We give a simple and provably correct replacement for the contested ``domain-extension'' in Step 9 of a recent windowed-QFT lattice algorithm with complex-Gaussian windows~\citep{chen2024quantum}. As acknowledged by the author, the reported issue is due to a periodicity/support mismatch when applying domain extension to only the first coordinate in the presence of offsets. Our drop-in subroutine replaces domain extension by a pair-shift difference that cancels all unknown offsets exactly and synthesizes a uniform cyclic subgroup (a zero-offset coset) of order $P$ inside $(\mathbb{Z}_{M_2})^n$. A subsequent QFT enforces the intended modular linear relation by plain character orthogonality. The sole structural assumption is a residue-accessibility condition enabling coherent auxiliary cleanup; no amplitude periodicity is used. The unitary is reversible, uses $\mathrm{poly}(\log M_2)$ gates, and preserves upstream asymptotics.

翻译：我们针对近期采用复高斯窗的窗口化量子傅里叶变换格算法（参见文献~\citep{chen2024quantum}）中备受争议的第9步“定义域扩展”操作，提出了一种简洁且可证明正确的替代方案。正如原作者所承认的，该算法在存在偏移量的情况下仅对第一个坐标进行定义域扩展时，会因周期性/支撑集不匹配而导致问题。我们提出的即插即用子程序通过“对移位差分”取代定义域扩展，该方法能精确抵消所有未知偏移量，并在 $(\mathbb{Z}_{M_2})^n$ 中合成一个阶为 $P$ 的均匀循环子群（即零偏移陪集）。后续的量子傅里叶变换通过简单的特征正交性来强制实现预期的模线性关系。该方案唯一的结构性假设是满足余数可访问条件以实现相干辅助清理，且无需利用振幅周期性。该酉变换具有可逆性，仅需 $\mathrm{poly}(\log M_2)$ 数量级的量子门，并完全保持上游算法的渐近复杂度特性。