We establish a local topological obstruction to flattening Berry curvature in spin-orbit-coupled Bose-Einstein condensates (SOC BECs), valid even when the global Chern number vanishes. For a generic two-component SOC BEC, the extended parameter space \(M=T^{2}_{\textrm{BZ}}\times S^{1}_{\phi _{+}}\times S^{1}_{\phi _{-}}\) carries a natural metric connection \(\nabla ^{C}\) whose torsion 3-form encodes the synthetic gauge fields. Its harmonic part defines a mixed cohomology class \(\begin{aligned} {}[\omega ]\in \bigl (H^{2}(T^{2}_{\textrm{BZ}})\otimes H^{1}(S^{1}_{\phi _{+}})\bigr )\oplus \bigl (H^{2}(T^{2}_{\textrm{BZ}})\otimes H^{1}(S^{1}_{\phi _{-}})\bigr ), \end{aligned}\) whose mixed tensor rank equals one. Using a general geometric bound for metric connections with totally skew torsion on product manifolds, we show that the obstruction kernel \(\mathcal {K}\) vanishes, yielding the sharp inequality \(\dim \mathfrak {hol}^{\textrm{off}}(\nabla ^{LC}) \ge 1\) . This forces at least one off-diagonal Riemannian curvature operator, preventing complete gauging-away of Berry phases even when the total Chern number is zero. This provides the first cohomological lower bound certifying locally irremovable curvature in SOC BECs beyond the Chern-number paradigm.
