math.NT — Number Theory
Finite Valuation Words and Terminal Obstructions for the Accelerated Syracuse Map
We develop a finite-word obstruction framework for the accelerated Syracuse map on positive odd integers. A finite valuation word determines an exact iterate formula and a unique realization class modulo a power of two; compatible infinite prefixes define a natural two-adic inverse-limit object together with least positive representatives. The first part 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. The second part uses these tools 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, successively reduced through a chain of geometric and combinatorial normal forms to a packet-surplus compensation target. The resulting reduction spine is lossless: Collatz follows from exclusion of nontrivial accelerated cycles together with a packet-surplus, or stronger record-ladder, compensation theorem for the unbounded branch. As a closed module, we include a closed class-exclusion theorem showing that no eventually periodic infinite low-surplus valuation chain is realized by a positive odd integer. On the bounded branch, we further record, as an explicit unproved roadmap step, that the terminal-ceiling strict-jump obstruction reduces to a single recurrent valuation-mismatch statement. This manuscript proves the architecture stated in its theorems, propositions, lemmas, and corollaries; it states the remaining terminal exclusions explicitly as problems, conjectures, or roadmap steps, and does not claim a completed proof of the full Collatz conjecture.