Strict Growth and Descent Thresholds for Bounded-Gap 132-Avoiders

We prove that the exponential growth constants of 132-avoiding permutations with adjacent value differences at most m are strictly increasing in m. This establishes the remaining strictness assertion in Nadler's growth conjecture, following the finite-state enumeration of Mayama and Akita. The comparison comes from shifting both endpoint thresholds in their state system. We extend it to enumeration with a positive weight for each descent. In the weighted model, there is a strictly decreasing sequence of positive thresholds of order m to the power minus two, below which the exponential growth is one. Above the appropriate threshold, each increase of the gap bound strictly increases growth. For every fixed positive descent weight t, the growth constants converge to (1 + sqrt(t)) squared. The two nontrivial state components have separated descent thresholds, of orders m to the power minus two and (log(m)/m) squared. The results combine a coefficientwise component embedding with elimination of the acyclic append transitions.