Convergence and Affine–Dyadic Boundary Dynamics in the General System Qn+x

Source abstract

We build on established modular analyses of the Collatz map, which show that odd iterates occupy only the residue classes 1, 3, 5 (mod 6) while even iterates are confined to {2, 4} (mod 6). From these constraints, the module–LCM iteration equation naturally emerges, demonstrating that all trajectories evolve strictly within this three-class modular subspace. Within this structure, the affine–dyadic boundary equation Qn + x = 2^t and the invariant ray 5·2^t identify the only admissible intersection capable of neutralizing affine expansion. Consequently, every valid orbit of the 3n + 1 map enters the Collatz even-division chain and ultimately terminates at the absorbing state n = 1.