Extended Computational Verification of the Ternary (4k±1)/3 Collatz-Type Map up to 10^11: A 128-Bit Exhaustive Certification

This article extends the exhaustive computational verification of the ternary (4k±1)/3 Collatz-type map from 10^9 to 10^11. The map is defined on the positive integers by T(k) = k/3 if k ≡ 0 (mod 3),T(k) = (4k−1)/3 if k ≡ 1 (mod 3),T(k) = (4k+1)/3 if k ≡ 2 (mod 3). A previous exhaustive computation verified that every starting value 1 ≤ n ≤ 10^9 reaches 1. The present work exhaustively examines the additional interval 10^9 < n ≤ 10^11, containing 99,000,000,000 starting values. Using a multithreaded C11 implementation with unsigned 128-bit integer arithmetic, every tested starting value was certified to descend into the previously verified interval [1,10^9]. The computation produced tested = 99,000,000,000certified = 99,000,000,000128-bit overflows = 0. The maximum number of iterations required to enter the verified base interval was 548, attained from n = 58,528,894,046. The largest value encountered was 1,030,251,791,629,144,029,921, arising from the starting value n = 72,496,238,260 at iteration 168. An earlier 64-bit implementation left 158 starting values unresolved solely because intermediate values exceeded the unsigned 64-bit range. Repeating the complete interval with unsigned 128-bit arithmetic eliminated this computational limitation and resulted in zero arithmetic overflows. Combining this certification with the previous exhaustive verification through 10^9 establishes that every starting value 1 ≤ n ≤ 10^11 has been computationally verified to reach 1 under this map. This is a finite exhaustive computational verification and does not constitute a proof of global convergence for all positive integers. Source code, machine-readable summaries, and reproducibility materials are included in the accompanying computational package.