Consider the comparision in positive integers $\small {rq-1 \over 2^B} \le q-1 $ where $\small B $ is the given parameter and r and q are residues modulo $\small 2^B$ (which also implies, that the lhs is assumed to be integer and which then puts restrictions on the relation of r and q) .
In my calculations I use $\small r=3^N (\mod 2^B) $and $\small q=3^{-N} (\mod 2^B) $ with $\small N \gt 0$ as free parameter to guarantee an integer lhs. (This implicitely restricts empirical results to less possible residue classes in r,q and thus possible solutions, but leave this for the moment).
My observation is, that the lhs is always at most equal to the rhs, which makes me curious. But I was not yet able to extract the key argument, why this is so. It is easy to see, that if $\small q=2^B-1 $ then also $\small r=q $ (from which then the equality in the equation follows) and also if $\small q=1 $ then $\small r=1$ and again the equality holds.
So my question is, how can I see that the given inequality holds always?