The Limit Structure of Goal-Free Evolutionary Formal Systems :From Minimal Single-Orbit Cores to the Inevitability of Targets

This repository develops a formal theory of goal-free evolution in finite-state systems. In contrast to goal-directed or optimization-based models, the systems studied here generate behavior through local step rules while filtering admissible trajectories exclusively via global, path-level constraints. No target states, attractors, or privileged subsets of the state space are referenced at any stage of legality or dynamics. The work constructs a hierarchy of formal systems (KFX-β through KFX-ω), beginning with a minimal non-unique single-orbit core and progressively relaxing structural constraints. These relaxations include group-cover dynamics, multi-orbit decompositions, and sparse inter-orbit transitions, all while preserving goal-free admissibility. A central result is a sharp boundary theorem: once specific structural symmetries are broken—such as phase coherence or invariant-preserving transitions—goal-like behavior becomes mathematically unavoidable. In this sense, targets are not introduced by definition but emerge as a formal necessity beyond a precise structural threshold. The analysis is invariant under state relabeling and representation choices, relying only on structural properties of paths and admissibility constraints. This makes the framework suitable for foundational discussions of evolution, learning systems, artificial intelligence, and dynamical models where “goals” are often implicitly assumed rather than formally specified.