Recall that the clause/variable ratio (α) is a critical parameter in random k-SAT problems, and the phase transition occurs when α ≈ 4.26 for 3-SAT. How does this relate to the expected number of solutions or the probability of satisfiability?

Consider the connection between the clause/variable ratio and the *density* of constraints in the problem. How does increasing α beyond 4.26 affect the likelihood of finding a satisfying assignment, and what does this imply about the *search space*?

Think about the *backbone* of a SAT problem—the set of literals that must be true in all solutions. How does the phase transition at α ≈ 4.26 influence the size or structure of the backbone, and what does this reveal about the *hardness* of the problem?