Topological Constraints, Computational Limits, and Invariant Stabilization in the Collatz Dynamical System

Repository: Zenodo Preprint Archive Abstract The Collatz conjecture (3n+1 problem) remains one of the most stubborn unresolved problems in modern number theory. Despite exhaustive empirical verification up to 2^{71} and beyond, a general analytical proof has eluded researchers for nearly ninety years. This paper examines the structural barriers preventing a universal proof—including inductive failure across 2^k - 1 binary structures, undecidability links to Turing's halting problem via John Conway’s cellular automata mappings, and Gödelian logical depth parallels to Goodstein’s theorem. Furthermore, we present an advanced architectural framework utilizing a discrete hexagonal lattice, a Mod 9 invariant, a 7-cycle periodic break, and the 3I pulse sequence (8\text{-}13\text{-}8\text{-}5\text{-}13\text{-}8) to bound phase-space divergence and ensure system stabilization below the 5184 frequency threshold.