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Stable-isotope variations exist among inner solar system solids, planets, and asteroids, but their importance is not understood. We report correlated, mass-independent variations of titanium-46 and titanium-50 in bulk analyses of these materials. Because titanium-46 and titanium-50 have different nucleosynthetic origins, this correlation suggests that the presolar dust inherited from the protosolar molecular cloud was well mixed when the oldest solar system solids formed, but requires a subsequent process imparting isotopic variability at the planetary scale. We infer that thermal processing of molecular cloud material, probably associated with volatile-element depletions in the inner solar system, resulted in selective destruction of thermally unstable, isotopically anomalous presolar components, producing residual isotopic heterogeneity. This implies that terrestrial planets accreted from thermally processed solids with nonsolar isotopic compositions.