Sub-Limit Dynamics: Boundary Interaction as the Structural Determinant in Finitely Constrained Generative Systems

Two independent lines of investigation have produced structurally parallel results. In the arithmetic setting, the morphological vocabulary of positional expansions of unit fractions consists of exactly six forms, with completeness governed by a criterion on the joint factorization of b and b−1. In the trajectory setting, boundary-constrained walks on de Bruijn graphs produce exactly seven dynamic regimes collapsing to three terminal outcomes, with feasibility determined by the interaction between accumulated history and residual action space. Both settings exhibit the same phenomena: finite vocabulary, early saturation, an efficiency paradox in which internal optimization is antagonistic to structural capacity, and a bottleneck located at the interaction between structure and boundary.

This paper establishes that these parallels are not analogies. We formulate three axioms — Finite Alphabet, Irreversible Consumption, and Observable Iteration — that are satisfied by both systems and prove that any system satisfying these axioms exhibits: (i) a finite morphological vocabulary, (ii) a saturation threshold strictly below the theoretical maximum, (iii) a bottleneck necessarily located at the structure-boundary interaction, and (iv) an efficiency paradox in which locally optimal strategies fragment residual capacity. We identify the mechanism of the efficiency paradox as a structural decoupling between global residue (strictly decreasing) and local maneuverability (oscillatory), and provide constructive witnesses establishing the independent necessity of each axiom. The framework is verified on three independent systems: positional arithmetic, constrained geometric trajectories, and a semantic game (Word Duel) in which LLM agents play under irreversible vocabulary consumption. Patent experiments further demonstrate that the bottleneck theorem has prescriptive power: the interaction surface identified by the axioms is the precise location where an engineering intervention prevents regime transition.