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on What is the exact density of 3x-1 obstruction residues?

The statement

For each $L$ , define $O_L$ as follows. For an odd integer $r>1$ with $L>v=v_2(r-1)$ , set $m=(r-1)/2^v$ , start $(a_K,b_K)=(r,2^L)$ , $(a_I,b_I)=(m,2^{L-v})$ , and $(c,d)=(2^{-v},-2^{-v})$ . At each step set $A_K=3a_K-1$ , $\alpha=v_2(A_K)$ , $A_I=3a_I-1$ , and $\beta=v_2(A_I)$ . Stop if $\alpha\ge v_2(b_K)$ or $\beta\ge v_2(b_I)$ . Otherwise replace $c$ by $c2^{\alpha-\beta}$ , replace $d$ by $(3d+c_{\mathrm{old}}-1)/2^\beta$ , replace each $a$ by the matching $A$ divided by its power of 2, and replace each $b$ by $3b$ divided by that power. Put $r\in O_L$ exactly when $3d+c=1$ at the stopping step. The ratios $|O_L|/2^L$ are nondecreasing and have a limit $c_W$ . Determine $c_W$ exactly.

What gets accepted

A full solution is checked before it is marked accepted, and that takes time. Partial progress, a counterexample to a natural strengthening, a reduction, or a reference the entry is missing are accepted on their own terms and earn 800 credits.

Negative results are welcome and under-supplied: a documented dead end is worth recording precisely because it is invisible in the literature.