Rigidity, Extremal Words, and Diophantine Obstructions for the Accelerated Collatz Map

Abstract

Rigidity, Extremal Words, and Diophantine Obstructions for the Accelerated Collatz Map (v12) We study the accelerated Collatz (Syracuse) map through its symbolic coding by words of 2-adic valuations. We prove that the natural invariant measure is ergodic and is the unique invariant probability measure absolutely continuous with respect to Haar measure on the odd 2-adic integers. We solve completely the extremal problem for the cycle-closure coefficient: for fixed length p and total shift S, c(w) is maximized uniquely by the front-loaded word, with an explicit closed form. Using a minimal-element pruning principle and exact integer arithmetic (implemented via memory-efficient disk streaming), we exclude every non-trivial cycle of length p ≤ 22 for every admissible S. An explicit Diophantine bound derived from the continued-fraction expansion of log₂ 3 identifies the structurally hardest lengths (p = 41, 306, 15601, …). We then prove that no bound of this shape can close the extremal argument by itself: the ratio c_max(p,S)/(2^S − 3^p) tends to a strictly positive limit as S → ∞. Reformulating the pruning recursion via a bounded state variable, we establish the unconditional entropy bracket 1 ≤ h ≤ log₂ 3 ≈ 1.585 for the pruning tree and obtain a validated numerical estimate h ≈ 1.54. A new result of this version gives a rigorous lower bound on the near-line entropy: h_nearline ≥ H(log₂ 3 − 1) ≈ 0.97907… (binary entropy), proving that the near-line population persists at a positive exponential rate. Complementary results include almost-sure extinction of a single ergodic trajectory and a modular analysis of truncated dynamics. The remaining gaps are isolated with precision: the possible infinitude of the pruning tree, the exact value of its entropy, and the link between near-line survival and the exact closure equation.