A Deterministic Framework Toward the Collatz Conjecture — A Precise, Exhaustive, and Finite Descent

This paper introduces a novel V2 deterministic approach to the infamous Collatz Conjecture, built on a refined shortcut iteration, a family of dynamically defined anchor points, and a carefully constructed residue-class sieve. At its core, the method leverages: a potential function blending logarithmic growth with the Hamming weight of binary representations, and a periodic structure over residue classesto rigorously quantify and limit the depth at which any integer must connect to the anchor tree. A key result shows that the exceptional set of integers avoiding anchors beyond depth ddd shrinks geometrically with ddd. Combined with a controlled potential drop at each iteration, this leads to the conclusion that every positive integer enters the anchor structure in finite time, and thus ultimately reaches 1. The shortcut stopping time is proven to obey a quadratic upper bound in log⁡2n, offering a powerful refinement over classical bounds. If validated, this framework would provide the first complete and deterministic proof of the Collatz conjecture. But even if future scrutiny reveals flaws, the tools developed here — residue sieving, anchor dynamics, and potential-based descent — offer a new mathematical lens worth exploring. This work is open for peer review and collaborative refinement.