A Unified Proof of the Collatz Conjecture The Invariant Structure of the 3n+1 Problem:Generalization to All Integers via Collatz Phase Expression (CPE) Theory

This paper presents a unified and constructive structural framework that resolves the long-standing open problem known as the \textbf{Collatz Conjecture} ($3n+1$ problem) for all nonzero integers, including both positive and negative cases. We introduce the \textbf{Collatz Phase Expression (CPE)}, a deterministic geometric representation of the Collatz map. CPE is constructed from a novel structural decomposition known as the \textbf{Alternating Binary Notation (ABN)}, a sign-alternating encoding of integers that reflects their topological structure. Each integer is decomposed into three fundamental unit types: Chain ($\mathbf{C}$), Single ($\mathbf{S}$), and Node ($\mathbf{K}$), which collectively describe the internal complexity of the sequence.

Within this framework, we define three key structural quantities: $\mu$ (non-zero unit count), $B$ (bit-length), and $\mathbf{H_K}$ (node complexity). We then establish two \textbf{Universal Structural Limits} that apply globally to all Collatz sequences:

\begin{enumerate}

\item A strict linear bound on bit-length growth: $B(F^m(n)) \leq B(n) + m$, which prohibits exponential divergence.

\item A self-regulating trade-off governing the node complexity $\mathbf{H_K}$, enforced by a Fundamental Inequality, which prevents unbounded complexity increases.

\end{enumerate}

These constraints guarantee that all positive Collatz sequences converge to the unique minimal-complexity state $\mathbf{H_K = 0}$, corresponding to the trivial cycle $\{1\}$. The same formalism extends to negative integers, for which we prove a universal upper bound on unit boundary complexity ($\mathbf{H_{CS} < 6}$), ensuring convergence to one of the known finite negative cycles.

This structural approach transforms the Collatz problem into a fully deterministic system governed by Internal Constraints, providing a new framework for analyzing discrete dynamical systems within the realm of \textbf{Phase Theory of Numerics}.