Feige's Random $3$-SAT Refutation Hypothesis

For every sufficiently large constant \(\Delta>0\),
there is no randomized polynomial-time algorithm \(A\) having both of
the following properties. First, for every satisfiable \(3\)-CNF
formula \(F\), \(A(F)\) never outputs \(\mathsf{UNSAT}\). Second, if
\(F_{n,\Delta}\) is formed on variables \(x_1,\ldots,x_n\) by choosing
\(m=\lfloor\Delta n\rfloor\) clauses independently and uniformly, each
clause using three distinct variables with independent uniformly random
signs, then
\[
  \Pr[A(F_{n,\Delta})=\mathsf{UNSAT}]\longrightarrow1
  \quad\text{as }n\to\infty,
\]
where the probability includes the randomness of both the formula and
\(A\).