Semigroup Pruning Algorithms and Length Density in Numerical Semigroups

Length density δ(n) quantifies the proportion of attained factorization lengths of n within the interval [minL(n), maxL(n)] of a numerical semigroup. We derive explicit asymptotic lower bounds in terms of the minimal and maximal generators, proving that δ(n) → 1 uniformly as n → ∞. The analysis exploits extremal length estimates and introduces a general pruning template for factorization trees, from which both ordered and unordered pruning algorithms follow. We show that unordered pruning is valid only when atoms commute or when commutation relations ensure equivalence of permutations, while in non-commutative or non-cancellative settings pruning must be constrained by right-ideal and prefix-ACCP conditions. These results provide a unified framework connecting asymptotics, algorithmic pruning, and structural invariants such as elasticity and delta sets. Numerical experiments support the theoretical error bounds and confirm stabilization phenomena, demonstrating pruning as both an algebraic and computational tool in factorization theory.