Ternary (5k±1)/4 Collatz-Type Map with Two Observed Attractors: Exhaustive Computational Verification up to 10^10 and Algebraic Structure

This work studies a residue-dependent reduced Collatz-type map on the positive odd integers. For an odd positive integer x, the affine numerator is defined by N(x) = 5x - 1 if x ≡ 1 (mod 4), and N(x) = 5x + 1 if x ≡ 3 (mod 4). The reduced map is obtained by dividing N(x) by its highest power of 2. Since N(x) is always divisible by 4, the map sends positive odd integers to positive odd integers. Two exact periodic attractors are immediately present: the fixed point 1 -> 1 and the three-cycle 7 -> 9 -> 11 -> 7. The article develops the algebraic structure of this dynamical system, including an exact Diophantine identity satisfied by every periodic orbit, inverse-branch formulas, basin structure, two-adic valuation statistics, and a probabilistic logarithmic-drift model. The main computational result is an exhaustive verification of every one of the 5,000,000,000 positive odd starting values below 10^10. The computation was performed using unsigned 128-bit arithmetic for orbit states and raw affine values, with a safety bound of 10^6 reduced iterations per trajectory. Every tested starting value entered one of the two displayed attractors. No unresolved trajectory and no additional terminal cycle were encountered in the verified range. The observed basin counts were: 3,599,424,753 starting values entering the fixed point 1, and 1,400,575,247 starting values entering the three-cycle 7 -> 9 -> 11 -> 7. These correspond to basin proportions of 71.988495060% and 28.011504940%, respectively. The maximum observed stopping time was 174 reduced iterations, attained at the starting value 8,793,787,045. The maximum reduced odd peak was 11,134,926,214,451, attained from the starting value 9,250,534,785. The maximum raw affine peak was 55,674,631,072,256. The maximum observed reduced peak/start ratio was approximately 3673.419845666274. It was attained from the starting value 2,750,717,625, whose reduced odd peak was 10,104,540,713,499. The largest observed two-adic valuation was 34. This occurs at the state 3,435,973,837, where 5 * 3,435,973,837 - 1 = 17,179,869,184 = 2^34. The article also derives the exact periodic-orbit identity and studies the natural two-adic valuation distribution. The associated negative logarithmic drift is treated strictly as a probabilistic heuristic and is not used as a proof of global convergence. Accordingly, the exhaustive computation constitutes a finite-range verification up to 10^10. It does not prove that every positive odd integer globally converges to one of the two observed attractors, nor does it prove that no other periodic orbit exists outside the verified range. The accompanying computational archive contains the source code, scan results, summary data, LaTeX source, compiled PDF, metadata, documentation, and SHA-256 checksums required for independent reproduction and verification of the finite computation.