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Lead halide perovskites such as methylammonium lead triiodide (CH₃NH₃PbI₃) have outstanding optical and electronic properties for photovoltaic applications, yet a full understanding of how this solution-processable material works so well is currently missing. Previous research has revealed that CH₃NH₃PbI₃ possesses multiple forms of static disorder regardless of preparation method, which is surprising in light of its excellent performance. Using high energy resolution inelastic X-ray (HERIX) scattering, we measure phonon dispersions in CH₃NH₃PbI₃ and find direct evidence for another form of disorder in single crystals: large-amplitude anharmonic zone edge rotational instabilities of the PbI₆ octahedra that persist to room temperature and above, left over from structural phase transitions that take place tens to hundreds of degrees below. Phonon calculations show that the orientations of the methylammonium …