Ultimate fault tolerance for exponential delay model

Prove the conjectured ultimate fault tolerance threshold for Nakamoto consensus under the exponential network delay model, in which honest block propagation delays are i.i.d. Exp(μ1) and the mining process is Poisson with rate μ2, namely that transactions become eventually safe (the safety violation probability tends to zero as the confirmation depth k → ∞) if and only if β < (1 − β)/(1 + (1 − β)κ), where β is the adversarial mining power fraction, α = 1 − β is the honest fraction, and κ = μ2/μ1. Establish rigorously that any transaction is safe after waiting sufficiently long when the inequality holds and that no transaction is safe otherwise.

Background

The paper analyzes security–latency for Nakamoto consensus assuming exponential network delays, introducing the safety parameter κ = μ2/μ1. For bounded delay models, ultimate fault tolerance thresholds are known; for the exponential delay model, the authors state a specific threshold as an unproven claim.

Appendix C presents the claimed threshold and explicitly notes that no proof is provided and that a different model would be needed to rigorously establish it. This sets a clear open problem to prove or refute the stated threshold.