It is true in an absolute, mathematical sense that the search space gets smaller when you fix a variable to a particular value. However, by fixing a variable (

It is true in an absolute, mathematical sense that the search space gets smaller when you fix a variable to a particular value. However, by fixing a variable (and effectively removing it), you are also removing potential solutions from that search space. In a sense, you could view the difficulty of a SAT problem as the fraction of valid solutions in the whole search space. By fixing a variable, you are decreasing both the number of valid solutions and the size of the total search space. But the fraction could end up being either greater or smaller, depending on the variable and the value you set it to.