Theorem 10, first claim — no spin-block algorithm is competitive within a factor less than $e/(e-1)$

Let $C>0$ be the context-switch cost of the spin-block problem. If a randomized on-line algorithm $A$ (a probability distribution over deterministic on-line algorithms, each choosing the blocking time for the next lock wait from the release times of the previous ones) is $c$-competitive against an oblivious adversary, that is, for some constant $a$ and every finite sequence $\sigma$ of lock waits

then

This is the lower-bound half of Theorem 10: no algorithm, deterministic or randomized, beats the factor $e/(e-1)\approx1.582$ against an oblivious adversary, while deterministic algorithms cannot beat $2$.

Formalization Note The statement quantifies over every randomized algorithm and every factor $c$; deterministic algorithms are the point masses. The additive constant $a$ is part of competitiveness, so the bound concerns long sequences of lock waits, not a single wait. On-line algorithms may use the release times of earlier waits, which only strengthens the lower bound. The hypothesis $C>0$ is the paper's "some large cost $C$".