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.
Proth,Sur la série des nombres premiers, Nouvelle Correspondance Mathématique4(1878), 236–240
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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.