Consistent truncation of eleven-dimensional supergravity on $S^8\times S^1$

Eleven-dimensional supergravity on $S^8\times S^1$ is conjectured to be dual to the M-theory matrix model. We prove that the dynamics of a subset of fluctuations around this background is consistently described by D=2 SO(9) gauged maximal supergravity. We provide the full non-linear uplift formulae for all bosonic fields. We also present a further truncation to the SO(3)$\times$SO(6) invariant sector and discuss its relation to the BMN matrix model at finite temperature. The construction relies on the framework of generalised Scherk-Schwarz reductions, established for E$_9$ exceptional field theory in a companion paper. As a by-product, we severely constrain the most general gauge deformations in D=2 admitting an uplift to higher dimensions.