A 5^r-Scaled Family of the Reduced Ternary (5k ± 1)/4 Collatz-Type Map: Exact Algebraic Conjugacy, Cycle Preservation, Orbit-Statistic Invariance, and Transfer of the 10^10 Finite Verification

This article develops the full 5^r-scaled family associated with the reduced ternary (5k ± 1)/4 Collatz-type map studied in the author's preceding work. For every integer r >= 0, consider the scaled state space S_r = 5^r N_odd, where N_odd denotes the positive odd integers. On this scaled domain, the corresponding affine numerators are 5x - 5^r and 5x + 5^r, with branch selection determined by x modulo 4, followed by complete removal of all factors of 2. The central result is an exact algebraic conjugacy between the base system and every scaled system on S_r. Under the scaling map Phi_r(y) = 5^r y, the reduced maps satisfy T_r(Phi_r(y)) = Phi_r(T_0(y)), and therefore, for every integer j >= 0, T_r^j(5^r y) = 5^r T_0^j(y). Thus every scaled trajectory is exactly 5^r times its corresponding base trajectory. This conjugacy preserves the complete dynamical structure. Branch sequences, two-adic valuation sequences, periodic orbits, minimal cycle lengths, transient times, capture times, first-descent stopping times, basin membership, and inverse-branch structure are preserved exactly. First-lower values, reduced peaks, and raw affine peaks are multiplied by 5^r, while normalized quantities such as peak/start ratios remain unchanged. In particular, the base fixed point 1 becomes the fixed point 5^r. The base three-cycle 7 -> 9 -> 11 -> 7 becomes 75^r -> 95^r -> 115^r -> 75^r. For example, the corresponding cycles for r = 1, 2, and 3 are 35 -> 45 -> 55 -> 35, 175 -> 225 -> 275 -> 175, and 875 -> 1125 -> 1375 -> 875. Periodic orbits in the scaled domain are therefore in one-to-one correspondence with periodic orbits of the base system. In particular, no additional periodic orbit can occur inside S_r unless a corresponding periodic orbit exists in the base system. The previously established exhaustive finite verification of the base map covers all 5,000,000,000 positive odd starting values below 10^10. By exact conjugacy, these results transfer directly to the corresponding scaled set {5^r y : 1 <= y < 10^10, y odd} for every fixed r >= 0. Hence, for each r, the transferred verification covers exactly 5,000,000,000 starting values in the scaled domain, extending up to 5^r * 10^10. This is a theorem-based transfer of the original computation and is not presented as a new independent exhaustive scan of all odd integers below 5^r * 10^10. The inherited basin statistics are unchanged: 71.988495060% of the verified starting values enter the scaled fixed point, while 28.011504940% enter the scaled three-cycle. The corresponding exact base counts are 3,599,424,753 and 1,400,575,247, respectively. The maximum capture time remains 174 reduced iterations and occurs at the scaled starting value 5^r * 8,793,787,045. The maximum reduced odd peak becomes 5^r * 11,134,926,214,451 and is attained from the scaled start 5^r * 9,250,534,785. The corresponding maximum raw affine peak is 5^r * 55,674,631,072,256. The maximum observed reduced peak/start ratio remains invariant at approximately 3673.419845666274. It is attained at the scaled starting value 5^r * 2,750,717,625, with reduced peak 5^r * 10,104,540,713,499. The maximum observed two-adic valuation remains unchanged at 34. The corresponding event occurs on the scaled trajectory starting from 5^r * 2,814,749,767, at the scaled state 5^r * 3,435,973,837, where the affine numerator is 5^r * 17,179,869,184 = 5^r * 2^34. Thus multiplication by the odd factor 5^r changes the absolute scale of the event but leaves its two-adic valuation exactly unchanged. The article further derives the scaled periodic-orbit identity, establishes exact scaling of inverse branches and inverse forests, and shows that the two-adic valuation distribution and logarithmic drift heuristic are identical to those of the base system. In particular, the normalized logarithmic drift remains log(5/8), independently of r. A crucial scope restriction is maintained throughout: the exact conjugacy is proved on the scaled domain S_r = 5^r N_odd. No corresponding equivalence is claimed for arbitrary positive odd integers that are not divisible by 5^r. The finite verification also does not constitute a proof of global convergence. Instead, the article proves that the global dynamical question for the entire 5^r-scaled family, when restricted to its natural scaled domains, is mathematically equivalent to the corresponding unresolved global question for the base system. The accompanying reproducibility archive contains the complete LaTeX source, compiled PDF, a finite consistency verifier for the scaling identities, machine-readable inherited base statistics and scaling rules, Zenodo metadata, documentation, and SHA-256 checksums.