A Computation of the Two-Adic Stuck Set for the Collatz Map

Let T denote the Syracuse map T(n) = (3n+ 1)/2ν2 (3n+1) on the odd positive integers, where ν2 is the 2-adic valuation. For each K ≥1, let SK be the set of odd residues r (mod 2K) for which the partial sums of log2 3·ν3−log2 2·ν2 along the orbit of r remain non-positive over all Syracuse steps resolvable within K bits of input. The set SK is the natural nite-precision approximation to the set of stuck 2-adic integers those whose Collatz orbits would, if they failed to terminate at 1, fail by drifting to innity rather than entering a non-trivial cycle. We compute |SK| by exact enumeration with 128-bit integer arithmetic for K= 3,4,...,30, and by Monte Carlo sampling for K = 33,50,60,70,80,90,100 (N = 2 ×108 samples per K). At K = 30 the exact enumeration gives |S30|= 11 759 296 survivors out of 229 = 536 870 912 odd residue classes, an elimination rate of 97.81%. We prove, via a Lundberg-type root-nding argument on the closed-form joint moment generating function of the cycle-type pair (X,c), that |SK|/2K−1 ≤poly(K)·2−α∗ K, where α∗ = 1 / ln 2 * [ln 2−ln(log2 3) + (log2 3−1 / log2 3) * ln(log2 3−1)] = 0.050 044... This gives the upper bound dimbox(F) ≤0.949 956 for the 2-adic limit set F of forever-stuck residues. We also establish the matching lower bound dimbox(F) ≥s∗ = 0.643 164... via a Hutchinson-type self-similar Cantor construction using stuck cycle types. A free t of the form |SK|/2K−1 ∼C·Kp ·2−αK to the data K ∈{30,33,50,60,70,80,90,100} gives α = 0.0542 ±0.0013 and p=−0.98 ±0.05, with the empirical exponential rate in close agreement with the rigorous theoretical value α∗. All numerical claims are reproducible from the accompanying open-source code.