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

We present a unified and structural framework to resolve the long-standing Collatz Conjecture (the $3n+1$ Problem) for all non-zero integers, encompassing both positive and negative numbers. The Collatz Conjecture is known for its highly irregular behavior despite its simple definition, making it notoriously difficult for traditional number-theoretic approaches (e.g., \cite{Lagarias1985, Chamberland2006}).

We introduce the \textbf{Collatz Phase Expression (CPE)}, a geometric representation of the Collatz map. CPE is constructed from a new structural decomposition called the \textbf{Alternating Binary Notation (ABN)}, which represents the topological structure of an integer as an alternating signed sequence.

Each integer is uniquely decomposed into three fundamental unit types: Chain (C), Single (S), and Node (K). Using this decomposition, we define the following structural quantities:

A. $\mu(N)$: The number of continuous non-zero structural units (sum of C and S units).

B. $H_K(N)$: The total count of $0$ symbols (K units), representing structural complexity.

C. $H_{CS}(N)$: The total sum of non-zero units (structural dimension).

We then establish two decisive structural constraints universally applicable to all Collatz sequences.

 C. \textbf{Strict Linear Upper Bound on Bit Length Growth}:

$B(F^m(n)) \le B(n) + m$

This structurally prohibits the exponential divergence that previous studies could not fully constrain (e.g., \cite{Lagarias2010}).

D. \textbf{Algebraic Contradiction of Node Complexity $H_K$ Growth}:

By combining an algebraic upper bound law (Law B) governing the increase of $H_K$ with the definition of the stability region (Law A), we show that non-convergent candidates (non-trivial cycles and infinite divergence) lead to an absolute algebraic contradiction (e.g., $1 \le -4/3$). This proves the impossibility of the perpetual increase of $H_K$.

These constraints guarantee that all positive Collatz sequences are forced to converge to the unique minimum complexity state $H_K = 0$, which corresponds to the trivial cycle $\{1,2,4,1,\dots\}$. The same formal treatment extends to negative integers, where we prove a universal upper bound on the unit boundary complexity ($H_{CS} \le 6$), ensuring convergence to the known finite negative cycles.