A Ternary (4k+1(2))/3 Collatz-Type Map with Two Attracting Cycles: Theoretical Analysis and Exhaustive Computational Verification up to 10⁹

This paper studies a ternary Collatz-type map based on the transformation (4k+1(2))/3(4k+1(2))/3 on positive integers not divisible by 3. For an integer n≢0(mod3)n≡0 3, the map first applies G(n)=⌊4n+23⌋,G(n)=4n+2/3, and then removes all powers of 3 from the result. The resulting compressed map is a 3-adic, Collatz-type dynamical system. The paper gives both theoretical and computational analysis of this map. The theoretical part includes a study of the 3-adic valuation v3v_3, its distribution under the Haar-measure model on Z3 Z_3, the expected number of removed factors of 3, and the corresponding average logarithmic drift. This gives a heuristic explanation for the contracting behavior of the system. The paper also analyzes the algebraic structure of the observed periodic cycles and discusses stopping times and orbit peaks. Two cycles are observed: C1=(1,2)C_1=(1,2) and C7=(7,10,14,19,26,35,47).C_7=(7,10,14,19,26,35,47). The first is the compressed cycle obtained from the states 1 and 2, while the second is a longer cycle in the same dynamical system. An exhaustive computational verification was performed for all admissible starting integers n≤109n≤ 10^9, excluding multiples of 3. This gives 666,666,667 admissible starting values. Every tested starting value reached one of the two observed cycles, with no unknown or unresolved orbit. Among the tested starting values, 21,785,111 (3.267766648366117%) reached C1C_1, while 644,881,556 (96.732233351633880%) reached C7C_7. The total number of iterations was 42,759,887,447, with a mean of 64.139831138430083 iterations. The maximum stopping time was 385 steps, attained by n=696,171,200n=696,171,200. The largest value reached during the computation was 134,701,251,885,711,310, starting from n=920,435,228n=920,435,228. The computational results strongly support the conjecture that every positive integer under this compressed ternary map eventually enters one of the two cycles C1C_1 or C7C_7. However, the global convergence statement is presented as a conjecture rather than a theorem. The 3-adic drift analysis is also used as a heuristic model and is not claimed as a proof of global convergence. The source code, computational data, and supporting materials are provided with this record to make the computational results reproducible.