Are generic spacelike singularities locally oscillatory, as the BKL picture predicts?
In plain words
Near the Big Bang or deep inside a black hole, each small region of space is predicted to stretch and squeeze chaotically in an endless sequence, a picture due to the Soviet physicists Belinskii, Khalatnikov and Lifshitz (BKL). This has never been proved for general solutions.
Precise statement
For generic 3+1 vacuum solutions with a spacelike singularity, prove that spatial derivatives become negligible near the singularity, so that each spatial point evolves as an independent homogeneous Bianchi IX (mixmaster) cosmology with infinitely many Kasner epochs. An answer is a proof of these asymptotics for an open set of data, or a counterexample.
What would settle it
A rigorous description of the asymptotics near a generic spacelike singularity in vacuum, confirming or refuting local mixmaster oscillations.
Status in the literature
Unverified note
Stable Kasner-like big bang formation is proved in non-oscillatory regimes such as stiff matter or high dimensions (Fournodavlos, Rodnianski, Speck 2020); the oscillatory vacuum case is open as of 2026.