Local Topological Constraints on Berry Curvature in Spin--Orbit Coupled BECs

We establish a local topological obstruction to the simultaneous flattening of Berry curvature in spin--orbit-coupled Bose--Einstein condensates (SOC BECs), which remains valid even when the global Chern number vanishes. For a generic two-component SOC BEC, the extended parameter space is the total space $M$ of a principal $U(1)_+ \times U(1)_-$ bundle over the Brillouin torus $T^{2}_{\mathrm{BZ}}$. On $M$, we construct a Kaluza--Klein metric and a natural metric connection $\nabla^{C}$ whose torsion 3-form encodes the synthetic gauge fields. Under the physically relevant assumption of constant Berry curvatures, the harmonic part of this torsion defines a mixed cohomology class $[\omega] \in \bigl(H^{2}(T^{2}_{\mathrm{BZ}}) \otimes H^{1}(S^{1}_{\phi_{+}})\bigr) \oplus \bigl(H^{2}(T^{2}_{\mathrm{BZ}}) \otimes H^{1}(S^{1}_{\phi_{-}})\bigr)$ with mixed tensor rank $r=1$. By adapting the Pigazzini--Toda (PT) lower bound to the Kaluza--Klein setting through explicit pointwise curvature analysis, we demonstrate that the obstruction kernel $\mathcal{K}$ vanishes for the physical metric, yielding the sharp inequality $\dim \mathfrak{hol}^{\mathrm{off}}(\nabla^{C}) \geq 1$. This bound forces the existence of at least one off-diagonal curvature operator, preventing the complete gauging-away of Berry phases even in regimes with zero net topological charge. This work provides the first cohomological lower bound, based on the PT framework, certifying locally irremovable curvature in SOC BECs beyond the Chern-number paradigm.