Eventually Periodic Low-Surplus Valuation Chains for the Accelerated Syracuse Map

We study infinite valuation chains for the accelerated Syracuse map via their exact finite-word residue classes and least positive odd representatives. Each finite valuation word determines a unique realization class modulo a power of \(2\), and compatible prefixes therefore define a natural \(2\)-adic inverse-limit object. We prove that an infinite chain is realized by a positive odd integer if and only if its least representatives eventually stabilize; in particular, infinitely many strict jumps exclude positive odd realization. As a consequence, no eventually periodic infinite low-surplus valuation chain is realized by a positive odd integer. Equivalently, every such chain has infinitely many strict jumps, and its associated \(2\)-adic limit fails to be a positive odd integer. The argument reduces a periodic tail to an affine period map and shows, via the low-surplus inequality, that its unique fixed point is necessarily negative. Subjects / optional classifications Mathematics; Number Theory; Dynamical Systems; Discrete Dynamics; 2-adic Dynamics; Collatz-type Problems Suggested MSC 2020, optional:11B83; 11A99; 37P99