A 5^r -Scaled Family of the Reduced 5k+1(3)/4 Collatz-Type Map: Exact Algebraic Conjugacy, Four-Attractor Preservation, and Transfer of the10^10 Finite Verification

This work develops the full 5^r-scaled family associated with the reduced residue-dependent 5k+1(3)/4 Collatz-type map on the positive odd integers. For every integer r ≥ 0, the scaled state space is S_r = 5^r N_odd, with affine branches 5x + 3·5^r for x ≡ 1 (mod 4) and 5x + 5^r for x ≡ 3 (mod 4), followed by complete removal of all powers of two. The central result is the exact algebraic conjugacy T_r(5^r y) = 5^r T_0(y), which shows that multiplication by 5^r preserves the branch sequence, complete two-adic valuation sequence, periodic-orbit structure, minimal cycle lengths, capture times, first-descent stopping times, basin membership, and normalized orbit geometry. Consequently, the four observed attractors of the base system scale exactly to the fixed point 5^r and the three cycles 31·5^r → 39·5^r → 49·5^r → 31·5^r, 37·5^r → 47·5^r → 59·5^r → 37·5^r, and 61·5^r → 77·5^r → 97·5^r → 61·5^r. The exact conjugacy also transfers the exhaustive finite verification of all 5,000,000,000 positive odd base starting values below 10^10 to every corresponding 5^r-scaled finite set. This transfer is a theorem derived from the base computation and is not presented as a new independent scan for each value of r. The accompanying computational package contains the LaTeX source, compiled PDF, verification code for the scaling identities, machine-readable inherited statistics and scaling rules, reproducibility documentation, Zenodo-oriented metadata, citation information, and SHA-256 checksums. The results establish exact equivalence between every scaled system and the base dynamical problem. They do not constitute a proof of global convergence: the existence of additional base cycles or exceptional divergent positive orbits beyond the verified finite range remains open.