Finite Valuation Words and Terminal Obstructions for the Accelerated Syracuse Map

This preprint develops a finite-word obstruction framework for the accelerated Syracuse map on positive odd integers. A finite valuation word determines an exact affine iterate formula and a unique realization class modulo a power of 2, while compatible infinite prefixes define a natural 2-adic inverse-limit object together with least positive representatives. The manuscript records the exact finite-word congruence machinery, finite-descent equivalence, terminal-surplus identity, terminal-ceiling break criterion, bad-prefix windows, activated density collapse, stabilization criterion, and strict-jump bookkeeping. These tools are used to isolate terminal obstruction classes for a hypothetical counterexample: a bounded branch leading to a nontrivial accelerated cycle and an unbounded eventual no-jump branch, which is reduced through geometric and combinatorial normal forms to packet-surplus and record-ladder compensation targets. As a closed module, the paper includes the class-exclusion theorem that no eventually periodic infinite low-surplus valuation chain is realized by a positive odd integer. The remaining terminal exclusions are stated explicitly as problems, conjectures, or roadmap steps. The manuscript does not claim a completed proof of the full Collatz conjecture. Field:Mathematics Subfields:Number Theory; Discrete Dynamical Systems; 2-adic Dynamics; Collatz-type Problems Suggested MSC 2020 codes:11B83; 11A99; 37P99