Asymptotic Analytic Framework for Pruning Dynamics in Numerical Semigroups

Abstract

This paper investigates asymptotic bounds for the length density of factorizations in numerical and more general atomic semigroups, introducing a unified algorithmic and analytic treatment through pruning dynamics. Building on the pruning paradigm previously developed for factorization trees, we establish finiteness, correctness, and length-preservation properties under flexible chain conditions such as Finite Factorization (FF)-monoid and prefix-Ascending Chain Condition on Prefixes (ACCP) hypotheses. We then extend these results through a series of major analytical developments that reveal deeper structural behaviour: asymptotic growth laws for normalized length density, categorical representations of pruning as functorial morphisms, entropy-based bounds on factorization complexity, probabilistic models for random pruning trees, spectral analysis of pruning operators, and topological compactification yielding continuum limits. Collectively, these results define an Asymptotic-Analytic Framework for Pruning Dynamics that unifies combinatorial, categorical, and analytic viewpoints of semigroup factorization. The framework establishes new links between algebraic finiteness, entropy growth, spectral stability, and topological convergence, thus extending classical length-density theory toward a continuous and dynamical formulation of semigroup complexity.