Cryptanalyzing FibonaRNG

Sorry about the title. This post is motivated by Steven’s comments to the “What is Randomness?” post, where he describes a way that the current paper-and-pencil cipher champion, FibonaRNG, could be broken. Rather than responding with more comments, I thought a whole new post on the issue would make more sense, since it’s going to be rather long. For those who prefer the short version: yes, what Steven says would work, but not very well, although it looks like it should. Read on if you prefer the long version.