Collatz Conjecture: A Reverse Construction Bijective Proof via Layered Directed Acyclic Graphs

This paper presents a bijective proof of the Collatz Conjecture using a reverse construction framework. By systematically generating all odd predecessors and analyzing their structure through modular decomposition and parity-controlled mappings, we demonstrate that every odd number follows a unique reverse path terminating in the base cycle. A set of closed-form equations defines each sequence, and their disjoint coverage guarantees completeness. The result is a comprehensive bijective mapping of the odd integers within a unified reverse construction, validating the conjecture. [email protected]