A 4^r-Scaled Family of the Ternary (4k−1(2))/3 Collatz-Type Map: Exact Algebraic Conjugacy, Cycle Preservation, and Transfer of the \(10^9\) Finite Verification

This paper develops an exact algebraic scaling theory for the ternary (4k−1(2))/3 Collatz-type map introduced and computationally studied in the author's preceding work. The base system is defined on the positive integers not divisible by 3. For every integer r >= 0, a scaled state space is obtained by multiplying the base domain by 4^r. The corresponding scaled map has the two affine branches 4x − 4^r and 4x − 2·4^r, followed by complete removal of all factors of 3. The central result is an exact algebraic conjugacy between the base system and every member of the 4^r-scaled family. If y is a state of the base system, then the corresponding scaled state is x = 4^r y, and the maps satisfy T_r(4^r y) = 4^r T_0(y). More generally, for every integer j >= 0, T_r^j(4^r y) = 4^r T_0^j(y). Consequently, corresponding trajectories have exactly the same dynamical structure. Periodic orbits correspond bijectively, cycle lengths are preserved, the sequences of removed 3-adic factors are unchanged, stopping times are preserved on corresponding trajectories, basin membership is transported exactly, and orbit peak values are multiplied by 4^r. In particular, the two fixed points 1 and 2 of the base system correspond to the scaled fixed points 4^r and 2·4^r. The observed base four-cycle 22 -> 29 -> 38 -> 50 -> 22 corresponds exactly to 22·4^r -> 29·4^r -> 38·4^r -> 50·4^r -> 22·4^r. An independent derivation from the algebraic cycle equation gives the same conclusion: every periodic orbit in a scaled domain is exactly 4^r times a periodic orbit of the base system. Therefore, the 4^r scaling does not introduce an independent periodic structure; it produces dynamically equivalent scaled copies of the base system. The exhaustive computation through 10^9 reported in the preceding paper tested 666,666,667 admissible positive starting values and found the two fixed points and the four-cycle above as the three observed positive attractors, with no unresolved or overflowed starting values. By the exact conjugacy theorem, these finite computational results transfer algebraically to the corresponding 4^r-scaled domains without requiring the same computation to be repeated independently at every scale. This transfer must be distinguished from a proof of global convergence. The scaling conjugacy and cycle correspondence established in this paper are exact algebraic results, whereas the statement that every positive base trajectory eventually reaches one of the three observed attractors remains a conjecture. The finite verification through 10^9 provides strong computational evidence but does not prove convergence for all positive integers. This work directly extends the base study: Farhad Banazadeh (2026), "A Ternary (4k−1(2))/3 Collatz-Type Map: Complete 3-Adic Reduction, Algebraic Cycle Analysis, and Exhaustive Computational Verification up to 10^9." Zenodo. DOI: 10.5281/zenodo.22662980.