A Ternary (7k±1)/6 Collatz-Type Map and Its (7k±7^r)/6 Scaled Family

This work develops the full 7^r-scaled family associated with the ternary reduced (7k +/- 1)/6 Collatz-type map. The base system acts on the positive integers coprime to 6. For n congruent to 1 modulo 6, the numerator 7n - 1 is formed, while for n congruent to 5 modulo 6, the numerator 7n + 1 is formed. All remaining factors of 2 and 3 are then removed completely. The resulting reduced map is denoted by T. For every integer r >= 0, the scaled state space is defined by D_r = 7^r D, where D is the base domain. The scaled family is obtained by replacing the affine corrections +/-1 with +/-7^r while simultaneously scaling the state space by the same factor 7^r. The central result of the paper is the exact algebraic conjugacy T_r(7^r n) = 7^r T(n). More generally, for every integer j >= 0, T_r^j(7^r n) = 7^r T^j(n). Thus the scaled system is not a new independent dynamical problem. It is an exact scaled copy of the base system on the domain D_r. The conjugacy gives a complete one-to-one correspondence between the dynamics of the base map and those of every scaled level r. Complete trajectories, residue branches, 2-adic and 3-adic valuation sequences, first-descent stopping times, total stopping times, minimal cycle lengths, basin membership, and normalized orbit ratios are preserved exactly. Absolute state values are multiplied by 7^r. Consequently, first-lower values and reduced peaks are also multiplied by 7^r, while the iteration at which they occur remains unchanged. The fixed point 1 of the base system is transported to the fixed point 7^r. More generally, every periodic orbit of the base system corresponds bijectively to a periodic orbit of the scaled system having the same minimal period, and every periodic orbit in the scaled domain reduces to a base periodic orbit after division by 7^r. The paper derives the complete scaled cycle identity and shows that, after division by 7^r, it reduces exactly to the corresponding Diophantine cycle identity of the base map. Hence the scaled system introduces no new cycle equation beyond the base problem. Exact inverse branches are also obtained. The predecessor structure of the scaled fixed point 7^r is the 7^r multiple of the predecessor structure of the base fixed point 1. Therefore the corresponding inverse trees are isomorphic under multiplication by 7^r. Because 7 is coprime to both 2 and 3, multiplication by 7^r does not alter the 2-adic or 3-adic valuations of corresponding affine numerators. The complete valuation sequence is therefore preserved step by step. The residue-density laws obtained for the base system transfer unchanged to every scaled level. The associated logarithmic drift model is also invariant under scaling. Corresponding states have exactly the same one-step normalized multiplier. The mean logarithmic increment, the geometric typical multiplier, and the ordinary expected approximate multiplier are therefore identical to those of the base map. These probabilistic observations remain heuristic and are not used as proofs of global convergence. An important structural point is that the conjugacy is valid on the scaled domain D_r, that is, for states of the form x = 7^r n with n in the base domain. It is not generally true that an arbitrary integer k under the map with correction +/-7^r is simply 7^r times the trajectory of k under the base map. The scaled state-space condition is essential. The paper also clarifies the appropriate asymptotic interpretation. A large value of x alone is not sufficient when r varies, because x may be large only because of the scaling factor 7^r. The relevant normalized quantity is x / 7^r, or equivalently the underlying base state n. The previously completed exhaustive computation for the base map tested all 33,333,333,333 admissible starting values n below 10^11 with gcd(n, 6) = 1. Every tested trajectory reached the fixed point 1, with no detected nontrivial cycle and no unresolved trajectory within the stated computational conditions. By exact conjugacy, this finite verification transfers immediately to every fixed r >= 0 without rerunning the full computation. For each r, all corresponding scaled starting values x = 7^r n, with n < 10^11 and gcd(n, 6) = 1, reach the scaled fixed point 7^r. The transferred finite set contains exactly the same 33,333,333,333 corresponding trajectories. First-descent stopping times and total stopping times are unchanged. First-lower values and reduced peaks are multiplied by 7^r, and the steps at which those records occur are preserved. In particular, the base record first-descent stopping time of 35 transfers to every scaled level with the same stopping time. The base record total trajectory length of 77 reduced steps likewise remains 77 at every scaled level. The largest reduced peak in the transferred finite data is the corresponding base peak multiplied by 7^r. The paper further proves that the global convergence problem for every scaled level is equivalent to the global convergence problem for the base map. If every base orbit converges to 1, then every orbit in D_r converges to 7^r for every r >= 0. Conversely, any additional base cycle or unbounded base trajectory would generate a corresponding scaled example at every level. The article therefore separates three logically distinct components: Exact algebraic results valid for every integer r >= 0. Finite computational results inherited from the previously completed base verification. Probabilistic interpretations based on residue-density and valuation models. The exact conjugacy theorem is global on the scaled domain and requires no computational approximation. However, it does not by itself prove global convergence of the base system, exclude all possible additional cycles outside the verified finite range, or rule out unbounded trajectories. The main conclusion is that the parameter r introduces no new independent dynamical problem. The entire (7k +/- 7^r)/6 scaled family is algebraically equivalent to the original reduced (7k +/- 1)/6 system through multiplication by 7^r.