A Unified Proof of the Collatz Conjecture The Invariant Structure of the 3n+1 Problem: Generalization to All Integers via Phase Expression Theory and Universal Structural Limits

This paper presents a uni ed and constructive framework that resolves the long-standing open problem, the Collatz Conjecture (3n+1), over all non-zero integers (positive and negative). We introduce the Collatz Phase Expression (CPE), which models the Collatz map as deterministic geometric transformations. The CPE utilizes the Alternating Binary Notation (ABN), a method that encodes integers as alternating-sign powers of 2, and decomposes this into structural units: Chain (R), Single (T), and Node (K). Using the CPEs characteristic quantities ( : unit count, HRT: complexity, B: bit-length), we establish two Universal Structural Limits. These constraints imply that innite divergence and non-trivial cycles are structurally impossible: 1. Astrict linear bound on the total bit-length, B(Fm(n)) B(n)+m, prevents exponential growth. 2. A deterministic, self-regulating trade-o for the local complexity HRT is enforced by the Fundamental Inequality, ensuring complexity cannot increase unboundedly. These structural constraints demonstrate that every positive sequence converges to the minimal complexity state (HK = 0), leading exclusively to the trivial loop 1 . Furthermore, the same formalism resolves the negative Collatz conjecture by establishing a universal structural upper bound (HRT < 6),whichguaranteesconvergence to the known nite loops (including 1 3 11 cycles). The CPE framework provides a deterministic lens for analyzing discrete dynamical systems.