The Cosine Ruler and the Selection Law: A Computing Instrument Built on the Fibonacci 60-Ring

This paper introduces a single instrument and the law that governs it. Place two equal circles sotheir boundaries cross on a 60-division ring; one integer setting, k, then fixes the distance betweentheir centers exactly, by the relation d = 2R·cos(6k°). As k runs from 1 to 14, this becomes asliding ruler — the cosine ruler — whose single setting determines a distance, a pair of crossingpoints, the two digits resting on them, and the count of digits trapped on each arc. Read this way,the Fibonacci ring becomes a small computing instrument: turn the dial to k and it returns a fixed,structured result. A simple counting rule — the selection law — then describes those results inclosed form. The instrument's outputs agree throughout with classical trigonometry, as they must;what is new is not the values but the mechanism that produces them on the ring. Every result isverified by exhaustive computation.