + r_1 \cos\theta_1)^2 d\theta_1^2 + r_1^2 d\theta_2^2 - \left((R_2 + r_2 \cos\phi_1)^2 d\phi_1^2 + r_2^2 d\phi_2^2\right) + \frac{\epsilon^2}{r^2} dr^2$ Let $\omega \in \Omega^{k,k}(X)$ be a smooth differential $(k,k)$-form mapped onto this product space. Under standard Hodge decomposition, $\omega = d\alpha + \delta\beta + \gamma$, where $\gamma$ is harmonic ($\Delta \gamma = 0$). Expanding $\omega$ into its Fourier-toroidal components along with a boundary/defect integral term gives: $\omega = \sum_{m,n \in \mathbb{Z}} A_{m,n} e^{i(m\theta + n\phi)} + \int \Phi_{\text{defect}}(r) \, dr$ What I've tried / Specific Questions If we impose a condition where the off-diagonal coupling terms vanish ($C_{m,n} \to 0$), forcing the boundary term $\int \Phi_{\text{defect}}(r) \, dr$ to zero identically: Does the vanishing of these off-diagonal terms $C_{m,n} = 0$ rigorously guarantee that the period integral over any closed $2k$-dimensional cycle $Z_{2k}$ collapses strictly to rational multiples: $\int_{Z_{2k}} \omega = \left(\frac{m}{n}\right) \cdot \lambda$ for $m, n \in \mathbb{Z}, n \neq 0$, where $\lambda$ is a fundamental flux/volume element? From an algebraic geometry perspective, what specific topological or analytical obstructions arise when attempting to map these discrete harmonic winding modes back onto subvarieties of a complex projective manifold $X$?