Abstract

The Collatz Conjecture has intrigued mathematicians for decades. It proposes that iterating the function C(n) = n/2 for even n or C(n) = 3n + 1 for odd n on any natural number ultimately leads to the cycle 1, 2, 4. Despite its simplicity, the conjecture’s unpredictable nature complicates proofs. Algebraic Inverse Trees (AITs) offer a novel approach by modeling Collatz sequences in reverse. AITs recursively track numeric pathways using C-1, enabling global analysis, anomaly detection, and convergence estimation. They exhibit topological equivalence with Collatz sequences, allowing for the transfer of key properties. By establishing mappings between AITs and Collatz sequences, discrete systems can exchange cardinal attributes, facilitating convergence proofs. This  multidimensional strategy combines inverse algebraic models and topological equivalences for an innovative approach to this historic puzzle.

Keywords: AIT, Topology, Transport, Homeomorphism, Reverse collatz function