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.
Christopher Clack (Tue,) studied this question.