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Zamolodchikov periodicity is a property of $T$- and $Y$-systems, arising inthe thermodynamic Bethe ansatz. Zamolodchikov integrability was recentlyconsidered as its affine analog in our joint work with P. Pylyavskyy. Here weprove periodicity and integrability for similar discrete dynamical systemsbased on the cube recurrence, also known as the discrete BKP equation. Theperiodicity part was conjectured by Henriques in 2007.

Periodicity and integrability for the cube recurrence