A Ternary (4k−1(2))/3 Collatz-Type Map: Exact 3-Adic Symbolic Dynamics, a Density-One First-Descent Theorem, Algebraic Cycle Analysis, and Exhaustive Computational Verification up to 10^11

This work studies the accelerated ternary Collatz-type map T(x) = (4x − r(x)) / 3^ν₃(4x − r(x)), where r(x) = x mod 3 ∈ {1, 2}, on the positive integers not divisible by 3. The paper develops an exact 3-adic description of the map. Explicit inverse branches are used to establish Haar-measure preservation and an exact independent and identically distributed valuation law, Pr(a = k) = 2 / 3^k, k ≥ 1. An exact multi-step identity is then derived and used to obtain a deterministic sufficient condition for descent in terms of the accumulated 3-adic valuation. A finite-horizon correspondence between 3-adic survivor cylinders and ordinary positive integers is proved. In particular, for each fixed horizon, the asymptotic relative natural density of integers having no first descent through that horizon is exactly equal to the Haar measure of the corresponding 3-adic survivor set. An exact-rational dynamic program computes these measures. At horizon m = 50, the resulting first-descent density is approximately 99.7628073795%. A Chernoff bound further yields a density-one first-descent theorem: among positive integers not divisible by 3, the set of starting values that eventually fall strictly below their initial value has relative natural density 1. The theoretical analysis is complemented by exhaustive computation for all 66,666,666,667 admissible starting values up to 10^11. Every tested orbit reaches one of the three known terminal cycles {1}, {2}, 22 → 29 → 38 → 50 → 22, with no additional positive cycle or unresolved trajectory detected in the verified range. For every admissible n with 50 < n ≤ 10^11, a strict first descent is found. The largest observed first-descent time is 355 reduced-map iterations, attained at n = 76,089,719,024. The accompanying computational pack contains source code, archived checkpoint data, exact-rational dynamic-programming material, validation results, audit reports, the extremal first-descent trace, aggregate results, and SHA-256 manifests intended to support independent verification and reproducibility. The exhaustive computation establishes a finite verification through 10^11; it does not constitute a proof of universal convergence. Global convergence of all positive admissible starting values to the three known terminal cycles remains an open conjecture for this map.