A Structural Solution to the Collatz Conjecture across All Integers: Perspectives on Integer Dynamics via Alternating Binary Notation and Collatz Phase Expression

We study Collatz-type dynamics in a unified manner across the positive and

negative integer domains, using a deterministic structural encoding of integers

rather than treating them as atomic points or relying only on congruence classes.

We introduce Alternating Binary Notation (ABN), a signed binary

representation with an alternation constraint, and its maximal block

decomposition, the Collatz Phase Expression (CPE), written as a sequence

of two types of blocks (Chains and Nodes). From CPE we extract a small set of

structural features, including the number of chains and the total lengths of

chain- and node-blocks.

The core observation is that one odd step of the accelerated Collatz map can be

interpreted as a finite set of local rewriting rules on CPE. These local rules

yield explicit inequalities controlling the structural features along

iteration. In particular, a global ``growth budget'' is at most linear, while a

certain ``resistive'' feature cannot remain nondecreasing indefinitely unless

the orbit is already in a terminal regime. This provides a reduction scheme in

which the positive-domain dynamics enter the classical $\{1,2,4\}$ cycle, and

the negative-domain dynamics (via the associated odd-to-odd $3n-1$ map) enter

one of finitely many periodic systems.