A 6^r-Scaled Family of the Reduced 6k ± {1,2}/5 Collatz-Type Map with Three Observed Attractors

This work develops a 6^r-scaled family of the reduced 6k ± {1,2}/5 Collatz-type map, extending the previously studied base system to an infinite family indexed by integers r ≥ 0. The construction is based on simultaneously scaling the state space and the residue-dependent affine corrections by 6^r, while retaining complete 5-adic reduction after each affine step. The central result is an exact algebraic conjugacy between the base map and every member of the scaled family. If T_0 denotes the reduced base map and T_r its 6^r-scaled counterpart, the scaling map y → 6^r y gives T_r(6^r y) = 6^r T_0(y), and consequently T_r^j(6^r y) = 6^r T_0^j(y) for every j ≥ 0. This identity follows from preservation of the relevant residue classes modulo 5, together with invariance of the complete 5-adic valuation under multiplication by 6^r. The conjugacy provides an exact correspondence between the dynamics of the base and scaled systems. Complete trajectories, branch selections, 5-adic valuation sequences, periodic orbits, minimal cycle lengths, transient lengths, stopping times, and basin membership are transported bijectively. Absolute reduced-state peaks are multiplied by 6^r, whereas normalized orbit ratios and the underlying dynamical structure remain unchanged. In particular, the three observed attractors of the base system—the fixed points 1 and 2, together with the 9-cycle 22 → 26 → 31 → 37 → 44 → 53 → 64 → 77 → 92 → 22, are transported to the scaled fixed points 6^r and 2·6^r, and to the corresponding 6^r-multiple of the 9-cycle. Numerical trajectory examples are included for all three basins to illustrate explicitly how starting values enter each of these scaled attractors. The work also transfers the exhaustive finite verification previously established for the base map. That computation tested all 80,000,000,000 reduced starting values n ≤ 10^11 with 5 ∤ n, and every tested orbit entered one of the three observed base attractors. Through the exact conjugacy theorem, all of these verified trajectories transfer rigorously to the corresponding 6^r-scaled finite sets, extending to absolute scale 6^r · 10^11 within the scaled domain. This transfer does not represent a new exhaustive 10^11-scale computation for each value of r; rather, it is an exact algebraic consequence of the base finite verification. The scaled family therefore introduces no independent periodic structure within its scaled domains: any periodic orbit in a scaled system corresponds, under division by 6^r, to a periodic orbit of the base system with the same minimal period. Likewise, any hypothetical exceptional or unbounded base trajectory would be transported to corresponding trajectories throughout the scaled family. The results establish an exact scaling symmetry of the reduced 6k ± {1,2}/5 dynamics and separate this theorem from the unresolved global convergence question. The finite verification through 10^11, the observed three-attractor structure, and the algebraic conjugacy provide strong finite and structural evidence, but they do not constitute a proof that every positive orbit is eventually captured. Global three-attractor convergence remains a conjecture equivalent across the entire 6^r-scaled family.