Recall from our previous post that the main security statement of Subspace consensus is the following:

“Given some honest storage \Omega, an always-online adversary must control or fake a storage of at least \Omega/\phi_c (1 + \lambda \Delta) in order to break the security (with high probability).”

This is an **asymptotic** result, meaning that the adversary can break the security with high probability no matter what the confirmation depth is. In practice, we may be interested in the following, non-asymptotic, question: Given certain fraction of adversary storage (owned or faked) and confirmation depth k, what is the probability that the adversary can break the security (by conducting a double-spending attack on a target transaction)? Intuitively, such probability should decrease with k.

Saeid (our research assistant) will help us understand such probability for two special cases (c \to \infty and c = 1), since these two cases can be reduced to some known results in the literature called **settlement bounds**.

**Correction**: Only one special case (c \to \infty) is discussed in the literature. To the best of my knowledge, the settlements bounds for c = 1 is still open.