[2025-10-13v3]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.pdf

This paper presents a \textbf{unified and constructive framework} that resolves the long-standing open problem, the \textbf{Collatz Conjecture} ($3n+1$), over all non-zero integers (positive and negative). We introduce the \textbf{Collatz Phase Expression (CPE)}, which models the Collatz map as deterministic geometric transformations. The CPE utilizes the \textbf{Alternating Binary Notation (ABN)}, a method that encodes integers as alternating-sign powers of 2, and decomposes this into structural units: Chain ($\mathbf{R}$), Single ($\mathbf{T}$), and Node ($\mathbf{K}$). Using the CPE's characteristic quantities ($\mu$: unit count, $H_{RT}$: complexity, $B$: bit-length), we establish two \textbf{Universal Structural Limits}. These constraints imply that infinite divergence and non-trivial cycles are structurally impossible: \begin{enumerate} \item A strict linear bound on the total bit-length, $B(F^m (n)) \le B(n) + m$, prevents exponential growth. \item A deterministic, self-regulating trade-off for the local complexity $H_{RT}$ is enforced by the **Fundamental Inequality**, ensuring complexity cannot increase unboundedly. \end{enumerate} These structural constraints demonstrate that every positive sequence converges to the minimal complexity state ($\mathbf{H_{K}=0}$), leading exclusively to the trivial loop $\{1\}$. Furthermore, the same formalism resolves the negative Collatz conjecture by establishing a universal structural upper bound ($\mathbf{H_{RT} < 6}$), which guarantees convergence to the known finite loops (including $\{-1, -3, -11\}$ cycles). The CPE framework provides a deterministic lens for analyzing discrete dynamical systems.