A $3$-adic Recurrence for the Fixed Points of the Josephus Function $J_4$
In the Josephus problem with stepsize four, the participants in a circle are eliminated one by one, every fourth person leaving, until a single survivor remains. A fixed point occurs when the survivor turns out to be the person who began in the last seat. The circle sizes with this property form the sequence 1; 21; 38; 51; 122; 163; 689; 919; 2,906; and so on, whose gaps fluctuate erratically. This paper explains the fluctuation and turns it into a recurrence. Between consecutive fixed points, the circle sizes at which the survivor falls exactly one or two seats short of the last one, the near-misses, group into alternating blocks of the two kinds, and the length of every block is the number of times three divides a simple quantity built from the circle size that precedes the block. Iterating these divisibility counts carries each fixed point to the next. Stepsize four is the first case in which two kinds of near-miss coexist, and the alternation they force is what separates it from the solved cases of stepsizes two and three. As a byproduct, the survivor's position for an arbitrary circle size can be computed by walking the near-misses of a single interval, in a number of steps proportional to their count, rather than stepping through every smaller circle as the defining recursion does.