The Cramér random model for prime gaps satisfies Gilbreath's conjecture almost surely, and the only deterministic obstructions are long zero-blocks or very long shallow two-valued blocks.
Granville,Harald Cramér and the Distribution of Prime Numbers, Scand
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Gilbreath's conjecture: a Cram\'er random model and a deterministic analysis
The Cramér random model for prime gaps satisfies Gilbreath's conjecture almost surely, and the only deterministic obstructions are long zero-blocks or very long shallow two-valued blocks.