= Solution
For a smooth configuration, neighboring sites lie on opposite sublattices and contribute <Spin coherent-state Berry phases> with opposite orientations. The terms $S\Gamma[\widetilde{\mathbf n}]$ and $S\Gamma[-\widetilde{\mathbf n}]$ therefore cancel pairwise modulo the quantized solid-angle ambiguity. Their smooth bulk contribution vanishes in the <continuum limit>.
Singular spacetime configurations can leave lattice Berry phases attached to hedgehog events, but those are outside the smooth sector used in this continuum derivation. Thus the order-parameter-only <Wess–Zumino term> may be dropped here.
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