On the 3n+1 Problem: Binary Carry Dynamics, Pipeline Decomposition, and Natural Density-1 Convergence

Version 3 update: This manuscript undergoes comprehensive revision and structural optimization. We supplement complete reasoning for the two-stage potential decay process, carry out full case checks for the potential descent property, add detailed demonstrations of martingale convergence, and calculate quantitative statistical deviation bounds. We clearly differentiate our natural density conclusion from Tao’s 2019 logarithmic density result, extend the pipeline analysis framework to the generalized 3n+C mapping family, and clearly point out the outstanding combinatorial obstacle that prevents the density-one result from being extended to a complete pointwise proof. Many omitted intermediate derivations are supplemented, error estimation data are supplemented, and the scope of research conclusions is standardized to fit preprint publication standards.