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A main contribution is the construction of a twisted Beauville-Voisin class $\\mathfrak{o}_{\\mathscr{X}} \\in \\operatorname{CH}_0(X)$ that extends fundamental results of O'Grady and Shen-Yin-Zhao \\cite{OG13, SYZ20} to twisted settings. This class enables:\n  1. A derived equivalence-invariant filtration $\\mathbf{S}_\\bullet(\\mathrm{D}^{(1)}(\\mathscr{X}))$ preserved under Fourier-Mukai transforms,\n  2. 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