Lemma 6 ([Root-flip tests]). For i≤m<L , on the event |J_m|≥1 , (A_m G_m) ≤1_Sm≠∅ q_m^ |J_m| .

Suppose A_m occurs, and fix j∈J_m . The nonroot parts at the indices in I_ j,m must kill S_m . If they did not, choose x∈S_m satisfying all of them. Extend x by the root assignment t , then flip root j . We check the clauses of B_m . A clause with no true root literal under t has a satisfied nonroot part, because (t,x) satisfies B_m . The flip does not change that part. A clause with a true literal on another root remains true. Every remaining clause has j as its unique true root under t . It cannot be a collision, by j∈J_m , so it is a test clause at an index in I_ j,m ; its nonroot part is satisfied by the choice of x . The flipped assignment therefore satisfies B_m with root values different from t , contradicting A_m . Conditional on G_m , the test parts belonging to distinct roots are disjoint independent families of proper (k-1) -clauses. For nonempty S_m , a family of s_ j,m ≤d tests kills it with probability q_ s_ j,m (S_m)≤q_d(S_m) . This also covers s_ j,m =0 , whose killing probability is zero. The necessary tests for all roots in J_m thus have conditional probability at most q_m^ |J_m| . Finally A_m requires S_m≠∅ .