We prove that a number of computational problems that ask for the largest sparse induced subgraph satisfying some property definable in \(\mathsf{CMSO}_{2}\) logic, most notably Feedback Vertex Set , are polynomial-time solvable in the class of \(P_{6}\) -free graphs. This generalizes the work of Grzesik, Klimošová, Pilipczuk, and Pilipczuk on the Maximum Weight Independent Set problem in \(P_{6}\) -free graphs [SODA 2019, TALG 2022], and of Abrishami, Chudnovsky, Pilipczuk, Rzążewski, and Seymour on problems in \(P_{5}\) -free graphs [SODA 2021]. The key step is a new generalization of the framework of potential maximal cliques . We show that instead of listing a large family of potential maximal cliques, it is sufficient to only list their carvers : vertex sets that contain the same vertices from the sought solution and have similar separation properties.