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

We study the accelerated Collatz (Syracuse) map T on the odd positive integers, T(n) = (3n+1)/2^v2(3n+1), through its symbolic coding by finite words of 2-adic valuations. First, using the classical conjugacy between the 2-adic extension of the (non-accelerated) Collatz map and the one-sided Bernoulli shift, together with the standard theory of induced transformations, we show that the natural invariant measure of T is ergodic and is the unique T-invariant probability measure absolutely continuous with respect to Haar measure on Z2. Second, we solve completely the extremal problem underlying cycle detection: for words of length p and total shift S, the affine coefficient c(w) appearing in the cycle-closure equation is maximized uniquely by the front-loaded word, with explicit value c_max(p,S) = 3^(p−1)(1+2^(S−p+1))−2^S. Third, we introduce a minimal-element pruning principle — the smallest element of a hypothetical cycle must lie below every partial trajectory of its own orbit — which, combined with exact integer arithmetic, excludes every non-trivial cycle of length p ≤ 17 unconditionally, for every admissible shift S, not merely a bounded window; a simpler direct enumeration separately and independently confirms p ≤ 12. Fourth, we prove a fully explicit and rigorous Diophantine bound, via the continued fraction expansion of log2 3, quantifying exactly how good a rational approximation S/p to log2 3 can be for every p — in particular showing that p = 41, 306, 15601, … are exactly the convergent denominators of log2 3, which explains structurally why they are the hardest cases. We then prove, precisely rather than merely arguing informally, why this bound cannot close the extremal approach: the ratio c_max(p,S)/(2^S−3^p) provably tends to (3/2)^(p−1)−1 > 0, not 0, as S→∞ for fixed p≥2, so no Diophantine input of this kind, however sharp, can ever exclude all sufficiently large S this way. We take one further step toward the remaining question — whether the pruning tree is eventually finite — by reformulating it in terms of a bounded ratio σⱼ: we prove an unconditional bracket 1 ≤ h ≤ 2 on the tree's entropy h, and obtain a validated, cross-checked numerical estimate h ≈ 1.54 via a population Monte Carlo method reaching depth 300, while proving the exact value is not yet determined. This clarifies precisely which gaps remain between this circle of results and a proof of the Collatz conjecture: one structural, one combinatorial, one measure-theoretic.