Summary: A formal ‘small tension’ expansion of \(D=11\) supergravity near a spacelike singularity is shown to be equivalent, at least up to 30th order in height, to a null geodesic motion in the infinite-dimensional coset space \(E_{10}/K(E_{10})\), where \(K(E_{10})\) is the maximal compact subgroup of the hyperbolic Kac-Moody group \(E_{10}(\mathbb R)\). For the proof we make use of a novel decomposition of \(E_{10}\) into irreducible representations of its \(\mathrm{SL}(10,\mathbb R)\) subgroup. We explicitly show how to identify the first four rungs of the \(E_{10}\) coset fields with the values of geometric quantities constructed from \(D=11\) supergravity fields and their spatial gradients taken at some comoving spatial point.