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.

References

We provide our conjecture for the ultimate fault tolerance for exponential delay model in Appendix~\ref{app-c}. The ultimate fault tolerance for exponential delay model is, \begin{align} \beta<\frac{1-\beta}{1+(1-\beta)\kappa}.\label{ult-tol-conj-eqn} \end{align} In other words, any transaction is safe after waiting long enough if ult-tol-conj-eqn is satisfied and no transaction is safe if ult-tol-conj-eqn is not satisfied. We do not provide any proof for the observation above as it is out of the scope for this paper and requires another model to rigorously prove it.

— Transaction Capacity, Security and Latency in Blockchains  (2402.10138 - Doger et al., 2024) in System Model; Appendix C

The queue proof uses the strict inequality $\beta_c\lambda\tau>1$, rather than treating equality of average supply and consumption as sufficient for an infinite busy period. The sufficient condition $\lambda_a<\rho$ is equivalent to $\beta<\beta_{\mathrm{pa}}(\lambda\Delta)$, and $\lambda_a\tau<1$ is equivalent to $\beta<1/(\lambda\tau)$ when $\tau>0$. The stock and permanence theorems therefore give security strictly below the displayed upper boundary. The thresholds match in the fully decentralized regime. We do not settle behavior at equality.

— Security Limits of Mining Before Validation in Nakamoto Consensus  (2609.29222 - Zhou et al., 24 Sep 2026) in Section 4, final paragraph of Section 4.1, “Comparison with ordinary private mining”