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We develop a piece of general theory of strict admissible lattice filtrations in triangulated categories and show that $\\mathcal{D}^b(X)$ has such a filtration $\\mathcal{L}$ where the lattice is the set of all birational decompositions $f \\colon X \\xrightarrow{g} Z \\xrightarrow{h} Y$ with smooth $Z$. 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